Theoretical and Technological Advancements in Nanotechnology and Molecular Computation: Interdisciplinary Gains

Theoretical and Technological Advancements in Nanotechnology and Molecular Computation: Interdisciplinary Gains Bruce MacLennan University of Tennessee Knoxville, USA

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Published in the United States of America by Information Science Reference (an imprint of IGI Global) 701 E. Chocolate Avenue Hershey PA 17033 Tel: 717-533-8845 Fax: 717-533-8661 E-mail: [email protected] Web site: http://www.igi-global.com Copyright © 2011 by IGI Global. All rights reserved. No part of this publication may be reproduced, stored or distributed in any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark. Library of Congress Cataloging-in-Publication Data Theoretical and technological advancements in nanotechnology and molecular computation : interdisciplinary gains / Bruce MacLennan, editor. p. cm. Summary: "This book compiles research in areas where nanoscience and computer science meet exploring current and future trends in areas such as, cellular nanocomputers, DNA self-assembly, and the architectural design of a nano-brain"-- Provided by publisher. Includes bibliographical references and index. ISBN 978-1-60960-186-7 (hardcover) -- ISBN 978-1-60960-188-1 (ebook) 1. Molecular computers. 2. Cellular automata. 3. Quantum computers. I. MacLennan, Bruce J. QA76.887.T44 2011 004.1--dc22 2010047292 British Cataloguing in Publication Data A Cataloguing in Publication record for this book is available from the British Library. All work contributed to this book is new, previously-unpublished material. The views expressed in this book are those of the authors, but not necessarily of the publisher.

Table of Contents

Preface . ............................................................................................................................................... xvi Section 1 Chapter 1 Optimal DNA Codes for Computing and Self-Assembly........................................................................ 1 Max H. Garzon, The University of Memphis, USA Vinhthuy Phan, The University of Memphis, USA Andrew Neel, The University of Memphis, USA Chapter 2 DNA Hash Pooling and its Applications................................................................................................ 15 Dennis Shasha, New York University, USA Martyn Amos, Manchester Metropolitan University, UK Chapter 3 Cellular Nanocomputers: A Focused Review........................................................................................ 28 Ferdinand Peper, National Institute of Information and Communications Technology (NiCT), Japan Jia Lee, Celartem Technology Inc., Japan Susumu Adachi, National Institute of Information and Communications Technology (NiCT), Japan Teijiro Isokawa, University of Hyogo, Japan Chapter 4 An Advanced Architecture of a Massive Parallel Processing Nano Brain Operating 100 Billion Molecular Neurons Simultaneously........................................................................................... 43 Anirban Bandyopadhyay, National Institute for Materials Science, Japan Subrata Ghosh, National Institute for Materials Science, Japan Daisuke Fujita, National Institute for Materials Science, Japan Ranjit Pati, Michigan Technological University, USA Satyajit Sahu, National Institute for Materials Science, Japan

Section 2 Chapter 5 Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing.......................................................................................................................... 75 Takashi Morie, Kyushu Institute of Technology, Japan Chapter 6 Application of Single Electron Devices Utilizing Stochastic Dynamics............................................. 100 Shigeo Sato, Tohoku University, Japan Koji Nakajima, Tohoku University, Japan Chapter 7 On the Reliability of Post-CMOS and SET Systems........................................................................... 114 Milos Stanisavljevic, Swiss Federal Institute of Technology EPFL, Switzerland Alexandre Schmid, Swiss Federal Institute of Technology EPFL, Switzerland Yusuf Leblebici, Swiss Federal Institute of Technology EPFL, Switzerland Chapter 8 Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs.......................................................................................................................................... 131 Takuya Kaizawa, Hokkaido University, Japan Mingyu Jo, Hokkaido University, Japan Masashi Arita, Hokkaido University, Japan Akira Fujiwara, NTT Corporation, Japan Kenji Yamazaki, NTT Corporation, Japan Yukinori Ono, NTT Corporation, Japan Hiroshi Inokawa, Shizuoka University, Japan Yasuo Takahashi, Hokkaido University, Japan Jung-Bum Choi, Chungbuk National University, Korea Chapter 9 Investigation on Stochastic Resonance in Quantum Dot and its Summing Network.......................... 140 Seiya Kasai, Hokkaido University & Japan Science and Technology Agency, Japan Chapter 10 A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation................... 149 Andrew Kilinga Kikombo, Hokkaido University, Japan Tetsuya Asai, Hokkaido University, Japan Takahide Oya, Yokohama National University, Japan Alexandre Schmid, Swiss Federal Institute of Technology (EPFL), Switzerland Yusuf Leblebici, Swiss Federal Institute of Technology (EPFL), Switzerland Yoshihito Amemiya, Hokkaido University, Japan

Section 3 Chapter 11 Simple Collision-Based Chemical Logic Gates with Adaptive Computing........................................ 162 Rita Toth, University of the West of England, UK Christopher Stone, University of the West of England, UK Ben de Lacy Costello, University of the West of England, UK Andrew Adamatzky, University of the West of England, UK Larry Bull, University of the West of England, UK Chapter 12 Toward Biomolecular Computers Using Reaction-Diffusion Dynamics............................................ 176 Masahiko Hiratsuka, Sendai National College of Technology, Japan Koichi Ito, Tohoku University, Japan Takafumi Aoki, Tohoku University, Japan Tatsuo Higuchi, Tohoku Institute of Technology, Japan Chapter 13 Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions................. 184 B. P. J. de Lacy Costello, University of the West of England, UK J. Armstrong, University of the West of England, UK I. Jahan, University of the West of England, UK N. M. Ratcliffe, University of the West of England, UK Chapter 14 Dynamics of Particle-Based Reaction-Diffusion Computing: Active vs. Passive, Attraction vs. Repulsion....................................................................................................................... 194 Jeff Jones, University of the West of England, UK Chapter 15 Towards Arithmetical Chips in Sub-Excitable Media: Cellular Automaton Models........................... 223 Liang Zhang, University of the West of England, UK Andrew Adamatzky, University of the West of England, UK Section 4 Chapter 16 Organization-Oriented Chemical Programming of Distributed Artifacts............................................ 240 Naoki Matsumaru, Friedrich Schiller University Jena, Germany Thomas Hinze, Friedrich Schiller University Jena, Germany Peter Dittrich, Friedrich Schiller University Jena, Germany

Chapter 17 Dominant Spin Relaxation Mechanisms in Organic Semiconductor Alq3.......................................... 259 Sridhar Patibandla, Virginia Commonwealth University, USA Bhargava Kanchibotla, Virginia Commonwealth University, USA Sandipan Pramanik, University of Alberta, Canada Supriyo Bandyopadhyay, Virginia Commonwealth University, USA Marc Cahay, University of Cincinnati, USA Chapter 18 The Synthesis of Stochastic Circuits for Nanoscale Computation...................................................... 279 Weikang Qian, University of Minnesota, USA John Backes, University of Minnesota, USA Marc D. Riedel, University of Minnesota, USA Chapter 19 Random Dynamical Network Automata for Nanoelectronics: A Robustness and Learning Perspective.............................................................................................. 295 Christof Teuscher, Portland State University, USA Natali Gulbahce, Northeastern University, USA Thimo Rohlf, Genopole, France Alireza Goudarzi, Portland State University, USA Compilation of References ............................................................................................................... 315 About the Contributors .................................................................................................................... 348 Index.................................................................................................................................................... 360

Detailed Table of Contents

Preface . ............................................................................................................................................... xvi Section 1 Chapter 1 Optimal DNA Codes for Computing and Self-Assembly........................................................................ 1 Max H. Garzon, The University of Memphis, USA Vinhthuy Phan, The University of Memphis, USA Andrew Neel, The University of Memphis, USA DNA has been re-discovered and explored in the last decade as a “smart glue” for self-assembly from the “bottom-up” at nanoscales through mesoscales to micro- and macro-scales. These applications require an unprecedented degree of precision in placing atom-scale components. Finding large sets of probes to serve as anchors for such applications has been thus explored in the last few years through several methods. We describe results of a tour de force to conduct an exhaustive search to produce large codes that are (nearly) maximal sets while guaranteeing high quality, as measured by the minimum Gibbs energy between any pair of code words, and other criteria. We also present a quantitative characterization of the sets for sizes up to 20-mers and show how critical building blocks can be extracted to produce codes of very high quality for larger lengths by probabilistic combinations, for which an exhaustive search is out of reach. Chapter 2 DNA Hash Pooling and its Applications................................................................................................ 15 Dennis Shasha, New York University, USA Martyn Amos, Manchester Metropolitan University, UK In this article we describe a new technique for the comparison of populations of DNA strands. Comparison is vital to the study of ecological systems, at both the micro and macro scales. Existing methods make use of DNA sequencing and cloning, which can prove costly and time consuming, even with current sequencing techniques. Our overall objective is to address issues such as whole genome detection, fragment detection and sample similarity. Because our method is similar in spirit to hashing in

computer science, we call it DNA hash pooling. To illustrate this method, we describe protocols using pairs of restriction enzymes. The in silico empirical results we present reflect a sensitivity to experimental error. Our method, performed as a filtering step prior to sequencing, may reduce the amount of sequencing required (generally by a factor of 10 or more). Even as sequencing becomes cheaper, an order of magnitude remains important. Chapter 3 Cellular Nanocomputers: A Focused Review........................................................................................ 28 Ferdinand Peper, National Institute of Information and Communications Technology (NiCT), Japan Jia Lee, Celartem Technology Inc., Japan Susumu Adachi, National Institute of Information and Communications Technology (NiCT), Japan Teijiro Isokawa, University of Hyogo, Japan Cellular Automata have their roots in von Neumann’s research on self-reproduction, but since their debut they have been used for a much wider variety of purposes. In recent years they have attracted attention as architectures for nanocomputers–computers to be realized by nanotechnology. Their highly regular structure is considered an important advantage in this context, because of the potential for fabrication by bottom-up techniques like molecular self-assembly. This article gives an overview of research on cellular automaton-based nanocomputers, and discusses their strong points and challenges. Chapter 4 An Advanced Architecture of a Massive Parallel Processing Nano Brain Operating 100 Billion Molecular Neurons Simultaneously........................................................................................... 43 Anirban Bandyopadhyay, National Institute for Materials Science, Japan Subrata Ghosh, National Institute for Materials Science, Japan Daisuke Fujita, National Institute for Materials Science, Japan Ranjit Pati, Michigan Technological University, USA Satyajit Sahu, National Institute for Materials Science, Japan Molecular machines (MM, Badjic, 2004; Collier, 2000; Jian & Tour, 2003; Koumura & Ferringa, 1999; Ding & Seeman, 2006) may resolve three distinct bottlenecks of scientific advancement (Bandyopadhyay, Fujita, Pati, 2008). Nanofactories (Phoenix, 2003) composed of MM may produce atomically perfect products spending negligible amount of energy (Hess, 2004) thus alleviating the energy crisis. Computers made by MM operating thousands of bits at a time may match biological processors mimicking creativity and intelligence (Hall, 2007), thus far considered as the prerogative of nature. State-of-the-art brain surgeries are not yet fatal-less, MMs guided by a nano-brain may execute perfect bloodless surgery (Freitas, 2005). Even though all three bottlenecks converge to a single necessity of nano-brain, futurists and molecular engineers have remained silent on this issue. Our recent invention of 16 bit parallel processor is a first step in this direction (Bandyopadhyay, 2008). However, the device operates inside ultra-high vacuum chamber. For practical application, one needs to design a 3 D standalone architecture. Here, we identify the minimum nano-brain functions for practical applications and try to increase the size from 2 nm to 20 μm. To realize this, three major changes are made. First, central

control unit (CCU) and external execution units (EU) are modified so that they process information independently, second, CCU instructs EU the basic rules of information processing; third, once rules are set CCU does not hinder EU-computation. The basic design of the proposed nano-brain is a dendrimer (Hawker, 2005; Galliot, 1997; Devadoss, 2001; Quintana, 2002; Peer, 2007), with a control unit at its core and a molecular cellular neural network (m-CNN, Rosca, 1993; Chua, 2005) or Cellular Automata (CA, Wolfram, 1983) on its outer surface (EU). Each CNN/CA cell mimics the functionality of neurons by processing multiple bits reversibly (Rozenberg, 2004; Li, 2004; Bandyopadhyay, 2004). We have designed a megamer (Tomalia, 2005) consisting of dendrimer (~10 nm) as its unit CNN cell for building the giant 100 billion neuron based nano brain architecture. An important spontaneous control from 10 nm to 20 μm is achieved by an unique potential distribution following r = a sin k θ, where r is the coordinate of doped neuron cluster, k is the branch number, θ is the angle of deviation and a is a constant typical of the megamer architecture. Section 2 Chapter 5 Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing.......................................................................................................................... 75 Takashi Morie, Kyushu Institute of Technology, Japan The single-electron circuit technology should aim at developing information processing systems using the intrinsic properties of single-electron devices. The operation principles of single-electron devices are completely different from that of conventional CMOS devices, but both devices should co-exist in the information processing systems. In this paper, according to a scenario for achieving large-scale integrated systems of single-electron devices, some single-electron devices and circuits utilizing stochastic operation for associative processing and a spiking neuron model are described. Chapter 6 Application of Single Electron Devices Utilizing Stochastic Dynamics............................................. 100 Shigeo Sato, Tohoku University, Japan Koji Nakajima, Tohoku University, Japan Single electron devices utilizing the Coulomb blockade phenomenon have attractive features such as extreme low power consumption, one by one electron flow controllability, small device size, etc. However, besides promising applications such as the current standard and charge detection, it is not easy to apply the single electron devices to conventional computational tasks due to its stochastic operation and low amplification capability. Therefore, it is important for us to consider suitable applications of single electron devices. In this paper, we show three applications such as a noise generator, a stochastic neural network, and a charge detector employing stochastic resonance. Trough these applications, we see the advantages of single electron devices and study the direction of applications.

Chapter 7 On the Reliability of Post-CMOS and SET Systems........................................................................... 114 Milos Stanisavljevic, Swiss Federal Institute of Technology EPFL, Switzerland Alexandre Schmid, Swiss Federal Institute of Technology EPFL, Switzerland Yusuf Leblebici, Swiss Federal Institute of Technology EPFL, Switzerland The necessity of applying fault-tolerant techniques to increase the reliability of future nano-electronic systems is an undisputed fact, dictated by the high density of faults that will plague the chips. The averaging and thresholding fault-tolerant technique that has proven remarkable efficiency in CMOS is presented for SET-based designs. Computer simulations demonstrate the superiority of this fault-tolerant technique over other methods, which is specifically the case when an adaptable threshold is used. Chapter 8 Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs.......................................................................................................................................... 131 Takuya Kaizawa, Hokkaido University, Japan Mingyu Jo, Hokkaido University, Japan Masashi Arita, Hokkaido University, Japan Akira Fujiwara, NTT Corporation, Japan Kenji Yamazaki, NTT Corporation, Japan Yukinori Ono, NTT Corporation, Japan Hiroshi Inokawa, Shizuoka University, Japan Yasuo Takahashi, Hokkaido University, Japan Jung-Bum Choi, Chungbuk National University, Korea A highly functional Si nanodot array device that operates by means of single-electron effects was experimentally demonstrated. The device features many input gates, and many outputs can be attached. A nanodot array device with three input gates and two output terminals was fabricated on a silicon-oninsulator wafer using conventional Si MOS processes. Its feasibility was demonstrated by its operation as both a half adder and a full adder when the operation voltage was carefully selected. Chapter 9 Investigation on Stochastic Resonance in Quantum Dot and its Summing Network.......................... 140 Seiya Kasai, Hokkaido University & Japan Science and Technology Agency, Japan Stochastic resonance behavior of single electrons in a quantum dot and its summing network is investigated theoretically. Dynamic behavior of the single electron in the system at finite temperature is analyzed using a master equation with a tunneling transition rate. The analytical model indicates that an input-output correlation has a peak as a function of temperature, which confirms the appearance of the stochastic resonance. The peak position and height depend on charging energy, tunnel resistance, and input signal frequency. It is also found that the correlation is enhanced by formation of a summing network integrating quantum dots in parallel. The present model quantitatively explains the stochastic resonance behaviors of the single electrons predicted by a circuit simulation (Oya, Asai, & Amemiya, 2007).

Chapter 10 A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation................... 149 Andrew Kilinga Kikombo, Hokkaido University, Japan Tetsuya Asai, Hokkaido University, Japan Takahide Oya, Yokohama National University, Japan Alexandre Schmid, Swiss Federal Institute of Technology (EPFL), Switzerland Yusuf Leblebici, Swiss Federal Institute of Technology (EPFL), Switzerland Yoshihito Amemiya, Hokkaido University, Japan We propose a bio-inspired circuit performing pulse-density modulation with single-electron devices. The proposed circuit consists of three single-electron neuronal units, receiving the same input and are connected to a common output. The output is inhibitorily fedback to the three neuronal circuits through a capacitive coupling. The circuit performance was evaluated through Monte-Carlo based computer simulations. We demonstrated that the proposed circuit possesses noise-shaping characteristics, where signal and noises are separated into low and high frequency bands respectively. This significantly improved the signal-tonoise ratio (SNR) by 4.34 dB in the coupled network, as compared to the uncoupled one. The noise-shaping properties are as a result of i) the inhibitory feedback between the output and the neuronal circuits, and ii) static noises (originating from device fabrication mismatches) and dynamic noises (as a result of thermally induced random tunneling events) introduced into the network. Section 3 Chapter 11 Simple Collision-Based Chemical Logic Gates with Adaptive Computing........................................ 162 Rita Toth, University of the West of England, UK Christopher Stone, University of the West of England, UK Ben de Lacy Costello, University of the West of England, UK Andrew Adamatzky, University of the West of England, UK Larry Bull, University of the West of England, UK We present a method that is capable of implementing information transfer without any rigidly controlled architecture using the light-sensitive Belousov-Zhabotinsky (BZ) reaction system. Chemical wave fragments are injected into a subexcitable area and their collisions result in annihilation, fusion or quasi-elastic interactions depending on their initial positions. The fragments of excitation both pre and post collision possess a considerable freedom of movement when compared to previous implementations of information transfer in chemical systems. We propose that the collision of such wave fragments can be controlled automatically through adaptive computing. By extension, forms of unconventional computing, i.e., massively parallel non-linear computers, can be realised by such an approach. In this study we present initial results from using a simple evolutionary algorithm to design Boolean logic gates within the BZ system.

Chapter 12 Toward Biomolecular Computers Using Reaction-Diffusion Dynamics............................................ 176 Masahiko Hiratsuka, Sendai National College of Technology, Japan Koichi Ito, Tohoku University, Japan Takafumi Aoki, Tohoku University, Japan Tatsuo Higuchi, Tohoku Institute of Technology, Japan This article investigates a possibility of constructing massively parallel computing systems using molecular electronics technology. By employing the specificity of biological molecules, such as enzymes, new integrated circuit architectures that are free from interconnection problems could be constructed. To clarify the proposed concept, we present a functional model of an artificial catalyst device called an enzyme transistor. In this article, we develop artificial catalyst devices as basic building blocks for molecular computing integrated circuits, and explore the possibility of a new computing paradigm using reaction-diffusion dynamics induced by collective behavior of artificial catalyst devices. Chapter 13 Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions................. 184 B. P. J. de Lacy Costello, University of the West of England, UK J. Armstrong, University of the West of England, UK I. Jahan, University of the West of England, UK N. M. Ratcliffe, University of the West of England, UK Under normal reaction conditions [AlCl3 0.28-0.34M and NaOH 2.5M A.Volford et al.] spontaneous spiral and circular travelling precipitate waves were observed. We constructed a phase diagram for the reaction and identified a large controllable region at lower aluminum chloride levels. We show that it is possible to selectively initiate travelling circular waves and other self-organized structures within this controllable region. In previous work initiation was undertaken before adding the outer electrolyte resulting in disorganized waves. However, marking the gel one minute after adding outer electrolyte resulted in cardioid waves. Increasing the time interval to two minutes caused a transition to single circular waves. If the gel is marked sequentially nested circular waves (target waves) are formed. These reactions were used to calculate simple and additively weighted Voronoi tessellations. The fine control of self-organization in precipitation reactions is of interest for the synthesis of novel and functional Materials. Chapter 14 Dynamics of Particle-Based Reaction-Diffusion Computing: Active vs. Passive, Attraction vs. Repulsion....................................................................................................................... 194 Jeff Jones, University of the West of England, UK Reaction-diffusion computing utilises the complex auto-catalytic and diffusive interactions underlying self-organising systems for practical computing tasks – developing variants of classical logical computing devices, or direct spatial embodiments of problem representations and solutions. We investigate the concept of passive and active approaches to reaction-diffusion computing. Passive approaches use front propagation as a carrier signal for information transport and computation. Active approaches can both

sense and modify the propagation of the underlying carrier signal. We also consider the differences in attraction and repulsion behaviour for both passive and active approaches. Using particle approximations of reaction-diffusion behaviour in chemical systems, and the plasmodium of Physarum polycephalum, we demonstrate the similarities and differences between the passive and active approaches using both attraction and repulsion behaviour. We provide examples of how the approaches can be used for complex spatially represented computational tasks. We note that the active approach results in secondorder emergent behaviour, exhibiting complex quasi-physical properties such as apparent surface tension effects and network minimisation which may have utility in future physical implementations of reaction-diffusion computing devices. Chapter 15 Towards Arithmetical Chips in Sub-Excitable Media: Cellular Automaton Models........................... 223 Liang Zhang, University of the West of England, UK Andrew Adamatzky, University of the West of England, UK We discuss a theoretical design of an arithmetical chip built on an excitable medium substrate. The chip is simulated in a two-dimensional three-state cellular automaton with eight-cell neighborhoods. Every resting cell is excited if it has exactly two excited neighbors, the excited cells takes refractory state unconditionally. A transition from refractory back to resting state also happens irrelevantly to a state of the cell neighborhood. The design is based on principles of collision-based computing. Boolean logic values are encoded by traveling localizations, or particles. Logical gates are realized in collisions between the particles. Detailed blue prints of collision-based adders and multipliers presented in the article pave the way to future laboratory experimental prototypes of general-purpose chemical computers. Section 4 Chapter 16 Organization-Oriented Chemical Programming of Distributed Artifacts............................................ 240 Naoki Matsumaru, Friedrich Schiller University Jena, Germany Thomas Hinze, Friedrich Schiller University Jena, Germany Peter Dittrich, Friedrich Schiller University Jena, Germany The construction of molecular-scale machines requires novel paradigms for their programming. Here, we assume a scenario of distributed devices that process in-formation by chemical reactions and that communicate by exchanging molecules. Programming such a distributed system requires specifing reaction rules as well as exchange rules. Here, we present an approach that helps to guide the manual construction of distributed chemical programs. We show how chemical organization theory can assist a programmer in predicting the behavior of the program. The basic idea is that a computation should be understood as a movement between chemical organizations, which are closed and self-maintaining sets of molecular species. When sticking to that design principle, fine-tuning of kinetic laws becomes less important. We demonstrate the approach by a novel chemical program that solves the maximal independent set problem on a distributed system without any central control—a typical situation in ad-hoc

networks. We show that the computational result, which emerges from many local reaction events, can be explained in terms of chemical organizations, which assures robustness and low sensitivity to the choice of kinetic parameters. Chapter 17 Dominant Spin Relaxation Mechanisms in Organic Semiconductor Alq3.......................................... 259 Sridhar Patibandla, Virginia Commonwealth University, USA Bhargava Kanchibotla, Virginia Commonwealth University, USA Sandipan Pramanik, University of Alberta, Canada Supriyo Bandyopadhyay, Virginia Commonwealth University, USA Marc Cahay, University of Cincinnati, USA We have measured the longitudinal (T1) and transverse (T2) spin relaxation times in the organic semiconductor tris (8-hydroxyquinolinolato aluminum) - also known as Alq3 - at different temperatures and under different electric fields driving current. These measurements shed some light on the spin relaxation mechanisms in the organic. The two most likely mechanisms affecting T1 are hyperfine interactions between carrier and nuclear spins, and the Elliott-Yafet mode. On the other hand, the dominant mechanism affecting T2 of delocalized electrons in Alq3 remains uncertain, but for localized electrons (bound to defect or impurity sites), the dominant mechanism is most likely spin-phonon coupling. Chapter 18 The Synthesis of Stochastic Circuits for Nanoscale Computation...................................................... 279 Weikang Qian, University of Minnesota, USA John Backes, University of Minnesota, USA Marc D. Riedel, University of Minnesota, USA Emerging technologies for nanoscale computation such as self-assembled nanowire arrays present specific challenges for logic synthesis. On the one hand, they provide an unprecedented density of bits with a high degree of parallelism. On the other hand, they are characterized by high defect rates. Also they often exhibit inherent randomness in the interconnects due to the stochastic nature of self-assembly. We describe a general method for synthesizing logic that exploits both the parallelism and the random effects. Our approach is based on stochastic computation with parallel bit streams. Circuits are synthesized through functional decomposition with symbolic data structures called multiplicative binary moment diagrams. Synthesis produces designs with randomized parallel components—and operations and multiplexing—that are readily implemented in nanowire crossbar arrays. Synthesis results for benchmarks circuits show that our technique maps circuit designs onto nanowire arrays effectively. Chapter 19 Random Dynamical Network Automata for Nanoelectronics: A Robustness and Learning Perspective.............................................................................................. 295 Christof Teuscher, Portland State University, USA Natali Gulbahce, Northeastern University, USA Thimo Rohlf, Genopole, France Alireza Goudarzi, Portland State University, USA

It is generally expected that future and emerging nanoscale computing devices will be built in a bottomup way from vast numbers of simple, densely arranged components that exhibit high failure rates, are relatively slow, and connected in an unstructured way. Other than that, there is little to no consensus on what type of technology and computing architecture holds most promises to go far beyond today’s topdown engineered silicon devices. Highly structured crossbar-like and cellular automata architectures have been proposed as possible alternatives to the von Neumann computing architecture, which is not generally well suited for emerging, massively parallel and fine-grained nanoscale electronics. While the top-down engineered semi-conducting technology favors regular and locally interconnected structures, emerging bottom-up self-assembled devices tend to have to be unstructured and heterogeneous because of the current lack of precise control over these processes. In this paper, we survey and assess two types of random dynamical networks, namely Random Boolean Networks (RBNs) and Random Threshold Networks (RTNs), as candidates for alternative computing architectures and models for future nanoscale information processing devices. In a high-level approach that is based on previous work, we illustrate that they have the potential to offer superior properties over highly structured crossbar- or mesh-like cellular automata architectures, such as an inherent and scale-invariant robustness, more efficient communication capabilities, manufacturing benefits for bottom-up self-assembled devices, and the ability to learn and solve tasks successfully. We also show that RBNs can learn and generalize. Our investigation is driven by the need for alternative computing and manufacturing paradigms to mitigate some of the challenges traditional approaches face. Compilation of References ............................................................................................................... 315 About the Contributors .................................................................................................................... 348 Index.................................................................................................................................................... 360

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Preface

COMPUTATION AND NANOTECHNOLOGY This volume collects revised or original articles from the inaugural year (2009) of the International Journal of Nanotechnology and Molecular Computation, which aims to publish high-quality, state-ofthe-art research in all areas of nanotechnology and molecular computation. Nevertheless IJNMC has a unique focus, which differs from other nanotechnology journals, because it seeks to publish work in the fruitful area where nanoscience and computer science meet. Since the rest of this Preface will address the importance of this research, I must emphasize here that this focus is only an orientation, not a limitation, and so the journal’s scope includes the full breadth of nanotechnology and molecular computation, as is apparent from the articles collected herein.

HIERARCHICAL ASSEMBLY Enormous progress has been made in recent years in the nanostructuring of materials, and a variety of techniques are available for fabricating bulk materials with a desired nanostructure. However, the higher levels of organization have been neglected, and nanostructured materials are assembled into macroscopic structures using techniques that are not essentially different from those used for conventional materials. For example, nanostructured materials may be shaped by machining or molding and assembled by conventional manufacturing techniques. Thus we may have self-assembly at the nanoscale and conventional manufacturing at the macroscale, but no systematic fabrication technology applicable to all scales. Is there an alternative? Fortunately nature provides a suggestive example, for embryological morphogenesis creates highly complex hierarchical systems, with structures ranging from the nanoscale within cells, up through multicellular tissues, to the level of gross anatomy. As a significant example, we may take the mammalian nervous system. The brain comprises a number of anatomical regions (the lobes), each comprising hundreds of smaller functional regions (e.g., Brodmann’s areas, computational maps), which are structured into macrocolumns, which in turn contain minicolumns, each with a half-dozen or so layers. The minicolumns comprise about one hundred neurons with dendritic trees of characteristic shape (and tens of thousands of synapses), all interconnected in specific ways. At the other end of the scale, the brain itself is part of a nervous system, which includes a highly ramified but organized network of nerves. Thus, embryological morphogenesis provides an inspiring example of how self-organized growth, differentiation, and interaction can produce these complex macroscopic structures from microscopic components.

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Similarly, the mathematical principles of morphogenesis may be applicable to the fabrication of complex hierarchically-structured artificial systems (MacLennan, 2010, in press-b). The physical realization of these mathematical principles is closely connected to computation, which I will consider next.

POST-MOORE’S LAW COMPUTING The reign of Moore’s Law is near its end, and so we must consider the implications of a new regime in computing. Whatever the specifics of new technologies, it is apparent that the quest for greater densities and greater speeds will dictate a closer assimilation between computational processes and the physical processes that implement them. This is because greater densities will require us to use fewer physical devices to implement each element of the computational state (e.g., each abstract number), and because achieving greater speeds will preclude us from using sequences of physical processes to implement elementary computational operations (such as addition). That is, the space and time scales of elementary computational processes will have to approach the space and time scales of the physical systems that realize them. Nanoscale physical processes have different characteristics from the computational processes on which contemporary computer technology is built (binary digital logic). For example, these physical processes are often continuous in both time and physical quantity. Even discrete phenomena (such as molecular interactions within a cell) may be best treated as continuous stochastic processes. Further, natural processes are fundamentally parallel and asynchronous. In traditional computer design, however, we have arranged physical processes so that they are synchronized and sequential — because we understand such systems better — but that is wasteful, and in the long run it will be more efficient to learn to take advantage of asynchronous parallel processes for computation. Contemporary computing, including logic design, is often done in a sort of abstract realm in which there are logical dependencies, but spatial arrangement is ignored. At small time and space scales, however, the spatial organization of computational elements becomes much more important. For example, the propagation time of signals is always relevant, but may be exploited for information processing. More generally, the spatial organization of computational elements becomes an essential element in information systems design. Also, molecular computation is not a purely abstract process, but we must consider steric factors, diffusion rates, etc., which may be treated as problems, but are better viewed as potential computational resources. Nondeterminism is pervasive at the nanoscale. For example, thermal noise is significant in magnitude and unavoidable, defects and faults are inevitable, and quantum phenomena are often relevant. We may view these characteristics as problems — as implied by such terms as “noise,” “defect,” and “fault” — or we may view them as sources of free variability, that is, of randomness, diversity, etc., which can be used for many purposes. Chemical reactions always have a nonzero probability of going backward, and so molecular computation cannot be assumed to go monotonically forward. Similarly, many other nanocomputational processes get their direction from energy differences, barriers, etc. and have some probability of reversal. Thus models of nanocomputation must take reversibility into account. Again, we can treat it as a problem to be solved (an imperfect realization of an idealized unidirectional process), or we can treat it as a free resource to be exploited. For example, simulated annealing and similar stochastic optimization

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algorithms depend on low-priority uphill transitions to escape from local minima (Kirkpatrick, Gelatt & Vecchi, 1983). Although we have been inclined to think of computation as an abstract process, nanocomputation requires us to recognize that computation is a dissipative or nonequilibrium physical process: potentially nonterminating computation cannot continue without a source of matter or energy. Therefore, in nanocomputation, much more so than in traditional computation, the flow of matter and energy becomes a central factor in information system design. The nanoscale presents novel problems for energy supply, but also new opportunities for sources of matter and energy, including electrical, chemical, optical, thermal, and vibrational energy. Similarly, there are novel problems and opportunities for dissipation of waste energy and matter. At the nanoscale, entropy is both a physical and information-theoretic quantity. Self-assembly is often treated as an equilibrium process (analogous to a terminating computation), but in a more general sense an assembled nanostructure can be a stationary state (or dynamic equilibrium) in a nonequilibrium process, as is often the case in natural, especially living, systems. In effect such a system is continually regenerating its stationary state, so that if it is perturbed (e.g., damaged) it will restore the stationary state. Therefore such systems may be self-healing without an explicit selfrepair mechanism. Further, since a stationary state may be a result of the system’s interaction with its environment, as well as of its internal dynamics, a change in the environment can cause a shift to a new stationary state. In this way many natural systems adapt automatically to changes in their environments. Therefore we can exploit the physical medium to implement implicitly important characteristics such as healing and adaptation. Finally, quantum effects are inevitable when dealing with small space scales, time intervals, energies, etc. Thus we need a fundamentally quantum-oriented approach to computation, including techniques to make productive use of quantum phenomena, such as tunneling, exchange interactions, entanglement, and superposition (the goal, of course, of quantum computation). All these characteristics of nanocomputation can be considered problems to be solved, and this is the approach taken in contemporary computer technology, where devices and circuits are designed to operate “correctly” (i.e., to implement binary digital logic) in spite of these physical characteristics. However, arranging physical processes to implement preconceived ideas of computation is costly in density, speed, and power. Therefore we should, as is sometimes said, “respect the medium” and view its characteristics as resources to be exploited rather than as problems to be avoided. For example, thermal noise may be used as a “free” source of variability, which is useful for many algorithms (e.g., simulated annealing, stochastic resonance: Benzi, Parisi, Sutera & Vulpiani, 1982; Kirkpatrick, Gelatt & Vecchi, 1983). Low precision real numbers may be represented directly by continuous physical quantities, rather than indirectly, as we do now, with a single number being represented by multiple bits, each implemented by operating one or more continuous physical devices (transistors) in saturated mode. Thus post-Moore’s Law computing should seek to make productive use of physical properties and processes in the computational medium. This increased dependence on physical properties might seem to turn computation into a kind of applied physics, but there is still an important role for computational abstractions. We can see this from the history of contemporary computing technology, for the same mathematical abstraction — Boolean logic — has been used as a model of computation since Boole’s Investigation of the Laws of Thought (1854), through successive generations of implementation technology, from the mechanical logic of Jevon’s logical piano (1869), through relays, vacuum tubes, discrete transistors, integrated circuits, and several generations of VLSI. This stable theoretical background has permitted a cumulative investment

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in Boolean logic and circuit design, providing continuity from one technological generation to the next, and saving us from having to reinvent computer design with each new technology. This is possible because Boolean logic is physically realizable, yet sufficiently abstract that it can be realized by a variety of physical systems. Therefore, in laying the foundation for post-Moore’s Law computing we should seek new models of computation that combine physical realism with sufficient abstractness to be implementable in a variety of physical media. Our models of computation need to be close to the underlying physical realization, but not so close that only one realization is possible. Therefore we should adopt as fundamental computational operations those processes that occur in a wide variety of physical systems or that can be fairly directly implemented in them. For example, diffusion is a common physical process, which occurs in a variety of media, from charge carriers diffusing in a semiconductor to molecules diffusing in a fluid, to cells wandering through mesenchyme, and it has proved useful for information processing and control in natural and artificial systems; therefore it is a good candidate as an operation in post-Moore’s Law computing. Fortunately nature provides many examples of the use of physical processes for information processing, and these can often be abstracted from their specific physical embodiment and realized in other physical systems. Examples include neural network models of computation, excitable media and reactiondiffusion systems used to control spatial organization, molecular regulatory circuits in cells, intracellular DNA/RNA computing, and embryological pattern formation and morphogenesis. Understanding these systems in information processing terms will show how common physical processes may be exploited to more directly realize information-processing functions, and thus show the way to post-Moore’s Law computing technologies.

COMPUTATIONAL CONTROL OF MATTER Nanotechnology and computation interact in another important way, but to see it we have to step back and look at computation from a general perspective. Computation uses physical processes to realize abstract processes. For example, in manual computation the beads on an abacus or the scales of a slide rule are manipulated to realize abstract mathematical operations, such as addition and multiplication. In an analog computer, electrical processes realize a system of differential equations, and in digital computers electrical switches implement binary logic, in order to process numbers, matrices, characters, sequences, trees, and other abstract objects. What distinguishes these physical processes as computation is that their purpose is to realize an abstract (mathematical) process, which is in principle multiply realizable, that is, realizable by any physical process that has the same abstract structure (MacLennan, 1994, 2004). There is a tendency to confuse the physical process of computation with the abstract processes it realizes, and to think of computation as an abstract process itself. This tendency is reinforced by the theory of computation, which is based on abstract (mathematical) machine models (such as the Turing machine), whose purpose is generally expressed as the evaluation of mathematical functions. Therefore it is important to recall that computation is a physical process during which real matter and energy are controlled. This is easiest to recognize in analog computers, but even in digital computation an abstract process governs the flow of currents, the movement of electrons, etc. Computation bridges the abstract and the physical (or, as we might say, the formal and the material). Normally we use the physical processes as a means of realizing an abstract purpose, but we can turn the

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tables and view the abstract process as a means of achieving a physical effect. This is already the case for a computer’s output transducers, which are intended to have a physical effect, but we can extend the idea so that the entire computation is designed for the sake of the corresponding physical processes. While the physical processes in an electronic computer might not seem very useful for purposes other than computing, other computing technologies, such as molecular computation, reorganize matter in ways that can be useful for nanotechnology. An example is algorithmic self-assembly, which can be implemented with DNA (e.g., Winfree, 1998; Reif, 1999; Rothemund, Papadakis, & Winfree, 2004; Rothemund & Winfree, 2000). In this context the idea of a general-purpose computation is especially intriguing, since it implies that a wide variety of physical processes could be programmed much like a digital computer is programmed, and provide a systematic methodology for nanostructure synthesis and control (MacLennan, 2002, 2003; von Neumann, 1966; Winfree, 1996). Computation applied to nanotechnology has different standards and tradeoffs from conventional computation. In traditional applications we want the computation to go as fast as possible (and to dissipate as little power as possible, and to store information as densely as possible), so the common goal is to move as little matter and energy as possible in each computational operation. (This progress can be traced from the movement of relay contacts and changes of magnetization in core storage, through vacuum tubes, discrete transistors, and CMOS, to a possible terminus in single-electron transistors.) We have strived to decrease the matter (and energy) of computation as much as possible in order to approximate pure (immaterial) form. However, if our purpose is to move matter and to create nanostructures with specified physical dimensions, then we may want our computations to move more matter rather than less. (Again, output transducers and actuators provide a familiar example from conventional computing, but here we consider the physical characteristics of the entire computation.) Similarly, we will want our computation to proceed at a rate compatible with its intended physical effect, even if that is slow by the standards of contemporary computation. Molecular computation, especially DNA self-assembly, provides one of the best contemporary examples of the computational control of matter for nanotechnological purposes (e.g., LaBean, Winfree & Reif, 2000; Reif, 1999; Rothemund, 2006; Rothemund, Papadakis, & Winfree, 2004; Rothemund & Winfree, 2000; Seeman, 1999; Winfree, 1998; Yan, Finkelstein, Reif & LaBean, 2003). Therefore IJNMC especially seeks papers reporting progress in molecular computation, but welcomes work on other computational and non-computational approaches to nanotechnology, as exemplified by the articles in this volume.

EMBODIED COMPUTATION This more intimate relation between information processing and physical processes is characteristic of embodied computation, which refers to the synergistic interaction of formal computation and its material embodiment (MacLennan, 2008, 2010, in press-a, in press-b; cf., Hamann & Wörn, 2007; Stepney, 2004, 2008). This concept is inspired by embodied cognition, an important recent development in cognitive science and robotics (Brooks, 1991; Clark, 1997; Iida, Pfeifer, Steels & Kuniyoshi, 2004; Johnson & Rohrer, 2007; Pfeifer & Bongard, 2007; Pfeifer, Lungarella & Iida, 2007; Pfeifer & Scheier, 1999), which addresses the critical role that the body and its physical environment play in cognition (in humans and

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other animals). One of the insights of embodied cognition is that there is much information that the brain does not have to represent because, in effect, the body and its environment represent themselves, and further that there are many information processing tasks that the brain does not have to carry out because they are effectively realized by physical processes in the body and its environment (Dreyfus, 1979, pp. 248–250, 253). As a consequence, the cognitive load on the brain is decreased enormously. Embodied computation generalizes this approach to all sorts of information processing and control. Thus embodied computation is an attractive strategy for post-Moore’s Law computing in that it supports a greater assimilation between computational and physical processes (MacLennan, 2008, in press-a). In embodied computation, many useful computational processes come “for free” as physical processes. For example, simulated diffusion has proved useful in a number of applications, including path finding, optimization, and constraint satisfaction (e.g., Khatib, 1986; Miller, Roysam, Smith & O’Sullivan, 1991; Rimon & Koditschek, 1989; Steinbeck, Tóth & Showalter, 1995; Ting & Iltis 1994); in effect it is massively parallel breadth-first search. However, simulating diffusion can be expensive on serial or modestly parallel computers, but it is simple to implement physically, and the parallelism comes for free as a consequence of the parallelism of the physical process. Therefore it is not surprising that nature exploits diffusion (of chemicals or the agents themselves) to solve complex information processing and control problems (e.g., Camazine, Deneubourg, Franks, Sneyd, Theraulaz & Bonabeau, 2001). Further, as mentioned previously, noise is unavoidable, especially at the nanoscale. We can view these stochastic processes negatively, as noise corrupting otherwise perfect representations, and which we strive to eliminate or mitigate, or we can “respect the medium” and exploit them as useful sources of free variability that can be applied to information processing (e.g., in stochastic resonance and simulated annealing: Benzi, Parisi, Sutera & Vulpiani, 1982; Kirkpatrick, Gelatt & Vecchi, 1983). Similarly, unavoidable “error” in the realization of idealized computational processes can be turned to our advantage. Again, nature is a useful model; for example, ants follow their trails imperfectly, and there is variability among ants in trail following, which maintains a certain degree of unbiased search and adaptability in their activity (Camazine & al., 2001; MacLennan, in press-b). In effect, this implements an adaptive control system that shifts resources between exploration (gathering information) and exploitation (using that information). Nature also provides informative examples of how the physical system may be its own representation, which are relevant to the application of computational ideas in nanotechnology. For example, stigmergy refers to the process wherein the “project” undertaken by one or more organisms embodies the information required to continue and complete the project (Camazine & al., 2001). The best-known example is wasp nest building (Bonabeau, Dorigo & Theraulaz, 1999). The partially completed nest itself provides the stimuli that guide the individual wasps in the construction process. Therefore there is no need for the wasps to have representations of the completed nest or of the current state of its construction, or to have an internal “program” for nest construction. In this way, relatively simple agents (with modest information processing capacity) can construct complex, functional structures. The greatest degree of integration between a computation and its realization occurs when the computation is not controlling some separate physical system, but rather is modifying or constructing the physical realization of itself. That is, the computer and the computation co-create each other. So stated, such a process might seem impossible, but it is the basis of embryological morphogenesis, in which embodied computation creates the physical substrate for later embodied computation. Cells signal each other in order to coordinate the creation and differentiation of new cells, which extend the morphogenetic process. Further, in later developmental stages, neural processes create the nervous system, including the brain.

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(Thus living systems are described as autopoietic, or self-making: Maturana & Varela, 1980; Mingers, 1994.) Similarly, in some DNA-based algorithmic self-assembly processes, molecular computation creates the physical structure that supports further computation and assembly (e.g., Barish, Rothemund & Winfree, 2005; Cook, Rothemund & Winfree, 2004; Rothemund & Winfree, 2000). Therefore we may conclude that the creation of complex hierarchical systems, with specific structures from the nanoscale up through the macroscale, and especially post-Moore’s Law nanocomputers, will require a close alignment of computational and physical processes. Thus, within the wider realm of nanotechnology and molecular computation, IJNMC seeks to publish research in the fertile ground where computation and nanotechnology overlap. Bruce MacLennan Editor-in-Chief, International Journal of Nanotechnology and Molecular Computation

REFERENCES Barish, R. D., Rothemund, P. W. K., & Winfree, E. (2005). Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Letters, 5, 2586–2592. Benzi, R., Parisi, G., Sutera, A., & Vulpiani, A. (1982). Stochastic resonance in climatic change. Tellus, 34, 10–16. Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm Intelligence: From Natural to Artificial Systems. Oxford, UK: Oxford University Press. Brooks, R. (1991). Intelligence without representation. Artificial Intelligence, 47, 139–159. Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, G., Theraulaz, J., & Bonabeau, E. (2001). Selforganization in Biological Systems. New York: Princeton University Press. Clark, A. (1997). Being There: Putting Brain, Body, and World Together Again. Cambridge: MIT Press. Cook, M., Rothemund, P. W. K, & Winfree, E. (2004). Self-assembled circuit patterns. In J. Chen and J. Reif (Eds.), DNA Computing 9 (pp. 91–107). Berlin & Heidelberg: Springer-Verlag. Dreyfus, H. (1979). What Computers Can’t Do: A Critique of Artificial Reason. New York: Harper & Row. Hamann, H., & Wörn, H. (2007). Embodied computation. Parallel Processing Letters, 17(3), 287–298. Iida, F., Pfeifer, R., Steels, L., & Kuniyoshi, Y. (Eds.) (2004). Embodied Artificial Intelligence. Berlin, Germany: Springer. Johnson, M., & Rohrer, T. (2007). We are live creatures: Embodiment, American pragmatism, and the cognitive organism. In J. Zlatev, T. Ziemke, R. Frank & R. Dirven (Eds.), Body, Language, and Mind, vol. 1 (pp. 17–54). Berlin: Mouton de Gruyter. Khatib, O. (1986). Real-time obstacle avoidance for manipulators and mobile robots. International Journal of Robotics Research, 5, 90–99.

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Kirkpatrick, S., Gelatt, C.D., Jr., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671–680. LaBean, T. H., Winfree, E., Reif, J. H. (2000). Experimental progress in computational by self-assembly of DNA tilings. In E. Winfree & D.K. Gifford (Eds.), DNA-based Computers V (pp. 123–140). Providence: American Mathematical Society. MacLennan, B. J. (1994). Continuous computation and the emergence of the discrete. In K.H. Pribram (Ed.), Rethinking Neural Nets: Quantum Fields and Biological Data (pp. 199–232). Hillsdale,NJ: Lawrence-Erlbaum. MacLennan, B.J. (2002). Universally programmable intelligent matter: A systematic approach to nanotechnology. In IEEE Nano 2002 Proceedings (pp. 405–8). Piscataway, NJ: IEEE Press. MacLennan, B.J. (2003). Molecular combinatory computing for nanostructure synthesis and control. In IEEE Nano 2003 (Third IEEE Conference on Nanotechnology) (p. 13_01 ff). Piscataway, NJ: IEEE Press. MacLennan, B. J. (2004). Natural computation and non-Turing models of computation. Theoretical Computer Science, 317, 115–145. MacLennan, B. J. (2008). Aspects of embodied computation: Toward a reunification of the physical and the formal (Tech. Rep. UT-CS-08-610), University of Tennessee, Knoxville, Department of Electrical Engineering and Computer Science. Retrieved July 28, 2010, from http://www.cs.utk.edu/~mclennan/ papers/AEC-TR.pdf MacLennan, B. J. (2010). Models and mechanisms for artificial morphogenesis. In F. Peper, H. Umeo, N. Matsui & T. Isokawa (Eds.), Natural Computing, Springer series, Proceedings in Information and Communications Technology (PICT) 2 (pp. 23–33). Tokyo: Springer. MacLennan, B. J. (in press-a). Bodies—both informed and transformed: Embodied computation and information processing. In Gordana Dodig-Crnkovic & Mark Burgin (Eds.), Information and Computation, World Scientific Series in Information Studies, Vol. 2. Singapore: World Scientific Publishing. MacLennan, B.J . (in press-b). Morphogenesis as a model for nano communication. Nano Communication Networks Journal, 2010. Maturana, H., & Varela, F. (1980). Autopoiesis and Cognition: The Realization of the Living (edited, R.S. Cohen & M.W. Wartofsky), Boston Studies in the Philosophy of Science, 42. Dordecht, The Netherlands: D. Reidel Publishing Co. Miller, M.I., Roysam, B., Smith, K.R., & O’Sullivan, J.A. (1991). Representing and computing regular languages on massively parallel networks. IEEE Transactions on Neural Networks, 2, 56–72. Mingers, J. (1994). Self-Producing Systems. New York: Springer. von Neumann, J. (1966). Theory of Self-reproducing Automata (edited & compl., A.W. Burks). Urbana, IL: University of Illinois Press. Pfeifer, R., & Bongard, J. C. (2007). How the Body Shapes the Way We Think — A New View of Intelligence. Cambridge, MA: MIT.

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Pfeifer, R., Lungarella, M., & Iida , F. (2007). Self-organization, embodiment, and biologically inspired robotics. Science, 318, 1088–1093. Pfeifer, R., & Scheier, C. (1999). Understanding Intelligence. Cambridge, MA: MIT. Reif, J. (1999). Local parallel biomolecular computing. In H. Rubin & D.H. Wood (Eds.), DNA-based Computers III (pp. 217–254). Providence, RI: American Mathematical Society. Rimon, E., & Koditschek, D. E. (1989). The construction of analytic diffeomorphisms for exact robot navigation on star worlds. In Proceedings of the 1989 IEEE International Conference on Robotics and Automation, Scottsdale AZ (pp. 21–26). New York: IEEE Press. Rothemund, P. W. K. (2006). Folding DNA to create nanoscale shapes and patterns. Nature, 440, 297–302. Rothemund, P.W.K., Papadakis, N., & Winfree, E. (2004). Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology, 2(12), 2041–2053. Rothemund, P. W. K., & Winfree, E. (2000), The program-size complexity of self- assembled squares. In Symposium on Theory of Computing (STOC) (pp. 459–468). New York: Association for Computing Machinery. Seeman, N. C. (1999). DNA engineering and its application to nanotechnology. Trends in Biotechnology, 17(11), 437–443. Steinbeck, O., Tóth, A., & Showalter, K. (1995). Navigating complex labyrinths: Optimal paths from chemical waves. Science, 267, 868–871. Stepney, S. (2004). Journeys in non-classical computation. In T. Hoare & R. Milner (Eds.), Grand Challenges in Computing Research (pp. 29–32). Swindon, UK: BCS. Stepney, S. (2008). The neglected pillar of material computation. Physica D, 237(9), 1157–1164. Ting, P.-Y., & Iltis, R.A. (1994). Diffusion network architectures for implementation of Gibbs samplers with applications to assignment problems. IEEE Transactions on Neural Networks, 5, 622–638. Winfree, E. (1996). On the computational power of DNA annealing and ligation. In R.J. Lipton & E.B. Baum (Eds.), DNA-based Computers (pp. 199–221). Providence, RI: American Mathematical Society. Winfree, E. (1998). Algorithmic self-assembly of DNA. Unpublished doctoral dissertation, California Institute of Technology, Pasadena. Yan, H., Park, S. H., Finkelstein, G., Reif, J. H., & LaBean, T. H. (2003). DNA-templated self-assembly of protein arrays and highly conductive nanowires. Science, 301, 1882–1884.

Section 1

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Chapter 1

Optimal DNA Codes for Computing and Self-Assembly Max H. Garzon The University of Memphis, USA Vinhthuy Phan The University of Memphis, USA Andrew Neel The University of Memphis, USA

ABSTRACT DNA has been re-discovered and explored in the last decade as a “smart glue” for self-assembly from the “bottom-up” at nanoscales through mesoscales to micro- and macro-scales. These applications require an unprecedented degree of precision in placing atom-scale components. Finding large sets of probes to serve as anchors for such applications has been thus explored in the last few years through several methods. We describe results of a tour de force to conduct an exhaustive search to produce large codes that are (nearly) maximal sets while guaranteeing high quality, as measured by the minimum Gibbs energy between any pair of code words, and other criteria. We also present a quantitative characterization of the sets for sizes up to 20-mers and show how critical building blocks can be extracted to produce codes of very high quality for larger lengths by probabilistic combinations, for which an exhaustive search is out of reach.

INTRODUCTION Until very recently, DNA has been considered just as primary storage of instructions for the makeup of living organisms. Over a decade ago, (Adleman, 1994) gave the successful demonstration of a DNA’s potential use for different purposes, more specifically, a solution to the Hamiltonian DOI: 10.4018/978-1-60960-186-7.ch001

Path Problem (HPP), a problem considered beyond reach for feasible solution on conventional computers. A computational equivalent of the well known traveling salesman problem (TSP), the Hamiltonian Path problem (HPP) calls for deciding the existence of a Hamiltonian path to traverse the edges a directed graph with two singled out as source and destination vertices. The problem calls for a Boolean decision whether there exists a Hamiltonian path joining the source

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Optimal DNA Codes for Computing and Self-Assembly

to the destination (i.e., a path passing through every vertex exactly once). Later in the 1990s, several authors (Winfree et al, 1998; Seaman, 1999; Benenson et al., 2001) realized a second, perhaps more important use of DNA, as a “smart glue” for self-assembly from the “bottom-up” of atomic and molecular components to form much larger and complex structures in the order of mesoscales and microscales. Even more recently, (Neel and Garzon, 2006; Garzon et al., 2003) have suggested yet a third use of DNA as a structure capable of indexing large data repositories in terascale memories searchable in feasible times, for example in the form of DNA microarrays or DNA chips (Draghici, 2003). All these applications require an unprecedented degree of precision in placing and operating on atomic-scale components, unachievable by traditional “top-down” assembly methods, such as microlithography. The specificity and selectivity of hybridization between (nearly) complementary small DNA oligonucleotides confers upon DNA a well recognized advantage to serve as a material to self-assemble such components. Several recent results have demonstrated the exquisite degree of control that DNA would offer on structures built on DNA scaffolds. Methods to find large sets of probes (or code sets, as will be referred to below) to serve as building blocks for such operations have thus been explored in the last few years. Although the problem of finding optimal sets has been shown to be computationally difficulty (NP-complete), even for very specific and very coarse measures of hybridization affinity (Phan and Garzon, 2008), a variety of methods have been tried to find such DNA codeword sets, as well as of techniques to measure their quality. These methods have produced good DNA codes that have sufficed for prototypes of the applications mentioned above. For example, (Phan and Garzon, 2008) used a new construction called shuffling that theoretically builds good codes of size nearly optimal (within a constant factor of the optimal set) from seed DNA codes of shorter length. The size of the codes is,

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however, sensitively dependent on the quality and size of the seed codes. The search for such codes is thus not only of theoretical importance, but also of practical significance since they are likely to be required for the full realization of self-assembly applications in practical settings. The gigantic size of the search spaces involved even for small lengths (exp(2,416) for 16-mers) is a primary road block for progress in this area, especially when nearly optimal performance is expected or desirable. In this paper, we describe results of a tour de force to conduct an exhaustive search to produce code sets that are large enough to be nearly maximal sets while guaranteeing high quality. The quality criteria and a summary of prior methods are described in Section 2. Section 3 shows how to solve the problem of actually producing the actual composition of the sets, unlike its counterpart in vitro. Consequently, we also present a quantitative characterization of these codes through an analysis of occurrence of shorter k-mer blocks. In Section 4 we summarize some of the implications of these analyses for the application of these code sets. Finally, we also discuss some of the remaining problems to these sets and mention two interesting problems for further research. We assume that the reader is familiar with basic facts about molecular biology; see (Watson et al., 2003; Wetmur, 1997) for background details. Some of the results presented here have been announced in preliminary or incomplete form in (Bobba et al., 2006) and (Garzon et al., 2006). Full results and complete analyses are presented here.

THE SEARCH FOR DNA CODES The smart part in the “smart glue” property of DNA is hybridization affinity (Watson et al., 2003). The fundamental difficulty with hybridization affinity is that it is an uncertain property determined not only by the specific sequence of oligonucleotides making up the strands, but also

Optimal DNA Codes for Computing and Self-Assembly

by a number of physical-chemical parameters such as temperature, salinity of the solvent, and other variables (Wetmur, 1997). Even if feasible models of hybridization affinity are found, finding optimal DNA codes for a given oligonucleotide length n remains a challenging problem because of the enormous size of the search space of all n-mers (exponential 4n) and the practically infinite size of possible candidate subsets (superexponential exp(2, 4n).) Originally, finding good code sets appears to be a straightforward extension of the analogous problem of finding error-correcting codes of maximal rate in information theory (Roman, 1995). Two differences make the problem much more difficult. First, the metric involved in determining the quality of the code, namely the Gibbs energy of duplex formation (SantaLucia et al., 1990) is not even a distance in the strict sense of the word, indeed not ever a metric remotely comparable to the Hamming distance, although the Hamming distance has been used to produce interesting small codes (Arita and Kobayashi, 2002). In fact, even finding a very large set of very short oligonucleotide (say 20-mers) is computationally difficult (Marathe et al., 2001). Recent efforts based on the thermodynamic estimate of free energy has only been able to produce sets of no larger than 400 strands of 12- to 20-mers (Tulpan et al., 2005). Second, the computation of the Gibbs energy is in itself a challenge (probably NP-hard in itself) which requires an approximation of its own. It was thus natural to turn to DNA itself for answers about good quality DNA codes. A method based on the Polymerase Chain Reaction has been implemented experimentally and shown to produce code sets of high quality in vitro (Chen et al., 2006; Bi et al., 2003; Deaton et al., 2002). The PCR Selection (PCRS) protocol of (Deaton et al., 2003) is described in some detail in the nest section. From a principled point of view, however, PCRS presents an obvious disadvantage, namely that the composition of the codes would require sequencing of each of the species available in the

product of the protocol, a massive operation in terms of time and cost for the estimated hundreds of thousands of species available even in the case of small oligonucleotides (20-mers). An alternative method to overcome this problem will be described in the following section.

The PCR Selection Protocol The Polymerase Chain Reaction (PCR) is a well established technique to amplify the number of copies of DNA strands by thermal cycles using a polymerase enzyme (Mullins, 2001). At low temperatures, DNA strands are prepared for copying by attaching primers (short ssDNA) to their 5’-end and 3’-ends. By supplying additional Taq polymerase and nucleotide bases, the ssDNAs will extend primers to form fully double stranded DNA (dsDNA). Heating the mix to a high temperatures then melts dsDNA and separates them into ssDNA. By repeating this basic round, new copies of original DNA are produced in high volume and accuracy in only a few cycles. The PCR Selection Protocol (PCRS) of (Deaton et al., 2002) is an in vitro protocol that was designed to make use of PCR to obtain good code sets from a seed set by adding a filtering phase based on evolution in vitro. In the experimental confirmation, the target strands in the seed set (in the prototype case, the full population of 20-mers) were flanked by carefully chosen 20-mer primers on both sides obtained using the model in (Deaton et al., 2002a). In the extension phase, the template strands that have not found sticking to (pseudo) complements are extended in a PCR cycle; in the filtering phase, those templates not amplified are kept in the set while the others are eventually filtered out of the set. After repeating this process for about four rounds, a code set of actual DNA strands has been experimentally confirmed to be obtained of high noncrosshybridizing (nxh) quality (Chen et al., 2006). Because the seed set is the largest possible and strands are only filtered out only when they have been caught crosshybridizing

3

Optimal DNA Codes for Computing and Self-Assembly

with other strands, the results obtained after an appropriate number of rounds can be expected to be a near maximal noncrosshybridizing set, not unlike the perfect codes well known in information theory with the Hamming distance (Roman, 1995) These conclusions have been confirmed in various experimental characterizations of their quality, which are described in detail in (Chen et al., 2006; Chen et al., 2004). It is truly remarkable how the PCR Selection protocol can thus perform an exhaustive search that cuts through the combinatorial explosion to produce good code sets in feasible time (a matter of days). It is worth pointing out that the protocol does obtain the code set in vitro, thus eliminating the cost of synthesizing a large number of strands in code sets designed by any conventional algorithmic means. On the other hand, this selection protocol (PCRS) presents some problems of its own in practice, some of which are addressed in this paper. First, the actual composition of the sequences and the quality of the full set cannot be subject to the kind of informatic analysis (Mount, 2001) and so be given the guarantees that designs and evolutionary methods in silico do provide. Even if one is willing to bear the cost, in terms of money, of sequencing a sample of the code set in vitro, it is forbiddingly infeasible in terms of time to do so for the whole set (an estimated 10,000-50,000 different species present in the final product of 20-mers, for example, according to (Chen et al., 2006).) What fraction of the original 420 20-mers has been retained? How to quantify on a more principled basis the noncrosshybridization quality of the set? Second, the flanking 20-mers used to enable PCR extension force an alignment of the strands and thus the codewords obtained are not guaranteed to be crosshybridization-free once the primers have been removed and hybridization in a frameshift is allowed, a situation likely to commonly occur in practical applications. In order to obtain a more principled understanding of these sets, we turn to analogous techniques in silico.

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Gibbs Energy Models Code search in silico presents two basic problems. First, how to quantify the hybridization affinity of two given oligonucleotides. Second, how to overcome the combinatorial explosion posed by the enormous sizes of the DNA spaces being searched, even for small oligonucleotides of 20-mers. In this subsection, we discuss the first problem and leave the second for the following Section. Basic facts about DNA indicate that the Gibbs energy of duplex formation is the most important factor in determining the hybridization likelihood between any pair of oligonucleotides. Of the various methods that have been proposed, we use in this section a computation based on a model that shortcuts the common dynamical programming algorithm in the nearest neighbor model by adding a penalty function for bulges (Deaton et al., 2002a). This model has been proven to work well in the solution of the encoding problem in vitro (Chen et al., 2006; Chen et al., 2004). The stringency conditions in the test tube (as determined by factors such as temperature, salinity, and kinetic effects) can be abstracted as an appropriate parameter tau representing the threshold under which hybridization will occur. We set this value at the standard of tau =-6 kCal/mole. Further discussion on this point can be can be found in (Garzon and Deaton, 2004). However, the dynamic programming algorithm still remains computationally expensive when dealing with large sets of oligonucleotides and simpler approximations are needed to reduce the search to a feasible task. Simpler approximations of the Gibbs energies have been introduced and rigorously analyzed by the authors (Phan and Garzon, 2008; Garzon et al., 1997), namely the h0 and the h-distance defined as follows for two n-mers x,y: h0(x,y) = min {H(x,y),H(x,y’)} hk(x,y) = mink {hk (x,y)},

Optimal DNA Codes for Computing and Self-Assembly

where hk = k + min { h0 (x,y[k]), h0 (x,y[-k])} (k>0), y’ is the Watson-Crick complement of y, H is the ordinary Hamming distance counting the numbers of identity matching bases in a perfect alignment of the arguments, and y[k] denotes the suffix (prefix, if k<0) after a shift of k positions off a perfect alignment with respect to x .The corresponding DNA spaces are metric spaces in the mathematical sense of the term, analogous to the standard Euclidean spaces found in elementary Calculus. This analysis shows the strong contrast in the topology of the corresponding DNA spaces compared to the Hamming spaces used in information theory to find error-correcting codes. Nevertheless, in these spaces, the problem of searching for DNA codes is analogous to the well known sphere-packing problem in ordinary Euclidean or Hamming spaces. Inclusion of any n-mer x as a codeword automatically excludes all other n-mers within a hybridization distance from x below tau, herein named the ball of radius tau centered at x. We are thus searching for code sets C of noncrosshybridizing n-mers of the largest possible size with the requirement that the minimum distance between every pair of n-mers is at least tau. Codes of high quality according to the h-distance can be further filtered by a Gibbs energy criterion, such as the well known nearestneighbor model (Deaton et al., 2001a; Santalucia, 1998; Santalucia et al, 1990) to produce codes of high quality for experiments in vitro.

SEARCH FOR DNA CODES IN SILICO Inspired by Adleman’s approach and the relatively high cost of molecular protocols, we have developed a computational environment to reproduce essentially equivalent systems in silico (Garzon et al, 2004). Software of this type, a virtual test tube (VTT) EdnaCo, was developed to understand biochemical reactions with DNA oligonucleotides for computational purposes (Garzon et al. 2004).

We next provide a high-level description of the software involved in the simulation. Further details can be found in (Blain and Garzon, 2004; Garzon, 2004). EdnaCo follows the complex systems paradigm (Bar-Yam, 1997) of entities (objects) interactions and emerging properties, i.e. instead of programming their entire behavior over time, only entities (originally DNA molecules) and individual interactions between pairs of them are programmed by the user. Conceptually, the VTT is spatially arranged as a 3D coordinate system in which molecules can move about. The tube moves molecules around to simulate motion of three different types, Brownian motion, according to a pre-determined schedule, or no motion at all. The entities are allowed to interact freely in a predetermined manner specified by the experimenter. Entities could be homogenous or heterogeneous and may represent any complex bio-molecules (herein referred to as DNA complexes.) Each molecule is located at a unique space coordinate at any given time. The VTT can also code for physical-chemical properties such as temperature, pressure, salinity, and pH that may vary across space and time and affect the way structures interact therein. Interactions between entities are programmed by the experimenter depending on the nature of the molecules being simulated. Multiple instances of an entity behave in the same manner, as a function of environmental factors such as temperature, pH, and such. All entities are capable of sensing the position of other entities up to a specified distance defined by a radius of interaction, a parameter in the simulation common to all entities. If two or more entities come within the interaction distance, an encounter is triggered between them. An encounter is resolved by appropriate software that may not affect the molecules at all (e.g., DNA molecules may not form a duplex if the temperature is too high), or may reflect an appropriate chemical reaction (e.g., formation of a DNA duplex and disappearance of the encountering single strands.)

5

Optimal DNA Codes for Computing and Self-Assembly

An interaction between two entities may be viewed as a chemical reaction or mechanical interaction between them. As a result of an interaction, existing entities may get consumed, their status may get changed, and/or new entities may, or may not, get created. Moreover, the concentration of entities may be manipulated externally by adding or removing entities to or from the VTT at any point of time. The running time of a simulation is divided into discrete time steps, or iterations. At every iteration, the state of the objects and the tube may change recursively, based on the current state resulting from previous changes, to reflect the interaction rules among themselves and/or with their environment. In the actual implementation of this conceptual model, EdnaCo is implemented by dividing the computer’s memory into a number of discrete segments, each running on a different processor. This way allows multiple interactions to take place at once. When entities move, they either change positions within a segment, or they may migrate across processor boundaries. Thus, the container of the VTT is a discrete 3D space residing inside a digital computer, the entities are instantiated as objects in classes in a programming language (C++), and the interactions are appropriate functions and methods associated with these objects. These methods either leave the objects unperturbed, or make the appropriate deletions and insertions to reflect the reactions they simulate. The communication between processors is implemented using a message-passing interface, such as MPI, on a cluster of personal computers or a high performance cluster (in our case, an IBM cluster containing 112 dual processors). The architecture of EdnaCo allows it to be scaled to an arbitrarily large number of processors and to be portable to any other cluster supporting C++ and MPI. The results of these simulations are thus guaranteed to be reproducible on any other systems running the same software as the original simulation. Further details of this simulation environment can be found in (Blain and Garzon, 2004; Garzon, 2004), while

6

examples of other applications can be found in (Neel and Garzon, 2006; Garzon et al., 2003). The striking property of the VTT is that the programming stops at the level of local interactions. Nothing else is programmed, every other observable is an emergent property of the simulation. Unlike other common simulations of biochemical phenomena (Schlick, 2002). In particular, if the molecules in Adleman’s original experiment are placed in the tube, they seem to be moved about randomly by Brownian motion. Nevertheless, encounters between vertex and edge molecules may create longer and longer paths and eventually a Hamiltonian path if one is possible, and indeed this is the case with a very high probability (reliability of 99.6% with no false negatives has been reported; see Garzon et al., 2004) In other words, the solution to an optimization problem has been captured in silico by a simulation of the most relevant properties of a natural phenomenon occurring inside a test tube containing DNA oligonucleotides with the appropriate characteristics. The scalability of this solution is clear in two different directions. First, the parallelism of EdnaCo provides the parallelism inherent in chemistry. Second, the size of problems solvable by this method is only limited by our ability to find sets of oligonucleotides large enough to code for the vertices and edges without causing any undesirable interactions between them. Although it was initially suggested that random encodings would prove sufficient (Adleman, 1994), further work had determined that care must be exercised in selecting oligonucleotides that are inert to crosshybridization with each other due to the uncertain and thermodynamic nature of DNA hybridization (Deaton et al., 1998). Although this problem has proven to be itself NP-complete for virtually every reasonable metric for hybridization (Phan and Garzon, 2008), similar methods can be used to provide nearly optimal solutions, both in vitro by the PCR Selection protocol of (Chen et al., 2006; Chen et al., 2004) and in silico by its simulation (Garzon et al., 2006). So-called DNA code sets,

Optimal DNA Codes for Computing and Self-Assembly

i.e., noncrosshybridizing (nxh) molecules) are now readily available to systematically solve large problems (order of tens to hundreds of thousands of noncrosshybriding oligonucleotides under 20-mers, for example) (Garzon et al., 2006), as described next.

Simulations of PCR Selection A simulation of the PCR Selection protocol was conducted on EdnaCo, as described above. In this section we present the results of an analogous exhaustive search in silico by running PCR Selection in simulation. The costs of this approach are only those associated with developing the required software, purchasing the required hardware, and the time required to run it. If the simulation is feasible to perform in silico, the full sequences of the code sets would actually be obtained and analyses could be performed of their composition and nature. The results in terms of knowledge of the actual composition of the set and their structure would be clearly superior, if it could be done. PCR Selection in simulation is not without its challenges. For example, the set of 20-mers contains about 1 trillion species and would hold a full terabyte of data if just one byte is associated to just one DNA species. Simulation of PCRS on a terabyte of data is no trivial task for any computer.

Therefore, it was necessary to do a preprocessing step in order to filter a number of a number of obvious undesirable candidates in a code set at the outset. Several filters were applied to the full seed set of n-mers in order to remove a priori those obvious candidate DNA species likely to be removed by PCR Selection. The filters remove from the full set (4n strands) those strands which are palindromes, contain hairpins (a folding of DNA onto itself), contain 4-mer runs of a single base, or are too close in h-distance to other strands already passed through the filter. Filtering the DNA sets can require from several minutes (for shorter strands) to several weeks (for larger sets.) The net gain is a filtered set (second bars) that is at least one order of magnitude smaller in size than the full set (left bars), as shown in Figure 1. The general fidelity of the software simulations has been verified and documented in (Garzon et al., 2004; Garzon et al., 2003) As a further validation of our simulation, we ran PCR Selection on a smaller subset of about 5,400 40-mers in order to determine the number of PCR rounds necessary to filter out all crosshybridizing pairs. The protocol converged to a stable product after four 4 rounds, which is in good agreement with the number of rounds considered to be necessary with runs of the analogous protocols in vitro. Figure 2 (left) shows that simulation of PCRS (referred to

Figure 1. Size of the PCR Selection product obtained from the filtered input sets (second bars in each group). The size of the resulting set (third bars) after filtering from the full seed set (left bars) can be estimated by the sub-power law 41.9sqrt(n)+0.3 (fourth bars). (Garzon et al., 2006)

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Optimal DNA Codes for Computing and Self-Assembly

Figure 2. Typical run of the simulation PCRS++ on a single processor (left). The code set stabilizes over 20 rounds. GC content of the PCRS++ products for all simulations (right). The average GC content hovers about 50% on the average and is always at least 45%. (Garzon et al., 2006)

as PCRS++) typically converges in practice in under 20 rounds to a stable set of 20-mers on filtered seed sets of about 5,000 species, which was heuristically determined to be the optimal value for parallelization of the job, despite the fact that, in principle, this randomized algorithm may take a very long time to converge to an ideal product subset of the seed set. Further validation of the results will be possible when a comparison can be made with the analogous characterizations of the product code sets to the available characterization of the products in vitro for 20-mers (Chen et al., 2006; Chen et al., 2004). Figure 1 also shows the results of filtering the full set of DNA n-mers of various lengths. The size of the remaining set of strands after filtering still grows exponentially with a power law exp(4, 1.9sqrt(n)+0.3). This suggests that at some point, filtering will yield DNA sets that still contain too

many DNA species. After projecting the curve forward, their intercept is estimated to be beyond DNA sets of length 60. This evidence does not suggest a limitation of our method since the Gibbs energy approximation of (Deaton et al., 2002a) breaks down beyond 60-mers. Rather, it suggests that filtering as described above may not be ideal for discovering a maximal set of strands of a specific length. Even so, a subset of strands obtained will far exceed the size of the best set know today, and by analogous arguments to the advantages of PCR Selection in vitro, produce sets that are of comparable order of magnitude to a maximal (perfect) code. Figure 3 shows the noncrosshybridizing quality of the code sets obtained by the simulation over a period of nearly 12 months, as measured by the pairwise Gibbs energy of strand pairs in the input filtered sets and across word pairs in the set

Figure 3. Quality of the PCRS++ seed sets before filtering (left) and product codes after filtering (right) as evaluated by the Gibbs energy model in (Deaton et al., 2002). (Garzon et al., 2006)

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Optimal DNA Codes for Computing and Self-Assembly

resulting from the PCRS products. The minimum energies are the most stringent criterion, and a value of -6 kCal/m is generally required to avoid crosshybridization between any pair of strands. Most sets obtained satisfy this condition on the average in under fifteen (15) rounds, and in the few that do not, an additional round of exhaustive filtering (not included in the results reported here for PCRS) was found to bring it to within this minimum criterion easily.

Characterization of Nearly Maximal Codes With the code sets in hand, we can proceed to do a number of analyses that would be impossible to perform if the results were obtained in vitro. For example, the quality of the sets in terms of GC-content (occurrence of single bases, i.e., GC– content across all words in the code) hovers about 46-50%, and the frequency of k-mer blocks for some values of k in the range 3-7 shows a distinct bias in the distribution of the blocks for all codes. This bias is typical and can be contrasted in Figure 4 in a comparison between the distribution in a 20-mer code and that of a code of the same size obtained by selecting the same number of 20-mer words at random. Certain larger blocks seem to occur with comparably the same highest frequency if the oligos

are long enough (for 13-mers, for example, GA, AG, AT, GG among 2-mers; GTA, GAG, GGT, GGA among 3-mers; and GAGG, AGGT, ATGA, AGGG among 4-mers.) The full distribution can be seen in Figure 5 in the range of 7 < n < 17. A closer analysis of these results indicates that the distribution of relative frequency of occurrence of the various blocks is a very important indicator of the noncrosshybridizing quality of a DNA code. To test this hypothesis, we used these blocks as a “combinatorial alphabet” to produce other codes of high quality by simple concatenation by the following method. For example, let’s suppose we want to build a 56-mer code with 500 words in it. A 56-mer consists of eight 7-mer blocks. The blocks were selected probabilistically from a subset Qf of 7-mer blocks with a probability of selection analogous to that given by the distribution in Figure 5 for 7-mer blocks in the 16-mer DNA maximal code. The words in each batch were selected from subsets Qf consisting of blocks occurring with a minimum frequency f depending on mu, the average Gibbs frequency among all 7-mer blocks, and std, the standard deviation of the same distribution (Figure 5), as described in Figure 6. The quality of the codes is directly related to the stringency in the choice of f. Thus, the best code is obtained with f = mu + 2std and has a Gibbs energies across pairs of codewords of min=-8.88 KCal/mole, ave=-0.63 KCal/m, and

Figure 4. Typical contrast between the distribution of k-mer blocks in a nocrosshybriziding (nxh) DNA codes of 20-mers obtained by simulation of the PCR Selection (left, k=3) and an analogous random code of 20-mers (right)

9

Optimal DNA Codes for Computing and Self-Assembly

Figure 5. Characteristic distribution of k-mer blocks occurring in nocrosshybriziding (nxh) DNA codes of size n obtained by simulation of the PCR Selection. protocol (PCRS) for n = 10, 13, 16 (top to bottom) and k = 4, 5, 6, 8 (left to right)

max=0.98 KCal/m, with a standard deviation of std=0.89 KCal/m. As mentioned above, typically, a maximum Gibbs energy of -6 KCal/m is considered sufficient for two strands to hybridize, so these codes can be considered “good” enough for a noncrosshybridizing code. Similar results were obtained for all the other codes generated by the same method using the probability distributions, as described in Figures 6 and 7. These results thus show that this new method can produce DNA codes of longer oligomers by shuffling the high frequency k-mer blocks occurring among code words for a given maximal code. For code words this long, an exhaustive search of the type described above for 16-mers is prohibitive or simply unthinkable. The codes can be produced in linear time in the size of the code set desired because the blocks and their frequency distribution are now known. We mention that this technique can also be combined with the shuffling

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technique in (Phan and Garzon, 2008) to produce longer and larger codes with the same blocks while preserving the quality of the codes. Whether this technique is applicable to any code of high quality, not just maximal codes, remains an open question.

CONCLUSION AND FUTURE WORK We have obtained nearly optimal sets of DNA codewords of very good noncrosshybridizing quality that are guaranteed to perform well under reaction conditions in wet test tubes by a tour de force, an exhaustive search that has lasted nearly 12 months (and yet has to bear full results for 20-mers). As a result, we have in hand or can produce a number of code sets of n-mers for values of n=4-10,13,16, 20. The search could be scaled for sets of longer oligonucleotides within

Optimal DNA Codes for Computing and Self-Assembly

Figure 6. Gibbs energy quality of four batches of ten DNA codes each consisting of 500 n=56-mers each obtained by probabilistic concatenations of 7-mer blocks in a set Qf of selected high frequency blocks occurring above a certain threshold f and sampled according to their frequency of occurrence in maximal 16-mer codes (shown in Figure 5.) Shown are statistics (maximum, average, minimum) of their energies, which on the average always remain in the noncrosshybridizing range (above -6 Kcal/m) in all four batches, although the lowest energies reveal that crosshybridizing words remain in all except the best batch 1. The quality of the codes is directly related to the minimum frequencies of the blocks: f = mu + 2sigma (top left, first batch), f = mu (top right, second batch), f=mu-0.5sig (bottom left, third batch) and f=minimum (no restriction, bottom right, fourth batch). The Gibbs energy model used is (Deaton et al., 2002a).

Figure 7. Gibbs energy of best code (top) and worst (bottom) codes in the four batches described above in Figure 6 (left to right, top to bottom.) The best (worst) of the batch is defined as the code with the largest (lowest, respectively) Gibbs energy among any pair of words in the code, as given by the model in (Deaton et al., 2002a).

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Optimal DNA Codes for Computing and Self-Assembly

feasible (but long) run times up to about 60-mers (when the Gibbs energy model used begins to fail), although now (near) maximal sizes will not be guaranteed. Arguments have been given that the size of these code sets may be in the order of magnitude of maximal sets. The method is constructive, not in the sense that it produces the actual coding wet strands as the PCR Selection protocol does in vitro (Chen et al., 2006; Chen et al., 2004), but in the complementary sense that the sequence and composition of the actual code words is now known, thereby bypassing a costly or impossible sequencing procedure on the product of PCRS in vitro. The availability of these codes and their characterization make possible realistic estimates of the scope of DNA structures for self-assembly and other applications. These codes support virtually any kind of application of biomolecular computing that requires the use of DNA molecules in a controlled way. They provide, for example, a good practical estimate of the capacity of a DNA memory (Neel and Garzon, 2006; Garzon et al., 2005). The searches have revealed a set of building k-mer blocks that can be used to produce “good enough” DNA codes of larger length by simple random concatenation very rapidly. As a result, careful analyses could be made of the structure and nature of the code and the abilities and limitations of storing information in DNA molecules. First of all, they provide some evidence that the size of the maximal codes seems to grow at an exponential rate O(csqrt(n)) for a given length n, for some constant c>1. Whether this result scales up for larger values of n is a question of interest of its own, either experimentally or theoretically. Second, the results of these analyses have furthermore identified a characteristic probability distribution in the occurrence of shorter k-mer (k near n/2) core of building blocks of frequent occurrence in high quality codes, which can be used to construct other codes of greater length by a shuffling operation, already known to produce nearly optimal results for arbitrary large codeword

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lengths (Phan and Garzon, 2008; Garzon et al., 2004). An exact theoretical characterization of this distribution also remains a question of interest for future work. Finally, these results also give some hope that structure of the Gibbs energy landscapes of DNA spaces of a given length (e.g. 20-mers) may be given a more geometric characterization, at least in some approximation, such as the h-distance.

ACKNOWLEDGMENT Partial support from the National Science Foundation grant QuBiC/EIA-0130385 is gratefully acknowledged. Special thanks to Russell Deaton (U of Arkansas) for making the C code for the Gibbs energy model available for the Gibbs energy estimates, as well as to Anne Condon (U of British Columbia) for making available the code to compute Gibbs energies used in the PCRS simulation in Section 3. We also thank Sujoy Roy for his help with the data and the simulations on the IBM HPC cluster at The U of Memphis.

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Bi, H., Chen, J., Deaton, R., Garzon, M., Rubin, H., & Wood, D. (2003). A PCR Based Protocol for In Vitro Selection of Non-Crosshybridizing Oligonucleotides. J. of Natural Computing, 2(4), 461–477.

Garzon, M., Blain, D., Bobba, K., Neel, A., & West, M. (2003). Self-Assembly of DNA-like structures in silico. Journal of Genetic Programming and Evolvable Machines, 4, 185–200. doi:10.1023/A:1023989130306

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Chen, J., Deaton, R., Garzon, M., Kim, J.-W., Wood, D.H., Bi, H., Carpenter, D., & Wang, Y.Z. (2006). Characterization of noncrosshybridizing dna oligonucleotides manufactured in vitro. J. of Natural Computing. Springer Netherlands 15677818, 165-181. Chen, J., Deaton, R., & Wang, Y.-Z. (2004). A DNA-based Memory with in vitro Learning and Associative Recall. Proc. 9th International Meeting on DNA based computing. Springer-Verlag Lecture Notes in Computer Science 2943, 145-156. Deaton, R., Chen, J., Bi, H., Garzon, M., Rubin, H., & Wood, D. H. (2002). A PCR-based Protocol for In Vitro Selection of Non-crosshybridzing Oligonucleotides. Proc. of the 8th International Meeting on DNA Computing. Lecture Notes in Computer Science, 2568, 196–204. doi:10.1007/3540-36440-4_17 Deaton, R., Chen, J., Bi, H., & Rose, J. (2002a). A Software Tool for Generating Non-crosshybridizing Libraries of DNA Oligonucleotides, Proc. DNA8. Springer-Verlag Lecture Notes in Computer Science, 2568, 252–261. Deaton, R., Garzon, M. H., Murphy, R., Rose, J., Franceschetti, D., & Stevens, E. Jr. (1998). The Reliability of DNA-based Computing. Physical Review Letters, 80(2), 417–420. doi:10.1103/ PhysRevLett.80.417 Draghici, S. (2003). Data Analysis for DNA Microarrays. Chapman & Hall/CRC.

Garzon, M., & Deaton, R. (2004). Codeword design and information encoding in DNA ensembles. Journal of Natural Computing, 3(33), 253–292. doi:10.1023/B:NACO.0000036818.27537.c9 Garzon, M., Neathery, P. I., Deaton, R. J., Murphy, R., Franceschetti, D., & Stevens, S. E., Jr. A New Metric for DNA Computing. Proc. 2nd Annual Genetic Programming Conference. J.R. Koza et al. (eds). Morgan Kaufmann, 230-237. Garzon, M., Phan, V., Roy, S., & Neel, A. (2006). In Search of Optimal Codes for DNA Computing. Proc. of DNA Computing, 12th International Meeting on DNA Computing, Springer-Verlag. Lecture Notes in Computer Science, 4287, 143–156. doi:10.1007/11925903_11 Garzon, M.H., Bobba, Kiran Neel, Andrew. (2003). Efficiency and Reliability of Semantic Retrieval in DNA-based Memories. Lecture Notes in Computer Science, 2943, 157–169. Garzon, M. H. (2004). In Paun, G., & Rozenberg, G. (Eds.), Biomolecular Computing in silico. Selected Collection of EATCS papers 2000-2003 (pp. 505–528). World Scientific. Garzon, M.H. and Yao, Han (eds.) (2008). DNA Computing. Proc. 13th International Meeting. Lecture Notes in Computer Science 4848, Springer-Verlag. Garzon, M. H., Blain, D., & Neel, A. (2004). Virtual Test Tubes for Biomolecular Computing. J. of Natural Computing, 3(4), 461–477. doi:10.1007/ s11047-004-2642-y

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Phan, V., & Garzon, M. H. (2008). (in press). On Codeword Design in Metric DNA spaces J. of Natural Computing. Springer Netherlands. Roman, J. (1995). The Theory of Error-Correcting Codes. Berlin: Springer-Verlag. SantaLucia, J. (1998). A unified view of polymer, dumbbell, and oligonucleotide DNA nearestneighbor thermodynamic. Proceedings of the National Academy of Sciences of the United States of America, 95(4), 1460–1465. doi:10.1073/ pnas.95.4.1460

Watson, J. D., Baker, T. A., Bell, S. P., Gann, A., Levine, M., & Losick, R. (Eds.). (2003). Molecular Biology of the Gene (5th ed.). New York: Benjamin Cummings. Wetmur, J. G. (1997). Physical Chemistry of Nucleic Acid Hybridization. Third Annual DIMACS Meeting on DNA Based Computers, American Mahematical Society Dimacs Series 48, 1-23. Winfree, E., Liu, F., Wenzler, L. A., & Seeman, N. C. (1998). Design and Self-Assembly of Two Dimensional DNA Crystals. Nature, 394, 539–544. doi:10.1038/28998

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 1-17, copyright 2009 by IGI Publishing.

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Chapter 2

DNA Hash Pooling and its Applications Dennis Shasha New York University, USA Martyn Amos Manchester Metropolitan University, UK

ABSTRACT In this article we describe a new technique for the comparison of populations of DNA strands. Comparison is vital to the study of ecological systems, at both the micro and macro scales. Existing methods make use of DNA sequencing and cloning, which can prove costly and time consuming, even with current sequencing techniques. Our overall objective is to address issues such as whole genome detection, fragment detection and sample similarity. Because our method is similar in spirit to hashing in computer science, we call it DNA hash pooling. To illustrate this method, we describe protocols using pairs of restriction enzymes. The in silico empirical results we present reflect a sensitivity to experimental error. Our method, performed as a filtering step prior to sequencing, may reduce the amount of sequencing required (generally by a factor of 10 or more). Even as sequencing becomes cheaper, an order of magnitude remains important.

INTRODUCTION Biologists often examine large and diverse populations of organisms (for example, molecules, microbes or plants). This is particularly the case in fields such as microbial ecology, which studies the interactions between living microorganisms (such as algae, or bacteria) and their environment. One of the most significant and challenging problems DOI: 10.4018/978-1-60960-186-7.ch002

in these areas of biology is to compare the species in different samples. This task is often made even more difficult by the fact that many “wild” organisms resist laboratory cultivation (and, thus, have unknown phenotypes and their genomes are unknown), or may be present in a population in relatively low numbers. The study of metagenomics has emerged in recent years (Zhou, 2003; Handelsman, 2004; Tringe et al., 2005; Huson et al., 2007) to perform what has been described as “environmental

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DNA Hash Pooling and its Applications

forensics”, including the quantification of relative abundances of known species, and the estimation of the number of “unknown” species in a given environment (Huson et al., 2007). The potential impact of this new field is huge, with applications ranging from medicine to agriculture and biotechnology. Furthermore, the insights gained will be of significant assistance in advancing our understanding of biodiversity in both new and familiar environments, such as frozen Antarctic lakes and the human gut (Handelsman, 2004). The rest of this article is organized as follows: We provide a brief overview of metagenomics, describing current techniques and practices. We discuss several shortcomings of existing methods, motivating the work presented in the following section. Here, we present our new method for population analysis, along with the results of in silico experiments. We conclude with a discussion of the implications of this work, and suggest possible future lines of enquiry.

AN OVERVIEW OF METAGENOMICS The term metagenomics was first used in print by Handelsman et al. (Handelsman et al., 1998) to describe the study of genetic information recovered from environmental samples. In line with the other “omics” (e.g., proteomics, metabolomics), the emphasis lies on sets of products and/or genes, the suffixes “-ome” and “-omics” being interpreted to imply totality or collectivity (Lederberg & McCray, 2001). According to Schloss and Handelsman, “metagenomics is the culture-independent analysis of a mixture of microbial genomes (termed the metagenome) using an approach based either on expression or sequencing” (Schloss & Handelsman, 2005). Only a tiny fraction (estimated at below 1%) of micro-organisms found in nature are amenable to isolation and culture for further study (Ammann et al., 1995). This means that we need alternative (“culture-independent”) strategies

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for studying the vast majority of microbes found in the wild, a significant number of which will be unknown to science (Ward et al., 1990). The possible implications of metagenomics are many and wide-ranging. As Handelsman argues, Microbiology has long relied on diverse methods for analysis, and metagenomics can provide the tools to balance the abundance of knowledge attained from culturing with an understanding of the uncultured majority of microbial life. Myriad environments on Earth have not been studied with culture-independent methods ... and they invite further analysis. Metagenomics may further our understanding of many of the exotic and familiar habitats that are attracting the attention of microbial ecologists, including deep sea thermal vents, acidic hot springs, permafrost, temperate, desert, and cold soils, Antarctic frozen lakes, eukaryotic host organs, the human mouth and gut, termite and caterpillar guts, plant rhizospheres and phyllospheres, and fungi in lichen symbioses. With improved methods for analysis ...and attraction of diverse scientists to identify new problems and solve old ones, metagenomics will expand and continue to enrich our understanding of microorganisms. (Handelsman, 2004)

Methods for Metagenomics Early efforts to characterise environmental samples used ribosomal RNA (rRNA). This class of RNA plays a major role in protein synthesis, and rRNA makes up at least 80% of the RNA molecules in a typical eukaryotic cell. The gene encoding one particular molecule - 16S rRNA - is non-coding - rather than “becoming” (or being translated into) a protein, the RNA molecule resulting from its transcription “assists” in protein synthesis. The gene is highly conserved, meaning that it has been preserved throughout the process of speciation through evolution (the implication being that it performs a fundamental role in basic

DNA Hash Pooling and its Applications

biochemical processes that are essential for life). This conservation across species gives 16S rRNA a degree of universality, since it is present in a diverse range of organisms. However, despite its highly conserved nature, 16S rRNA also has the property of hypervariability, which also makes it useful to biologists. The average 16S rRNA molecule is 1,500 nucleotides long, which gives ample space for variation between (groups of) organisms. The evolutionary distance between any two organisms can therefore be calculated by aligning their 16S rRNA sequences (for example) and calculating the number of differences (Pace, 1997). We can use these known gene sequences to identify microbes in an environmental sample. Any unknown variants of the gene must therefore correspond to unknown organisms. Early work on using 16S rRNA to study microbial ecology was performed by Pace and colleagues (Lane et al., 1985; Olsen et al., 1986) (see also (Head et al., 1998)), and further advances came with the development of the polymerase chain reaction (PCR) (Mullis and Faloona, 1987). The “universality” of 16S rRNA means that if we wish to amplify (and thus clone) DNA from a population of microbes, we may use a pair of sequences encoded within it as primers for PCR, without advance knowledge of which specific organisms are present in the sample (Medlin et al., 1988; Weisburg et al., 1991; A.L. et al., 1992). Although 16S rRNA sequencing is a powerful and well-used technology, doubts have been raised over its resolution and absolute applicability (Janda and Abbott, 2007). By amplifying a sample, we may “drown out” relatively rare sequences, may not differentiate between subspecies that have common ribosomal RNAs but differ elsewhere (perhaps by incorporating foreign DNA easily, as is the case for Bacillus subtilis), and will miss viruses (which do not contain the gene). Until recently, metagenomic analysis involved the extraction of DNA from an environmental sample, cloning of the DNA into a suitable “vector”,

insertion of the vector into a host bacterium and then screening the resulting transformed bacteria (Handelsman, 2004). Screening may occur on the basis of gene expression using microarrays (Zhou, 2003), using some other trait, such as antibiotic production (Schloss and Handelsman, 2003), or simply via sequencing. A combination of recent advances (which are largely computational) have changed the landscape substantially. Shotgun sequencing (Fleischmann et al., 1995) has been applied with great success to metagenomics. This method of sequencing randomly shears long DNA strands into shorter fragments, which are then individually sequenced and then reassembled in silico into a “consensus sequence”. The assembly of genomes from complex communities historically “demands enormous sequencing expenditure for the assembly of even the most predominant members” (Tringe et al., 2005). Because of this difficulty, borne out by initial studies by Tringe et al. (Tringe et al., 2005), the authors decided to employ an alternative, “gene-centric” approach that does not attempt to attribute genes obtained to any particular genome. They obtained their initial dataset by taking four sets of samples, one from agricultural soil, and three from whale carcasses. Samples were then partitioned into bacteria, archea or eukaryotes using PCR-amplified rRNA libraries. Genomic small-insert libraries were then shotgun sequenced from each sample (100 million base pairs from the soil and 25 million base pairs from each whale sample). These sequences, derived from different population members, were termed “Environmental Gene Tags” (EGTs), since they may encode regions of functional genes that are necessary for survival in a particular environment. Different environment types will exhibit unique EGT “fingerprints”, containing genes derived from many different genomes. The study showed that two whale carcasses, located 8000km apart; nontheless had very similar EGT patterns. Thus,

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one may determine the type of environment from this fingerprinting technique. Other studies of note that have applied shotgun sequencing to metagenomics include an examination of the ecology of the Sargasso Sea (Venter et al., 2004) and drainage in an acid mine (Tyson et al., 2004). For a brief introduction to the field and a review of recent work, the reader is directed to (Eisen, 2007). In a recent study, Huson et al. present an approach (which they call MEGAN, for MEtaGenome ANalyzer) (Huson et al., 2007) to the problem of genomic assembly in which the authors compare sequenced data to existing databases. Specifically, the set of DNA sequences obtained by random shotgun sequencing from the environmental sample is run against known sequences using BLAST. The resulting meta-data is then provided as input to the MEGAN package, which estimates and explores the taxonomical content of the data set. This may be a good technique to obtain the most abundant species in a sample, but will have difficulty locating rare sequences of interest. One of the themes of our proposed approach is to hunt systematically for signs of a genome of interest. Even for a relatively simple community study on the drainage region of an acid mine, roughly 15 million bases were sequenced in order to obtain the required metagenomic data (Tringe et al., 2005), at a cost of approximately $150,000. A soil study, requiring at least 50 million bases, might then cost half a million US dollars. Recent developments such as pyrosequencing (Ronaghi, 2001; Elahi and Ronaghi, 2004; Ahmadian et al., 2006) and other high throughput sequencing efforts have reduced the cost of sequencing substantially, facilitating large-scale metagenomics (Mardis, 2008), such as the recent study of catastrophic collapses of honey bee colonies (Cox-Foster et al., 2007). This emerging technology permits groups to consider for the first time, hugely ambitious projects, such as sequencing the human biome (the entire ecosystem of the human body) (Blow, 2008).

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These are still substantial efforts that attempt to understand an entire ecosystem. However, such efforts may not be necessary in order to answer potentially simpler questions such as: “What is in common between these two samples?” In the next section, we consider the notion of “pre-processing” a sample in order to reduce the complexity of any analysis.

SAMPLE PRE-PROCESSING Fortunately, sequencing is not always necessary as a first step. Molecular techniques that work at the whole sequence level may be used to reduce the initial complexity of a sample population. One tool commonly employed is “GC fractionation” (Holben and Harris, 1995), which works along the principles of a molecular “sieve”, sorting strands according to their relative GC content (guanine and cytosine being heavier than their counterparts adenine and thymine). This may be effective when trying to partition a sample into eukaryotic and bacterial sets, since eukaryotic DNA tends to have a much lower GC content (e.g, we selected two complete bacterial genome sequences, A (Escherichia coli K12) and B (Shigella boydii Sb227) for early studies; each of these had a GC content of roughly 51%, while the human genome is made up of around 45% GC and that of the mouse roughly 44%). However, such a relatively crude tool rapidly proves ineffective when dealing with shorter sequences, where we may only possess genomic fragments within our sample. For bacterial sequence A, when taking 200 random consecutive sequences of length 50,000, we obtained a GC content ranging from 46.7% to 53.3% with the 90% confidence interval ranging from 47.3% to 52.8%. Preliminary work on estimating the complexity of a heterogenous population of DNA strands (without using sequencing) is reported in (Faulhammer et al., 1999). This article, motivated in part by the authors’ earlier work on DNA-based

DNA Hash Pooling and its Applications

computing (Adleman, 1994), reports initial experimental investigations into the use of basic laboratory methods (combined with probability theory) to estimate the complexity of a tube of strands. Faulhammer et al. digested their initial tube with a set of restriction enzymes with recognition sites differing in sequence and of length 4≤k≤8. The contents of the tube were then visualised in a gel, and the number of distinct bands observed used to obtain an estimate of the number of different strands. This basic approach (the use of restriction enzymes to digest a population sample, followed by analysis of the fragment size) also underpins an early variant of the well-known technique of DNA fingerprinting (Jeffreys et al., 1985). Restriction fragment length polymorphisms (RFLPs) (Dowling et al., 1990) provide a technique by which organisms may be differentiated by comparing the patterns obtained by digesting a certain portion of their DNA. If two organisms differ in the distance between restriction sites, the length of the fragments produced will differ (i.e., be polymorphic) when the DNA is digested. However, this method is generally only useful when the population sample is relatively homogeneous (e.g., one wishes to distinguish between members of the same species). A related technique, involving amplification of 16S rRNA followed by restriction enzyme digestion, has been used to detect pathogens in spinal fluid (Lu et al., 2000) and characterize the diversity of model communities (Liu et al., 1997). Our proposed technique uses some of the same basic laboratory methods, but it differs from all of the previous approaches in several important ways: 1. We avoid reliance on 16S rRNA. 2. Depending on the exact problem, our process can be simulated in silico or performed in vitro. 3. We use multiple levels of restriction enzyme digestion to achieve reasonable purity.

In the rest of this article we present and evaluate a simple and powerful technique called DNA hash pooling. We conclude with a discussion of plans for future theoretical and experimental work.

DNA HASH POOLING In computer science, hashing (Knuth, 1998) maps a relatively small set from a large domain (e.g., 10,000 integers ranging in value from 0 to one billion) to a small domain (e.g,. the set of integers from 1 to 5000) through a mathematical hash function. Applications of hashing include cryptography, error correction, authentication and identification. A typical hash function is modulus (i.e., remainder). For example, 7 mod 5 = 2 because 2 is the remainder after dividing 7 by 5. For the same reason, 28 mod 5 = 3. A hash data structure based on “mod 5” will map 28 to bucket (or pool) 3, 7 to pool 2, 12 to pool 2, 59 to pool 4, and so on. There are many variants of hashing, some of which entail hashing each pool resulting from the first hash function in order to get “purer” pools, and then using the combined hash results to generate an item “label”. For example, using a second hash function, based on “mod 7”, 28 would map to 0 and 53 to 5. Thus the full “label” of 28 would be (3, 0) because 28 mod 5 = 3 and 28 mod 7 = 0. By contrast, the label of 53 would be (3, 4) because 53 mod 5 = 3 and 53 mod 7 is 4. Associated with each unique label is a pool having a relatively small number of distinct values. DNA hash pooling or hash pooling for short is the analogous operation on DNA. The “hash functions” in this scenario correspond to biological operations that give rise to distinctive and quantifiable “fingerprints” (e.g., measurement of GC content followed by digestion by a set of restriction enzymes). The label components correspond to the “values” obtained by application of the hash functions (e.g. GC content and fragment length).

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DNA Hash Pooling and its Applications

In silico, our method involves simulating these operations on known sequences (typically though not necessarily of entire genomes) and characterizing different portions of those sequences from the result(s). In vitro, our method involves performing the bench-based operations and sequencing only those pools that are likely to be of interest. For concreteness, this article focusses on hash pooling based solely on restriction enzymes. This focus is for illustration purposes only. Any experimental technique that separates portions of DNA deterministically into buckets would suffice, including partial sequencing and hybridization. The basic operations for restriction enzymes are: 1. Apply a six base-pair (base pair) restriction enzyme to a sequence, yielding a set of fragments. 2. Partition those fragments based on length (perhaps approximately), using a technique such as gel electrophoresis. 3. Apply a four base pair restriction enzyme to a selected subset of partitions and separate on length again. 4. Sequence selected lengths. Each resulting pool is therefore associated with a label consisting of two lengths, the first based on a six base pair restriction enzyme and the second based on a four base pair restriction enzyme (Figure 1). The procedure may be described for K stages in pseudo-code as follows: hash(stage j, sample s, label L, K) r: = restriction enzyme for stage j frags:= apply r to s mysamps:= partition frags on length for each t in mysamps tlabel:= L concat (r, length(t)) if (j < K) hash(j+1, t, tlabel, K)

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else (t, tlabel) is a member of the final pool end for

Pseudo-code for K stage hash pooling based on restriction enzymes. The initial call on an initial sequence orig would be hash (1, orig, null, K), where null is the empty label. For example, consider the genomic sequence of bacterium A (E. coli K12). If we cut A using the enzyme SmaI (recognizing CCCGGG), take the pool corresponding to length 264, cut that pool with RsaI (GTAC) and take the pool of length 32, we get a pool having label (264, 32). It happens to have a single member with the sequence CTATCCGCTCAATGAGTCGGTCGCCATTGCCC. By contrast, the pool with label (770, 207) has three different sequences. For some applications, we will want pools having singletons (i.e., a set with only a single element) in order to obtain a pure sequence without the need for cloning. One may object that separating fragments by length entails a certain inaccuracy imposed by the laboratory technique; a reasonable estimate of this error may be plus or minus 10 base pairs, if we assume the use of specific sorting techniques (Heiger et al., 1990). In this case, in order to obtain a pure sample, we may be interested in finding a

Figure 1. Two-stage hash pooling

DNA Hash Pooling and its Applications

pool whose label has no “10 base pair-neighbors”. The labels L and are 10 base pair-neighbors if (i) the first component of L and the first component of are different but differ by 10 or less; or (ii) the first component of L and are the same but the second components differ by 10 or less. For E. coli K12, the labels (188, 59) and (188, 106), for example, have no 10 base pair-neighbors. Experimentalists have told us different stories about accuracies. Some think that the 10 base pair criterion is reasonable provided one works carefully and slowly. Others think it is not. We have rerun some of the experiments below assuming that we could measure lengths up to an accuracy of 50 base pairs. The results do not substantially change.

EXPERIMENTS Having presented our formal framework, we can now present several applications and our in silico empirical results.

Genome Detection The first question we ask is the following: given a tube, T, of unknown DNA (perhaps from an environmental sample) and a genome whose sequence is known, are “reasonably sized” porFigure 2. Genome sequence detection

tions of that genome present in T, even if in small concentrations? (Figure 2.) A “reasonably sized” portion is a sequence of length at least 200,000 base pairs (or roughly 5% of the length of a bacterial genome.) This might be used for the detection of bacterial pathogens in food, for example. In what follows, we used bacterium A, E. coli K12, which is often used as an indicator organism in the detection of facal contamination.

Genome Detection A Our in silico experiment involved the following steps: 1. Compute the candidate set of A, consisting of possibly non-singleton pools having no 10 base pair neighbors. There were 3,567 candidates. This gives us a “comparison library” of pools. It is important to note that this step is purely computational and can be computed just once for any combination of known genome and restriction enzyme set. 2. We simulate the unknown sample T by taking a 200,000 consecutive base pair subsequence of A (with the start position taken uniformly at random) and combining it with a sequence of length four times that of A (generated pseudo-randomly to have the same GC content as A). 3. We then compute the resulting candidate set of hash pools, based on no 10 base pair neighbors. Having tried this 20 times, we found, on average, 2,000 pools in the second candidate set. On average the size of the the two sets had common labels only 71 times. When labels were equal, 99.8% of the time there was a match of the sequence and the sequence came from that 200,000 consecutive base pair subsequence, giving a precision of 99.8% (only 3 times out of 1436 were the labels the same but the sequences dif-

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DNA Hash Pooling and its Applications

ferent over the 20 attempts). Of the 200,000 base pairs in the common sequence, more than 24% are found (standard deviation 9%). This required sequencing under 50,000 base pairs on the average with a standard deviation of under 20,000. Since the test sequence is over 18 million base pairs long, this is more than a factor 100 reduction in necessary sequencing. When applying this in a laboratory setting, there is the significant question of whether this operation may require many separate DNA extractions and applications of a restriction enzyme. Fortunately, the answer is no. For each of the 20 tests, first the six base pair restriction enzyme was used. This gave a collection of fragment lengths. On average only 5.8 of those lengths had no 10 base pair neighbors and had lengths similar to the lengths of the candidates from A (typical lengths were between 7,000 base pairs and 39,000 base pairs). So on average only 5.8 fragment lengths required extraction. Of those, 4.7 (on average) yielded matching sequences. So, if this were done in vitro, approximately 70 common strands would be found using one application of SmaI and under six applications of RsaI. Virtually all (99.8%) tested strands would be shown to be equal.

Genome Detection B A variation of experiment Genome Detection A is to embed the 200,000 consecutive base pair subsequence of A into a related bacterium B (Shigella boydii Sb227). In that case, when labels were equal, 33% of matching labels (1921 out of 5877 matching labels) led to matching sequences among the 200,000 consecutive base pair subsequence. This method gives reasonable coverage: an average of 62,000 base pairs among the 200,000 base pairs are covered (standard deviation of 11,000). Total sequencing costs were again modest: 146,000 on the average with a standard deviation of 10,500. As a further variation, consider the case where the 200,000 consecutive base pair subsequence of A to be placed in B as above, but length measure-

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ment is accurate to only 50. This means that we would not consider labels unless they had no 50 base pair neighbors. This gave 100% precision, but found only 136 matching labels. The coverage was 4.3% (8,600 base pairs found) with a standard deviation of 3.1. Only 8,600 base pairs were sequenced as well with the same standard deviation. This shows that less accurate read lengths can reduce coverage but preserve (in fact increase) precision. This experiment shows that in silico hash pooling on a known genome can identify pools to look for in a sample, such that those pools have a strong likelihood of containing a subsequence of the known genome. Thus, we can see this method as an improvement over random sampling, and can be used even if the bacterium of interest is relatively rare in the sample.

Sequence Query Here we address a closely related question to our first experiment: given a query sequence, is that sequence present, at least in part, in the tube? This might be used to look for the presence of a pathogen, for example. In fact, an experiment similar to that used to address the first question allows us to measure the effectiveness of our approach. Suppose the sample under scrutiny contains A plus a lot of other assorted DNA (e.g. the full genome of A amongst a pseudo-random sequence four times the length of the A sequence and having the same GC content). Then, 20 times, we take a random query subsequence of length 200,000 from A and see if we can find matching parts in the sample tube. The sample tube (A sequence plus a random sequence four times A in length with no 10 base pair neighbors) has 3,516 candidate pools. The average 200,000 base pair subsequence of A has about 200 candidates. In our 20 experiments, whenever two labels are equal, the corresponding sequences matched 100% of the time (precision of sequence matching given label match of 100%).

DNA Hash Pooling and its Applications

This is not guaranteed to hold always of course, but again shows that even without sequencing one can be quite sure that sequences will match if labels match. As in the first experiment, the six base pair restriction enzyme would cut the fragments into certain lengths, but, on the average, only 2.3 of those lengths (ranging from 10,000 base pairs to 30,000 base pairs) would have the properties that (i) they had no 10 base pair neighbors and (ii) they matched the candidates from the 200,000 base pair query sequence. Thus, on the average, under three extractions need to be taken and then digested by the four base pair restriction enzyme. Consider now the negative case when the query sequence was nowhere present in the sample, On the average, after cutting with the six base pair restriction enzyme, on the average, under one of those lengths had the properties that (i) they had no 10 base pair neighbors and (ii) they matched the candidates from the 200,000 base pair query sequence. When extracted and digested by the four base pair restriction enzyme, there were no matching labels (other than a single label whose final fragment length was only 4). So this technique does not throw up false positives.

Similarity Discovery in the Field Here we consider the problem: given two tubes of DNA, do they contain strands that are the same or very similar? This might be useful when comparing samples of unsequenced genomes, but rRNA genes are not enough. In this case, we cannot compute candidate pools that have no 10 (or 50) base pair neighbors using known genomes. Instead, we have to measure them. Sometimes we may not know whether the sample contains known sequences. If it does, we can use the techniques in the Subsequence Detection subsection above to find out which known genomes each sample contains and then see which are the same. Let us assume, however, that the sample contains no known genomes (or that we want to

detect commonalities besides those among known genomes). Our strategy will be to choose the most likely pairs to study by focussing on “unusual” labels (Figure 3). We therefore performed the following in silico experiment 20 times: 1. Take a 200,000 base pair sequence, target, with the same GC content as A, plus a random sequence four times the size of A (4×4.7Mb=≈20Mb), with the same average GC content as A. 2. For the second sample, we use the same 200,000 base pair sequence target plus another random sequence four times the size of A, with the same average GC content. Thus the target in each sample is 200,000 base pairs long, just 1% of the roughly roughly 20 million for the entire sequence present in each tube. In which pools should we look for common strands? That is, is it better to look at pools where the six base pair restriction enzyme has cut strands of length approximately 4,000 (the expected value) or much longer? The in silico answer is obvious in retrospect: to find common strands, the best pools to look at are ones corresponding to long lengths when cut by the first restriction enzyme. Thus the procedure is this:

Figure 3. Sample comparison

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DNA Hash Pooling and its Applications

1. Cut each sample with the six base pair restriction enzyme, then find all lengths that are the same (within an accuracy of 10 base pairs). 2. On the upper quartile of those lengths (approximately 235 of them), apply the four base pair restriction enzyme. Of those fragments that have the same lengths (within 10 base pairs) for both the first and second restriction enzymes, between 4% and 7% are the same sequence over the 20 experiments that we tried. Other quartiles are about a factor of 10 less good. An even better approach is to look at dectiles. Considering only the upper 1/10 of the lengths from the 6-base pair restriction site (from 10,210 to 24,550) gives a hit ratio of 13% on the average (standard deviation of 2%) and only 94 lengths (standard deviation under 1) from the six base pair restriction enzyme require an application of the second restriction enzyme. The upper dectile covers an average of 18% of the 200,000 base pair common string (standard deviation 6%). The upper dectile requires sequencing only about 330,000 bases on the average (standard deviation 18,000). Since sequence A is 4.5 million base pairs by itself, this represents more than a factor of 10 reduction in sequencing. If we have already identified known genomes that the two tubes share in common, then we should avoid labels that correspond to those. This allows us to avoid resequencing known commonalities.

Field Similarity B As above, we take a 200,000 base pair sequence, target, with the same GC content as A, plus a random sequence four times the size of A (4×4.7Mb=≈20Mb), with the same average GC content as A. For the second sample, we use the same 200,000 base pair sequence target, but embed it into bacterium B. The same method works. In this case, the upper dectile yields a percentage of

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hits (successful sequence matches/label matches) of 62% on the average with a standard deviation of 16%. That is, the high dectiles gives even higher precision when embedding the target in sequence B (Shigella boydii) than in a random sequence. This comes at a cost of coverage. The upper dectile covers an average of only 12% of the 200,000 base pair common string (standard deviation 4%). The upper dectile requires sequencing only 70,000 bases on the average (standard deviation 14,000). This is a factor of 100 reduction of sequencing compared with sequencing the two genomes. When the samples are large (contain many different genomes) and the results of the cuts by the larger restriction enzymes show no separation, the best labels are the ones that have few neighbors (the long cuts from the 6 base pair restriction enzymes). To determine whether a pool from one sample might match a pool having the same “good” label as one from the other label, one could hybridize the two samples. If they match, then it is worthwhile to sample them.

CONCLUSION DNA hash pooling is a method to simplify many problems in metagenomics. It gives the experimenter the ability to query for known sequences and genomes in a sample or to find common sequences from unknown genomes in two or more samples even if the identified sequences are rare. The version of the technique described in this article involves a small number of steps of the form: extract DNA of a certain length, apply a restriction enzyme to it, and measure the lengths of the results. In most cases, a few cuts with restriction enzymes can reduce sequencing to the best candidates. The main future work we anticipate is to validate the technique and then extend the method as new application scenarios present themselves.

DNA Hash Pooling and its Applications

ACKNOWLEDGMENT Shasha’s work has been partly supported by the U.S. National Science Foundation under grants IIS-0414763, DBI-0445666, N2010 IOB0519985, N2010 DBI-0519984, DBI-0421604, and MCB-0209754. This support is greatly appreciated. We are also grateful to Laura Landweber, Rob Knight and David A. Hodgson for helpful advice and discussions.

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This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 18-32, copyright 2009 by IGI Publishing.

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Chapter 3

Cellular Nanocomputers: A Focused Review

Ferdinand Peper National Institute of Information and Communications Technology (NiCT), Japan Jia Lee Celartem Technology Inc., Japan Susumu Adachi National Institute of Information and Communications Technology (NiCT), Japan Teijiro Isokawa University of Hyogo, Japan

ABSTRACT Cellular Automata have their roots in von Neumann’s research on self-reproduction, but since their debut they have been used for a much wider variety of purposes. In recent years they have attracted attention as architectures for nanocomputers–computers to be realized by nanotechnology. Their highly regular structure is considered an important advantage in this context, because of the potential for fabrication by bottom-up techniques like molecular self-assembly. This article gives an overview of research on cellular automaton-based nanocomputers, and discusses their strong points and challenges.

INTRODUCTION Research into nanocomputer architectures have increasingly attracted attention in recent years, driven by the realization that improvements in integration densities can only be sustained at an unchanged pace if new approaches are adopted. As top-down fabrication methods like optical lithography are gradually facing their technological and economical limits, alternatives are

called for. Bottom-up fabrication methods are still in their early stages of development, but, being based on the self-assembling properties inherent in molecules, they offer much promise for nanocomputers. With expected changes in fabrication method, there will also be changes in the architectures of the resulting computers: it is unlikely that the complicated structures of von Neumann computer architectures can be produced by top-down methods in the nanometer-scale regime. Rather, future computers are expected to

DOI: 10.4018/978-1-60960-186-7.ch003 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

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have very regular, repetitive, structures, or–alternatively–random structures. How can these structures be employed for meaningful computational tasks? It will require the allocation of hardware resources to tasks according to some directives or algorithm. When hardware needs to be (re)configured for a computation, regular structures offer a profound advantage over random structures, because regularity provides us with information about a structure, which translates into more control. Though random structures are easier to fabricate than regular structures, they are harder to reconfigure in a controlled way, as a result of which configuration will likely be limited to an initial phase, directly after fabrication (Stan, Franzon, Goldstein, Lach, and Ziegler, 2003). Since regular structures do better in this regard, this is a key motivation for the research on cellular automata for nanocomputer architectures. This article gives an overview of such research. After giving an informal definition of cellular automata, we discuss the complexity of cells in such architectures–an important issue, because of its relation with the efficiency of physical implementations. This is followed by a short overview of cellular automata used for VLSI implementations of specific applications. General-purpose cellular automata take over the remainder of this article, claiming their place as architectures that have a significant potential for manufacturing by bottomup techniques. After a short history on cellular automaton based nanocomputers, we describe our recent work on asynchronous cellular automata in this framework. These models carry the local character of cellular automata one step further by relaxing the need for all cells to be timed by a global clock signal. We discuss how the randomness of asynchronous updating is not necessarily an impediment to achieve a deterministic computation process, and how it can be efficiently used. Fault- and Defect-tolerance will also be discussed, before we finish with a future outlook on cellular automaton based nanocomputers.

CELLULAR AUTOMATA A Cellular Automaton is a collection of finite automata (each called a cell) organized in a regular grid (called a cellular space), which may be of any finite number of dimensions. The cells are usually displayed in the form of tiles in the case of 1-dimensional or 2-dimensional cellular automata, and in the form of cubes in the case of 3-dimensional cellular automata (Imai, Hori, & Morita, 2002). Each cell can be in one of a finite number of states from a state set. The states of cells are updated according a Transition function, which takes as input the states of the cells in the cell’s Neighborhood at time t and results in the updated cells having new states at time t+1. Well-known neighborhoods in cellular automata are the von Neumann neighborhood–consisting of the cells up to a certain orthogonal distance from a cell– and the Moore neighborhood–consisting of the cells up to a certain orthogonal or diagonal distance from a cell. Some definitions in literature do not include the center cell itself in the neighborhood. A transition function is usually described by a rewriting system consisting of a set of update rules. A typical rule has as its Left-Hand-Side (LHS) the states of a cell and its neighborhood, and has as its Right-Hand-Side (RHS) the updated state of the cell. An important feature of cellular automata is whether they are totalistic or not. A totalistic cellular automaton uses the counts of the number of cells in certain states in a cell’s neighborhood as the guideline to determine the cell’s update. In the Kaleidoscope of Life cellular automaton (Adachi, Lee, Peper, & Umeo, 2008), for example, a cell’s state is updated to 1 if there is a total of four cells having state 1 in a distance-2 Moore neighborhood from the cell (excluding the cell itself), otherwise the cell’s state becomes 0. Another–better known, but slightly more complicated–totalistic cellular automaton is the Game of Life (Conway, Guy, & Berlekamp, 1982). Cellular automata derive their popularity from their local character. The update of a cell’s state

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is only dependent on the states of the cells in the neighborhood at each time step, and not of any other cells’ states. This makes them very suitable models not only to study certain physical phenomena, but also for hardware implementations of computers, since their local character allows interconnection wires to be short. When all cells are updated at the same time, the cellular automaton is called synchronous. A synchronous mode of updating requires a temporary storage of the cell space to retain the old states of cells while the cells are updated. Since all cells of a synchronous cellular automaton are updated at each time step, there should be update rules defined for all possible combinations of states that can occur in the neighborhoods of the cells. The opposite of synchronous updating is asynchronous updating. In asynchronous updating, one cell is randomly selected at each time step as a candidate to be updated. The update is conducted according to a certain update rule if the states of the selected cell’s neighborhood match with the LHS of this rule. If no rule is found that matches, no update is done to the cell. Asynchronous updating does not require a temporary storage of the cellular space. Rather, each cell is updated independently from the others, with its state becoming immediately available as input to the updates of other (neighboring) cells. There may also be mixed versions of synchronous and asynchronous updating methods. For example, cells may be divided in two groups, like in a chessboard pattern, whereby in each step only the cells in one group are updated, followed in the next update step by the cells in the other group. Yet another update method would, for example, randomly select a set of cells in each step, all of which would be updated simultaneously. The prevailing method in most cellular automaton models is synchronous updating.

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COMPLEXITY OF CELLS To be competitive with CMOS-based computers, nanocomputers need to be scalable by several orders of magnitude beyond CMOS, and they should be competitive in terms of speed, energy consumption, etc. Though the extent to which cellular automaton-based nanocomputers satisfy these requirements depend on the actual technology used, there appears to be no fundamental reason why they would not qualify in principle. Their regularity potentially allows the manufacturing of huge numbers of cells, probably even extending beyond the 1014 range, which offers unprecedented availability of parallel computational resources. This massive parallelism is likely to more than compensate for the overhead of cellular automata, which may exceed a factor 10. Compared to a VLSI circuit that is optimized with regard to the number of devices, a cellular automaton with similar functionality fares worse, but it can afford this overhead if its competitors fail to deliver the above huge number of devices. Complexity of cells will be an important factor determining the success of cellular automata as nanocomputer architectures, because it is directly related to the efficiency of physical realizations. The complexity of a cell is not only determined by its number of states, but by the number of update rules describing the transition function. Centralized storage of a table of update rules is no option if efficient physical implementations are aimed for, because it violates one of the key tenets of cellular automata, i.e., locality. That leaves the size of the rule table as the main factor determining the complexity of cells. An attractive alternative to storage of the rule table in each cell is to find an efficient representation through inherently available physical interactions implied by the structure of a cell, but this requires update rules to closely reflect physical reality–a restriction that may be hard to achieve. How do we measure a cell’s complexity? A straightforward way is to count the number of

Cellular Nanocomputers

bits needed to represent the information about cell state and update rules. It may be important, though, to consider the overall costs, since compression of this information would result in lower bit costs, while it would require decoding hardware in a cell, which increases complexity. Absent a concrete physical implementation, it tends to be easiest to settle for a cost measurement in terms of the number of bits to represent the cell states added to the number of bits to represent the update rules, under the assumption that decoding of this information is done through a straightforward and simple scheme. The asynchronous cellular automaton of Lee, Adachi, Peper, and Morita (2003), for example, has cells that can be in five states, which require three bits to represent. The von Neumann neighborhood of a cell in this model requires each transition rule to encode in its LHS the states of a cell itself and the states of four neighboring cells, and in the RHS the new state of the cell. Given that there are 58 transition rules in this model, this results in a total of 3 + 58 × 3 × (1 + 4 + 1) = 957 bits that are required to encode the functionality in a cell. A similar calculation yields 100 bits of information for the asynchronous cellular automaton of Peper et al. (2004) and 62 bits of information for the Brownian cellular automaton of Lee and Peper (2008).

SPECIAL-PURPOSE OR GENERAL-PURPOSE? Though cellular automata were initially proposed with computational universality in mind (Neumann, 1966), they have since found frequent use for specific applications. Such applications have in common that they can be matched efficiently to the regular and localized structure of cellular automata. Image processing is especially suitable for cellular automaton-based algorithms, since images can be easily mapped to the cellular space of a 2-dimensional cellular automaton. These

kinds of applications started in the mid-1950s and have continued for several decades after that. Examples include medical image processing (Preston, Duff, Levialdi, Norgren, & Toriwaki, 1979) in which operations like filtering, thinning, and skeletonizing are conducted by (small-scale) cellular automata. More recent has been the use of cellular automata to watermark images (Mankar, Das, & Sarkar, 2007). Cellular automata are considered an attractive hardware structure for VLSI implementation in this application because of the simplicity and homogeneity of the cells, the huge parallelism, and the constant distance of communication wires between cells. The regular structure of cellular automata also forms an ideal base for the implementation of a wide variety of functional units in VLSI. Efforts along this line have started in the early 1960s and have continued up to recently. Examples include the 1-dimensional iterative multiplier of Atrubin (1965), modulo arithmetic units structured after 1-dimensional (Pries, Thanailakis, & Card, 1986) or 3-dimensional (Tsalides, Hicks, & York, 1989) cellular automata, a memory controller based on 1-dimensional cellular automata (Wasaki, 2008), and systolic arrays based on 2-dimensional cellular automata (Kung & Leiserson, 1979). Another application of cellular automata is in the generation of test patterns for Built-In Self-Test (BIST) of VLSI chips (e.g. (Hortensius, McLeod, Pries, Miller, & Card, 1989; Dasgupta, Chattopadhyay, Chaudhuri, & Sengupta, 2001)). Testing of chips requires a representative set of inputs, and 1-dimensional cellular automata implemented onchip are very suitable for this purposes, requiring relatively small hardware resources. An overview of various applications of cellular automata can be found in the article by Ganguly, Sikdar, Deutsch, Canright, and Chaudhuri (2003). General-purpose computation is less common on cellular automata implemented by VLSI due to the high overhead of hardware relative to the obtained functionality. This will likely be a less important issue in nanocomputers, since the overhead could

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still be acceptable if it can be compensated for by the low-cost availability of cells in the huge quantities that can be expected with bottom-up fabrication. The next section gives an overview of nanocomputers designed with general-purpose computation in mind. Consequently, we expect the use of cellular automata to be not merely limited to specific applications, but also include general-purpose computation. To this end, cellular automata need to be Turing-universal, i.e., they should be able to compute any computable function.

SHORT HISTORY OF CELLULAR AUTOMATON BASED ARCHITECTURES The use of cellular automata for nanocomputers originates in research to implement cellular automata through simple physical mechanisms. Most prominent in this context is the work of Fredkin, Toffoli, and Margolus from the early 1980s on, who aimed for computation inspired by physical models (Fredkin & Toffoli, 1982; Margolus, 1984; Fredkin, 2005). A major theme in this research is reversibility of computation and the associated reduction of energy consumption promised by it. One accomplishment from these research efforts was the design and construction of the Cellular Automaton Machine (CAM) for efficiently running cellular automaton simulations (Toffoli, 1984)–a machine that was implemented by conventional technology. The same era of the early 1980s also saw efforts by Carter (1984) to design computers based on molecular arrays. Carter was a chemist and he focused particularly on how certain operations of cells could be realized by means of the cascaded exchange of single and double bonds in molecules. The work is relatively unknown by computer scientists, but it may be one of the first systematic efforts toward the design of nanocomputers.

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The Quantum-dot Cellular Automaton (QCA) is another remarkable model that has been proposed with the aim of realizing nanocomputers (Lent, Tougaw, Porod, & Bernstein, 1993; Tougaw & Lent, 1994; Porod, 1998). It has cells consisting of four quantum dots, of which two contain (ideally) an electron each. The four dots are placed near the corners of a cell, which has a square form. Due to Coulomb interactions inside a cell, the two electrons inside a cell have the tendency to maximize their mutual distance. This results in two possible polarizations of a cell, with the two electrons residing in dots in opposite corners. These polarizations are interpreted as signals 0 and 1, respectively. On a larger scale there are also Coulomb interactions between cells through the electrons in them. These interactions can be used for the propagation of signals in the model, and for logical operations between signals, such as a NOT-gate or an AND-gate. The QCA model is not a cellular automaton in the true sense, since it assumes that cells are laid out following the design of a circuit. This may even include layouts in which a cell is placed with an offset of half a cell at the side of another cell (Tougaw et al., 1994). The QCA architecture promises extremely low power consumption, as its operation does not involve flows of electrons, but mere tunneling of electrons between the quantum dots within a cell. The problem with this model, however, is that it is very sensitive to noise, which imposes a low operating temperature on it. An alternative design of a QCA is based on submicron magnetic dots, and it is expected that this may allow operation up to room temperatures (Cowburn & Welland, 2000). A 2-dimensional cellular automaton using optical signals to trigger transitions of cell states has been proposed by Biafore (1994). This model has very simple cells, because it employs physical interactions rather than explicit rules for its transitions. By using optical clock signals of different wave lengths supplied in a specific order, this design gives control (at least in theory) over which cells interact as neighbors and when these

Cellular Nanocomputers

interactions occur. The underlying model used in the design is the Billiard Ball Model of Fredkin et al. (1982). Cellular Nonlinear (or Neural) Networks (Chua & Yang, 1988), abbreviated as CNN, have been motivated by the need to process analogue-valued signals, like for example sensing signals. CNN models have analogous states and continuous transition functions, which describes the dynamics of a CNN in terms of the variables associated with a cell and neighboring cells. The transition function may also include (depending on the model) weighted feedback operators and weighted synaptic input operators (Chua & Roska, 2002). CNNs find applications in image processing, the solution of partial differential equations, processing sensory information from visual or other sensors, etc. Apart from these applications, it has also been shown that the CNN model is computationally universal (Chua, Roska, & Venetianer, 1993), as it can be configured such as to simulate the Game of Life (Conway et al., 1982), which has been proven Turing-universal. CNN have been used as the basis for proposals of nanocomputer architectures, due to their potential for simple physical realizations, like in (Yang, Kiehl, & Chua, 2001; Kiehl, 2006), in which the logic states of such cellular automata are expressed in terms of the electrical phases in a dynamical physical process. A possible implementation of such a model is by tunneling phase logic (Yang et al. 2001). A nanoscale cellular automaton that has actually been physically realized is reported by Heinrich, Lutz, Gupta, and Eigler (2002) and Eigler et al. (2004). It is based on the physical interactions between CO molecules that are arranged on a copper surface such as to form so-called Molecule Cascades. Some of the configurations of CO molecules are unstable, but that makes it possible to predict with a high probability into which configurations they will change. By arranging the CO molecules in suitable ways, a wide variety of functionalities can be realized,

such as signal crossings, AND-gates, etc. These types of systems have been experimentally shown to be able to conduct simple computations, such as 3-bit sorting, but unfortunately they are very slow. Faster interactions between molecules on a surface have been shown by Bandyopadhyay and Acharya (2008), but these systems have not yet been shown to be able to compute, though they appear to have such potential. The cellular automata discussed above utilize the physical interactions inherent in their design to conduct transitions. This results in extremely simple cells, but it also restricts the functionality of cells to a limited set of possible transitions. Additional functionality, such as the correction of errors or the configuration of the cells into certain states, is hard to implement through such simple rules. For this reason some researchers advocate models with cells that are more complex, but still sufficiently simple to potentially allow huge numbers of them to be fabricated by bottom-up techniques and organized in arrays at nanometerscale integration densities. One approach along this line is the Cell Matrix (Durbeck & Macias, 2001). Each cell in this model consists of less than 100 bytes memory and a few dozen gates, but this is sufficient to be able to configure a cell for a simple operation like a logic gate or a one-bit addition, etc. A somewhat similar line of thought is followed by Beckett (2008), who proposes a cellular automaton that is based on Double-Gate transistor technology.

RANDOMNESS AND CELLULAR AUTOMATA: THE NATURAL WAY? The cellular automata models in the previous section implicitly assume synchronous timing, and this mode of updating is the most frequently studied in the literature. This may be because of historical reasons–the cellular automaton initially proposed by von Neumann (1966) in the 1950s was synchronous–but it may also be partly due

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to synchronous models being easier to grasp and design. A synchronous mode of timing, however, has drawbacks: it is associated with a range of problems of which the most prominent are high power consumption and heat dissipation. These problems, which tend to get worse at higher integration densities, have been a motivating factor behind research efforts on asynchronous timing, which emphasizes timing governed by a local exchange of control signals, such as handshaking through Request and Acknowledge signals. Not needing a global clock signal, such schemes facilitate an increased level of locality, which has obvious advantages for physical implementations. Nanocomputers based on asynchronously timed cellular automata have started to attract interest for this reason. The updating method of a cellular automaton has a significant influence on its behavior. Phenomena appearing in synchronous cellular automata tend to be smoothed out when an asynchronous update method is used instead, due to the randomness associated with it. The randomness is also the reason that it tends to be more difficult to design an asynchronous cellular automaton and to prove its correctness, as compared to a synchronous cellular automaton with similar functionality. So, how do we compute on asynchronous cellular automata? Most approaches to conduct computations somehow use synchronization between cells on strictly local scales. The first attempt in this direction by Nakamura (1974) uses a simulation of a synchronous cellular automaton on an asynchronous cellular automaton that keeps cells approximately in pace with each other through a counting mechanism in each cell. This involves a significant overhead, and is less suitable for nanocomputer implementations. A more efficient way is the simulation of an asynchronous circuit on an asynchronous cellular automaton. This frees us from the need to keep all cells approximately synchronized, and rather focuses on the synchronization of small areas of the cellular space where signals happen to be.

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The latter approach involves the design of transition rules such that the cellular space behaves like an asynchronous circuit. How does such a circuit look like? While the circuits in synchronous architectures are typically standard logic circuits involving AND, OR, and NOT gates, different functionalities are required for asynchronous circuits, since the absence of a clock signal needs to be compensated for by a local synchronization functionality. The asynchronous circuits used for computation on the asynchronous cellular automata of Peper, Lee, Adachi, and Mashiko (2003), Lee et al. (2003), Peper et al. (2004), Adachi, Peper, and Lee (2004), and Lee, Adachi, Peper, and Mashiko (2005) are delay-insensitive. Such circuits are robust to delays in their operations or in their wires. Following is a short description of some important delay-insensitive circuit primitives. The first primitive is a Fork (Figure 1(a)), which is basically a fan-out of a wire: upon receiving a signal from its single input wire, it produces one signal on each of its two output wires. The second primitive is the Merge (Figure 1(b)), which redirects signals from either of its two input wires into its single output wire. The third primitive is the Tria (Figure 1(c)), which has three input wires and three output wires, organized alternatingly. The Tria produces a signal on an output wire as a result of a signal on each of the two input wires at both sides of the output wire. When there is only a signal available on one input wire, the signal remains pending, until a signal on another input wire arrives. This type of behavior, also called Join-functionality, facilitates synchronization of signals on a local scale. The fourth primitive is the Sequencer (Figure 1(d)), which can be viewed as two wires running through the primitive and one control wire that determines whether a signal on one of the two wires may pass through the primitive. In case both wires contain an input signal, an arbitrary choice is made as to which one may pass through. The Sequencer is then said to conduct arbitration.

Cellular Nanocomputers

Figure 1. Universal set of primitives for delay-insensitive circuits. (a) Fork, which is a fanout that duplicates every input token from input wire I to its two output wires O1 and O2. (b) Merge, which merges streams of tokens from input wires I1 and I2 into one output stream O. (c) Tria, which joins two input tokens into one output token as follows. Upon receiving an input token from each of the two wires Ii (i ∈{1,2,3}) and Ij (j∈{1,2,3}\{i}), it outputs a token to the wire O6-i-j. If there is only a token on one input wire, it remains pending, until a token on a second input wire arrives. (d) Sequencer, which arbitrates the passage of tokens from wire Ii to wire Oi (i∈{1,2}). For each token on input wire c, exactly one token on the input wires may pass to the corresponding output wire.

These four primitives form a universal set, which means that any arbitrary delay-insensitive circuit can be constructed from them, in the same way as any arbitrary synchronous logic circuit can be constructed from for example AND-gates and NOT-gates. A universal set is not unique: in fact, many different universal sets exist, all of which–due to their universality–can be used to construct the above universal set consisting of the Fork, Merge, Tria, and Sequencer. The trick of designing simple asynchronous cellular automata for general-purpose computation is to find a universal set of delay-insensitive primitives that allows easy mapping on the cellular automaton model. The set used can make a huge difference in the number of required update rules, which directly impacts on the complexity of the cells. The details of the cellular automaton, such as neighborhood model, update model, etc., also tend to make a difference as to the complexity of cells. As examples of some models with different number of rules we mention the asynchronous cellular automaton of Adachi et al. (2004), which is totalistic and has a Moore neighborhood and which requires 85 rules; alternatively, the model of Lee et al. (2003), which is not totalistic and which has a von Neumann neighborhood, requires

58 rules. Both models use slightly different (but similar) sets of circuit primitives. Due to their robustness to signal delays, delayinsensitive circuits combine very well with asynchronous cellular automata, since the requirement no longer holds that signals must arrive at certain locations at certain times dictated by a central clock, as in synchronous circuits. This takes away concerns about variations in the operational speed of cells and considerably simplifies the design of configurations representing circuit elements embedded in the cellular space. To avoid the randomness in the timing of transitions interfering with proper deterministic operation of an asynchronous cellular automaton, we need to observe some principles in their design. The first principle states that series of transitions that are critically dependent on the order in which cells are updated need to be serialized. In other words, dependencies should be designed into update rules, such that they can only be applied in well-defined orders. One way to accomplish this—and this brings us to the second design principle—is to temporally block a cell’s update until its neighborhood matches the LHS of an update rule. In other words, a cell will be made waiting until conditions are right in terms of its

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state and the states of its neighboring cells to conduct a transition on it. It is the absence of update rules that facilitates the blocking of transitions at appropriate times. The art of using the second design principle is thus a matter of leaving out update rules rather than adding them. These two design principles may not always suffice, since transitions may still be unpredictable due to the randomness of the system. For example, some transitions may be unintended: they may occur as a side effect of rules intended for other purposes in the design. For these reasons, there is a third design principle, that is, the use of update rules that are the reverse of some other update rules. This latest design principle resolves many of the problems that can not be dealt with by only the first two principles. It facilitates the return to a previous state of the cellular automaton if an incorrect transition was conducted, which is a possibility that cannot be excluded because of the randomness of transitions. Some asynchronous cellular automata designed according to these principles can be found in the articles by Adachi et al. (2004), Lee et al. (2003), and Lee et al. (2005). Unfortunately, many of these models still require tens of update rules, which may hinder efficient physical implementations. To further reduce the number of rules, alternative models have been proposed, called Self-Timed Cellular Automata (STCA) (Peper et al., 2003, 2004), in which cells are subdivided in four partitions, each representing a part of a cell’s state information. With this model, a reduction to merely six rules is possible, while retaining Turing-universality (Peper et al, 2004). Even less update rules–as few as three–are achieved in the so-called Brownian cellular automaton model of Lee et al. (2008), which exploits properties of random fluctuations. In this model, signals are allowed to fluctuate forward and backward on wires embedded on the cellular space. Notwithstanding their randomness, fluctuations serve a useful purpose: they drive a random search process in the state space of the

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circuit simulated by a cellular automaton. This search process facilitates an important mechanism. Reverse update rules as mentioned above in the third design principle for asynchronous cellular automata are no longer necessary, because they are inherent in the backward motion caused by fluctuations. This is one factor contributing to the model’s simplicity. Another reason for the simplicity is the sharply reduced complexity of the universal set of delay-insensitive primitives on which the cellular automaton is based, as it turns out that the searching ability of random fluctuations can be used to take over part of the functionality of the primitives. Apart from the obvious advantages of the resulting reduction in cell complexity to implementations of nanocomputers, fluctuations themselves will be an important factor in a physical sense. The usual strategy to deal with fluctuations (and noise) in electronic circuits is to suppress them, but if fluctuations can be exploited, circuits working in a regime with sharply reduced signal / noise-ratios may become possible, with obvious implications for a reduced power consumption. How do asynchronous cellular automata compare with their synchronous counterparts? The many different orders in which cells can be updated in an asynchronous cellular automaton tend to require an increased number of update rules in the design. On the other hand, unlike its synchronous counterpart, an asynchronous cellular automaton does not need a rule for every possible combination of states in a cell’s neighborhood; this contributes to a reduction in the number of update rules. It is hard to draw general conclusions about which update method requires more rules: it depends on specific circumstances of the model. An asynchronous mode of updating for cellular automata appears certainly competitive to synchronous updating, witness the small numbers of update rules we have been able to achieve.

Cellular Nanocomputers

FAULT-TOLERANCE AND DEFECT-TOLERANCE Fault-tolerance, the ability of a system to recover from a failure of a temporary character, will be an important property of nanocomputers, since the reliability of their components is expected to be limited due to a number of factors. First, the individual behavior of electrical carriers, particles, molecules, or other nanometer scale features gains importance due to their decreased numbers involved in the operation of a device. In other words the “Law of Large Numbers” according to which device operation can be analyzed in terms of the averaged behavior of carriers or other elements ceases to hold (Birge, Lawrence, & Tallent, 1991; DeHon, 2004). Second, this individual behavior is highly probabilistic, as it is rooted in thermodynamics and quantum mechanics. Approaches to fault-tolerance tend to center around the implementation of redundancy in efficient ways. Early work in the context of cellular automata is reported by Nishio and Kobuchi (1975) with a model that can correct at most one error in 19 cells. Since this model requires a cell neighborhood of 49 cells, however, cells are quite complex, suggesting that they will be very errorprone in physical implementations. The practical value of this work is thus limited, as is the related work by Harao and Noguchi (1975). Better fault-tolerance is obtained by Gács (1986) and Gács and Reif (1988) with synchronous cellular automata, and by Wang (1991) and Gács (2001) with asynchronous cellular automata simulating synchronous cellular automata. According to these schemes, cells are organized in blocks that perform a fault-tolerant simulation of a second cellular automaton, which on its turn is also organized in blocks, simulating even more reliably a third cellular automaton, and so on. This results in a hierarchical structure with high reliability at the higher levels. The cells in these models, however, are extremely complex: they contain detailed information regarding their hi-

erarchical organization, which impends efficient physical implementations. Yet another approach uses error correcting codes through a scheme in which computation takes place in the encoded space, whereby errors are corrected locally, and encoding and decoding is only necessary at the beginning and the end, respectively, of the computation. This line of thought is followed by Spielman (1996) to realize a fault-tolerant computing scheme with improved reliability, which is, however, not a cellular automaton. The scheme has been an inspiration for schemes based on STCA cellular automata, like the article by Isokawa et al. (2004). A more efficient STCA based on this scheme (Peper et al., 2004) uses the STCA’s cell partitions as small memories that store information encoded by an error correcting code. Up to one third of the memory’s bits can—if corrupted—be corrected by this method. Defect-tolerance, the ability of a system to operate correctly if one or more of its parts experience a permanent defect, will also be an important property of nanocomputers, since defects are almost certain to occur in bottom-up manufacturing. Defects being permanent, they need to somehow be isolated from non-defective parts to prevent them from affecting a system’s correct functioning. Strategies to cope with defects thus need to detect them and then configure the system around them. In the context of cellular automata, configuration (or reconfiguration) means that there is a mechanism to set the cells in certain states such that subsequently computations can be carried out. Reconfiguration is self-contained if through the use of merely update rules a computational structure can be configured on the cellular space. Such reconfiguration functionality resembles selfreproduction of configurations in cellular automata (Neumann, 1966). Unfortunately, implementing self-reproduction on cellular automata tends to give a large overhead in terms of update rules. To cope with this problem, an outside mechanism may be used to set cell states, though this comes

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with the added problem of how this mechanism can be physically implemented and controlled. In the asynchronous cellular automaton of Isokawa, Abo, Peper, Kamiura, and Matsui (2003), defects are detected and isolated from non-defective cells through waves of cells in certain states propagating over the cellular space, leaving defective cells, which are unable to adhere to the state changes, standing out to be detected as defective. Isokawa et al. (2007) take this approach one step further by additionally scanning the cell-space for defect-free areas on which circuits can be configured. Being self-contained, this scanning process uses techniques resembling self-reproduction functionality. These approaches are off-line, which means that reconfiguration takes place while the cellular automaton is not in the process of computation. An on-line approach to defect-tolerance is followed by Isokawa, Kowada, Peper, Kamiura, and Matsui (2006), in which configurations called random flies move around randomly in the cellular space, and stick to configurations that are static. A configuration unable to compute due to defects is static in this model and will thus be isolated by a layer of random flies stuck to it. Key to the functioning of this random fly scheme is the cellular automaton’s timing being asynchronous, since this provides the randomness required. Being highly experimental, the above approaches require an overhead in terms of update rules that may be comparatively high, so more research is required to make them practical for nanocomputers.

OUTLOOK AND DISCUSSION As it has become increasingly difficult to continue improvements in integration densities according to the lines drawn in previous decades, new methods and architectures are being considered. Cellular automata occupy a promising position in this respect, due to their regularity, which may facilitate efficient physical implementations and manufac-

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turing methods. Many challenges remain, though, for cellular automaton based nanocomputers. An important challenge will be the development of bottom-up methods to mass-manufacture such computers. If such methods eventually become available, will technology have progressed to the extent that less regular architectures will be within reach as well? If so, that may indicate doom for cellular automata, since the huge numbers of cells they will posses may no longer be sufficient to compensate for their overhead. Somewhat related is the question as to the path from current architectures to cellular automaton based architectures. It is conceivable that top-down manufacturing will not suddenly be replaced by bottom-up methods, but that a gradual transition will take place, in which the latter assume an increasingly prominent role over time. Such a development may be accompanied by an increase in the number of cells over time, coupled with a decrease in complexity of those cells. This trend has already started in a certain sense, with the ongoing increase in the number of processors in multi-core architectures. To facilitate the trend towards simple cells, it will be important to investigate the limits of cell complexity. In other words, how simple can cells be, without the cellular automaton losing the ability to support general-purpose computation? Fault- and defect-tolerance, and–related to them– reconfiguration ability, will play an important role in this story. Though their realization in cellular automata will very likely increase the complexity of cells, the question is to what extent? Can they be realized by distributing the functionality for fault- and defect-tolerance over many cells, in the spirit of cellular automata, rather than using more complex cells that support much of this functionality on their own? The mode of timing will also play a role in cellular automaton based nanocomputers. If timing is synchronous, some mechanism is required to deliver clock signals to cells. This is not infeasible, since optical signals, for example, may be used to

Cellular Nanocomputers

this end, but such solutions come with the need to make cells sensitive to such signals, which creates overhead. Alternatively, asynchronous update protocols appear to deliver a more “natural” way, but their implementation is not trivial either, since problems like read- or write- conflicts and arbitration issues involving neighboring cells are likely to occur. There is also the question of how to conduct I/O to a cellular automaton consisting of a huge array of cells. One may envision that such operations have similarities to those for semiconductor memories, but this will bring with it the question how the required mechanisms can be controlled by the cells themselves. The alternative that we mentioned, i.e. using self-reproduction techniques to spread information over the cell space through the use of update rules implemented in the cells themselves, may be a solution, but it carries a high price in terms of cell complexity. This solution will also need to address the question how the cells at the borders of the array can receive the required information from the outside world. Address decoding through multiplexers implemented by crossbar arrays based on nanowires (Williams & Kuekes, 2001) may offer solutions to this end, but still many challenges remain. Finally, even if cellular automaton based nanocomputers are general-purpose, they may favor certain applications, but which will that be? The huge parallelism of cellular automata may provide a clue: problems that can be divided in a large number of small subproblems will carry the torch in this respect. Such problems are likely to occur in Artificial Intelligence, search and optimization, pattern recognition, and image processing.

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This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 33-49, copyright 2009 by IGI Publishing.

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Chapter 4

An Advanced Architecture of a Massive Parallel Processing Nano Brain Operating 100 Billion Molecular Neurons Simultaneously Anirban Bandyopadhyay National Institute for Materials Science, Japan Subrata Ghosh National Institute for Materials Science, Japan Daisuke Fujita National Institute for Materials Science, Japan Ranjit Pati Michigan Technological University, USA Satyajit Sahu National Institute for Materials Science, Japan

ABSTRACT Molecular machines (MM, Badjic, 2004; Collier, 2000; Jian & Tour, 2003; Koumura & Ferringa, 1999; Ding & Seeman, 2006) may resolve three distinct bottlenecks of scientific advancement (Bandyopadhyay, Fujita, Pati, 2008). Nanofactories (Phoenix, 2003) composed of MM may produce atomically perfect products spending negligible amount of energy (Hess, 2004) thus alleviating the energy crisis. Computers made by MM operating thousands of bits at a time may match biological processors mimicking creativity and intelligence (Hall, 2007), thus far considered as the prerogative of nature. State-of-the-art brain surgeries are not yet fatal-less, MMs guided by a nano-brain may execute perfect bloodless surgery (Freitas, 2005). Even though all three bottlenecks converge to a single necessity of nano-brain, futurists and molecular engineers have remained silent on this issue. Our recent invention of 16 bit parallel DOI: 10.4018/978-1-60960-186-7.ch004 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Advanced Architecture of a Massive Parallel Processing Nano Brain

processor is a first step in this direction (Bandyopadhyay, 2008). However, the device operates inside ultra-high vacuum chamber. For practical application, one needs to design a 3 D standalone architecture. Here, we identify the minimum nano-brain functions for practical applications and try to increase the size from 2 nm to 20 μm. To realize this, three major changes are made. First, central control unit (CCU) and external execution units (EU) are modified so that they process information independently, second, CCU instructs EU the basic rules of information processing; third, once rules are set CCU does not hinder EU-computation. The basic design of the proposed nano-brain is a dendrimer (Hawker, 2005; Galliot, 1997; Devadoss, 2001; Quintana, 2002; Peer, 2007), with a control unit at its core and a molecular cellular neural network (m-CNN, Rosca, 1993; Chua, 2005) or Cellular Automata (CA, Wolfram, 1983) on its outer surface (EU). Each CNN/CA cell mimics the functionality of neurons by processing multiple bits reversibly (Rozenberg, 2004; Li, 2004; Bandyopadhyay, 2004). We have designed a megamer (Tomalia, 2005) consisting of dendrimer (~10 nm) as its unit CNN cell for building the giant 100 billion neuron based nano brain architecture. An important spontaneous control from 10 nm to 20 μm is achieved by an unique potential distribution following r = a sin k q , where r is the co-ordinate of doped neuron cluster, k is the branch number, θ is the angle of deviation and a is a constant typical of the megamer architecture.

INTRODUCTION: ESSENTIAL FUNCTIONS OF NANO-BRAIN The core architecture of a nano-brain whether operating in a nano-factory, functioning as a nano-surgeon or nano-computer could be the same. The architecture may consist of a control unit connected to all execution parts following one-to-many communication principle. In our recently described proto-nano-brain (Bandyopadhyay, 2008), we have demonstrated this principle in practice. The principle states that, if a large number of molecules are connected radially to a single molecule then by tweaking the central molecule one can logically control all radially connected units at a time. To control the logic operation of a large assembly, we need to control only the central molecule, which we name the central control unit (CCU). Currently, the CCU can send only one instruction without any external interference. The reason is that a CCU, which is a molecular switch could be excited to a particular state, and only other possible transition of this molecule would be returning back to the normal

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state. Therefore, remotely, without any human interference CCU can send only one instruction. We wish to develop it in such a way that it is able to send a series of logical instructions to the execution units (EU) during its operation. Only then, the complete architecture would execute series of operations one after another by itself, independent of any external stimuli or human interference (Koumura, 1999). This is important as it is not practical to instruct the control unit of a nano-factory several times for completing the task, or instruct the control unit of a nano-computer at every stages of its derivation of a math problem, or advise the control unit of a nano-surgeon its next move during a brain operation. The fundamental element that constitutes a molecular nano-brain is a molecular neuron. A neuron is an analogue switch. Beyond a threshold voltage, continuous increase of applied bias should generate more than two conducting states in a neuron-like molecule. Unfortunately, almost all reported practical single molecule switches are binary (Chen, 1999). We reported the first 2-bit single molecule switch operating reversibly

Advanced Architecture of a Massive Parallel Processing Nano Brain

between four conducting states (Bandyopadhyay, Miki, 2006). Since then we have tested several multi-electron processing organic systems and invented as high as 4-bit molecular switch operating reversibly between 16 distinct conducting states (Simic Glavasky, 1989). Compared to an analogue switch, 16 choices may appear low, however, because of its simultaneous response in coherence with all neighbors, it can process massive information compared to the existing processors. Recently we have demonstrated such a massively parallel computation on an organic molecular layer (Bandyopadhyay, Pati, 2010). Since molecules are redox active, it has been possible to generate a density of electrons on the molecular assembly. By destabilizing the molecular assembly, we allowed these excess electrons doped in the molecules to rearrange themselves and reach equilibrium. We have replicated natural phenomenon like diffusion and evolution of cancer cells on this molecular layer in addition to the classical computing constructs like logic gates, Voronoi decomposition etc. We have recently demonstrated that even by rotating polaron across the benzene rings, Rose Bengal molecule can modulate relative rotation of its perpendicular planes and generate ten distinct conductivities and six unique electronics features (Bandyopadhyay, Sahu, 2010). Multi-level switches are essential to build nanobrain because of two fundamental reasons. The first rational originates from the fact that, while designing the hardware of a nano-brain, we adopt one major paradigm shift for information processing and acquiring the output data. Information is processed horizontally through the surface, and it may be a euclidian or a non-euclidian surface. However, the output is always accessed vertically to this surface. If the directions for information processing and information acquisition (input and output) are the same then the acquired information is transported along a particular circuit path by executing logical operations one after another. Then, the information processing noise that is mixed with the output signal is the product of

all noise functions generated at every step on the processing path (Figure 1 a). In contrast, for a vertical observer sitting above an analogue computer (Hopfield, Tank, 1987), every action executed at a particular molecule/pixel/CA cell/processing unit on the planar surface is mapped directly to the output, therefore the total output noise is solely contributed locally by one processing unit. Our recently realized molecular computing substrate is one such example (Bandyopadhyay, Pati, 2010). If noise is the first reason for adopting multi-level molecular switches, then the second reason would be adaptability, surviving computation process under defect or extreme environmental interference. Note that the plan for a versatile decision-making machine would be based on Cellular Automaton (Wolfram, 2002) or complete analogue electron/ molecule diffusion resembling computing (Andy, 2006). The practical device could be realized by fabricating a compatible in-plane processing circuit wherein it is possible to measure output signal at different spatial locations on a surface. This is well known that, any multilevel switching operation is possible to realize alternatively using binary switches. This particular assumption is true only for those devices where one switch interacts with only another one at a time. If we have a 2 D assembly of molecular switches, then each conducting state of a molecule that is assigned to a particular logic state (0, 1, 2, 3..) would interact differently with its neighbor. The physical environment thus becomes comprehensively versatile as it can take decisions in multiple different ways in a particular neighborhood. Therefore, for 2 D template based molecular or diffusive computing it is essential to increase the number of logic levels. To create an adaptable hardware, multi-level switches are essential. By adaptation, an organic computing substrate sets particular global rules for the decision-making process and all local cell state transition rules for updating a state operate within the framework of those global rules (Kohonen, 1989). Therefore, to mimic adaptation on an organic monolayer or in a 3 D molecular as-

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Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 1. Fundamentals of Major conceptual shift: a) Molecular monolayer is represented as 2 D array of CNN cells. Noise function for the vertical measurement is Y(t ) and for horizontal measurement through an organic monolayer, it is the product of all junctions in the probabilistic path each contributing as W(t ) (top). Nano-cell project has addressed horizontal noise in an innovative way (see Husband, 2002, Dutta, 1981). A Gaussian expanding noise is plotted at A, B, C points for three different rates of expansions, the triangular geometry does not change, and therefore decision remains the same. For vertical measurements, surface propagating local noise have minimal effect on the final decision; b) Schematic presentation of an conventional hardware is denoted as A and an adaptable hardware is denoted as B. A is composed of binary switch, therefore if there is a permanent change on the processing circuit, the computation is lost, i.e. it cannot adapt. B is composed of multilevel switches therefore, even after permanent changes; there are possibilities (top). Line profile of potential for an adaptable hardware is measured as an average spectrum (below, left). Four examples of potential profiles, -bold plotted pattern denoted as C depicts the line profile marked in A, or background potential. The dotted profile crosses this line, therefore is not allowed; c) Potential profile of a nano-brain in the configuration space, undergoing self-organization, arrow denotes an input logic pattern, A, B, C are self-organization, D is solution of a problem; d) Schematic presentation of transition from existing concept of linear communication to radial communication in a disk shaped object, and then in a sphere (left). The schematic presentation of functional parts of a 3D nano-brain hardware, it could be a nano brain seed of ~15 nm or a megamer of ~ 20 μm. Here, A is CCU, B is the boundary of brain seed, C is the symmetry transformation region, and D is the CNN region (right)

sembly, it is desirable to encode particular feature that would inhibit spontaneous transition to particular logic states. Normally, restricted conducting states require higher energy or large confor-

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mational change in its structure. The substrate survives its computing by isolating those higher energy states, by connecting minimum energy states of the rejected ones we get a background

Advanced Architecture of a Massive Parallel Processing Nano Brain

logic map. Due to an extreme environment such a situation could naturally occur in the 2 D computing substrate. An evolving pattern of logic states would continue to evolve acquiring values higher than the minimum energy surface defined by the background logic map. Since all solution sets generated on this surface for the information processing in the post-evolution period would have higher energy than the background logic map (Figure 1b), therefore the computing substrate would generate an alternative solution not the exact one we desire. However, there is a significant probability that the alternative solution would not change the decision for the molecular machine. Self-organisation is an essential however not a sufficient condition for a learning hardware. By self-organisation, processing units (like molecules, CA cells) re-arrange themselves in such a way that the input information encoded as a pattern of cellular states is re-distributed among processing units in a new pattern. Instead of directly processing the input information or a given problem, bio-processors use self-organisation to convert sensor-captured information into a signal essentially acceptable to the hardware for further processing. This is similar to classical Neumann computer where an encoded problem is converted into machine language prior to computation. An integrated molecular assembly undergoes a structural/electronic global relaxation process to reach a new energy minimum of the evolving pattern (Figure 1c). If structural relaxation is correlated with a spontaneous logical modification of the input pattern, then the self-organization process evolves with time as the hardware responds to different patterns distinctly. This is important as one can specifically identify the role of particular parameters to encode biological learning process into an electronic hardware. Note that, self-organization does not necessarily mean that the processing units should physically re-assemble themselves into a different conformation/arrangement. Rather, it could preferably be the reconstruction of a logical map formed by a transformation

of information among the processing units by an exchange of electrons, or other quasi particles. Based on the discussions above, we conclude that an efficient one-to-many communication may require a 3 D spherical architecture. The reason is that 360o rotation of our disk-shaped 2 D nanobrain (Bandyopadhyay, 2008) around its planar axis is a sphere (Figure 1d). Therefore, our earlier realization of one to 16 channel communication could not be increased further because of the restriction of space; i.e. we cannot put more than 16 processing units side by side, if we use the same CCU molecule. Therefore, only option left for increasing the multi-channel communication further is to build a 3 D architecture. The basic information-processing unit of such spherical assembly is a molecular neuron, essentially a single molecule multilevel switch. In literatures, three basic criteria have been suggested to design the processing circuit of genuine unconventional computers (Dewdney, 1984; Benenson, 2001; Păun, G., 2000). First, there should be a wireless network of elementary processing units as heating is one of the most important problem for the future generation processors (a Pentium IV processor has ~10 km wiring/cm2). Second, the processor should be massively parallel capable of computing more than several hundreds of bits at a time. Third, it should spontaneously evolve logical decisions or execute an emergent computation. One example for such computation is information processing through a cellular neural network (CNN, Rosca, 1993; Chua, 2005; Wolfram, 1983). A possible way to realize a practical CNN would perhaps be to identify the governing equation or cellular automata (CA) rules of an organic monolayer composed of molecular neurons. Note that there is a distinct difference between the CNN and the CA architecture. In CNN system, the cell state transition is a temporal evolution function. In a CA system, however, the cell state transition does not follow any equation, either all CA cells update their state globally at a time (synchronous cellular automata) or cells update their states in-

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Advanced Architecture of a Massive Parallel Processing Nano Brain

dependent of time whenever the condition is met or CA transition rules are satisfied (asynchronous cellular automata). Therefore, in a 2 D or 3 D molecular architecture, if there exists well-defined CA rules that enables the monolayer performing a meaningful computation, we can say that the molecular neuron functions as a CNN cell or CA cell. Then by changing the assembly of these molecular neurons we can modulate a euclidian spherical information processing surface and multiply the computing power of a planar CA. All molecular neurons on the surface of the 3 D architecture should be connected to the central core. The central core should function as CCU and the wired architecture would define the global relaxation function controlling self-organisation and adaptation, which in turn would control the learning process.

BASIC DESIGN PRINCIPLES Molecular Neurons Recently it is shown that the molecules following particular design criteria can exchange one or more electrons and undergo reversible phase transition by changing its structural symmetry (Pati 2008). Cyclic voltammogram (CV, Chambrier, 2006) can map these reversible phase transitions and one can measure quantitatively the number of electrons spent to induce these phase transitions into a molecule. By studying a typical Fe-terpyridine linker molecule Pati et al have shown that change in orbital symmetry and phase transition leads to negative differential resistance (NDR) peaks (Pati, 2008). The 2,3,5,6-tetramethyl-14-benzoquinone (DRQ) molecule exhibits two NDR peaks in its current voltage (IV) spectrum (Figure 2a). Using these two NDR peaks, we have demonstrated a reversible 2 bit information processing (Bandyopadhyay, 2003). Simulations show that the molecule undergoes phase transitions twice by exchanging one and two electrons

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reversibly. Theoretically, if we find a molecule that undergoes multiple distinct phase transitions by changing its symmetry and experimentally the same molecule shows multiple reversible peaks in the CV, then it could be a suitable candidate for single molecule multilevel switching (Figure 2b). These particular kinds of molecules should exhibit multiple NDR peaks in the IV characteristics as current across the molecule would increase as soon as a particular phase transition takes place generating a higher conducting conformer. Then the current would decrease with an increasing bias, leading to a distinct relaxed phase metastable conformer, which would possibly sustain prior to another phase transition (if required). To confirm that such a large number of NDR peaks are not noise, rather a molecular response we test random access memory (RAM) operations with millions of consecutive write-read-erase-read pulses (Bandyopadhyay, 2003).

Step-function Molecules or Q-dots A multiple NDR effect in a single molecule mimics the electronic response of a circuit where similar numbers of resonant tunneling diodes (RTD, Potter, 1988) are connected in series. However, CNN theorists have proposed a few alternate IV characteristics of CA cells. One of them is saturated bi-polar synaptic feature I (V ) = 1 / 2(V + T − V − T ) , see figure 2 (Chua, 2005). As no literature exist for the practical realization of such electronic behavior using a single device, here we propose a new methodology. The step function has two basic characteristics. First, the device should show conductance at zero bias (ZBC, Liang, 2002). The ZBC phenomenon could be realized when a device demonstrates the Kondo effect. The Kondo effect is normally realized in magnetic materials at very low temperatures. In single molecules, the Kondo effect could perhaps be realized even in room temperature. Q-dots localize spin density; there-

Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 2. Molecular neurons, design and realization: a) Four reversible electronic states of DRQ molecule; b) Cyclic voltammetry and phase transition relationship, only those states differed by more than 24 mV are distinct in room temperature; c) Schematic presentation of step function (top), an integrated spectrum of step function is identical to Kondo response (below). 2T is the total width of bias variation where a ZBC is observed; d) Current voltage characteristics of a Fluorocene molecule that shows spectrum identical to the Rose Bengal molecule. Its conductance spectrum (top left) and the height profile of Rose Bengal on Au (111) at different bias are shown (top right). Single molecule STM image (below), molecule is 1.1 nm long and 1.3 nm wide; e) Conceptual memristor’s equivalent circuit

fore, an artificial molecule composed of Q dots is a potential alternative to the specially designed molecules (Cronenwett, 1998). Apart from ZBC, the single molecule or Q dot would require one additional feature in its IV characteristics. Output current should remain independent of external bias or remain constant outside the ZBC region. This is hardly possible to realize ideally in a practical device since,

higher electric field tend to inject more carriers into the device. Statistically several reported devices reporting the Kondo effect show that within a particular range beyond ZBC, output current remains constant (Temirov, 2008). The constant current output beyond ZBC region may be a natural phenomenon in these devices; since ZBC is induced by scattering of carriers with the localized spin part of a system and a neutral region

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Advanced Architecture of a Massive Parallel Processing Nano Brain

exists both sides of the ZBC (outside ΔV region in Figure 2c) before the onset of the natural tunneling regime. These features, notably the ZBC and nearly constant currents at both sides of the ZBC peak are observed in a particular conformer of Rose Bengal molecule (Bandyopadhyay, Sahu 2010). Our resent studies on Rose Bengal derivatives (Bandyopadhyay, Sahu, 2010; Figure 2d) show that, particular conformer of this molecule exhibit a particular step function like IV characteristic in room temperature. Three other synaptic feature which could be replicated in designing the CNN cells is linear bipolarity, which is devised as a resistor, a rectifier (Chua, 2005) and a memristor or variable resistor (Figure 2e, Strukov, 2008; Chua, 1971). Single molecule analogue of a variable resistor could be realized in a conducting polymer chain (Terada, 2000). The single molecules doped with donoracceptor group may be used for operating as a molecular rectifier (Metzger, 2000). The single molecule memristor is not invented yet, though several thin film devices have been reported in the literatures that demonstrate comprehensive performance (McGinness, 1974; Potember, 1979; Mukherjee, 2005).

Cellular Neural Net of Molecular Neurons Organic molecules functioning as neurons or exhibiting step-function in the IV spectrum could be used to build cellular neural net or CNN. The basic criterion in order to use the molecular neuron as a CNN cell is that, in the molecular assembly, all molecules should retain their reversible switching feature of multiple conducting states. In addition, once a set of logic states are written by applying electrical bias to a set of molecules, the written logic pattern should change reversibly following particular rules characteristic of the molecular assembly. The conventional approach to build molecular assembly is to grow a monolayer on an atomic flat metal surface by e-beam evaporation

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of molecules in an ultra-high vacuum (UHV, 10-8 Torr) chamber, heating the molecule above melting point. Another approach is dropping micro/ nano molar molecular solution on a fresh atomic flat substrate. The monolayer is deposited on the atomic flat surface such that the logic state distribution pattern is read accurately by measuring a tunneling image using scanning tunneling microscope (STM). Analyzing a series of images, we identify common rules that govern transport of logic states on the surface, change in patterns and spontaneous creation/destruction particular logic states. We name these rules as CA rules and the literature is rich in analyzing the methods of computation using CA rules (Figure 3a, Wolfram, 2002). Recently, we have demonstrated such computation practically (Bandyopadhyay, Pati 2010). We have deposited bilayer of a 2-bit switching molecule, 2,3-Dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) on the Au (111) surface in an UHV chamber in particular conformation. Note that, for molecular neurons, it may be useful to examine crystal structures of the parent and related molecules and symmetry of the deposited monolayer, since conformer associated with different conducting states may give rise to structurally distinct monolayers, following distinct CA rules. Ordered molecules in a well-defined symmetry mimic a classical mathematical model of CNN. By imaging the surface continuously for hours we have determined CNN rules for DDQ (-1) monolayer on the Au (111) surface (Figure 3b). A classical organic monolayer is a CNN compatible monolayer if particular features are encoded. These features are as follows. First, the electron-density distribution pattern (or logic map) evolve consistently on the surface, it should not spread out completely over the monolayer and not even localize disappearing permanently from the surface. In both cases, CA rules do not allow versatile computation. Second, logic states should preferably depend on the number of electrons trapped in the molecule not on its conformational

Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 3. Cellular Neural Network (CNN), or CA monolayer fabrication: a) Molecular monolayer is first converted into a matrix of binary trinary or tetranary bits and then between two matrices basic changes at the smallest clusters of cells are determined, these basic rules are CA rules of the monolayer; b) STM image of negatively changed DDQ (-1) monolayer scale bar is 20 nm, scanned at 0.05 nA tip current and 0.88 V tip bias. A cluster of molecules change its direction by a particular angle, so we define one orientation as P and the other as Q. Elementary changes between P and Q is the CA rule; c) STM image of the organic monolayer of Rose Bengal molecule at four particular biases for a fixed tip current 0.05 nA. One molecule is 1 nm in length and 0.32 nm in width inside the monolayer; d) Rose Bengal single molecule device between Au (111) electrodes (left). Molecular model of a Rose Bengal monolayer, on a mono-atomic linear chain surface

changes. Then information processing would be re-distribution of electrons. Since in the monolayer conformational changes are probabilistic event, it might cause unwanted errors during computation, -invalidating the CA rules. Third, packing of molecules should be such that following a sequence of changes in orientation a cluster of molecules would return to the same state. This particular phenomenon allows a few neighboring

molecules to execute unique local CA rules and allows the system to respond to global changes at a much faster rate, for instance deleting entire pattern from the CNN surface. To demonstrate a practical application, we have fabricated a monolayer of Rose Bengal (RB) molecule in which the electronic property of the particular RB conformer that demonstrates step function in the IV measurement is pronounced.

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Advanced Architecture of a Massive Parallel Processing Nano Brain

The only possibility for such a response is that in the confined state, the molecule can switch only to a single conformer state. Therefore, the surface could serve as a potential CNN template (Figure 3 c, d). Irrespective of the electronic response of a nano-size CNN cell (constant resistor, variable resistor (memristor), rectifier, or multilevel NDR switch) when packed in the form of a molecular assembly, the integrated system should function as a cellular neural network. Note that the difference between CNN and classical CA, is that a CNN follows a particular equation to determine the next state of a cell therefore even neighboring cells would be temporarily independent of each other, whereas CA has a particular set of rules to do the same synchronously or in an asynchronous way. In this particular respect, building a CNN and a CA architecture would be essentially different in construction principle. While designing a 2 D molecular template for a CA architecture, coupling of cellular cells require to be dependent on neighbors for some particular aspects, and at the same time, independent of neighbors for some other particular aspects.

Spherical Assembly To create a standalone processor that can operate remotely, we propose the need to construct a practical CNN or CA on the surface of a stable spherical assembly. One option for the spherical assembly is to use dendrimers (Galliot, 1997; Devadoss, 2001; Quintana, 2002) or dendritic molecules (Hawker, 2005; Peer, 2007) as the core architecture and then replace the end groups with molecular neurons. Alternatively, molecular neurons may be anchored covalently/non-covalently with the dendrimer end-groups. The number of end-groups increases with the higher generations of dendrimer generally to the power of two as branching increases by two fold inside the dendrimer (Figure 4a). Chemists can modify the surface of the dendrimer with a distinct number of

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molecular neurons constructing a CNN/CA. For the CNN/CA to operate on the sphere, all molecular neurons on the upper surface should be packed densely in a similar way to the organic monolayer on a planar surface (Bandyopadhyay Pati 2010). Otherwise, a universal equation determining the next state of a cell (CNN), or the discrete CA rules cannot be determined accurately (see DRQ neurons on G2 PAMAM dendrimer movie, http://in.youtube.com/watch?v=5S8UW_3bxlg, Figure 4 b, c). There is also a serious doubt over the over the possibility of existence of any such rules or equations, if surface molecular neuron density is not properly optimized. Our dynamics simulations of such structures show that the entire architecture is extremely robust as these structural fluctuations are not localized only to the surface rather its a global phenomenon, i.e. the entire dendrimer architecture takes part in the relaxation process. Therefore, for a particular size of a molecular neuron, there is always limiting number of molecular neurons that could be attached to a dendritic surface. Note that for a particular kind of molecular motors, a threshold value of dendrimer-generation (say G8 or G9) is also necessary, which gives rise to an atomically packed CNN surface. An ordered atomic pack is essential for the well-defined transport of electrons or logic states throughout the surface. However, it appears a practical possibility that the surface neurons exchange electron/information even directly with the core part of the dendrimer which would modulate the computation in a much more complicated way than expected. The CA rules if a CA surface is generated on the dendrimer surface or a universal cell state transition equation if we generate the CNN on the spherical surface would be different from the rules/equation of CNN created by the same neurons on the planar surface. Dynamics of the dendritic branches would determine the global relaxation of a molecular assembly. Alternatively, therefore, it is also a strong possibility that the entire 3 D architecture will more predictably re-

Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 4. Dendritic assembly design and its architecture: a) The architecture of a PAMAM dendrimer. Shaded regions within the dotted area are doped multilevel molecular switches or molecular neurons; b) Examples of generation 2 (G2) PAMAM dendrimer with molecular switches on its surface, the surface is covered with DRQ 2 bit molecular switches, architecture is energy minimized using density functional computation; c) Hydrogen bonding and van-der Waals surface plot of the dendrimer covered with molecular neuron; d) Quadrupole moment contour image of a material. Dark shade denotes negative and lighter shade denotes positive charge region; e) STM image of G 6 PAMAM balls on an Au(111) surface, most of the balls are assembled at the step edges, image is taken at 0.91 V bias and 0.05 nA tip current; f) If the dendritic architecture has only one symmetry then CV appears as the left spectrum. All redox active sites accept electrons at once at a single bias, and releases all at some other bias. Two distinct structural changes are visible. If the dendrimer has multiple distinct symmetries then equivalently number of redox peak increases, for every single peak a distinct conformer exists. Note that the dendrimer may take more than one electron corresponding to a peak

sponse as a 3 D cellular automata due to the spontaneously robust dynamic behavior explained above. Note that with the increase of dendrimer generation, surface molecular interaction increases and the end groups collide less with the dendrimer’s core branches that comes outside during dynamic relaxation. Therefore, the magnitude of electric bias for reading and writing of bits on the molecular neurons assembled on the dendrimer surface would be significantly different from the monolayer constructed on the atomic flat metallic/semi-conducting surface. Also, encoding a particular state on the spherical surface

of the dendrimer will not be like writing a bit on a single molecule as potentially isolating all neighbors during writing a state is nearly an impossible task. Our studies have shown that the entire architecture would change during writing a bit. Therefore, writing process would be tricky, so will be understanding the fundamental cell state transition rules. There are two major factors that would determine the origin of CA rules or CNN governing equation on a spherical dendrimer surface. The most important factor is the ratio of orbital conjugation and non-conjugation regions of the

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Advanced Architecture of a Massive Parallel Processing Nano Brain

dendritic branches in the spherical core (Devadoss, 2001). If the core is conducting and forms polarons as electric pulse is applied to write a particular state on the CNN surface, polaron mediated conduction throughout the core may modify the logic pattern written previously. Moreover, during computation there is a strong probability that the quasi particles like polarons would modulate the evolving patterns. Our rigorous simulation of the dendritic architectures with various fractional ratio’s of conjugated/non-conjugated regions have shown that more is the conjugation region of the dendritic architecture less robust is the dynamism. It appears that there exists a critical ratio for enabling a computing in the computing architecture. The second factor is the quadrupole moment (Figure 4d). The extremely dynamic architecture does not remain ideally a sphere during operation, the symmetry of the architecture changes continuously during its computation and information processing. Therefore, the quadrupole moment of the system may play a major role in deriving the final solution pattern on the CNN/CA surface. We have deposited non-conjugated PAMAM dendrimers on an Au (111) surface (Galliot, 1997; Devadoss, 2001; Quintana, 2002). By fixing the STM tip at the vicinity of the top surface of a single dendrimer we have sent multiple random pulses to test Random Access Memory (RAM). We have studied 2, 5, 6, 7 and 8 generation PAMAM dendrimers with amine or COOH end-groups and have analysed the output current considering the complete system as a single unit (Figure 4e). Continuous RAM operation has detected more than 80 conducting states in each dendrimer system suggesting that even in the non-conjugated systems, the quadrupole moment originating from a particular symmetry enables the generation of distinct conducting states. Until now, we have not been able to map the logic pattern distinctly on the surface, which represents a fraction of entire statistical response since only the upper hemisphere is visible to the STM scanning (Figure 4e). To resolve the problem we place a dendritic

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architecture into a nano-gap junction (Khondaker, 2002) and by applying a horizontal electric bias across the dendrimer ball we tune the fundamental conductance level of the system. In presence of this electric field, a special provision made for scanning the dendritic surface vertically so that the conducting states are probed from the top. This dual characterization technique not only helps to identify the pristine conducting states of the dendrimer, an additional gating effect resolves the sub-conducting levels, -apparently not visible in the two probe measurements. Calculation further shows that these sub-conducting levels are strongly correlated to its relaxation symmetry. We have already shown experimentally that the simultaneous three-electrode measurement (Bandyopadhyay, Nittoh, 2006) of conductance variation is more reliable than the conventional two terminal measurements for mapping the dynamic relaxation of a nano-brain. Dendrimer-like spherical assemblies may trap several electrons at a time, since a sphere has a particular set of symmetry (Figure 4f; Balzani, 1998; Nielsen, 2000; Takada, 1997; Denti, 1992) and the entire architecture oscillates between these symmetries separated by a very low energy. Any molecular assembly with one fixed symmetry cannot sustain distinct patterns of different energies on the outer surface. At a very particular reduction/oxidation bias, all neurons would switch simultaneously to a fixed conducting/logic state without any control as a large number of electrons probably equal to the number of neurons on the surface would be trapped into the system or move out of the system at a time. Therefore, a perfect symmetry may be undesirable for the construction of a nano brain. The more is the possibility for a symmetry induced distinct phase transition of the dendritic architecture, the more versatile would be the information processing on its surface end groups. The multiple reversible electron exchange process of a dendritic architecture could be recorded using cyclic voltammogram (CV) similarly as reversibility of multiple conducting states in

Advanced Architecture of a Massive Parallel Processing Nano Brain

a single molecule redox switch is probed using CV or current voltage characteristics. However, simply by increasing an asymmetry in the architecture would generate another important problem. Externally, encoding a logic pattern on a spherical surface would be a reliable technique only if the surface is perfectly spherical. The reason is that prior to the real testing of CNN/CA performance, the governing equation for a CNN’s information processing or the existence of CA rules are required to be revealed. Only way to do that is to cover the surface with several vertical electrodes fixed at tunneling distances apart. These electrodes for commercial application would be static. If the spherical surface fluctuates extensively a reliable characterization can not be carried out, as static electrodes and end groups might couple physically. Therefore, a considerable number of structural symmetry of the dendritic architecture is essential to map the evolution of logic states on the dendritic surface. However, these structural transitions should not change the global spherical limit of the dendritic architecture.

The Central Control Unit (CCU) or Core of the Sphere Ideally, a little change in potential or conformation at the central region would generate massive changes in the surface logic pattern. As every neuron on the sphere is connected directly to the center, we have modulated the arrangement of molecular switches to in a particular pattern so that different kind of potential distribution could be created around the center of the spherical assembly (Figure 5a). A key principle to the CCU coding pattern is the modulation of polaron transport across the conjugated dendritic branches inside the sphere. If switching molecules are trapped at the center of the sphere then the polaron transport length is maximum, equal to the radius of the sphere (Figure 5b). Therefore, if an entire logic pattern on the spherical CNN surface are required to be modu-

lated at once then a multilevel switch has to be trapped at the very center of the sphere. Depending on the required modulation area of the CNN surface, molecular switches should be trapped at particular branching locations (Miklis, 1997). The smaller the surface area to be modulated, the further would be the trapping location from the sphere center. Since conductance level, logic state and number of electrons in a molecule are closely associated, therefore, the 3 D potential map calculated from the molecular switches located at the core in the dendrimer branches would be the CCU code (Figure 5a). The relaxation dynamics of the CCU region would continuously set new kinds of patterns on the CNN surface one after another as shown in the Figure 5 a series. Each pattern would correspond to particular set of CA rules or a CNN governing decisive equation. Therefore, particularly designed CCU coding enables a spherical assembly to carry out series of decision-making computations one after another, spontaneously. For computation, we need to write an input pattern on the CNN surface (Figure 5c). During encoding the input pattern on CNN surface, CCU code might change because of the pulse applied. Note that the molecule located at the center of the sphere would require the maximum bias, and the molecules located at higher concentric radii from the center would require lower bias for encoding a conducting state. Therefore, two distinct patterns, one in the CCU and another on the CNN could be encoded simultaneously. For that particular purpose, we need to generate a database and isolate central and surface encoding process. For a better accuracy in pattern encoding, different kinds of molecular switches could be used as structurally distinct molecular switches have distinct threshold biases (Figure 5a). The difference between switching biases would enable one to identify the parameters that allows selective encoding in the core and in the surface.

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Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 5. Design and analysis of Central Control Unit: a) Basic molecular architecture of the dendrimer core and schematic presentation of the potential density distribution in the CCU. Five molecular neuron examples A, B, C, D, E are presented here. Depending on the molecular conducting state; the CCU core changes its potential. The schematic presentation is based on theoretical simulation of five molecules, DDQ (A), Rose Bengal (B), DNA (C, only one periodic unit is used), OH/NH2 doped C60 (D) and phthalocyanine (E); b) Variation in polaron transport length L with angle θ, L is determined by measuring the distance between negative potential boundary surface of CCU and the outer CNN surface. In practice, polaron transport length does not follow this principle accurately but this particular scheme provides a consistent output. We fix a neuron molecule or CNN cell on the spherical surface, and angle that it makes with another CNN cell on the surface with respect to the center is the angle θ. L variation for two particular CCU potential cases are plotted here, each CCU potential generates a particular set of rule, say set A (having n distinct rules), set B (having another n distinct rules, n is a natural number); c) If a point charge moves through the surface of the CCU, the potential measured is a function of θ, CCU(θ). Similarly we get another potential function CNN(θ), and these two potentials are correlated with a function of L, f(L), which measures CCU initiation strength in determining the CNN rules; d) Pristine role of CCU is extracted from the mixed response of a nano brain using basin of attraction (BA) method. Each point on BA plot is a m×n logic matrix

OPERATIONAL ALGORITHM Dynamics of Cellular Automata/CNN and CCU It is not possible to analyze dynamics of such a large architecture using fundamental quantum mechanical methods, since complexity of dynamics would require an astronomically large computational power and time. Therefore, we follow an alternate method of analyzing the CNN dynamics in terms of CA dynamics known as basin of attraction (BA, Wuensche, 1991; Martin, 1984; Pitsianis,

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1989). Until now, we are not able to make very particular changes in the mathematical model of BA dynamics formulation so that it isolates the structural and electronic relaxation of a CNN/CA hardware. Since the BA dynamics formulation maps the time evolution of CNN/CA logic pattern’s distinction at an identity space, comparison of this map with the monolayer would provide important differences caused by the spherical shape and the CCU dynamics together. However, as stated above we need to isolate the role of the outer surface molecular switches/machines and CCU from the mixed electronic response so that

Advanced Architecture of a Massive Parallel Processing Nano Brain

we can map the effect of the CCU on the global performance of the CNN. One possible way to do that is to construct two spherical assemblies one with CCU and one without the doped switches (no CCU). Comparing BA profiles for these two different molecular architectures we can identify role of the CCU comprehensively (Figure 5d). Using statistical response, we need to develop two sets of grammatical rules to decipher the map of these dynamics simulation, one for the CCU and another for the CNN/CA surface. The BA dynamics of cellular automation is analyzed by identifying a cellular automation transition function. Using this function, all possible states and trajectories are determined and plotted as global behavior of the system. The space-time pattern generated by CA rules would also become a function of CNN pattern if simplified spatio-temporal way and that would represent its dynamic relaxation. Two kinds of symmetries, one rotational and another bi-lateral survive on the spherical surface. A correlation between the structural symmetry of the spherical assembly and the symmetrical emergence of CNN’s equation or CA rules is an important tool to manipulate continuous relaxation process of the CCU. Its relaxation changes the CA rules operational on the surface CNN/CA. Finally, time dilation between the occurrences of two relaxations requires to be tuned to develop a temporal control on multiple consecutive patterns evolved on the CNN surface (Figure 5c, 6a).

Spontaneous Relaxation of the Sphere Spontaneous relaxation of the molecular assembly is different from the self-organization. The spontaneous relaxation originates from the nature of structural asymmetry and a 3 D potential profile of the central core (CCU). While self-organization may be localized entirely on the surface, without changing the potential profile of CCU, a sponta-

neous relaxation would restructure the potential profile of CCU automatically and in turn would redefine the CNN governing equation or the associated CA rules (Figure 6a). To understand the spontaneity of the relaxation process, we excite the spherical molecular assembly to a higher temperature or forcefully induce an extra energy to the system, and then observe the dynamic relaxation to different local energy levels. The rate of relaxation decreases as the system reaches to a global energy minima (Figure 6b). Understanding the spontaneous relaxation process is extremely important, as our objective is to run the massive parallel processing on the CNN/CA surface continuously executing a series of operations one after another. Spontaneous generation of new logic patterns in hardware is a primary requirement for creativity, but here our concern is not to evaluate the degree of spontaneity. Instead, we need to define spontaneity as a predictable event within the stochastic framework. In other words, a global relaxation of the assembly would be a part of the continuous relaxation process on the CNN/CA surface. As soon as a global relaxation takes place, local relaxation on CNN/CA surface would continue to switch the assembly to a local energy minimum. Finally, CNN/CA surface relaxation would induce a global relaxation of the entire assembly (Figure 6a). This is necessary, as we cannot allow the time difference to change between the two consecutive relaxations. E. Behrman’s brain model (Behrman, 2006), driven by Hameroff and Penrose’s proposed quantum computation, also suggest a stable local minima (objective reduction) similar to our global minima, following quantum mechanical fluctuations. Time delay of the order of seconds is required between the two global relaxations for practical applications. This is achieved by connecting the two global relaxations with a series of essential local relaxation processes. Since a number of multilevel switches can construct the CCU, we

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Figure 6. Translation operation between CCU and CNN: a) The schematic potential variation of a nano-brain (left) and the potential of CCU (right) with the generalized time co-ordinate. The potential variation of CCU, appears as staircase compared to CNN potential variation. Fluctuation in input pattern on the CNN surface does not vary more than ΔE, CNN input pattern is so selected that it reaches equilibrium at < Δt, each step of staircase correspond to particular potential distribution of CCU, so that till computation finish, the dendrimer core does not change; b) Demonstration of spontaneous relaxation dynamics of a (RB/PAH)15 supramolecular assembly, packed in a ITO and Al coated sandwiched device, a possible staircase like degradation is observed. Capacitance variation in a continuous bias loop shows threshold charge storage, permanently. This phenomenon may prohibit continuous use of nano brain (it will be tired). c The structure of potential distribution in the CCU determines ΔE and Δt. If any section of potential distribution in the CCU is symmetric, and built with many such parts (left), then ΔE is lower and Δt is higher, however, if largely asymmetric, say connected by very small region between two large potential spheres (right end), then ΔE goes higher and Δt goes lower

convert the 3 D map of logic states into an energy map, and study its relaxation process. Conformational change is essentially a slower process and the process becomes even slower with the increase of design complexity and the assembly volume. Higher is the dendrimer generation, lower is the self-diffusion of atoms in the dendritic chain during dynamic relaxation (see Fricks law, Frenkel 1996). Mean square deviation of the constituent atoms remains nearly constant at higher than 6 generation. Therefore, the volume distribution of potential is an important factor for encoding CCU program. Structural relaxation studies have further shown that an isolated large negative po-

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tential region at different parts inside the sphere redistributes the logic states in such a way that effectively the total electric field is also minimized. Moreover, the global relaxation is initiated by a very particular design where two differently charged lobes are connected by a small channel (Figure 6c). Therefore, using this particular design trick, we can tune the number of global relaxation we require for a particular complex job where a number of independent works need to be done one after another. In addition, the time dilation between two relaxations is entirely determined by density of charge distribution inside the sphere.

Advanced Architecture of a Massive Parallel Processing Nano Brain

Self-Organization and Learning Input Pattern As soon as a distinct pattern is written on the surface of the spherical molecular assembly, the pattern undergoes certain changes and the architecture stabilizes generating a modified logic pattern on the CNN/CA surface. However, our preliminary study on such a spherical assembly shows the emergence of particular changes in the logical pattern evolution, when particular kind of logical operations are performed repeatedly on the CNN/CA surface (Figure 1b). For the hardware, this is hysteresis, and for the software, i.e. the logic pattern, this is learning. The system is therefore called adaptive. Hysteresis driven by local environment of the structure initiates certain changes so that particular rules are favored in comparison to others. When new kinds of patterns are processed repeatedly on the CNN surface, previously existing hysteresis is removed and a new kind of hysteresis is generated in the system. In other words, one kind of learning is erased and a new experience is learned (Figure 1b bottom). We feed multiple sets of logical input and output patterns into neural network package analyzing the learning process in terms of adaptive linear element, perceptron and learning matrices formalisms (by determining f(L), Kohonen, 1989). Note that, simple step function based devices or memristors are mathematically simpler than the multilevel molecular neurons, but no learning process has been detected in the monolayers of memristor kind molecules in our preliminary studies.

Mutually Co-Existing Spontaneity and Impulsive Operation In certain cases, nano-brain should execute a complete operation spontaneously and in some cases, it should follow computational instructions blindly. These are contradictory requirements in the same hardware. It is already mentioned that

a very particular kind of potential distribution enables the assembly to induce a global structural relaxation. If this is entirely a materials property then we cannot perform any computation that requires only one global relaxation of a particular kind. Impulsive operation means computation that occurs between two global relaxations (Figure 6a), and that is the time domain where a very particular kind of computation is performed blindly. Distribution of multilevel switches inside the spherical assembly is not the absolute condition for encoding a particular sequence of global relaxation, we also need to apply certain large electric pulses to activate CCU potential profile inside the assembly (Figure 5c). Therefore, we can select particular kind of impulsive computation in the wide bandwidth of energy spectrum, or a set of impulsive computation wherein spontaneous and impulsive computation may co-exist (Figure 6 a, b).

COHERENCY OF THE ARCHITECTURE DURING OPERATION: FACTUAL PARAMETRIC ANALYSIS Size, Dimension Relation of a Nano-Brain In a nano-brain, computation may emerge at nanoscale but its complete architecture should have a size that can be interfaced using existing nanotube or nanowire electrode based characterization setups. Otherwise, nano-brain would face the same fate as molecular (Hipps, 1991) or sub-molecular (Ami, 2002) electronics is facing today. Say, we have an atomically flat crystalline spherical Au (111) ball of 1 cm2 then on top of this surface we can adsorb 100 billion molecules each having 1 nm2 area in a spherical monolayer formation. 100 billion is possible on a sphere of 1 cm2 area if we can interface 1 atomically sharp electrode in 100 nm2 area, thereby 1 billion electrodes around this

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surface. Using one billion electrodes, one can write one billion bits at a time on this sphere and read same amount of data at a time. Therefore, it is potentially a 1 billion bits parallel processor. Biological cells have dimension of the order of 50 μm. To transport cells in our body, nano-brain carriers should be ~ 1 mm, however, to carry out operation inside a cell, its size should be ~ 10-20 nm or size of a virus. It has to withstand immense dynamic movement of the species inside a living body. This is more valid in a nano-factory where a considerable physical force would be exerted to the nano-brain during operation. Therefore, if nano-brain seed is sufficient for the need, we may use it as is; however to meet the requirement of complexity we may need to create architectures varying from 20 nm to ~ 50 μm. We can assemble many such nano brain seeds following the same design of nano brain seed as explained in Figure 1d (right) having various radius, however, here the basic construction unit would be the nano brain seed, with different distinct performances. Undoubtedly, they would provide much better flexibility in operation to the giant nano brain. Here we concentrate on dendritic nano-brain architectures since they are simplest model to study, however, the one-to-many principle could be realized in several unique assemblies, for an example icosahedral virus geometry (Casper, 2004). A dendrimer has total nbc numbers of branches, nbc = nc [(nbG +1 − 1) / (nb − 1)] , where nc and nb are the functionality of core and monomer, G is the generation number. If we plot the number of neurons that could be attached to the surface (nt = nc nbG ) with its size ( µ r 2 , where r is the radius), number of neurons varies particularly with its radius and soon reaches to a singular point. The number of neurons or end groups could be increased until end-groups are atomic distances apart. Beyond that limit (~103 neurons or 10 generations), no growth is possible. If we wish to assemble 100 billion neurons with-

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out using metal core or hollow sphere, on a ~10 nm nano brain seed, 103 concentric spherical layers of dendrimer are required to be fused (considering one nano brain seed volume ~ 350 nm3), see mechanism of such fusion in the Figure 7a. Final size of the fused megamer brain would be ~ 10 μm. There are several ways to create megamer (Tomalia, 2005). The megamer should grow in such a way that it would again have a CCU and a surface CNN region, but nano brain seed as the basic unit. If we use different kinds of nano brain seeds with unique dendritic chain inside the megamer (Figure 7c), or different generations of dendrimers (Figure 7d), or different kind of molecular switches doped dendrimers, or spherical hollow assembly directly (Figure 7e) then there are enough freedom to manipulate CCU and CNN/CA properties by tuning the synthesis of the megamer. Here we suggest one alternate method of creating megamer, apart from direct chemical synthesis (Tomalia, 2005). After synthesizing the optimum generation of dendritic architecture or final nano seed with CNN/CA on its top surface, we try to assemble many such seeds using layer-by-layer self-assembly (LBLESA, Decher, 1997; Bandyopadhyay, 2003). We use nano brain seeds with negative and positively charged surface (Figure 7a). For this purpose, CNN cell molecules are anchored with multiple COOH or NH2 groups so that when alternate layers of nano-seeds are grown, heating up the complete architecture at ~ 250-300o C, forms acetamide or –COONH- linkage between any two dendrimers. Thus, a giant single molecular sphere is created. Here 103 cycles (1500 bilayers) of alternate ESA deposition is required, as each bilayer growth saturates in ~20 minutes in solution; it requires automated deposition for 6 days. For fully automated fabrication, we adsorb the nano-brain seed at the top of vertically grown carbon nanotube (CNT) film (Wei, 2002), so that automated deposition produce several nano-brains at a time each placed at the top of a CNT (1.5 nm

Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 7. Translation region construction connecting CCU and CNN: a) To construct large assembly of dendrimers or megamers, each dendrimer nano seeds are to modified such that some of them are negatively charged with end group –COO-, and some of them are positively charged with NH3+; b) Construction of a megamer nano brain. Dendrimers with particularly designed cores are arranged very uniquely at the central region to construct the CCU, and similarly, another kind of dendrimer that generates large number of distinct conducting states, processes RAM and ROM operations are assembled on the top of the megamer, to construct the CNN. Different layers of concentric spherical electrode assemblies’ cover 50 μm diameter nano-brain; c) Modulating the electronic transport through the dendritic chain is possible by doping redox active groups at various possible locations in the chain, the very central region could also be changed by replacing the core with different molecules or functional groups; d) The complete megamer architecture could be built using dendrimers of different generations eventually leading to a complete sphere; e) The simplest nano-brain architecture could also be constructed on a microsphere, made by metallic or semiconducting nanoparticles, or even surfactants/designer molecules; f) A table comparing nano brain seed and a megamer brain is presented

in diameter). Among several other possibilities, we can also grow micelle architecture (Dou, 2003) using the basic seed dendrimer. Growing a giant

H bonded network (Satake, 2005; Huang, 2006) is another choice of growth starting from the basic seed assembly.

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Figure 8. Symmetry and potential of translation region: a) The basic pulse array sent to a nano brain seed/megamer to test memory and information processing ability of a device (left, top). If the dendrimer responds to the input pulse positively, both read current pulses would differ (left, bottom). The measurement circuit used for this purpose has a function generator and an oscilloscope (middle) attached to the STM. To detect all processable states, a continuous array of pulses with varying magnitude is applied (right); b) Different possible symmetry structures that coherently connect CNN and the CCU. Co-ordinates of the highly conducting dendrimers, (highly doped with conductance switches, and denoted as shaded balls) are located following an equation r = a sin k q . If the periodicity constant a varies, then we derive significantly different arrangements of dendrimers. Energetically allowed values of k for these arrangements fall into Fibonacci/Hemchandra series (top). If a is constant, then solutions fall into the L. Grandi’s lotus/rose series (bottom)

Writing a Pattern on CNN Without Destroying CCU Configuration and Vice Versa The top CNN surface of a megamer or nano brain seed would continuously change the input pattern with time, to derive the solution of a problem written in terms of logic pattern. Note that operational mechanism of nano brain seed and megamer would have significant differences (Figure 7 f). We can follow two distinct protocols to operate the spherical assembly selectively between CNN/CA and CCU. One of them is to write the CNN/CA pattern so that the potential pattern in the CCU is written simultaneously as an effect of same sequence of pulses (Figure 8a). As CCU and CNN/CA have different relaxation time constant, and respond to

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different bias magnitudes, particularly designed array of pulses is effective in creating two distinct codes one at CCU other at CNN/CA. Alternately, we can write CCU patterns first, since it requires much higher bias, and then write CNN/CA input pattern, as it requires much lower bias. In any molecular CNN, encoding an input pattern requires several tricky approaches. As soon as part of a targeted input pattern is written on the surface, evolution of the pattern following CA rules starts instantly. An effective input pattern processed by CNN is very different, as it is changed to a combination of partly evolved and partly targeted input pattern. This is a universal problem and true for any self-assembled CNN cells. One way to confirm that CNN is processing the targeted input pattern effectively is to start writing

Advanced Architecture of a Massive Parallel Processing Nano Brain

from a rather different pattern, which would change in course of time in such a way that when final part of the targeted input matrix is encoded, the final pattern at that instant would be the targeted input pattern. One key issue for this method is matching the dynamics of pattern evolution with the writing speed. Computation on the CNN surface is generally uncontrolled and starts spontaneously. By proper choice of input pattern one can initiate spontaneous change in pattern by scanning once at a higher bias. However, to minimize STM influence, spatial gap is created between different parts of the input pattern so that temporal evolution of pattern is controlled accurately. Note that in a megamer, the required electric bias would be much higher for writing CCU and CNN/CA codes, rather than that for the nano-brain seed. However, writing the potential pattern in megamer brain would be much easier than the nano brain seed since a finite potential barrier between two nano brain that seeds in the megamer would prohibit an uncontrolled communication among neighbors.

Coherent Control of CNN by CCU A small potential profile change in the CCU can change the set of CA rules under operation on the spherical surface. The size of a typical nanobrain seed would be ~ 12-15 nm. The tunneling mechanism is effective around ~ 6 nm separation, therefore a nano-brain would have a quasi-charge (polaron/soliton, Goodson, 2005; Wu, 2006) dominated transport mechanism in the dendritic trees. As the soliton moves with the velocity of sound, CCU to surface CNN communication of a nano-seed takes a few pico-seconds. CCU potential distribution governs transport of polaron waves from the central region to all directions. Simultaneously, change in surface potential distribution during evolution of CNN pattern is influenced by waves propagating through dendritic tree. Coherent control of CCU is modulated by large

number of switches doped in the branching trees connected between CCU and the CNN surface neurons in the nano brain seed. However, since our megamer nano brain size is ~ 10 μm in diameter growth of molecular assembly from 15 nm to 10 μm would die out all codes written in the CCU. Therefore, we have developed particular design, translating the CCU code to the final CNN surface. Building such a giant architecture follows particular rules. First, the entire architecture is conjugated so that polaron/soliton transport survives throughout the assembly. However, the system may not operate faster than megahertz frequency (106 Hz). Second, the basic pattern features of the CCU potential map in the space between nano-brain seed and outer CNN surface is created by arranging different nano brain seeds (operates in different bias region in particular geometric shape and pattern. Third, different nano brain seeds are arranged in the architecture so that they do not screen the CNN or any potential distribution created below the surface CNN. Our recent study has shown that if CCU potential distribution is created by arranging more conducting nano brain seeds (operates at very low bias region) in a geometric pattern following r = a sin k q , where r is the co-ordinate of doped neuron cluster, k is branch number, θ is angle of deviation, a is constant particular of a dendritic architecture then the CCU code can CCU strongly influence the CNN surface even at a separation of 20 μm (Figure 8b). Fourth, since all top layers of the megamer nano brain are covered entirely with nano brain seeds; they create a 3 D cellular neural network of nano brain seeds. However, they follow a new set of CA rules or CNN governing equation. For a nano brain seed, the CNN/CA surface consists of multilevel molecular switches, or molecular neurons, however, for the megamer giant brain, CNN surface consists of nano-brain seeds, each can take thousands of decisions. Therefore, we may nearly match the biological neuron performance in an artificial hardware.

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SPECIFIC DESIGN MODIFICATIONS FOR NANO-BRAIN OPERATING IN A NANO-FACTORY OR INSIDE OUR BODY OR OPERATING AS A SUPERCOMPUTER Nano-Brain Operating In Human Body as a NanoSurgeon or Nano-Doc Most essential requirement for building a nanosurgeon is to design and synthesize molecular machines operational while floating in the body fluid. Till now, molecular engineers have not adopted any universal solvent for probing their machine performance so that an integrated nanobrain is functional in that solution when coupled to those machines. Every different molecular machine operates in a specialized solvent, and there is no guarantee that they would continue to operate if that solution is replaced with the body fluid. Once the problem is resolved, nano-brain is all set to couple with the existing molecular machines on the CNN/CA surface and the final architecture would be ready to be injected in a body. However, some molecular machines are driven by electron injection, some of them are pH dependent and some of them even dependent on the ion concentration of the solution (Bertrand, 2000; Schular, 2000). If molecular machines are coupled to the CNN/CA surface, the CA operation or pattern based computing is practically stopped or modified significantly. The logic states are written on the molecular machines as instructions and the surface potential distribution is changed as the CCU relaxes the system following its inherent programming. The nano-brain functioning as surgeon does not require the CNN to compute the evolved input pattern encoded at the beginning. Inside a human body, a nano-brain is considered as foreign element and antibodies attack the assembly destroying its functionality. Therefore, we propose to cover nano-brain with hormones/ enzymes known to our body by forming giant

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enzyme complexes with the nano-brain (Perham, 2004). To form multienzyme complexes, those particular hormones/enzymes are chosen which specifically act in those targeted place where the nano-surgeon needs to operate. As hormones/ enzymes uses lock-and-key trick to reach to a very particular part of our body, nano-surgeon would be transported automatically to that very region. Covering and uncovering of the nano-brain should also depend on pH variation or Na/K ion concentration, so that they are covered with hormones/ enzymes once again automatically as soon as antibodies attack them during operation. Covering nano brain with micro-capsules is an alternative method where LBL-ESA deposition technique is used to cover the assembly, and it is released at particular pH similar to a drug delivery (Figure 9a; Sukhorukov, 2001). Finally non-antibody ligand could also be a possible choice to anchor with nano brain or it could be captivated inside the ligand-bound nanocarriers (Peer, 2007).

Nano-Brain Operating in a NanoFactory Control Unit The supercomputing megamer nano-brain has surface CNN/CA operating under particular governing equations or CA rules. When a CNN/CA surface is covered with the maximum possible number of machines, it is nano-doc. In contrast, nano-FCU has to exhibit both kinds of signal processing. To run a nano-factory we need to analyze multiple parallel decisions on the CNN/CA surface. In addition to this, multiple smaller conical/ spherical/tubular assemblies covered with large number of molecular machines are attached to the CNN surface to function as working arms of FCU. Versatile synthesis of dendrites allows fusing two distinct branching of dendrites into a single structure (Galliot, 1997). The primary sphere is the major computing CNN/CA based CA system, and a number of conical/spherical/tubular assemblies covered with large number of molecular machines connected to this primary sphere translate that

Advanced Architecture of a Massive Parallel Processing Nano Brain

Figure 9. Design modifications for medical application, factory building and supercomputing: a) Design of a nano-surgeon, surface covered with drugs or operating machines, and camouflage of nano-surgeon is done by covering with enzyme, microcapsules and pH dependent materials; b) Design of a nanofactory control unit, particular regions covered with machines interface with products, these interface region are potentially most active region of the CNN surface; c) For the nano brain to function as a supercomputer, a particular region is selected for the measuring output (cone shaped electrodes), for the nano brain seed, interfacing with multiple electrodes is nearly impossible, however, in a megamer brain one may attach one electrode in every ~150 nm2 area

decision into a well defined sequence of jobs (Figure 9 b; Kurzynski, 2006). Following this particular design, one can analyze and simultaneously execute operations of multiple distinct nano-surgeons together for creating a delicate product with multiple functionalities. Similar design requirement is also demanded by L. Behera’s quantum brain model (Behera, 2006). While modeling eye-activities, an implicit requirement of two parallel brains has been established. One is a quantum brain, triggering wave-packets to reproduce experimental observations and another one is classical brain processing the eye-sensor data. Here, smaller conical/spherical/tubular assemblies covered with large number of molecular machines functions equivalent to the quantum brain, and the nano-brain seed functions as the classical brain. Therefore, nano-FCU may also function as nano-assembler with robotic arms.

Nano-Brain Operating in Massive Parallel Supercomputer For using nano-brain as a standalone supercomputer, the spherical assembly has to be covered with a stable spherical cover with thousands of nano-needle electrodes to write input patterns on the surface (Figure 9c). Nano-needle electrodes must not move relative to each other during operation. The CNN surface should translate reliable output pattern to the electrode arrays. Supercomputing using the nano-brain means generating differential equation starting from a simple set of pattern (Toffoli, 1984; Biafore, 1994; Margolus, 1984; Wolfram, 1989), which change with time following solution of the differential equation. By changing the CCU potential profile, we instruct the assembly to carry out a particular kind of mathematical operation. Computation does not necessarily means generating a pattern that produce say a primary number as the solution of a problem. Rather, the solution would be a

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generalized pattern reflecting all possible solutions that could arise because of this particular problem. Generalized features are more pronounced than the specific features of a problem, for this kind of computation. Mathematical operators and operational tools for the computation process are written in the nature of potential energy distribution in the CCU. Importantly Christopher Davia’s brain model (Davia, 2006) also concludes that spatio-temporal pattern of the traveling waves inside our brain is responsible for computation. Solitons, traveling waves and non-dissipative robust waves maintain structure and energy during computation of our brain. However, according to him this condition is valid till they are propagating in the relevant environment. This particular condition enables the system to generate versatile decision-making and global co-operation in biological computation. The CCU potential profile mimics modulation of polaron/soliton length, which is equivalent to Davia’s constraint condition.

CONCLUSION: THREE KEY CONCEPTUAL ADVANCEMENTS AND A REVIEW OF CONSCIOUSNESS In this article, we have discussed three key factors to develop our primitive 17 molecular nano-brain to an advanced mm-brain composed of 100 billion neurons. In our previous proto-nano-brain, CCU could not take any independent decision; however, CCU potential profile can now take series of decisions in principle and instruct execution units on the CNN/CA accordingly. Prior to our work on practical cellular automaton, CNN models were built using constant resistor, variable resistor or memristor, and rectifier. For the first time, we propose to use multilevel switches to construct CNN. Note that multi-valued logic is essential to construct an adaptable learning hardware (Perkowski, 2002). Our conceptual spherical CNN/CA is apparently first of its kind with distinct advantages over planar CNN/CA. The spherical architecture could be interfaced more efficiently with the outer world than the

Figure 10. An example of simplest nano factory: We are showing here eight conventional machines connected to our nano brain (Bandyopadhyay, Acharya 2008)

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2 D disk shaped nano factory proposed earlier (Figure 10); surface area information processing is infinite in principle, computation is explicitly emergent than the planar surface as evolution of a pattern essentially considers all bits over the surface equally. Our third and final advancement is in linking a giant CNN/CA architecture to a nano-brain seed using a unique symmetric potential distribution. Note that this is a major paradigm shift from earlier proposed CA based computers (Hillis, 1984). Two significant experiments, dendritic expression of microtubule assisted protein (MAP2) in rats carried out by Nancy Woolf (Woolf, 2006) and the role of MAP-tau over-expression in the learning and memory of Drosophila studied by Mershim et al (Mershim, 2006) show that consciousness originates from the very communication between atom determining protein function. At higher scale consciousness is translated from proteins to cells and then from cells to tissue. These formulations are strongly supported by tracking activity of living human brains using electroencephalography (ECG) and magnetic resonance imaging (MRI). We are studying similar electronic and magnetic responses during nano-brain operation, since redox active atom’s activation to dendritic chains and thus communication between CCU and CNN may provide artificial BA map similar to the MRI scan of the real brain. The Hameroff’s argument (Hameroff, 2006), that consciousness is secondary response of a metastable pattern generated by synaptic pulses has close similarity to our studies. The correlation between BA dynamics for CCU and CNN maps the metastable state and the conscious output. Alternate to this spherical nano-brain, a helical assembly could also be constituted (Kornyshev, 2007), since microtubules may play major role in neuro-cognition and permanent memory in our brain (Woolf, 2006).

ACKNOWLEDGMENT Authors acknowledge Dr. John Liebeschuetz, The Cambridge Crystallographic Data Center, 12 Union Road, Cambridge, CB2 1EZ, UK for providing dendrimer structures and critical review of the work. Authors also acknowledge Prof. Michael J Cook from UEA, Dr. Jonathan Hill and Dr. Y. Wakayama from NIMS for their contribution in several results presented here. This work is part of the Japanese Patent filed JP-2006-19552. A.B acknowledges Kakenhi Grant from JSPS Young Scientist (A) number 21681015. Competing interest statement. The authors declare that they have no competing financial interest.

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Section 2

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Chapter 5

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing Takashi Morie Kyushu Institute of Technology, Japan

ABSTRACT The single-electron circuit technology should aim at developing information processing systems using the intrinsic properties of single-electron devices. The operation principles of single-electron devices are completely different from that of conventional CMOS devices, but both devices should co-exist in the information processing systems. In this paper, according to a scenario for achieving large-scale integrated systems of single-electron devices, some single-electron devices and circuits utilizing stochastic operation for associative processing and a spiking neuron model are described.

INTRODUCTION Many single-electron devices and circuits that realize CMOS-like digital logic were proposed so far (Takahashi, Ono, Fujiwara, & Inokawa, 2002). However, the single-electron circuit technology should aim at developing computing systems that perform information processing by using singleelectron phenomena (Morie, & Amemiya, 2006). Because their operation principles are completely different from that of conventional CMOS devices, we have developed a scenario for achieving large-

DOI: 10.4018/978-1-60960-186-7.ch005

scale integrated systems of single-electron devices as summarized in Figure 1. Single-electron devices should be used for massively parallel processing with a huge number of devices because of their large packing density and ultra-low power dissipation. The processing speed is improved by parallel operation. Few logic stages, a small fanout, regularity and repeatability in the circuit architecture overcome the interconnection complexity and lowers the background-charge effects. Parallel operation and such circuit architecture may make deep logic processing more difficult. Therefore, the use of ultra-small CMOS devices is essential. Singleelectron devices should be used for simple func-

Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 1. Single-electron circuit system design scenario. Morie et al. (2002).Nanosci. Nanotech, 2, 343, 2002.

tional circuits with few logic stages, and ultrasmall CMOS devices are used for multistage logic circuits. To overcome the design difficulty, it is important to use large output capacitance as a buffer for each circuit component. Here, “large” means a capacitance value that can store a few tens of electrons. Because of this output buffer, the operation of the circuit component is not affected by the following stage circuit, and thus modularized circuit design is applicable. The output capacitance is also used for an interface between SET and CMOS circuits. The information generated by massively parallel processing in SET circuits is collected and integrated into the output buffer and is transferred to the CMOS circuits. The gate capacitance of an ultra-small MOS device can be used as the buffer capacitance. In order to reduce the sensitivity to capacitance and background charges, new information-processing concepts and models also are required in addition to solutions regarding the averaging of information or redundancy configurations. The functionalities created by the single-electron operations and available for new information processing systems are as follows: (1) multiinputs circuits using multi-gate configuration in single-electron devices, (2) non-monotone inputoutput functions due to multi-peak (oscillatory) characteristics of single-electron operation, (3)

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stochastic operation due to stochastic tunneling phenomena, (4) energy minimization. Circuits and architectures using these functionalities for associative processing and spiking neurons are reviewed in this paper.

SINGLE-ELECTRON CIRCUITS FOR STOCHASTIC ASSOCIATIVE PROCESSING Concept of Stochastic Associative Processing Associative processing or associative memory extracts a pattern similar to the input key pattern from the memorized patterns. Conventional associative memories achieve deterministic association; the same input key pattern leads to the same association result. However, the human often associates different outputs from the same key. This property may be expressed as chaotic behavior in highly nonlinear dynamical systems. As another model, stochastic associative processing can be considered, and the stochastic property is achieved in single-electron devices effectively. There are some associative processing models also in the artificial neural network field. One is the associatron, a historical neural network model of associative processor (Nakano, 1972) and an-

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

other famous one is related to Hopfield networks (Hopfield, 1982 & 1984). However, these associative processing models are not considered in this paper, but the conventional associative processing architecture is used. In the conventional associative processing, the input key pattern is compared with all memorized patterns, and the memorized pattern most similar to the input is deterministically extracted. In contrast, stochastic associative processing does not always extract the most similar pattern, but the association probability depends on the similarity of the pattern to the key; i.e., the more similar pattern is extracted with larger association probability. When this concept is applied to digital data processing, its mathematical formalization is as follows. Let us define the key pattern given by the external system as A, and memorized patterns as Qk (k =1,…M). All patterns consist of N bit binary data: A = {aj; j = 1, ..., N},

(1)

Qk = {qjk; j = 1, ..., N, k = 1, ..., M},

(2)

aj ∈ {0,1}, qj ∈ {0, 1},

(3)

and uk is the number of unmatched bits between A and Qk, which is referred to as a Hamming distance. In addition, let us define k1 as the suffix of the most similar Qk to A, which means that uk1 ≤ uk, for all k ≠ k1, and k2 as that of the second, and so on; i.e., uk1 ≤ uk2 ≤ uk3 ≤ ....

define the probabilities of extracting Qk as Pk, it is expected that Pk1 ≥ Pk2 ≥ Pk3 ≥ ....

(5)

By repeating numerous extraction trials, we can obtain the order of k in similarity. The stochastic associative processing offers an intelligent information processing that differs from the conventional deterministic approach. Unique and useful applications of this processing are sequential stochastic association (Yamanaka, Morie, Nagata, & Iwata, 2001) and clustering for vector quantization (Morie, Matsuura, Nagata, & Iwata, 2002).

Stochastic Associative Processor Architecture Using Nanostructures On the basis of the processing model described above, a processor architecture is described. A pattern data is often referred to as a word in digital processing. Therefore, the associative processor has M word-comparators WCs and a winner-takeall unit (WTA) as shown in Figure 2. Each WC has N bit-comparators BCs. Bit-level comparisons Figure 2. Architecture of associative memory

(4)

The associative processor described here stochastically extracts Qk1. In order to achieve this operation, random fluctuation is added in the calculation of similarity. Because the calculation sometimes occurs a mistake, and therefore it sometimes extracts Qk2 or Qk3 and so on. if we

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

between A and Qk are performed at all BCs in each WC in parallel. Each WC unifies the results from the N BCs, the WTA compares all the outputs of WCs and selects the most similar word. The bit-comparator can readily be realized by using exclusive-OR (XOR) or exclusive-NOR (XNOR) logic operation, where the former outputs logical ’1’ only if the both inputs have the different logical bits, and the latter outputs ’1’ only if the both inputs have the same logical bit. In both cases, the output of the WC expresses the Hamming distance uk or N−uk, respectively. The associative processor extracts the memorized pattern having the shortest Hamming distance. Random fluctuation to achieve stochastic associative processing is naturally generated by the stochastic behavior of the single-electron devices, if a BC consists of single-electron devices. This fits to our scenario described because BCs are regularly arranged in the circuit. Figure 3 shows the architecture of a stochastic associative processor that uses nanostructures (Yamanaka, Morie, Nagata, & Iwata, 2008). The BCs consist of nanostructures and perform bitcomparison with random fluctuation. Thus, the input pattern is stochastically compared with the memorized patterns by WCs. The comparison

Figure 3. Architecture of the stochastic associative processor

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result by each WC is expressed as the total number of electrons released from the BCs belonging to the WC. The electrons are collected at each capacitor (Co1, Co2,..., CoM). The results of all the WCs are compared in the WTA, which deterministically selects the memorized pattern evaluated as that most similar to the input. The WTA circuit can be constructed by CMOS devices because it operates deterministically. In the following sections, some single-electron circuits and nanostructures for BCs and WCs with random fluctuation are described.

Word-Comparator Circuit Using SET Logic Devices Circuit Configuration The first example of stochastic BC/WC circuits is shown in Figure 4 (Saen, Morie, Nagata, & Iwata, 1998). The BC circuit consists of twoinput CMOS-like SET inverter INV and two-input

Figure 4. Word comparator using single-electron circuit. Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

SET switch SW. The two inputs are capacitively coupled, and are fed into the gate of the CMOSlike SET inverter. Let us define Ijk as the output current of the BC associated with electrons passing through tunnel junction TJ. By adjusting the device parameters, this circuit operates as follows. Input voltages Va and Vb correspond to data bits aj and qjk, respectively. If Va =Vb, then the voltage of node P, VP, is always. Output current Ijk is zero in this case. If Va ≠ Vb, then an electron goes in and out of node P at random; that is VP oscillates. Current Ijk oscillates accompanying electrons passing through TJ when VP = Va. Figure 5 schematically shows this situation. The frequency and duty-ratio of the oscillations are determined by the probability of the electron transition and depend on the device parameters. Thus, in this BC, if two inputs are equal: aj =qjk, then the output of the j-th BC is ‘0’. Otherwise it stochastically oscillates between {0,1}. All current outputs Ijk of BCs are summed up at capacitor Co and each WC outputs the result as a voltage. If both terminals of capacitor Co are shorted at intervals of sampling time ts by switch SW1, output voltage Vk at the end of each interval is

Figure 5. Schematic figure for explaining the relation between Vp and Ijk, where Va=1 is assumed. From Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

Vk =

1 Co

N

∑ ∫ j =1

ts 0

I jk (t )dt,

(6)

where Co is set proportionally to ts in order to obtain an appropriate voltage of Vk. If A =Qk; i.e., aj= qjk for all j, then obviously Vk =0. When A≠Qk for all k, in order to extract the most similar pattern, there are the following two matching methods.

Voltage-Domain Matching If we set ts longer than the average period of the oscillation in Vp, Vk is statistically proportional to the number of unmatched bits, uk, in the voltage domain. Thus, the order of similarity in Qk, ki, expressed in Eq. (4), can be obtained by comparing Vk for all k with the ramped reference voltage as in conventional analog sorting circuits. The fluctuation of Vk decreases when making ts long. If ts is long enough, the order of similarity can be obtained almost deterministically, which is the same as with ordinary associative processor. In contrast, if ts is set on the order of the average period of the oscillation, the fluctuation of Vk is very large and association becomes stochastic.

Time-Domain Matching If ts is set shorter than the average period of the oscillation in Vp, there exist sampling intervals during which Vk=0. The average span between the sampling events at which Vk = 0 statistically increases with increases in uk because Vk=0 only if Ijk=0 for all j. Thus, if the output of the ki-th WC, Vki, becomes zero earliest in the consecutive sampling intervals, the associative processor extracts Qki. The probability that Qki is most similar to A is a maximum in this case. Let us estimate the time dependence of the probability of detecting Vk=0 in the sampling events with a parameter of uk. Consider a BC with different inputs and a typical Vp change as shown

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 6. Schematic figure introducing the probability of detecting Ijk=0, where Va=1 is assumed. Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

in Figure 6, where Va=1 is assumed; therefore, Ijk ≠ 0 when Vp=1, and vice versa. Here, T and a are defined as the average period and the ratio of the interval when Vp=0 in oscillation of Vp, respectively. When a sampling event starts within the time span of aT−ts, Ijk is always zero in this sampling interval. Since the relation between Vp oscillation and sampling timing is random, the probability that Ijk is always zero in a sampling interval is statistically estimated at Probjk =

aT − ts T

=a −

ts

.

T

(7)

n −1

Probk (t ) = ∑ j =0

n= t

ts

≥ 1.

uk    ts     1 − a −     T   

j

u

 k a − ts  .   T  

(10) (11)

Figure 7 shows the time dependence of Probk (t) with a parameter of uk, where T= 36 ns, a=0.5, ts=8 ns. Thus, it is confirmed from Figure 7 that the order of similarity in Qk, ki expressed in Equation. (4), is stochastically obtained in the time domain.

In a WC, the probability of detecting Vk=0 is statistically estimated at u

k  ts     Probk = a −  . T  

(8)

The probability that Vk becomes zero at time t for the first time is t

uk  t   ts   s   Probk (t ) = 1 − a −     T    

u

 k a − ts  .   T  

(9)

Thus, the probability that Vk becomes zero at least once by time t is

80

Figure 7. Time dependence of Probk (t) with a parameter of uk. Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 8. Simulation results about the dependence of Ijk on Va, b in the bit-comparator. Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

Figure 9. Waveforms of Ijk and detection timing depending on uk. Saen et al. (1998). IEICE Trans. Electron., E81-C, 30, 1998.

on uk. Here, ts=10 ns is assumed. If only the first bit (j=1) is unmatched, the first time whenVk=0 is T1. If the first and second bits (j =1, 2) are unmatched, the first time whenVk=0 is T2. If the first to third bits (j=1, 2, 3) are unmatched, the first time whenVk=0 is T3. We can obviously see that T1< T2< T3. As described above, a WC including BCs with random fluctuation can be constructed by using SET logic circuits.

Simulation Results Figure 8 shows the SET circuit simulation results about the dependence of Ijk on Va,b in the BC. It can be seen that Ijk oscillates randomly when Va ≠Vb. The black areas indicate that Ijk oscillates at a high frequency, which is the effect of electrons passing through the switch SW. In this simulation, parameters were: Vdd=6.5mV, Vbias = 2.3mV, Ca=55aF, Cb=55aF, Cbias=10aF, C1=6 aF, C2 =9 aF, Cs1=10 aF, Cs2=9 aF, Ct1=1aF, Ct2=2 aF, CL=22 aF, Rt1=45 MΩ, Rt2=1MΩ, and the ambient temperature was 1 mK. Figure 9 shows the waveforms of Ijk when aj ≠ qjk, j=1, 2, 3, and detection timing depending

Word-Comparator Circuit Using Nanodots on a MOSFET Gate Bit-Comparator Using Dynamic Single-Electron XNOR Gate The above BC circuit can be simplified by applying another operation principle, in which only the SW part is used. A concrete nanostructure image can be illustrated because of the simple circuit architecture (Yamanaka, et al., 2000). Because of the periodic input-output characteristics of SET devices, a dynamic XNOR gate can be constructed easily by using a SET and a capacitor as shown in Figure 10(a). Here, VCo is assumed to be reset at a certain voltage level before operation.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Electrons pass through the SET to the capacitor Co only when Va= Vb =L or H, although the output voltages in both cases are not exactly equal, as shown in Figure 10(b). In order to equalize both output voltages, a complementary configuration of SETs (CS) is introduced, as shown in Figure 10(c). Figure 10(d) and (e) show simulation results of the waveforms of VCo, where Co is 500 aF and 5000 aF, respectively. In both simulations, the operations are repeated fifty times when the Figure 10. Dynamic exclusive-NOR gate using a SET and a capacitor: (a) basic circuit configuration and (b) transient behavior of VCo; (c) the gate using complementary configuration and (d) (e) its stochastic behavior (simulation results of 50 trials). Yamanaka et al. (2000). Nanotechnology, 11 154, 2000.

same inputs are applied, and all waveforms are represented by the gray lines in the figures. Because of the stochastic event in each electron tunneling, the rise time of voltage VCo fluctuates in every operation, even if the same inputs are applied. The fluctuation shown in Figure 10(e) is smaller than that shown in Figure 10(d) because the stochastic characteristic is eventually averaged as the number of electrons accumulating in the capacitor increases. Thus, the XNOR gates with larger Co operate more deterministically, and those with smaller Co operate more stochastically. Larger Co obviously leads to a longer circuitdelay time, and a Co that is too small makes integrated circuit design difficult. The appropriate capacitance value depends on the application. Although operation temperature of 1 mK was assumed in the simulations, it was verified that the circuit with the same parameters as shown in Figure 10(a) can operate properly up to around 3 K. Moreover, it was also verified that the circuit can operate up to around 100 K if the capacitance values except Co is reduced to one tenth, where thermal noise is used for stochastic operation.

Stochastic Associative Processing Using Dynamic Operation A BC with random fluctuation can be constructed by using the CS shown in Figure 10(c). The WC is constructed by connecting nodes Nc of the plural CS’ with the common output capacitor Co. The number of CS’ that pass electrons to the capacitor Co increases as the Hamming distance decreases, resulting in a shorter rise time for the voltage VCo. However, even when the Hamming distance is at its shortest, the output voltage does not always increase rapidly because of the stochastic characteristics in CS’ as shown in Figure 10(d) or (e). In this architecture, it is assumed that the WTA selects the WC output that reaches the threshold voltage most rapidly as shown in Figure 11. Because the output capacitance Co is large enough, e.g. 500 aF, the input stage of the WTA circuit

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 11. Dynamic winner-take-all operation

can be constructed by sub-100nm MOS devices, whose gate capacitance is less than 1 fF. A device structure image of the WC is shown in Figure 12. Isolated islands of SETs are regularly arranged on a capacitor plate, which corresponds to an electrode of Co and also a gate electrode of an ultra-small MOS device, where the gate capacitance acts as Co.

Simulation Results The basic association operation was confirmed by simulation of a digit pattern association. Each memorized pattern consisted of seven segments and represented a digit number of {0,1,…, 9}. Because the ON/OFF state of each segment corresponded to a bit data, the memorized data consisted of ten vectors, each of which had seven binary elements. The simulation of the system was performed as follows; (a) Input bit data Va

and memorized data Vi (i=1,2, …, 9) were supplied to BCs as voltage signals. (b) Operation of the WC that consisted of single-electron devices was simulated by the Monte Carlo simulator. (c) Operation of the WTA circuit was simulated as an ideal black-box that simply selected a winner from outputs of the word comparators. Two examples of output-voltage changes in the WCs when the input pattern was ‘5’ are shown in Figure 13(a). Because of the stochastic property, different patterns become winners; one is the pattern most similar to the input pattern ‘5’, and the other a second similar pattern ‘6’. Figure 13(b) shows the association probabilities of all memorized patterns where the input pattern is ‘5’, and Figure 13(c) is another simulation result, where the input pattern is not included in the memorized patterns. Both simulation results show that the association probability of the memorized pattern increases as the Hamming distance from the input pattern decreases.

Word-Comparator Circuit Using Coulomb Repulsion in Nanodots The circuits described above use SETs, through which electrons flow from the power supply to the ground, although some electrons stay at isolated nodes temporarily. In contrast, the circuits described below use electrons that are always confined in a nanodot array. Obviously, the Cou-

Figure 12. Device structure image of the word comparator. Yamanaka et al. (2000). Nanotechnology, 11, 154, 2000.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 13. Simulation results: (a) output voltage changes in word comparators, (b),(c) association probabilities for stored patterns, where the association is repeated 100 times.Yamanaka et al. (2000). Nanotechnology, 11 154, 2000.

of the term, Coulomb repulsion, as an effect for electrons confined in a nanodot array. Two types of BC/WC circuits using the Coulomb repulsion effect are described below. They can be constructed using a nanodot array arranged on a gate electrode of an ultrasmall MOSFET.

Bit-Comparison Principle Using Coulomb Repulsion in Nanodots Let us assume a string of nanodots as shown in Figure 14(a), put an electron, eM, at one of the three dots D1, and represent a bit (0 or 1) of the input and memorized data by whether an electron is put at each end dot D2 or not (Morie, Matsuura, Miyata, Yamanaka Nagata, & Iwata, 2000). When the corresponding bits of both data are matched, because Coulomb repulsion is symmetric, electron eM is stabilized at the center; otherwise it is offcenter. In order to detect the position of electron eM, there are two possible detection circuits: circuit-A that detects the center position and circuit-B that detects the off-center position as shown in Figure 14(b) and (c), respectively. By the Coulomb repulsion effect, the bit matching result reflects whether electron eR tunnels to node No. Consequently, electrons whose number is equal to that of the matched bits (circuit-A) or that of the unmatched bits (circuit-B) are accumulated in Co. To ensure the stabilizing processes, control voltage Vg is used. A three-dimensional (3-D) arrangements of nanodots realizing these circuits are shown in Figure 15, where common terminals to the ground and control voltages are omitted. The capacitance Co corresponds to the gate capacitance of an ultrasmall CMOS transistor.

Simulation Results lomb blockade phenomena, which are the operation principle of SETs, are based on the Coulomb repulsion effect, but we restrict here the meaning

84

Single-electron circuit simulation was performed, where parasitic capacitance between ground and nanodots, Cg, and that between the secondneighbor nanodots, Cd, are considered. Because

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 14. Principle of bit-comparison using Coulomb repulsion (a), and two word-comparator circuits (b) (c). Morie et al. (2000). Superlattices and Microstructures, 27, 613, 2000.

it was found that circuits A and B have almost the same characteristics, only the simulation results for circuit-A are described here. The operation temperature ranges for feasible capacitance values are shown in Figure 16(a). The upper limit of operation temperature gradually lowers with increasing Cg and Cd. In order to operate the circuit at higher temperature, one has to scale down all the values of capacitance, and at the same time, scale up the applied voltages. For room temperature operation, tunnel junction capacitance of 0.01 aF is required as shown in Figure 16(b) although this value is very difficult to realize. Setting margin of Co is shown in Figure 16(c) as a function of the word length (the number of the connected BCs). There exist minimum values of Co for the correct operation, and the values increase as the word length increases. This is because some electrons eR cannot tunnel to node No because of the Coulomb blockade effect by other eR’s. If the parasitic capacitance is negligible, there do not exist the upper limits of Co. However, Co should be as small as possible because the sensitivity to one electron in the output voltage e /Co decreases with increasing Co.

Multi-Nanodot Word-Comparator Circuit and Structure Using Thermal-Noise Assisted Tunneling The nanodot WC circuits described above operate only at very low temperature for practical junction capacitance. The multi-nanodot WC circuit and structure described here can operate even at room temperature with a junction capacitance around 0.1 aF by using tunneling processes assisted by thermal noise (Morie, Matsuura, Nagata, & Iwata, 2002). In the stochastic associative processing operation, the association probability distribution can be controlled by changing the detection timing of the electron position.

85

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 15. 3-D structure images of a word-comparator

Figure 16. Simulation results for circuit-A: (a) operation temperature range with one BC, (b) output voltage changes at room temperature, and (c) setting margin for Co. Morie et al. (2000). Superlattices and Microstructures, 27, 613, 2000.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Circuit and Structure Let us assume nanodot structures constructed on a MOS transistor gate electrode as shown in Figure 17. Each nanodot structure consists of a pair of one-dimensional (1-D) dot arrays: Av (D v1,Dv2, Dv3) and Ah (D1,…,Dn, Dc, Dn,…,D1), where n is the number of dots at a side of Ah, and it should be more than 4 for proper opeation described below. The array Ah has dot De outside of each end. The capacitance Co corresponds to the gate capacitance of an ultrasmall MOS transistor. A bit (1 or 0) of the input and memorized data is represented by whether or not an electron is placed at each end dot De. Alternatively, an appropriate voltage corresponding to a bit may be applied directly at De. Bias voltages are applied to the plate Pc over Dc (Vpc), to the nodes outside of De (Ve), and to the backgate of the MOS transistor (Vbg). An electron eM is introduced at the center dot Dc of the 1-D array Ah. Electron eM can move along

array Ah through tunnel junctions Cj, but it cannot move to either of De’s or to Dv3 through normal capacitors C1 or C2. Each nanodot structure works as an exclusive-NOR logic gate (bit comparator) with random fluctuation, as explained below. By applying appropriate bias voltages Vpc, Ve, and Vbg, the profile of the total energy as a function of the position of eM along the 1-D array Ah has a minimal value at Dc, as shown in Figure 18. For 1-1 state, where electrons are placed at both De’s, the energy at D1 rises, and thus eM is most strongly stabilized at the center position. Therefore, the difference between 0-0 state and 1-0 (or 0-1) state is important for correct bit-comparator operation. In the two states, the energy profile has another Figure 18. Schematics of total energy profile of 1-D dot-array structure. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

Figure 17. Multi-nanodot circuit and a structure image. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

minimal value at D1. The energy barrier height for eM located at Dc is approximately determined by the total capacitance for eM and bias voltages. The greater number of serial capacitance connections causes higher energy barriers, and the energy differences can be much larger than the thermal energy at room temperature even if the tunnel junction capacitance Cj is around 0.1 aF. The energy barrier at the ‘0’ side in 1-0 (0-1) state becomes lower than that in 0-0 state because of the Coulomb repulsion force of the electron placed at the opposite De, as shown in Figure 18. Thus, eM in 1-0 (0-1) state can more easily overcome the barrier when assisted by thermal noise at non-zero temperature and it then moves to D1 at the ‘0’ side. As a result, there exists a certain time span t0 within which eM in 1-0 (0-1) state moves to D1 while eM in 0-0 state stays at Dc. After spending t0, the vertical dot array Av detects whether or not eM stays at Dc, by changing the bias voltages if necessary. Only if eM stays at Dc, array Av is polarized and an electron is induced at the gate electrode of Co. (In order to achieve stable polarization, at least three dots are required in Av). The total number of electrons induced at the gate electrode is proportional to the number of matched bits; this reflects the gate voltageVo, and it can be measured by the source-drain current of the MOS transistor. Thus, the Hamming distance can be measured by this MOS transistor with nanostructure arrays. If this detection process starts just after t0, the most accurate bit comparison operation is achieved, although some statistical fluctuation remains. However, if the detection timing (td) is shifted from t0, an arbitrary amount of fluctuation can be introduced in the bit comparison result. Thus, controlled stochastic association can be achieved, which is necessary in order to apply the stochastic association model to various types of intelligent information processing effectively. The bit-comparator circuit composed of nanodot arrays works only at non-zero temperatures because at 0 K, eM can never escape from Dc, the valley of the energy profile. Furthermore, the

88

time span t0 depends on the operating temperature. Conversely, for a given t0, an appropriate amount of thermal noise is required; if the thermal noise is too small, electron eM in 1-0 (0-1) state cannot escape from Dc, thus no bit comparison is achieved. On the other hand, if thermal noise is too large, eM in both 0-0 state and 1-0 (0-1) state escapes from Dc, and thus the two states cannot be distinguished. In this sense, it can be considered that this circuit utilizes a stochastic resonance effect (Bulsura, A. R., & Gammaitoni, L., 1996) by thermal noise.

Simulation Results We analyzed the proposed circuit shown in Figure 17 by using a Monte Carlo single-electron simulator, where the tunnel junction capacitance Cj is 0.1 aF and tunnel resistance Rt is 5 MΩ, and other parameters are shown in Figure 17. In this case, the dot diameter is assumed to be around 1 nm. The bias voltages applied were Vpc=0 V, Ve=1.15 V, Vbg=0 V for the eM stabilization process, and Vpc=0V, Ve=1.8V, Vbg=3 V for the eM position detection process. Figure 19 shows the total energy profiles at the ‘0’ state side of Ah as a function of the position of eM, where the energy when eM is located at Dc is defined as zero. It is confirmed that the barrier height for eM at Dc is larger than the thermal energy at room temperature (26 meV), and the barrier height in 1-0 (0-1) state is lower than that in 0-0 state. Figure 20 shows the relationship between operating temperature and time (tM) required until eM moves to D1. The closed and open circles indicate tM in many trials at 1-0 state and 0-0 state, respectively. Because the moving process assisted by thermal noise is purely stochastic, tM scatters over a wide range. However, time span t0, defined in the previous section, can be determined from these simulation results. In order to determine t0 precisely, we measured tM for 100 simulations for 1-0 and 0-0 states with different seeds for random number generation.

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 19. Energy profiles for electroneM at the ‘0’ state side in 1-D dot array structures. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

The 100 data obtained for tM were sorted in increasing order and numbered from 1 to 100. The assigned number means the number of electrons that move to D1 within the corresponding tM. Therefore, the relationship between tM and the assigned number can approximately be considered as the probability that eM moves to D1 as a function of time. The results obtained at room temperature (300 K) are shown in Figure 21. The optimum t0 Figure 20. Relation between operating temperature and time when eM moves to D1. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

is obtained as the time having the smallest overlap between the two states; it is about 1 µs. It should be noted here that t0 depends on tunnel resistance Rt. If lower tunnel resistance is available, t0 becomes shorter. From Figure 21, we can obtain the probability of wrong detection, that is, the conditional probability that a given 1-0 state is detected as 0-0 state or vice versa. For example, when the detection timing is 0.1µs, the probabil-

Figure 21. Probability that eM moves to D1 as a function of detection timing. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

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ity that eM moves to D1 at 1-0 state is only 20%. This means that wrong detection occurs with a probability of 80%. By using this effect, we can add fluctuation to the bit comparison operation. However, in the above case, it must be noted that fluctuation can be added only in1-0 state. Bit-matched (1-1 and 0-0) states always answer correctly. Therefore, a comparison between patterns with a shorter Hamming distance is performed more deterministically. This means that a stochastic association operation cannot be achieved. An easy way to overcome this difficulty is to reverse the input bit pattern. Although this leads to a deterministic comparison between patterns with a longer Hamming distance, but such patterns are seldom associated, and thus it hardly affects the stochastic association operation. Figure 22(a)-(c) show association probability distributions as a function of the Hamming dis-

Figure 22. Association probability distribution as a function of Hamming distance for various detection timing td. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

tance for some td. In these simulations, the input pattern was (1,1,1,1), and the memorized reference patterns were (1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0), and (0,0,0,0). These reference patterns were reversed when they were applied to the multi-nanodot circuit, for the reason described above. With the number of electrons indicating the results of Hamming distance evaluation with fluctuation, the reference pattern having the smallest evaluation result became the winner. If two or more reference patterns had the same number of electrons, we determined the winner stochastically. The simulations were repeated 100 times with different seeds for random number generation. The number of trials when a given reference pattern becomes a winner is approximately proportional to the probability that it is associated. The simulation results shown in Figure 22 confirm that, as td becomes further apart from t0 (=1 µs in this case), the association probability distribution becomes flatter. Thus, the association probability distribution is controlled by changing td. Figure 23 shows the time dependence of voltage Vo as a parameter of the Hamming distance for a 4-bit word comparator at room temperature,

Figure 23. Time dependence of voltage Vo as a parameter of the Hamming distance. Morie et al. (2002). Nanosci. Nanotech., 2, 343, 2002.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

where the voltage for a distance of 0 bits is defined as 0 V. The voltage changes are proportional to the Hamming distance, and the voltage difference per bit is larger than 1 mV, which is large enough to detect with a CMOS circuit.

A MULTI-NANODOT FLOATINGGATE MOSFET CIRCUIT FOR SPIKING NEURON MODELS

Some Remarks about Nanostructures for Stochastic Associative Processors

As another application of the nanodot circuits described above, we introduce a spiking neuron model here. In order to realize brain-like information processing functions, such as association, perception and recognition, one very important and challenging approach is to mimic brain functions and structures. In the brain, a neuron receives many electric impulses via a few thousand synapses, and it outputs spike pulses. A typical neuron has three parts: dendrites, a soma, and an axon. Pulse signals called spikes are fed into the dendrites via synapses, the effects of the inputs are gathered up at the soma, and a spike pulse is output from the axon (Maass, & Bishop, 1999). Until the mid-1990s, pulse-rate coding models, which use analog values as averages of pulse events, were studied intensively in both artificial neural networks (Rumelhart, McClelland, & the PDP Research Group, 1986) and their VLSI implementation. The approaches to analog VLSI implementation treat such analog values directly in the voltage or current domain (Howard, Schwartz, Denker, Epworth, Graf, Hubbard, Jackel, Straughn, & Tennant, 1987; Shima, Kimura, Kamatani, Itakura, Fujita, & Iida, 1992; Morie, & Amemiya, 1994), whereas the approaches to digital VLSI implementation represent such analog values only by a set of digital bits (Yasunaga, Masuda, Yagyu, Asai, Shibata, Ooyama, Yamada, Sakaguchi, & Hashimoto, 1993; Kondo, Koshiba, Arima, Murasaki, Yamada, Amishiro, Shinohara, & Mori, 1994; Saito, Aihara, Fujita, & Uchimura, 1998). Another approach, called pulse-stream neural networks or pulse-density modulation (PDM), also focuses on the rate of pulse events (Murray, & Tarassenko, 1994;, Hirai, & Yasunaga, 1996). In contrast, since the mid-1990s, computational neuroscientists have focused on more-realistic

In these architectures described above, if it is difficult to represent one bit by one nanodot, a redundant architecture can be applied easily, where plural BCs represent one data bit. Such an architecture based on a majority decision principle has an advantages of robustness against effects of random background charge. For realizing these nanostructures described above, the basic technology of nanocrystalline floating-dot MOSFET devices, which are closely related to these nanostructures, has been reported (Kohno, Murakami, Ikeda, Miyazaki, & Hirose, 2001; Tiwari, Rana, Hanafi, Hartstein, Crabb´e, & Chan, 1996; Ohba, Sugiyama, Koga, Uchida, & Toriumi, 2000). Fabrication technology using self-organization may also be applied (Huang, Tsutsui, Sakaue, Shingubara, & Takahagi, 2000). Furthermore, well-controlled self-assembly processes using biomineralization technology (Kubota, Hashimoto, Takeguchi, Nishioka, Uraoka, Fuyuki, Yamashita, & Samukawa, 2007) would be utilized to fabricate the nanostructures. Although the use of digital data is assumed in the above sections, analog data can be treated in the same circuit by using pulse-width modulation (PWM) signals, which have a digital amplitude and a analog pulse width (Iwata, & Nagata, 1996; Iwata, Morie, & Nagata, 2001). Instead of a Hamming distance, a Manhattan distance, the summation of the absolute value of difference, is evaluated by using these nanostructures. The clustering algorithm using stochastic association for vector quantization (Morie, et al., 2002) can use this distance evaluation approach.

Neural Network Models

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

spiking neuron models, which treat spike pulses directly (Maass, et al., 1999 & 1997). In principle, the computational power of spiking neuron models is superior to that of the conventional rate coding models (Maass, et al., 1999 & 1997). In this section, a single-electron circuit based on a multi-nanodot floating-gate MOS device for spiking neurons is described (Morie, Matsuura, Nagata, & Iwata, 2003).

Spiking Neuron Models In spiking neuron models, information is represented by spatiotemporal patterns in spike pulse trains. A simple spiking neuron model, called the Spike Response Model (SRM) (Maas, 1993), is shown in Figure 24. A spike pulse inputted to a neuron via a synapse generates a post-synaptic potential (PSP). There are two types of synapses: excitatory and inhibitory. Respectively, these have positive and negative synaptic weights and generate an excitatory PSP (EPSP) and an inhibitory PSP (IPSP). The PSP temporarily increases or decreases according to whether the synaptic connection is excitatory or inhibitory, respectively.

The typical time course of a PSP is approximated by a so-called α-function: α(x) = Axexp(−x), x = (t − t0 − Δax)/τs, for t − t0 > Δax,

(12)

where A is a constant, t0 is a firing time, Δax is the axonal transmission delay, and τs is a time constant (Maas, 1993). In this function, the rise time and the fall time are related to each other; however, for various applications it is desirable for them to be independently changeable. The neuron’s internal potential I(t) is equal to the spatiotemporal summation of all PSPs generated by the input spike pulses. If I(t) exceeds a certain threshold th, the neuron emits a spike pulse and I(t) is reset to the resting level. There exists a refractory period after firing, in which the neuron cannot fire even if many spikes are inputted. The property of the spiking neural network consisting of SRM neurons depends on the relation between the average of interspike intervals tISI and the PSP decay time constant τPSP. As shown

Figure 24. Schematic of a simple spiking neuron model, the Spike Response Model (SRM). Morie et al. (2003). IEEE Trans.Nanotechnology, 2, 158, 2003.

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Figure 25. Schematic explaining that the spiking neural network property depends on the relation between the average of interspike intervals tISI and the PSP decay time constant τPSP: (a) τPSP>> tISI, (b) τPSP <
The spiking neural networks can process various types of information by using various PSP decay constants. This means that rate-coding-type integrators (conventional analog neural networks) and coincidence detectors can coexist in the same network. The two features can be combined for various applications. Therefore, τPSP must be controlled in the VLSI implementation of SRM neurons.

Multi-Nanodot Floating-Gate MOS Device for Spiking Neurons

in Figure 25, if τPSP is much greater than tISI, then PSPs are integrated during τPSP, in which case the neuron acts as an integrator. In contrast, if τPSP is much smaller than tISI, only those PSPs generated by spike pulses inputted within the time interval τPSP are integrated. In that case, the neuron acts as a coincidence detector of spike timing. That is, the neuron detects whether or not plural input spikes arise within a certain time span comparable to τPSP. The latter type of neuron enables higher-order and faster intelligent information processing functions such as the brain might perform. It is indicated that the computational power of networks composed of spiking neurons is superior to that of conventional analog neural networks based on the rate coding models (Maas, 1993). For example, spiking neurons can detect temporal patterns irrespective of a common additive constant, and they can compute weighted summations. Furthermore, they can approximate any continuous functions more efficiently.

Because one real biological neuron has a few thousand synapses, the key for very-large-scale integration of neural networks is the design of a small synapse circuit. Thus, the realization of the synapse functions should be more focused. The synapse circuit of SRM neurons has to implement a generation of controllable PSPs as well as a synaptic weighting. A nanodot circuit and structure for realizing the synapse of the SRM model is shown in Figure 26, which is very similar to the multi-nanodot structure shown in Figure 17 described in the previous section. The circuit parameters used in the following simulations are indicated in this figure. These parameters are assumed as an example in the case of the minimum size for nanometer-scale structures. Multi-nanodot structures are constructed on a MOSFET gate electrode. Each nanodot structure consists of a one-dimensional (1-D) dot array: Ah (D1,…,Dn, Dc,Dn’,…, D1’), where n(= n’) is the number of dots at a side of Ah. In the following simulations, n=5 is assumed. The center dot Dc is capacitively coupled with the MOS gate via one nanodot Dv. The capacitance Co corresponds to the gate capacitance of the MOSFET. It is assumed that only one electron eM exists in the array Ah. Electron eM can move along array Ah only through tunnel junctions, and it cannot move outside of Ah through normal capacitors C1 or C2.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 26. Single-electron circuit and nanostructure for the SRM neuron. The circuit parameters used in the simulations are also indicated. Morie et al. (2003). IEEE Trans.Nanotechnology, 2, 158, 2003.

Spike pulses are inputted at nodes IN that are capacitively coupled to one end of Ah. In order to achieve a high probability that eM exists at center dot Dc when no spike pulse is inputted, appropriate bias voltages are applied to the top plate Pc (Vp), to the end electrode Pe (Ve), and to the back gate of the MOS transistor (Vbg). For example, assume that Vp=Vbg=0 V and Ve< 0 V. The baseline voltage of the input signal VH is also set at 0 V or slightly less than 0 V. Thus, the profile of the total energy as a function of the position of eM along 1-D array Ah has two peaks, as shown in Figure 27, because of the charging energy of eM itself. The minimal values of the energy are located at Dc, D1 and D1 . Figure 27 also shows simulation results. The energy barrier height for eM staying at Dc is ap-

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Figure 27. Schematics of the total energy profile in 1-D nanodot array and simulation results, where Vp=Ve=Vbg=VH=0 V, and VL=-2.7V. Morie et al. (2003). IEEE Trans.Nanotechnology, 2, 158, 2003.

proximately determined by the total capacitance for eM and the bias voltages. The energy differences can be larger than the thermal energy at room temperature if the tunnel junction capacitance Cj is around 0.1 aF. Thus, in the stationary state without input pulses, electron eM almost always stays at the center dot Dc, although thermal noise sometimes causes eM to move to edge position D1 and back to Dc. When a spike pulse is inputted, if it has appropriate pulse width and amplitude voltage, eM moves quickly to D1. Then, after the pulse signal ceases, eM moves slowly back to the center dot Dc because of thermal-noise-

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

assisted tunneling through the energy barrier between nanodots. This behavior of eM creates a PSP. The energy barrier can be larger than the thermal energy at room temperature, and thus the 1-D nanodot array can operate at room temperature. The position of eM affects the gate voltage Vo through dot Dv; i.e., when eM stays at center dot Dc, Vo is reduced because of capacitive coupling between Dc and the gate electrode. Because the above process occurs stochastically, Vo fluctuates due to the position of eM. However, if one adopts a configuration where the same input pulse is applied to a number of arrays, or if one uses a low-pass filter in the detection part, an analog PSP is obtained from the MOSFET. In the former case, this means that the interconnection wires for input can be much wider than the interval between nanodot array sets. This condition is preferable for VLSI fabrication technology. Furthermore, by changing the circuit parameters such as Ve and Vp locally, the profile of PSP generated by each input pulse can be controlled. PSPs generated in plural nanodot arrays are summed on the gate capacitance of the MOSFET. If one adopts models that ignore the difference in the transmission delay caused by the spatial distribution of synapses on dendritic trees, excitatory and inhibitory PSPs are separately integrated, each by a different MOSFET having a nanostructure on the gate, and then the inhibitory contribution is subtracted from the excitatory one. The averaging process can also reduce various nonidealities in nanostructures, such as additional charge effects due to background offset, parasitic and/or surplus charges, and the effect of device parameter fluctuation. In single-PSP operation, the addition of charges comparable to the elementary charge causes a fatal error. However, even if some dot arrays do not work correctly by additional charges, the averaging process by many dot arrays can decrease the effect. Thus, the proposed device and circuit will operate successfully even if the fabrication technology is not yet fully mature.

Simulation Results The proposed circuit shown in Figure 26 was analyzed by using a Monte Carlo single-electron simulator. In the simulations, the tunnel junction capacitance Cj was assumed to be 0.1 aF, which means that the dot diameter is around 1 nm. The tunnel resistance Rt was assumed to be 5 MΩ. The operation temperature was set at 300 K. Figure 28(a) shows the transition probability of eM between Dc and D1 as a function of time according to an input voltage change. Those data were obtained as the same way for obtaining Figure 21 in the previous section. By replotting Figure 28(a), one can obtain the time dependence of the probability that eM stays at D1 when a pulse is applied, as shown in Figure 28(b). The time course of PSP (Vo) is approximately proportional Figure 28. Electron transition probability as a function of time with baseline voltages VH = 0 and -0.1 V: (a) transition probability of eM between Dc and D1 as a function of time, and (b) time dependence of the probability that eM stays at D1 when a pulse is applied. Morie et al. (2003). IEEE Trans. Nanotechnology, 2, 158, 2003.

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Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Figure 29. Time dependence of MOS gate voltage (Vo); (a) one-input case, (b) two-input case, (A): electron eM comes back to center-dot (Dc), (X): first electron comes back toDc, (Y): second electron comes back to Dc; (c) four-input case, a fitted α−function curve is indicated by the solid line. Morie et al. (2003). IEEE Trans.Nanotechnology, 2, 158, 2003.

to this probability. The time constant of PSP can be controlled by the baseline voltage of the input (VH) as shown in Figure 28. Figure 29(a), (b) and (c) show the time dependence of Vo when a pulse is applied in the case of one, two and four nanodot array(s), respectively. Voltage Vo fluctuates due to the position of the electron, but the average voltage roughly repre-

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sents a PSP; i.e., the exponential decay characteristic of a PSP is realized in Vo. In Figure 29(c), a fitted α-function curve is indicated by a solid line. Although an α-function expresses only a typical PSP time course, the fit is reasonably good. It is verified from Figure 29(b) and (c) that the summation of plural inputs can also be realized.

Single-Electron Devices and Circuits Utilizing Stochastic Operation for Intelligent Information Processing

Thus, the function of the synapse part, which is the generation of arbitrary PSPs, can be realized by using a MOSFET with nanostructures. The decay constant of PSPs is controlled by the input baseline voltage VH and the bias voltage Ve, which affect the potential barrier height. The functions of the soma part in the neuron, such as thresholding and refractoriness, are realized by MOS circuits, including the base MOSFET of nanostructures.

CONCLUSION This paper reviewed single-electron devices and circuits utilizing stochastic operation for associative processing and a spiking neuron model. These circuits are not based on conventional CMOS operation, but they can co-exist with conventional digital systems. Only sophisticated nanotechnology may construct these functional circuits, and at the same time, they should be promising applications of post-CMOS nanoelectronics.

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Hopfield, J. J. (1984). Neurons with Graded Response Have Collective Computational Properties Like Those of Two-state Neurons. Proceedings of the National Academy of Sciences of the United States of America, 81, 3088–3092. doi:10.1073/ pnas.81.10.3088 Howard, R. E., Schwartz, D. B., Denker, J. S., Epworth, R. W., Graf, H. P., & Hubbard, W. E. (1987). An Associative Memory Based on an Electronic Neural Network Architecture. IEEE Transactions on Electron Devices, ED-34, 1553–1555. doi:10.1109/T-ED.1987.23118 Huang, S., Tsutsui, G., Sakaue, H., Shingubara, S., & Takahagi, T. (2000). Electrical Properties of Self-Organized Nanostructures of Alkanethiol-Encapsulated Gold Particles. Journal of Vacuum Science & Technology B Microelectronics and Nanometer Structures, 18, 2653–2657. doi:10.1116/1.1318190 Iwata, A., Morie, T., & Nagata, M. (2001). Merged Analog-Digital Circuits Using Pulse Modulation for Intelligent SoC Applications. IEICE Trans. Fundamentals. E (Norwalk, Conn.), 84-A, 486–496. Iwata, A., & Nagata, M. (1996). A Concept of Analog-Digital Merged Circuit Architecture for Future VLSI’s. IEICE Trans. Fundamentals. E (Norwalk, Conn.), 79-A, 145–157. Kohno, A., Murakami, H., Ikeda, M., Miyazaki, S., & Hirose, M. (2001). Memory Operation of Silicon Quantum-Dot Floating-Gate Metal-OxideSemiconductor Field-Effect Transistors. Japanese Journal of Applied Physics, 40, L721–L723. doi:10.1143/JJAP.40.L721 Kondo, Y., Koshiba, Y., Arima, Y., Murasaki, M., Yamada, T., Amishiro, H., et al. (1994). A 1.2GFLOPS Neural Network Chip Exhibiting Fast Convergence. IEEE Int. Solid-State Circuits Conf. (ISSCC), (pp. 218–219).

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Kubota, T., Hashimoto, T., Takeguchi, M., Nishioka, K., Uraoka, Y., Fuyuki, T., Yamashita, I., & Samukawa, S. (2007). Coulomb-Staircase Observed in Silicon-Nanodisk Structures Fabricated by Low-Energy Chlorine Neutral Beams. J. Appl. Phys., 101, 124301.1-9. Maass, W. (1997). Networks of Spiking Neurons: The Third Generation of Neural Network Models. Neural Networks, 10, 1659–1671. doi:10.1016/ S0893-6080(97)00011-7 Maass, W. (1997). Noisy Spiking Neurons with Temporal Coding have more Computational Power than Sigmoidal Neurons. M. C. Mozer, M. I. Jordan, & T. Petsche (Eds.), Advances in Neural Information Processing Systems, 9, 211. The MIT Press. Maass, W., & Bishop, C. M. (1999). Pulsed Neural Networks. Cambridge, MA: MIT Press. Morie, T., & Amemiya, Y. (1994). An Allanalog Expandable Neural Network LSI with On-chip Back Propagation Learning. IEEE Journal of Solid-state Circuits, 29, 1086–1093. doi:10.1109/4.309904 Morie, T., & Amemiya, Y. (2006). Single-Electron Functional Devices and Circuits. M. Rieth & W. Schommers (Eds.), Handbook of Theoretical and Computational Nanotechnology, (pp. 239–318). American Scientific Publishers. Morie, T., Matsuura, T., Miyata, S., Yamanaka, T., Nagata, M., & Iwata, A. (2000). Quantum Dot Structures Measuring Hamming Distance for Associative Memories. Superlattices and Microstructures, 27, 613–616. doi:10.1006/spmi.2000.0874 Morie, T., Matsuura, T., Nagata, M., & Iwata, A. (2002). A Multi-Nano-Dot Circuit and Structure Using Thermal-Noise Assisted Tunneling for Stochastic Associative Processing. Journal of Nanoscience and Nanotechnology, 2, 343–349. doi:10.1166/jnn.2002.100

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Morie, T., Matsuura, T., Nagata, M., & Iwata, A. (2002). An Efficient Clustering Algorithm Using Stochastic Association Model and Its Implementation Using Nanostructures. T. G. Dietterich, S. Becker, & Z. Ghahramani (Eds.), Advances in Neural Information Processing Systems, 14, 1115–1122. MIT Press, Cambridge, MA. Morie, T., Matsuura, T., Nagata, M., & Iwata, A. (2003). A Multi-Nanodot Floating-Gate MOSFET Circuit for Spiking Neuron Models. IEEE Transactions on Nanotechnology, 2, 158–164. doi:10.1109/TNANO.2003.817221 Murray, A. F., & Tarassenko, L. (1994). Analogue Neural VLSI — A Pulse Stream Approach. London, UK: Chapman & Hall. Nakano, K. (1972). Associatron – A Model of Associative Memory. IEEE Transactions on Systems, Man, and Cybernetics, SMC-2, 380–388. doi:10.1109/TSMC.1972.4309133 Ohba, R., Sugiyama, N., Koga, J., Uchida, K., & Toriumi, A. (2000). Novel Si Quantum Memory Structure with Self-Aligned Stacked Nanocrystalline Dots. Ext. Abs. of Int. Conf. on Solid State Devices and Materials (SSDM), (pp. 122–123). Rumelhart, D. E., & McClelland, J. L.PDP Research Group. (1986). Parallel Distributed Processing. Cambridge, MA: MIT Press. Saen, M., Morie, T., Nagata, M., & Iwata, A. (1998). A Stochastic Associative Memory Using Single-Electron Tunneling Devices. IEICE Trans. Electron. E (Norwalk, Conn.), 81-C, 30–35. Saito, O., Aihara, K., Fujita, O., & Uchimura, K. (1998). A 1M Synapse Self-Learning Digital Neural Network Chip. IEEE Int. Solid-State Circuits Conf. (ISSCC), (pp. 94–95). Takahashi, Y., Ono, Y., Fujiwara, A., & Inokawa, H. (2002). Silicon Single-Electron Devices. Journal of Physics Condensed Matter, 14, R995– R1033. doi:10.1088/0953-8984/14/39/201

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Tiwari, S., Rana, F., Hanafi, H., Hartstein, A., Crabb’e, E. F., & Chan, K. (1996). A Silicon Nanocrystals Based Memory. Applied Physics Letters, 68, 1377–1379. doi:10.1063/1.116085

Yamanaka, T., Morie, T., Nagata, M., & Iwata, A. (2001). A CMOS Stochastic Associative Processor Using PWM Chaotic Signals. IEICE Trans. Electron. E (Norwalk, Conn.), 84-C, 1723–1729.

Yamanaka, T., Morie, T., Nagata, M., & Iwata, A. (2000). A Single-Electron Stochastic Associative Processing Circuit Robust to Random Background-Charge Effects and Its Structure Using Nanocrystal Floating-Gate Transistors. Nanotechnology, 11, 154–160. doi:10.1088/09574484/11/3/303

Yasunaga, M., Masuda, N., Yagyu, M., Asai, M., Shibata, K., & Ooyama, M. (1993). A SelfLearning Digital Neural Network Using WaferScale LSI. IEEE Journal of Solid-state Circuits, 28, 106–114. doi:10.1109/4.192041

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 1-28, copyright 2009 by IGI Publishing.

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Chapter 6

Application of Single Electron Devices Utilizing Stochastic Dynamics Shigeo Sato Tohoku University, Japan Koji Nakajima Tohoku University, Japan

ABSTRACT Single electron devices utilizing the Coulomb blockade phenomenon have attractive features such as extreme low power consumption, one by one electron flow controllability, small device size, etc. However, besides promising applications such as the current standard and charge detection, it is not easy to apply the single electron devices to conventional computational tasks due to its stochastic operation and low amplification capability. Therefore, it is important for us to consider suitable applications of single electron devices. In this paper, we show three applications such as a noise generator, a stochastic neural network, and a charge detector employing stochastic resonance. Through these applications, we see the advantages of single electron devices and study the direction of applications.

INTRODUCTION The semiconductor integrated circuit technology has formed the basis of today’s information oriented society and supported its continuous progress. However, several problems such as high power density and quantum effects have been serious recently. On the other hand, recent development of nanotechnologies realizes the use of a nano-scale device for us. On such a tiny tunnel junction, electron tunneling is restricted DOI: 10.4018/978-1-60960-186-7.ch006

by the Coulomb force between electrons, and this phenomenon is called Coulomb blockade. A single electron device based on the Coulomb blockade is one of the candidate devices which can overcome the difficulties of conventional LSIs. It has various advantages such as extreme low power consumption, high charge sensitivity, fine electron flow controllability, small area occupancy, and so on. Representative applications are charge detection, quantum computing, and current standard, etc. On the other hand, its low gain property and stochastic operation could be main obstacles when we apply it to a conventional

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Application of Single Electron Devices Utilizing Stochastic Dynamics

logic circuit. Therefore, a single electron device should be utilized for some specific fields where we can take its advantages reasonably. In this paper, first we briefly show the theoretical background and simulation methods of single election devices. Next, we introduce three application examples of single electron devices successively, and finally discuss those advantages in order to view the effectiveness of the single electron electronics. The first application is a random number generator. Though Coulomb blockade phenomenon is understood by the classical electromagnetism, electron tunneling is a quantum stochastic process, so that a single electron device has a stochastic property intrinsically. By constructing a symmetrical bistable circuitry with complementary single electron transistors (CSETs) proposed by Tucker (1992), and setting its initial state to an unstable equilibrium point, a random number is obtained. The second application is a hardware neural network. A huge number of neurons are required for its practical use, and it is said that artificial neural networks (ANNs) comprising about 103 neurons are necessary for practical applications such as pattern recognition and classification problems (Dayhoff, 1990). We introduce a single electron neural network using stochastic logic proposed by Gaines (1969), in which a digital or an analog value is converted to the firing rate of a stochastic pulse sequence. Since multiplication can be done with a single AND gate, one can reduce the circuit size greatly. Furthermore, its power consumption is kept reasonably low thanks to the low power property of single electron transistors. The last one is related to charge detection. Stochastic resonance (SR) has been known as a nonlinear phenomenon whereby the addition of noise can enhance the detection capability of weak stimuli. An optimal amount of added noise results in the maximum enhancement, whereas further increases in the noise intensity only degrade detectability. The detection of weak signals in noisy environment becomes an important subject in the context of the development of

nanotechnologies. Especially, it could be a critical subject for the practical application of quantum devices. As is well known, a single electron device operates as a good charge detector. Hence, we introduce the stochastic resonance with a single electron turnstile and discuss its performance as a charge detector.

THEORETICAL BACKGROUND AND SIMULATION METHODS Coulomb Blockade Operation of single electron devices is based on the Coulomb blockade phenomenon on ultra small tunnel junctions (Likharev, 1988; Averin & Likharev, 1991; Grabert & Devoret, 1992). Let us consider a junction whose electrostatic capacity is C. Its charging energy is given as E = Q2/ 2C, where Q is the charge on the junction. When an electron tunnels, the energy change is given as

(Q − e )

2

∆E =

2C

−

Q2 e2 = − eV , 2C 2C

(1)

where e and V(= Q/C) denote the elementary charge and the voltage across the junction, respectively. The second term in the right side can be viewed as the work done by a voltage source. The fact that an electron tunneling accompanies energy dissipation gives the relation ΔE < 0. Therefore tunneling does not occur if −

e e
(2)

Therefore, this is called Coulomb blockade. There are two conditions required for us to observe a Coulomb blockade with a real device.

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Figure 1.

The charging energy must be larger than thermal noise as given as e2 >> kBT , 2C

(3)

where kB is the Boltzmann constant and T is the temperature, respectively. Also, the tunnel resistance RT must be larger than the resistance quantum RK as given as RT >> RK =

h , e2

(4)

where h is the Planck constant. In general, a tunnel junction with capacitance of the order of aF shows the blockade property at room temperature.

Single Electron Transistor Among various devices utilizing the Coulomb blockade, a single electron transistor (SET) is the most important basic device. Figure 1 (a) shows an SET composed of a gate capacitor C0 and two tunneling junctions C1, C2. The efficient energy level of electrons in the center island is quantized

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by the Coulomb blockade as shown in Figure1 (b), so that the current flow can be controlled by changing the gate voltage U. Tucker (1992) has proposed the complementary SET (CSET) shown in Figure 2 (a), which is composed of two SETs with different biases. Unlike the SET in Figure1, each SET in Figure2 has two capacitors, where the additional capacitor is called a back capacitor. When a SET is biased by a negative voltage source via its back capacitor, it is called p-SET according to the fact that its characteristic is similar to a PMOS. On the other hand, an n-SET is biased by a positive voltage source. Therefore, a CSET composed of a p-SET and an n-SET works as an inverter similar to a CMOS inverter as shown in Figure 2 (b).

Simulation Method As same as conventional CMOS LSIs, the numerical simulation of single electron devices is important for implementing a desired function. Let us see that the simulation method of single electron circuits. Electron tunneling is a quantum mechanical phenomenon, and we know only the average tunneling rate, which is obtained as a function of a free energy change and the temperature of the environment. The average rate of the electron tunneling on the i-th junction is given as

Application of Single Electron Devices Utilizing Stochastic Dynamics

Figure 2.

Γi =

1

∆Fi

e RT exp (∆Fi / kBT ) − 1 2

,

(5)

where ΔFi denotes the free energy change (Likharev, 1987). To incorporate the randomness of a tunnel event, a noise is employed in the Monte Carlo simulation. Time to the tunneling event is calculated as follows

characteristic. It also cannot treat rare events properly, since any event with large ttunnel cannot be chosen as long as other frequent events exist, but it certainly occurs behind the scenes. On the other hand, simulation using the master equation evaluates probabilities of all states. The master equation is given as dPi dt

= ∑ Γij Pj − ∑ Γ ji Pi , j

(7)

j

−1

ttunnel

  1   1 ∆Fi  = ln    2  ,  r  e R exp (∆F / k T ) − 1 i B  T  (6)

where r(0 < r < 1) is a uniform random number (Kirihara, Kuwamura, Taniguchi, & Hamaguchi, 1994). In actual simulations of a circuit composed of SETs, possible state changes are evaluated by ΔF and ttunnel. Then a single tunnel event having the minimum ttunnel is chosen as an actual event, and the circuit state is updated. By repeating this procedure, one can know the circuit behavior. The Monte Carlo simulation is easy to execute and widely used. It, however, has some drawbacks. Because one can confirm only a transient behavior affected by a random number, any averaging process is always required for obtaining a static

where Pi and Γij denote the probability of state i and the transition probability between states i and j (Averin & Likharev, 1991; Fonseca, Korotkov, Likharev, & Odintsov, 1995). It is obvious that rare events can be treated in this formula. The necessity to store information about all states causes resource hungry calculation. The calculation cost depends strongly on the number of each junction charges we consider. Therefore, one should use the Monte Carlo method or the master equation method for different purposes. Any hybrid of these two methods could be efficient. (Amakawa, Majima, Fukui, Fujishima, & Hoh, 1998).

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Application of Single Electron Devices Utilizing Stochastic Dynamics

RANDOM NUMBER GENERATOR A conventional CMOS-based random number generator composed of shift registers generates a pseudo-random number sequence called an Msequence. However, it requires a large number of registers in order to obtain good randomness. On the other hand, the circuit shown below can easily generate a true random number sequence due to the randomness of electron tunneling process, and occupies smaller circuit area because it is constructed with ultra small tunnel junctions.

Circuit Design Single electron devices have intrinsic stochastic characteristic. A random number is obtained by detecting the motion of an electron, which results in a small electrical signal. Figure 3 (a) shows a single electron random number generator (Akima, Sato, & Nakajima, 2004). The circuit is composed of two CSETs and two SET switches. The two CSETs work as amplifiers in a closed loop. Nodes 1 and 2 are initialized with zero voltage before operation with the SET switches. After the SET switches turn off, the circuit becomes unstable due to its bistable property. The voltages of Node 1 and Node 2 fluctuate randomly. If the voltage of Node 1 increases slightly, it is amplified by Figure 3.

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the second CSET and then the voltage of Node 2 decreases. Next the voltage of Node 2 is amplified by the first CSET. Consequently, the voltage of Node 2, which is the output of the proposed circuit, reaches −VDD in some period. By repeating the preceding process, a random pulse sequence is obtained.

Simulation Results Figure 3 (b) shows a typical simulation result of the output pulse sequence. We regard the output voltage greater (or less) than 0 [V] as a HIGH (or LOW) output. The parameters are chosen so that high gain and stable operation can be achieved. The duration time of the HIGH or LOW output signal should be set longer than the time constant of a CSET, which is about 1.5 [ns]. And hence, the duration time of the HIGH or LOW output signal is set to 5 [ns] in the simulation for reliable operation. The initial duration time of the ZERO output should be set longer than the period in which electrons are charged or discharged on the output capacitances through the SET switches. According to the simulation results, 2.5 [ns] is enough to return the voltages of Node 1 and Node 2 to zero voltage for our circuit parameters. The proposed circuit works successfully. We have confirmed that the average HIGH output ratio of

Application of Single Electron Devices Utilizing Stochastic Dynamics

100 ensembles of 10000 pulses is 49.99% and its power spectrum is almost white.

Effect by Device Fluctuation It is important to consider the fluctuation of device parameters for practical use. The randomness of the output signal depends on the input-output characteristics of a CSET because two CSETs with feedback loop work as a bistable circuit. If a p-SET and an n-SET have the complementary characteristics, a CSET has the ideal input-output characteristics so that the threshold is equal to GND by bipolar biasing, and the output of the circuit becomes random. Since the fluctuation of the output capacitance changes only the time constant of the CSET, the randomness is not affected. However, the fluctuation of gate capacitances and tunnel capacitances change the CSET characteristics and result in the degradation of the randomness. A capacitance value is given as a function of device size, insulator thickness, and dielectric constant. Both device size and insulator thickness depend fabrication process parameters, and it is not easy to control them precisely. And hence, we should consider the fluctuation of capacitances. Another factor causing degradation is the fluctuation of tunnel resistances. A tunnel resistance value is the function of the insulator thickness and the device size. In order to obtain the relation between the tunnel resistance value and the insulator thickness, we have to know the physical mechanism of tunneling process and specify the materials both of the insulator and the electrodes.

Operation Speed It has been reported that a single electron transistor has 1/f power spectral charge noise (Kenyon, Lobb, & Wellstood, 2000). This is because that SET’s conductance is changed as an electron moves into or out of a defect level existing near the SET’s island. The random number generator proposed by Uchida et al. utilizes the similar phenomenon

by introducing an electron pocket (Uchida, Tanamoto, Ohba, Yasuda, & Fujita, 2002). On the other hand, the proposed circuit utilizes the fluctuation of a CSET’s output caused by thermal noise. The power spectrum of thermal noise is constant in all frequency range (white spectrum), and that of charge noise is in inverse proportion to frequency. According to the by Kenyon et al. (2000), thermal noise at 4.2 [K] is about 45 times as large as charge noise even at 10-1 [Hz]. Then, the proposed circuit utilizing thermal noise would be hardly affected by charge noise at the operation frequency, which is about 109 [Hz]. While the operation temperature should be low enough to satisfy the condition Equation (3), the operation speed should be limited slower as the operation temperature drops. This is because that time to the tunneling event becomes long as the operation temperature drops, as shown in Equation (5). For example, we have confirmed that the operation speed of the proposed circuit at 1.0 [K] should be dropped to about 1/10 as compared with that of 4.2 [K].

STOCHASTIC NEURAL NETWORK It is not easy to integrate such large scale ANNs with conventional CMOS technology. By using a single electron device, higher integration can be achieved thanks to its low power consumption and small area occupancy. One can design a single electron neural network in two ways, which are analog and digital. In general, analog circuits allow high speed operation because calculations can be implemented with current or voltage directory. However, it is difficult to keep high accuracy because current and voltage are influenced by noise and device fluctuations. While with digital circuits, noise immunity is high and accuracy increases because complex calculations can be achieved by Boolean logic. However, its circuit area tends to be large in proportional to the complexity of calculations required. Though

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Application of Single Electron Devices Utilizing Stochastic Dynamics

Figure 4. Stochastic encoding circuit

analog implementation has an advantage to digital implementation in terms of circuit area consumption, it would be difficult to operate a large scale single electron analog circuit correctly because its noise immunity is low and reliability is poor. Furthermore, it is difficult to construct a single electron analog multiplier with good linearity. It is quite important to construct a small multiplier circuit because synapse circuits, which execute multiplication, consume large circuit area. In this section, as an intermediate solution we introduce a neural network composed of SETs using stochastic logic. By using stochastic logic (Gaines, 1969) multiplication can be done with a single AND gate, and the circuit area can be reduced greatly.

Stochastic Logic A neural network utilizing stochastic logic, which was proposed by Kondo & Sawada (1992), is one of the pulse neural networks and the neurochip using stochastic logic with conventional CMOS technology has been reported (Sato, Nemoto, Akimoto, Kinjo, & Nakajima, 2003). One can convert

a digital or an analog value to a stochastic pulse sequence by stochastic logic, and various complex operations can be done with basic logic gates. In order to convert a binary digital value X stored in a register to a stochastic pulse sequence, it is compared with a uniform random number R. A comparator outputs a pulse when X is greater than R as shown in Figure4, where X and R have the same range [0, Xmax], and R is generated every unit time. The firing probability Pf of the comparator output is equal to X / Xmax. The output value, which is obtained by accumulating the pulse Na times, follows the binomial distribution having the expectation E and the variance V given as  X  E X  =  max  N a Pf = X ,  N a  2

 X  1 V X  =  max  N a Pf (1 − Pf ) = X (X max − X )  N a  Na

(9)

where Xmax / Na is a normalization constant. The pulse sequence has a noise with the variance given by Equation (9). This noise depends on the accumulation time Na. Figure 5 shows a multiplication of two independent stochastic pulse sequences using a single AND gate. The output firing probability PC is nearly equal to the product of the input firing probabilities PA and PB.

Figure 5. Stochastic multiplier using a single AND gate

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(8)

Application of Single Electron Devices Utilizing Stochastic Dynamics

Figure 6. Neuron circuit of a stochastic neural network

Circuit Configuration In order to implement a full connected neural network in which n neurons are linked to each other, n2 synapses are required. By employing time-division multiplexing, the number of synapses can be reduced to n in exchange for the decrease of operation speed. Figure 6 shows the neuron circuit of a stochastic neural network. The circuit operation is as follows: 1. A synaptic weight wij, which is stored in a register, is converted to a stochastic pulse sequence by a stochastic encoding circuit. 2. Multiplication between wij and xj is executed with an AND gate, where xj is the output of the neuron j. 3. The pulse output of the AND gate is accumulated Na times with an up-down counter. 4. By repeating the above procedure 1 to 3 for all other neurons, the membrane potentials of all neurons are obtained. 5. Each membrane potential stored in the updown counter is moved to a register. 6. The value stored in the register is converted to a stochastic pulse sequence as the neuron output.

n × Na time steps are required for a state update of the network. The saturation property of the firing rate can be implemented by setting the upper limit of the random number less than the maximum value Xmax. Therefore, a nonlinear function required for a neuron is obtained naturally with a comparator. The sub-circuits required for a single electron stochastic neural network are as follows; a comparator (see Figure7), an up-down counter, a register, and a random number generator (RNG). Single electron basic logic gates can be constructed easily by replacing MOSFETs with SETs. However, simple replacing does not work well in some circuits having feedback loops. The single electron logic gate composed of SETs has a relatively large active region, where a signal cannot be unambiguously interpreted as either high or low. If such gates are used in the feedback loop configuration, racing between feedback signals occurs. In such cases, one should use a circuit having narrow active region though it may cost large circuit area (Akima, Yamada, Sato, & Nakajima, 2004). Note that a large number of uncorrelated random numbers are required for encoding operations, but they can be given by using the RNG shown in the previous section. We have confirmed the successful of operations

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Figure 7.

of these sub-circuits. Our results (Akima et al., 2004) indicate that 16,000 SETs are required for constructing the stochastic neural network composed of 8 neurons. It is obviously small when we compare it with a full digital implementation.

STOCHASTIC RESONANCE IN A SINGLE ELECTRON TURNSTILE Stochastic resonance (SR) is a nonlinear phenomenon whereby the addition of noise can enhance the detection of weak stimuli. Although a large number of phenomena ranging from physics and engineering to biology and medicine have been studied (Moss, Ward, & Sannita, 2004; Oya, Asai, Kagaya, Hirose, & Amemiya, 2006), the essential ingredients for SR consist of a threshold, subthreshold stimulus and noise in nonlinear systems. Experimental observation of SR in a bistable system involved a Schmitt trigger circuit (Fauve & Heslot, 1983) and a radio frequency superconducting quantum interference device (rf-SQUID) (Rouse, Han, & Lukens, 1995; Hibbs et al., 1995). It may be naturally expected that SR in a single electron circuit can be realized due to the duality between an rf-SQUID and a single electron box (SEB) (Katsumoto, Sato, Iye, 1999). An SEB can

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be utilized for quantum computation (Nakamura, Pashkin, Yu, & Tsai, 1999), and SR with an SEB must be useful for detecting small charges under noisy environment. An rf-SQUID is composed of a Josephson junction, a superconducting loop inductance, and a bias current source in parallel. The number of flux quantums nf trapped in the loop is controlled by changing the bias current. On the other hand an SEB is composed of a tunnel junction, a gate capacitance and a bias voltage source in series. The number of excess electrons ne trapped at an island, which is the electrically isolated region between the tunnel junction and the gate capacitance, is controlled by changing the bias voltage. Though a perfect duality seems to exist, it is not true in terms of bistability. While an rf-SQUID has bistability related to nf due to hysteresis property of a superconducting loop with Josephson junctions, an SEB does not have bistability related to ne because one tunnel junction does not show any hysteresis property. We consider a single electron turnstile (Geerligs & Mooij, 1991) which is given bistability with four tunnel junctions in series.

Application of Single Electron Devices Utilizing Stochastic Dynamics

Single Electron Turnstile Figure 8 shows a single electron turnstile and its stability diagram. The stability diagram is the two-dimensional map of the stable states of a circuit (Wasshuber, 1998). The horizontal- and vertical-axes represent two bias voltages, namely the normalized gate voltage and the normalized bias voltage, respectively. At the operating points surrounded by each diamond in Figure 8 (b), the change of n, which signifies the number of excess electrons in the central island, leads to increase of free energy. Then electron tunneling is inhibited and the circuit remains stable states. Thus the gray regions, where two neighboring diamonds are overlapped, correspond to bistable regions: two different stable states with different n are realized exclusively. If the gate voltage Vg(t) = Av sin(2πfst) + Vg0 takes the operating point cut across the bistable region periodically like A → B → A→ Λ in Figure 8 (c), n varies as 0 → 1 → 0→ Λ. Consequently such single electron transportation synchronized with the frequency fs can be realized, and thus the

current Is = efs flows. This property can be utilized for a current standard. Notice that the periodic motion of an electric charge Qs(t) = Aq sin(2πfst) polarized in the central island corresponds to the ac-component Vs(t) = Av sin(2πfst), where Av = Aq/Cg and Cg is the gate capacitance. While the periodic motion of an outside electric charge near the central island produces the synchronized current flow in the same manner even without the ac-component of Vg. Therefore a single electron turnstile can be used as a detector of the periodic motion of electric charge.

Stochastic Resonance with a Single Electron Turnstile Let us consider Qs(t) and n(t) as the input and output signals, respectively. The bistability related to n(t) leads to SR with the help of some external noises such as fluctuation of voltages, fluctuation of background charge motion. We suppose the circuit operates at the point (Vg0, Vb0 − ε) as shown in Figure 8 (c), where Vg0 = e/(Cg + Cext),

Figure 8. Single electron turnstile (a) and its stability diagram at zero temperature (b). The stability diagram enlarged near the bias point (c). An alternating gate voltage Vg(t) swings the operating point

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Vb0 = 2/2(C + Cext), e is the elementary charge, C is the tunnel capacitance, Cext ≡ C(C/2 +Cg)/(3C/2 + Cg) and ε is a small positive constant. The input charge signal Qs(t) forces the operating point of the circuit to fluctuate with the amplitude Aq/Cg and the frequency fs around the bias point. When a Gaussian charge noise QN(t) with a cutoff frequency fc is added to Qs(t), the input gate voltage is given as Vg(t) = Vg0 + (Qs(t) + QN(t))/Cg ≡ Vg0 + Vs(t) + VN(t). Switching of n(t) occurs at Vg = Vt0(n = 0 → 1) and Vg = Vt1(n = 1 → 0), where

can occur with the help of the noise VN(t) = QN(t)/ Cg; this is the source of SR. Usually SR property is evaluated by calculating SNR as a function of noise amplitude, and the SNR profile is calculated by integrating the power spectral density over the peak centered at the signal frequency around fs, and dividing by the mean noise power, as given as  SG + N ∆  , SNR = 10 log   N∆   

(11)

Vt0 = 2Vg0 - 2Vb0(C + Cext)/ (Cg + Cext), Vt1 = 2Vb (C + Cext)/ (Cg + Cext)

(10)

and Vb is the source voltage. Note that the signal frequency fs has to meet the ``adiabatic limit” 1/fs >> τt, where τt is the tunneling time of electrons, in order to get the circuit settle in equilibrium states n = 0 or 1. The signal Vs(t) = Qs(t)/Cg modulates the operation point of the system periodically and lowers the effective potential barrier between the two stable states n = 0 or 1. Even if the amplitude of the signal is not large enough to make the potential barrier disappear (subthreshold signal), switching

where S and N are the power of the signal and noise, respectively, G is the processing gain, and Δ is the width of frequency bins (McNamara & Wiesenfeld, 1989). The SNR obtained by numerical simulation results are shown in Figure 9 together with theoretical curves, where three different amplitudes of Aq are considered (Akima, Sato, Nakajima, 2007). Parameters are chosen as follows; C = 1.0[aF], Rt = 100[kΩ], Cg = 0.5[aF], Vg0 = 160.218[mV], Vb = 50[mV], T = 30 [mK], fs = 100[MHz], τN = 125[ps], fc = 4[GHz], and τt is given as several tens picoseconds on average. The SR effect, wherein the SNR passes through

Figure 9. SNR versus noise profile. The simulation results are shown with points, where Dv denotes the variance of the Gaussian noise VN. Two theoretical curves are shown for reference (Akima et al., 2007)

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a maximum at certain noise strength, has been confirmed clearly from these results. These results could be applied to detection of the periodic motion of very small electric charge under noisy environment. One possible application is qubit detection in quantum computers (Nakamura et al., 1999; Kane et al., 2000) instead of using an SET or an rf-SET (Schoelkopf, Wahlgren, Kozhevnikov, Delsing, & Prober, 1998). Even though it detects not quantity but the motion of electric charge, the superior figure of noise tolerance is highly attractive for detecting weak signals.

Akazawa and Amemiya (1997). In some cases, the stochastic behavior of neurons enhances the performance of an ANN. The same holds for the stochastic resonance with a single electron turnstile as seen in the last application. Though a single electron device itself has good sensitivity for charges, stochastic resonance is helpful for the charge detection in noisy environment. Quantum devices have good potential for calculation as known in quantum computation. However, a quantum signal is very weak in general. Therefore, the importance of the charge detection with single electron devices will increase in future.

CONCLUSION

REFERENCES

Unfortunately, the most suitable application has not been know at present, and it is important for us to search applications, where any advantage of single electron devices can be utilized well. In this paper, we have introduced three application examples, where we pay much attention to the properties of single electron devices such as stochastic behavior, small device size and low power characteristics suitable for integration, and good sensitivity for charges. Thanks to the intrinsic stochastic behavior, the proposed RNG has the simplest structure and can generate a true random pulse sequence. It could be useful for various applications such as numerical simulations, cryptography, neural networks and so on. On the other hand, the stochastic property could be an obstacle when we try to use a single electron device for analog computation. Huge integration is required for implementing an ANN, and the low power property of single electron devices is well appropriate. Therefore, the proposed method employing the stochastic logic is one of solutions for huge integration of artificial neurons, in which the stochastic behavior of SETs does not lose the reliability of the circuit. If we do not need thousands of neurons, it is interesting to utilize the stochastic property intentionally as proposed by

Akazawa, M., & Amemiya, Y. (1997). Boltzmann machine neuron circuit using single-electron tunneling. Applied Physics Letters, 70(5), 670–672. doi:10.1063/1.118329 Akima, H., Sato, S., & Nakajima, K. (2004). Single Electron Random Number Generator. IEICE Trans. Electron. E (Norwalk, Conn.), 87C(5), 832–834. Akima, H., Sato, S., & Nakajima, K. (2007). Stochastic resonance in a single electron turnstile. quant-ph, 0710.2718. Akima, H., Yamada, S., Sato, S., & Nakajima, K. (2004). Single Electron Stochastic Neural Network. IEICE Trans. Electron. E (Norwalk, Conn.), 87-A(9), 2221–2226. Amakawa, S., Majima, H., Fukui, H., Fujishima, M., & Hoh, K. (1998). Single-Electron Circuit Simulation. IEICE Trans. Electron. E (Norwalk, Conn.), 81-C(1), 21–29.

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Averin, D. V., & Likharev, K. K. (1991). Single electronics: A Correlated Transfer of Single Electrons and Cooper Pairs in Systems of Small Tunnel Junctions. In B. L. Altshuler, P. A. LEE, & R.A.Webb (Eds.), Mesoscopic Phenomena in Solids (pp. 173-271). Amsterdam: Elsevier. Dayhoff, J. E. (1990). Neural Network Architectures: An Introduction. New York: Van Nostrand Reinhold. Fauve, S., & Heslot, F. (1983). Stochastic resonance in a bistable system. Physics Letters. [Part A], 97(1-2), 5–7. doi:10.1016/03759601(83)90086-5 Fonseca, L. R. C., Korotkov, A. N., Likharev, K. K., & Odintsov, A. A. (1995). A numerical study of the dynamics and statistics of single electron systems. Journal of Applied Physics, 78(5), 3238–3251. doi:10.1063/1.360752 Gaines, B. R. (1969). Stochastic computing systems. In Tou, J. F. (Ed.), Advances in Information Systems Science (Vol. 2, pp. 37–172). New York: Plenum Press. Geerligs, L. J., & Mooij, J. E. (1991). Charging Effects and ‘Turnstile’ Clocking of Single Electrons in Small Tunnel Junctions. In Ferry, D. K., Barker, J. R., & Jacoboni, C. (Eds.), Granular Nanoelectronics (pp. 393–412). New York: Plenum Press. Grabert, H., & Devoret, M. H. (Eds.). (1992). Single Charge Tunneling. New York: Plenum Press. Hibbs, A. D., Singsaas, A. L., Jacobs, E. W., Bulsara, A. R., Bekkedahl, J. J., & Moss, F. (1995). Stochastic resonance in a superconducting loop with a Josephson junction. Journal of Applied Physics, 77(6), 2582–2590. doi:10.1063/1.358720

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Kane, B. E., McAlpine, N. S., Dzurak, A. S., Clark, R. G., Milburn, G. J., Sun, H. B., & Wiseman, H. (2000). Single-spin measurement using single-electron transistors to probe two-electron systems. Physical Review B: Condensed Matter and Materials Physics, 61(4), 2961–2972. doi:10.1103/PhysRevB.61.2961 Katsumoto, S., Sato, H., & Iye, Y. (1999). Duality between Single-Electron Phenomena and Flux Quantization in Mesoscopic Superconductors. Japanese Journal of Applied Physics, 38(1B), 350–353. doi:10.1143/JJAP.38.350 Kenyon, M., Lobb, C. J., & Wellstood, C. (2000). Temperature dependence of low-frequency noise in Al-Al2O3-Al single-electron transistors. Journal of Applied Physics, 88(11), 6536–6540. doi:10.1063/1.1312846 Kirihara, M., Kuwamura, N., Taniguchi, K., & Hamaguchi, C. (1994). Monte Carlo study of single-electronic devices. Ext. Abst. of Int. Conf. on Solid State Devices and Materials, 328-330. Kondo, Y., & Sawada, Y. (1992). Functional abilities of a stochastic logic neural network. IEEE Transactions on Neural Networks, 3(3), 434–443. doi:10.1109/72.129416 Likharev, K. K. (1987). Single-Electron Transistors: Electrostatic Analogs of the DC SQUIDs. IEEE Transactions on Magnetics, MAG-23(2), 1142–1145. doi:10.1109/TMAG.1987.1065001 Likharev, K. K. (1988). Correlated discrete transfer of single electrons in ultrasmall tunnel junctions. IBM Journal of Research and Development, 32(1), 144–158. doi:10.1147/rd.321.0144 McNamara, B., & Wiesenfeld, K. (1989). Theory of stochastic resonance. Physical Review A., 39(9), 4854–4869. doi:10.1103/PhysRevA.39.4854

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Moss, F., Ward, L. M., & Sannita, W. G. (2004). Stochastic resonance and sensory information processing: a tutorial and review of application. Clinical Neurophysiology, 115(2), 267–281. doi:10.1016/j.clinph.2003.09.014

Sato, S., Nemoto, K., Akimoto, S., Kinjo, M., & Nakajima, K. (2003). Implementation of a New Neurochip Using Stochastic Logic. IEEE Transactions on Neural Networks, 14(5), 1122–1127. doi:10.1109/TNN.2003.816341

Nakamura, Y., Pashkin, Yu. A., & Tsai, J. S. (1999). Coherent Control of Macroscopic Quantum State in a Single-Cooper-pair Box. Nature, 398(6730), 786–788. doi:10.1038/19718

Schoelkopf, R. J., Wahlgren, P., Kozhevnikov, A. A., Delsing, P., & Prober, D. E. (1998). The Radio-Frequency Single-Electron Transistor (RFSET): A Fast and Ultrasensitive Electrometer. Science, 280(5367), 1238–1242. doi:10.1126/ science.280.5367.1238

Oya, T., Asai, T., Kagaya, R., Hirose, T., & Amemiya, Y. (2006). Neuronal synchrony detection on signle-electron neural network. Chaos, Solitons, and Fractals, 27(4), 887–894. doi:10.1016/j. chaos.2005.04.059 Rouse, R., Han, S., & Lukens, J. E. (1995). Flux amplification using stochastic superconducting quantum interference devices. Applied Physics Letters, 66(1), 108–110. doi:10.1063/1.114161

Tucker, J. R. (1992). Complementary digital logic based on the Coulomb blockade. Journal of Applied Physics, 72(9), 4399–4413. doi:10.1063/1.352206 Uchida, K., Tanamoto, T., Ohba, R., Yasuda, S., & Fujita, S. (2002). Single-Electron RandomNumber Generator (RNG) for Highly Secure Ubiquitous Computing Applications. IEDM. Tech. Digest., 177-180. Wasshuber, C. (2001). Computational SingleElectronics. Wien: Springer-Verlag.

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 29-42, copyright 2009 by IGI Publishing.

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Chapter 7

On the Reliability of PostCMOS and SET Systems Milos Stanisavljevic Swiss Federal Institute of Technology EPFL, Switzerland Alexandre Schmid Swiss Federal Institute of Technology EPFL, Switzerland Yusuf Leblebici Swiss Federal Institute of Technology EPFL, Switzerland

ABSTRACT The necessity of applying fault-tolerant techniques to increase the reliability of future nano-electronic systems is an undisputed fact, dictated by the high density of faults that will plague the chips. The averaging and thresholding fault-tolerant technique that has proven remarkable efficiency in CMOS is presented for SET-based designs. Computer simulations demonstrate the superiority of this fault-tolerant technique over other methods, which is specifically the case when an adaptable threshold is used.

INTRODUCTION The advent of ubiquitous electronic appliances in the modern information society has founded its success on the premise of using highly reliable components in every development level. Based on the mature CMOS fabrication process, larger and faster, but also power-hungry and very complex integrated circuits have been fabricated on the assumption of very reliable operation of their constituting modules, from the atomic elements such as transistors, and passive components such as capacitances, routing lines, etc., to complex DOI: 10.4018/978-1-60960-186-7.ch007

modules made of several thousand of transistors, such as arithmetic and logic unit modules, memory blocks. Also, the availability of reliable electronic design automation (EDA) tools and efficient design-flows has been assumed. The vast majority of microelectronic developments presented nowadays uses the well-established CMOS process and fabrication technology which exhibit high reliability rates. The hypothesis of reliable components has mostly been adopted in the development of electronic systems fabricated in the past four decades. Several indicators show that future fabrication processes will exhibit increased failure rates and degraded fabrication yield. This Chapter focuses on the construction of

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On the Reliability of Post-CMOS and SET Systems

reliable nanoelectronic systems made of intrinsically unreliable atomic constituting elements. The main sources of faults are described in Section Reliability and yield in post-CMOS fabrication technologies. In Section single electron transistors (SETs), the basic concepts and modes of operation of SETs is described. Section Techniques for improving reliability focuses on currently applied methods for increasing design reliability. The averaging and thresholding technique is presented as a method enabling increasing the reliability of nanoelectronic systems in Section Averaging and thresholding for increasing reliability. This work focuses on the specific case of singleelectron transistor-based designs. The analytical framework of the averaging and thresholding technique using single-electron transistor-based designs is presented. The proposed fault-tolerant averaging and thresholding technique is compared with other standard fault-tolerant techniques in Section Comparison of different fault-tolerant techniques, showing its clear superiority in terms of fault resilience.

RELIABILITY AND YIELD IN POST-CMOS FABRICATION TECHNOLOGIES Some typical physical defects in VLSI chips include: • •

•

Process defects: missing contact windows, parasitic transistors, oxide breakdown, etc. Material defects: bulk defects (cracks, crystal imperfections), surface impurities, etc. Aging defects: dielectric breakdown, electromigration, etc.

Errors are traditionally categorized into three main groups: permanent, intermittent and transient errors according to their stability and concurrence. Permanent errors are irreversible physical changes

in a chip. The most common sources for this kind of errors are the manufacturing processes. Permanent errors also occur during usage lifetime of the circuit, especially when the circuit is old and therefore wears out. Intermittent errors are occasional error bursts that usually repeat themselves every now and then, i.e. are not continuous as permanent errors. These errors are caused by unstable or marginal hardware, and are activated by environmental changes such as temperature or supply voltage change. Transient errors are temporal single malfunctions caused by temporary environmental conditions which can be external phenomenon such as radiation or noise originating from the other parts of the chip. Sources of errors can be classified according to the phenomenon causing the error. Such origins are for instance: the manufacturing process, physical changes during operation, internal noise caused by other parts of the circuit and external noise originating from the chip environment.

SINGLE ELECTRON TRANSISTORS The end of the ITRS roadmap for classical CMOS devices and circuits envisions the emergence of future nanotechnologies and nano-devices, and also evidences many new related challenges. Logic design at present is solely applied to microelectronics. The process of transferring circuits and systems to nanoelectronics and relevant hybrid technologies (e.g., molecular electronics) has already started. Very fundamental and technological differences between nanoelectronic devices and microelectronic devices exist, those latter possibly in the nanometer size domain. Even though CMOS devices are reaching below 50nm dimensions, these devices rely on enhanced but standard CMOS fabrication processes, and hence do not formally participate to nanoelectronic devices. Novel physics, integrated with design methods and nanotechnology, leads to far-reaching revolutionary progress. One of the devices that

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Figure 1. Simplified structure of a MOSFET (a), compared with that of a SET (b)

has attracted a lot of attention of the researching community is the single electron transistor. Single-electron tunneling devices (SETs) are three-terminal devices where the electron transfer through the device is controlled with a precision of an integer number of electrons. An electron can tunnel from and to an island or quantum dot through a tunneling barrier which is controlled by a separate gate, based on Coulomb blockade. The electron island can accommodate only an integer number of electrons. This number may be as high as a few thousand. A single-electron transistor is composed of a quantum dot connected to an electron source and to a separate electron drain through tunnel junctions, where the electron injection is controlled by a gate electrode. Single electron transistors can be used to implement logic circuits by operating on one or more electrons as a bit of information (Likharev, 1999). The simplified structure of a SET is compared with that of a MOSFET in Figure 1. Indeed, the device is reminiscent of a typical MOSFET, but with a small conducting island embedded between two tunnel barriers, instead of the usual inversion channel. The current-voltage characteristics of the SET are shown in Figure 2, as a function of different gate voltage levels. At small drain-to-source voltages, there is no current since the tunneling rate between the electrodes and the island is very low. This suppression of DC current at low voltage

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levels is known as the Coulomb blockade. At a certain threshold voltage, the Coulomb blockade is overcome, and for higher drain-to-source voltages, the current approaches one of its linear asymptotes. A very significant property of the single-electron transistor is related to the fact that the threshold voltage as well as the drain-to-source current in its vicinity are periodic functions of the gate voltage. The physical reason for this periodicity lies in the fact that the conditions that govern the tunneling of charge between the electrodes and the isolated island can be established for consecutive, discrete states that correspond to the existence of integer multiples of an electron charge on the island. Still, it is evident that the device can be operated as a switch controlled by the gate electrode, capable of performing logic functions. The dimensions of the conductive island and the tunneling junctions need to be in the order of a few nanometers to a few tens of nanometers. While larger device dimensions allow observable device operation at very low temperatures, the dimensions may need to be reduced to subnanometer levels in order to achieve Coulomb blockade near room temperature (Likharev, 1999). It is estimated that the maximum operation temperature for 2nm SETs is 20 K, with an integration density of approximately 1011 cm-2 and an operating frequency in the order of 1 GHz (Chen, Korotkov & Likharev, 1996). Various logic applica-

On the Reliability of Post-CMOS and SET Systems

Figure 2. Typical current-voltage characteristics of a C-SET (capacitive input SET), displaying the Coulomb blockade region for low source-drain voltage values (adapted from Likharev, 1999)

tions of SETs, including inverters (Heij, Hadley & Mooij, 2001; Ono et al., 2000; Mahapatra, Ionescu & Banerjee, 2002; Uchida et al., 2003), OR, NAND, NOR and a 2-bit adder (Asahi, Akazawa & Amemiya, 1998; Kasai & Hasegawa, 2002; Takahashi et al., 2000; Uchida et al., 2003), have been demonstrated. The ability of the SET to operate with discrete charge levels makes it possible to construct functions that are based on multiple quantized input output levels, in contrast to the classical Boolean functions operating with two discrete levels. However, due to the high impedance required for Coulomb blockade, a SET gate would not be able to drive more than one cascaded gate. This has two implications. First, SET logic would have to be based on local architectures, such as cellular arrays and cellular nonlinear networks (CNNs) (ITRS, 2007). Second, although SETs may not be suitable for implementations of logic circuits, they could be used for memories. SETbased memory structures have been proposed and experimentally demonstrated (Durrani, Lnine & Ahmed 2000; Mizuta et al., 2001; Stone, Ahmed & Nakazato 1999). While offering a number of potential advantages in terms of very high integration density and

extremely low power dissipation, SET devices also have serious limitations in terms of output drive capability, speed and fan-out, which would restrict their large-scale integration and interfacing with other system components. Hence, the design of SET-CMOS interface circuits is already gaining importance, and future systems will likely be based on a hybrid SET-CMOS architecture, where intensive logic or memory functions are performed by very dense, regular arrays of SETs, and the interface functions among blocks are realized in classical, high-speed CMOS components (Han & Jonker, 2002; Uchida et al., 2003; Ziegler & Stan, 2003). One of the most significant difficulties of designing complex functions using SETs is the inherent sensitivity of their characteristics to background charge fluctuations (Likharev, 1999). This effect is the result of permanent or transient random variations in local charge due to fabrication irregularities, leakage, or external perturbations such as noise. Background charge effects may permanently or temporarily disrupt device function, rendering one or more SETs inoperative within a functional block in a random manner. To ensure reliable operation and to reduce the sensitivity of devices to background charge effects (especially

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On the Reliability of Post-CMOS and SET Systems

at room temperature), the device dimensions must be reduced to sub-nanometer levels, which is not expected to be feasible at large scale in the foreseeable future. In order to tackle this problem, besides the endeavor to develop novel computing schemes such as the multi-value SET logic, fault-tolerant architectures implemented at higher levels of circuits and systems may be a direction of investigation (Han & Jonker, 2002). A more likely scenario is that the functional blocks be designed with a certain degree of fine-grained, built-in immunity to permanent and transient faults, such that they are capable of absorbing a number of errors and still be able to perform their functions.

TECHNIQUES FOR IMPROVING RELIABILITY Improving the reliability of CMOS has mostly been considered at device-level through several decades of size scaling. Fabrication technologies have included new materials and process recipes. The fabrication methodologies themselves have evolved and adapted to the new thinner sizes to be fabricated. Nevertheless, fabrication yield shows a significant decrease in very-deep submicron technologies. It is expected that the defect density in future nano-electronic systems be significantly higher than what is observed in current CMOS. Fabricating reliable systems out massively unreliable atomic devices is the challenge in the development of future multi-billion transistor system-on-chip. Guidelines in the design of new hardware blocks have been used mostly to guarantee homogeneous design within a development group, and guaranteeing functionality by standardization of design techniques and methodologies. More recently, these ad-hoc techniques have started incorporating reliability directives at various levels of abstraction. For example, the systematic usage of multiple via connections has been prescribed in

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response to the increasing defect density of vias in very-deep submicron technologies. All efforts in improving the reliability of devices and fabrication methods are necessary, but are not going to guarantee a reliability continuum between CMOS and nano-electronics, at least in the first generations. Consequently, reliability of nano-electronic systems must be considered as a central element of all abstraction levels in the development process (Cotofana et al., 2005). Improving reliability figures is achieved by exploiting the specificities of nanoelectronic devices and system as countermeasures to their inherent weaknesses. Redundancy has been advocated to provide an efficient solution to reliability issues since the 1960’ (von Neumann, 1956), and can be declined as hardware redundancy (n-tuple presence of weak blocks), time redundancy (re-execution of sensible sequences), information redundancy (usage of correction codes), or hybrid methods (Bahar et al., 2007). The redundant usage of potentially weak modules exploits the abundant transistor count available in nano-electronic systems, and their very low-power intrinsic operation. Several solutions have been proposed, targeting the available CMOS technology, which can be adapted to future technologies, such as triple modular redundancy, von Neumann multiplexing, or reconfiguration (Nikolic, Sadek & Forshaw, 2002).

AVERAGING AND THRESHOLDING FOR INCREASING RELIABILITY Four-Layer Architecture for Improved Reliability Based on Redundancy A four-layer fault-tolerant hardware architecture, named 4LRA in the following (Figure 3) is used in order to offer a solution to the previously presented issues (Schmid & Leblebici, 2004). The architecture described in the following has been

On the Reliability of Post-CMOS and SET Systems

Figure 3. Block-diagram of the four-layer architecture (adapted from Stanisavljevic, Schmid & Leblebici, 2005)

applied at the gate, or extended gate level. It can be applied hierarchically in a bottom-up way, and combined with other high-level fault absorption techniques. Data flows in a strictly feed-forward manner through the four layers. The input terminals are located in the first layer, and can accommodate binary or multiple-valued logic inputs. The second layer consists of a number of redundant Boolean units that process the expected system function. The redundancy factor R can be adapted to the desired reliability level. The third layer consists of an averaging and rescaling hardware unit that performs a weighted average of the second layer outputs, and range compression of the result. The output of the third layer is in the form of a multiple-valued logic function, where the number of possible states equals to R+1. The fourth layer is a threshold unit used to extract a binary output from the third layer output signal. The details of this architecture have already been presented by the authors in earlier publications (Schmid & Leblebici, 2004), (Stanisavljevic, Schmid & Leblebici, 2005). The averaging/thresholding circuit used in layers three/four exhibits analog behavior. Adaptable thresholding is necessary to adapt the

4LRA to the actual faulty transfer function surface. Static errors can be recovered applying the proposed circuit architecture. This work has also been extended to the study of delay faults which can be recovered using the proposed 4LRA. The 4LRA has been thoroughly presented in the previous work by the authors, focusing on nanoelectronic devices and systems, also including very-deep submicronic CMOS technologies. A version of the averaging and thresholding layers has also been presented by the authors (Mahapatra et al., 2003), and is depicted in Figure 4.

Monte Carlo Simulations of C-SET and Derivation of a Method for the Statistical Analysis of Fault-Tolerant Techniques Simulations of hybrid architectures (SET-CMOS) tend to be slow and inaccurate. Therefore, all designs that have been evaluated are implemented as capacitive input SET (C-SET) (Tucker, 1992). The C-SET design is based on the SET inverter (Iwamura, Akazawa & Amemiya, 1998; Uchida et al., 2003), which consists of two “complemen-

119

On the Reliability of Post-CMOS and SET Systems

Figure 4. Circuit-level description of the averaging-thresholding hybrid circuit consisting of SETs circuits driving a MOSFET restoring stage (adapted from Mahapatra et al., 2003)

tary” SET transistors (equivalent to nMOS and pMOS transistors), in addition to bias and input capacitors. Manufactured C-SET inverters and multi-input C-SET gates (Heij, Hadley & Mooij, 2001; Takahashi et al., 2000) are supporting this approach. Pure C-SET simulation offers significant speed and accuracy advantage over simulations of hybrid architectures (SET-CMOS). A further reduction of complexity and computation time compared to simulations of complete fault-tolerant circuits is expected when using the analysis of the statistical properties of faulttolerant techniques. The analysis of fault-tolerant techniques such as averaging or four-layer architecture, which inherently use analog signals, is not possible using a single probability value to describe the fault-tolerance of each device in the circuit. Analog behavior here refers to a fault-tolerant architecture that operates with analog, continuous values of signals. The analysis of fault-tolerant techniques uses probability density functions (PDFs) of

120

one single unit within the so-called redundant layer (a logic layer, in Figure 1 for the four-layer reliable architecture). The output PDF of the averager/4LRA is obtained using the PDF of the unit output. Finally, the probability of error is derived from the output PDF of the averager/4LRA. An alternative approach proposed in (Martorell, Rubio & Cotofana, 2005; Martorell, Cotofana & Rubio, 2006; Martorell, Cotofana & Rubio, 2008) uses the mean and variance of the conditional output signal. The probability of error can only be estimated using this approach, but results are comparable to results obtained in this work. The logic layer of redundant NAND gates as units, together with averaging and thresholding is shown in Figure 5. The averaging and thresholding operation is performed mathematically, taking into consideration a hypothetical ideal averager/ thresholder, as opposed to a hardware realization shown in Figure 4. PDFs can be acquired from the analysis of the distribution of all the faulty states in the given circuit, as well as the impact of every single fault on the circuit output. However, state analysis becomes too complex for post-CMOS nano-devices. Therefore, PDFs are obtained by the means of simulation, or modeled using Gibbs distribution and the approach described in (Bahar, Mundy & Chen, 2003; Bahar, Mundy & Chen, 2004). A custom Monte-Carlo simulator is used in this work to acquire conditional output values in the presence of geometric variations and construct PDFs. The analysis of the sensitivity to variations was carried out using MATLAB-based modules (Sulieman & Beiu, 2004), simultaneously with SIMON (Wasshuber, Kosina & Selberherr, 1997) (see Figure 6). The module computing the sensitivity to variation operates as described in the following. Random variations are applied on C-SET elements (capacitors and tunneling junctions). A modified capacitor value is computed from a normal distribution N(C0,σr∙C0) centered around nominal value (C0) and with relative standard deviation

On the Reliability of Post-CMOS and SET Systems

Figure 5. Logic layer with NAND gates as units and ideal averaging and thresholding

σr. The new circuit (with modified capacitors) is subsequently simulated using SIMON, considering all the possible input vectors. The entire procedure including varying the capacitors’ values and performing simulations is repeated 10’000

times as a loop in MATLAB, while data is collected, in the form of data points. In the following analyses of reliability of fault-tolerant techniques, we assume fault-free averaging/thresholding gates. The nature of the analyzed circuit is irrelevant in terms of calcula-

Figure 6. 2-input NAND implementation using C-SET technology drawn in SIMON (Wasshuber et al., 1997)

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On the Reliability of Post-CMOS and SET Systems

tion procedure and complexity. A 2-input NAND Boolean gate (Figure 6) is used as a reference circuit and its output PDF is evaluated. The applied circuit parameters are CG=2 aF (1 aF for each nSET gate) for the gate capacitance, Cj=1 aF for all the junction capacitances, Cb=5.5 aF for all the bulk capacitances and CL=16 aF for a load capacitance. The supply voltage is Vdd=10 mV. Simulations are performed at a 1K operating temperature (as in Iwamura et al., 1998). In this Section, a statistical method is presented and verified using data obtained by the means of MC simulations. PDFs are constructed by analyzing circuit outputs obtained from MC simulations for each input vector, applying numerous fault patterns. For a circuit having n inputs there are 2n different input vectors and 2n corresponding output values. Let Yi, i Î {1,..., 2n } be random variables that correspond to output values of a circuit. PDFs that correspond to those random variables are marked with hi, i Î {1,..., 2n } . These variables can be divided into two sets, H1 consisting of output values corresponding to inputs vectors that produce a logic-1 output, and H0 consisting of output values corresponding to inputs vectors that produce a logic-0 output. Let Ymin1 be a random variable that corresponds to ymin=minH1, and Ymax0 be a random variable that corresponds to ymax=maxH0. Two additional PDFs of interest are PDFs that correspond to random variables Ymin1 and Ymax0, named worst case logic-1 and worst case logic-0 in further explanations (marked as hmin1 and hmax0, respectively). All PDFs are continuous, since the output voltage of a faulty circuit is only limited by the power rails and may potentially take any value. The individual values of the capacitances used in MC simulations of circuits obey a continuous normal distribution, and therefore the number of different possible output values is unlimited. In order to maintain the generality of the approach, the mathematical apparatus presented hereafter will use continuous functions. On the other hand,

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the actual calculations implemented in MATLAB with custom scripts are performed on discrete data sets. The worst case logic-0 and logic-1 PDFs for the analyzed logic gate are depicted in Fig. 7 and 8. According to the aforementioned definition, the worst case logic-0 represents the highest value of the output in the presence of variations, which is expected to be a logic-0 level in the absence of variations (and accordingly for the worst case logic-1). Values located along the x-axis are plotted in relative units of Vdd in Figure 7 through Figure 11. The plotted PDF values are continuous and interpolated using a 100 points histogram. The nonmonotonic nature of the curves and pronounced peeks are due to the fact that some circuit states are more common than the others, especially states around 0, Vdd/2 and Vdd which is expected due to the quantum nature of SET devices. The common occurrence of values around Vdd/2 is highly beneficial to the case where averaging is used in order to improve reliability, as it is shown further on. Without a fault-tolerant architecture, this kind of output would be classified as erroneous. However, averaging significantly helps in improving the performance. We assume that random variables which follow these PDFs, are independent. A small correlation Figure 7. PDF of gate output for the worst case logic-0 (h0)

On the Reliability of Post-CMOS and SET Systems

Figure 8. PDF of gate output for the worst case logic-1 (h1)

Figure 9. PDF of averager output for the worst case logic-0 ( h0*3 )

between them exists, but the error introduced by this hypothesis is not relevant, as will be shown in the following. Probabilities

represent the probabilities of faulty output of the gate for logic-0 and logic-1 (Figure 7 and 8), respectively. This assumes that the threshold which determines the correctness of the gate operation is set to Vdd/2. Finally the probability of the fault of a single gate is given as

1

PE 0 =

∫h

max 0

(x )dx , and

0.5 0.5

PE 1 =

∫



(1)

hmin 1 (x )dx

PE _ gate = PE 0 + PE 1 − PE 0PE 1,

(2)

0

where PE0 and PE1 are taken from (1). Notice the difference between hmax0 and hmin1 and accordFigure 10. PDF of averager output for the worst case logic-1 ( h1*3 )

Figure 11. PDF of 4LRA output (hTH)

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On the Reliability of Post-CMOS and SET Systems

ingly PE0 and PE1 in the case of the analyzed NAND gate. PE1 is significantly larger than PE0 mostly because there are three input vectors that produce a logic-1 and only one that produces a logic-0. This fact can be exploited to reduce the probability of failure by setting an optimal fixed threshold in 4LRA. In (2), the independence of the random variables represented with hmax0 and hmin1 is explicitly assumed. The expression of the fault-tolerance of 4LRA is derived in the following. In order to do so, we observe how PDFs are transformed in every layer. The analyzed 4LRA has three redundant units in logic layer, similar to triple modular redundancy. The PDFs of each output value after the averaging operation (marked as hi *3 , i Î {1,...2n } ) corresponding to an input vector are the PDFs of a sum of three identical and independent random variables (hi), which is represented by the 3-fold convolution (Papoulis & Pillai, 2002) in (3), where t represents the output value. In order to simplify the calculation without losing generality we suppose that the averager is not performing rescaling of the output and that its output value range is between 0 and 3Vdd (averager with three inputs ranging between 0 and Vdd). ∞

hi*3 (t ) = hi * hi * hi =

∞ 



∫  ∫ h (x )h (y − x )dx  h (t − y )dy i

−∞ −∞

i

i

(4)

We assume that hi *3 are mutually independent for each i Î {1,..., 2n } , as they are the result of convolution of independent random variables. Since hi are not mutually independent, mutual independence of hi *3 can be considered just as an approximation that allows further analytical developments. However, it will be shown in the results at the end of this Section that an error introduced by this approximation is acceptable. Let Yi *3 , i Î {1,..., 2n } be random variables that

124

correspond to PDFs hi *3 . Similarly to previous paragraph, Yi *3 can be divided into two sets, H1 that corresponds to logic-1 outputs, and H0 that correspond to logic-0 outputs. The worst case logic-1 PDF that corresponds to the minimum of *3 , and the worst case logic-0 H1 is marked hmin 1 PDF that corresponds to the maximum of H0 is *3 . Following the derivation of the marked hmax 0 PDF of a maximum (minimum) of two independent random variables (Ibe, 2005; Papoulis & *3 *3 and hmin can be obtained as: Pillai, 2002), hmax 0 1 t   h (t ) = ∑ h (t ) ∏  ∫ h j*3 (v )dv  −∞  j ∈H 0 , j ≠i  i ∈H 0 . ∞    *3 *3 *3 hmin 1(t ) = ∑ hi (t ) ∏  ∫ h j (v)dv    j ∈H 1 , j ≠i  i ∈H 1 t *3 max 0

*3 i

(5) The corresponding PDF for logic-0 and logic-1 *3 (h and hmin ) are represented in Figure 9 and 1 10. Due to the convolution operation, and the non-monotonicity (existence of peaks) of initial PDFs some pronounced local maxima are observed. In the general case of redundancy factor R, the function becomes R-fold convolution. Rfold convolution for high redundancy factors (in practice R>20) converges to a normal distribution with the same mean, and the variance that is R times smaller than the initial PDF’s variance (5), according to central limit theorem (Papoulis & Pillai, 2002). *3 max 0

1 R  1 R  σ2 E  ∑ X i  = µ, Var  ∑ X i  =  R i =1   R i =1  R (6) In (6), X1, …,Xn represent independent random variables with a mean µ, and a variance σ2, and whose PDFs are hmax0 and hmin1. Taking into account that only outputs above the threshold (VTH) are correct, we mark

On the Reliability of Post-CMOS and SET Systems

3

PE*03 =

∫h

*3 max 0

∫h

*3 min 1

(t )dt, and

VTH VTH

PE*13 =

(7)

(t )dt,

0

the probabilities of a faulty output of the averager for a logic-0 and a logic-1 (Fig. 9 and 10), respectively. Numerical values obtained from (7) in expression (8) are used to acquire the probability of failure of the averager (PE_AVG). PE _ AVG = PE*03 + PE*13 − PE*03PE*13

(8)

In (8) the independence of the random variables *3 *3 and hmin is explicitly represented with hmax 0 1 assumed. The threshold in the fourth layer has been considered to be fixed in the developments presented to this stage. However, we differ two cases: 1) VTH=Vdd/2 and 2) VTH=Vopt, where Vopt is the optimal threshold voltage chosen to minimize the probability of failure (also marked in Figure 9 and 10). The numerical solution of the equation dPE _ AVG / DVTH = 0 yields Vopt. These two cases are addressed in the following as AVG and AVG-opt respectively. An adaptable threshold in 4LRA, that can have a different value for any fault pattern, can correct the output of the averaging layer (Figure 3) if the worst case value for logic-1

at the output of the averaging layer is higher than the worst case logic-0, hence the difference between the random variables for worst case logic-1 *3 and a worst case logic-0 (whose PDFs are hmax 0 *3 and hmin ) is positive. The PDF of a difference of 1 two independent random variables is given in (9) and depicted in Figure 11 (Papoulis & Pillai, 2002). ∞ *3 *3 hTH (t ) = hmin (t ) * hmax (−t ) = 1 0

∫h

*3 min 1

*3 (x )hmax (x − t )dx 0

−∞

(9)

hTH is defined in the range [-3Vdd, 3Vdd] and the probability of failure of 4LRA (PE_4LRA) uses the values of hTH for negative random variables difference (the range [-3Vdd, 0]). PE_4LRA is expressed in (10) and is also illustrated in Fig. 11. 0

PE _ 4LRA =

∫h

TH

(t )dt

(10)

−1

PDFs acquired from the MC tool are plotted in Figure 7 through Figure 11. The standard deviation of variations is 10%. Applying (1)-(9), different probabilities are calculated and summarized in Table 1 (denoted as “calculated” in Table 1), for three values of standard deviation of variations (8%, 10% and 12%). Also, results from MC simulations are evaluated in each itera-

Table 1. Probabilities of Error (PE) of Different Architectures €€€€€€€€€€σ of variations €€€€€€€€€€calculated €€€€€€€€€€simulated €€€€€€€€€€calculated €€€€€€€€€€simulated €€€€€€€€€€calculated €€€€€€€€€€simulated

8% 10% 12%

€€€€€€€€€€PE for gate

€€€€€€€€€€PE for AVG

€€€€€€€€€€PE for AVGopt

€€€€€€€€€€PE for 4LRA

0.326200

0.037346

0.003090

0.000024

0.332000

0.036400

0.002800

0.000000

0.408047

0.076582

0.020463

0.000495

0.408900

0.077100

0.019800

0.000300

0.496637

0.143108

0.050493

0.005001

0.509700

0.130700

0.048300

0.004300

125

On the Reliability of Post-CMOS and SET Systems

tion, directly applying averaging and thresholding and are given for the purpose of verification (denoted as “simulated” in Table 1). Small values (<10%) of the relative error between calculated and simulated results for gate probability of error confirm the hypothesis of independence between hmax0 and hmin1. Similarly small values of the relative error between calculated and simulated results for probability of error of AVG architecture confirm the hypothesis of mutual independence of hi *3 , i Î {1,...2n } . Higher values of relative error between calculated and simulated results for probability of error of 4LRA are due to insufficient sample size for MC simulations (10’000 iterations). The simulated value was assumed to be equal to 0 when the probability of error was smaller than 10-4.

COMPARISON OF DIFFERENT FAULT-TOLERANT TECHNIQUES After the initial simulations verifying the statistical method for the analysis, a second set of simulations was carried out to acquire PDFs of different gates for distinct error densities (variation values). In the following analysis, the reliability

of AVG, AVG-opt and 4LRA architectures has been evaluated using the approach described in previous Section and compared to the reliability of a single gate. In all evaluations, a fault-free averaging/thresholding is assumed. The gates used for comparison are a 2-input NAND (described in previous Section) and a full adder (FA). A well known implementation of FA using inverting MAJ-3 gates (Iwamura et al., 1998; Tucker, 1992) is used (Figure 12). The applied circuit parameters are: CG=1 aF for each gate capacitance, Cj=1 aF for all the junction capacitances, Cb=5.5 aF for all the bulk capacitances and CL=16 aF for the load capacitance. The supply voltage is Vdd=10 mV. The applied standard deviation of the variability ranges from 1% up to 15%, and differs for different circuits. In Figure 13 through 15, the probability failure of different fault-tolerant realizations is plotted versus the standard deviation of the variability for NAND gate, Cout output of FA and S output of FA respectively. An advantage of the 4LRA can be observed in terms of the failure probability. In the case of the NAND gate, the AVG-opt configuration shows significantly better results compared to AVG. The fault tolerance capability of AVG-opt is compa-

Figure 12. (a) MAJ based SET FA (MAJ-SET). (b) MAJ gate based on SET inverter (Iwamura et al., 1998; Tucker, 1992)

126

On the Reliability of Post-CMOS and SET Systems

Figure 13. Probability of failure of a NAND gate for different fault-tolerant architectures plotted versus the standard deviation of variations

Figure 15. Probability of failure of Sum output of FA gate for different fault-tolerant architectures plotted versus the standard deviation of variations

rable to 4LRA in this case. Considering the low overhead of AVG-opt realization compared to 4LRA promotes AVG-opt as the best choice. However, in the case of a FA, where the output values for logic-0 and logic-1 are equally probable, the advantage of AVG-opt vs. AVG is not prominent. The overall improvement in reliability of the analyzed fault-tolerant techniques compared to a non-reliable gate is in the range of 100-10000. Also notice that for the same standard deviation of variations, the probability of failure

is increasing and improvement in reliability is decreasing for more complex gates (order of complexity: NAND, Cout output of FA, S output of FA). In Figure 16, the probability of failure for AVG and 4LRA fault-tolerant realization is depicted with respect to different redundancy factors (R=3 and R=5) for S output of FA. The values gave been calculated using the generalized approach from previous Section. It can be noticed that reliability

Figure 14. Probability of failure of Cout output of FA for different fault-tolerant architectures plotted versus the standard deviation of variations

Figure 16. Probability of failure of Sum output of FA gate for AVG and 4LRA realizations and different redundancy factors plotted versus the standard deviation of variations

127

On the Reliability of Post-CMOS and SET Systems

significantly improves with higher redundancy factors (R=5) and that already for R=5 the AVG realization becomes more reliable than the 4LRA realization for R=3, at higher level of variations.

CONCLUSION The averaging and thresholding fault-tolerant technique which has been thoroughly studied for CMOS circuits is presented to increase the reliability of SET-based designs. The theoretical framework based on probability density functions extracted in each layer through the output is presented for the construction of SET faulttolerant gates. The superiority of the four-layer architecture in terms of fault-tolerance over other standard methods is demonstrated, using small clusters such as a NAND Boolean gate and a full-adder cell. The proposed method proves to be specifically efficient when the adaptable threshold is used. The four-layer architecture has proven its fault-tolerance efficiency in large CMOS designs; applying the technique to large SET-based systems and assessing its benefits is the next step to be taken.

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On the Reliability of Post-CMOS and SET Systems

Iwamura, H., Akazawa, M., & Amemiya, Y. (1998). Single-electron majority logic circuits. IEICE Trans. Electronics. E (Norwalk, Conn.), 81-C, 42–48. Kasai, S., & Hasegawa, H. (2002). GaAs and InGaAs single electron hexagonal nanowire circuits based on binary decision diagram logic architecture. Physica E, Low-Dimensional Systems and Nanostructures, 13(2-4), 925–929. doi:10.1016/ S1386-9477(02)00236-9 Likharev, K. K. (1999). Single-electron devices and their applications. Proceedings of the IEEE, 87(24), 606–632. doi:10.1109/5.752518 Mahapatra, S., Ionescu, A. M., & Banerjee, K. (2002). A quasi-analytical SET model for few electron circuit simulation. IEEE Electron Device Letters, 23(6), 366–368. doi:10.1109/ LED.2002.1004237 Mahapatra, S., Pott, V., Ecoffey, S., Schmid, A., Wasshuber, C., Banerjee, K., et al. (2003). SETMOS: A novel true hybrid SET-CMOS high current coulomb blockade oscillation cell for future nano-scale analog ICs. In IEEE International Electron Devices Meeting (pp. 29.7.1-29.7.4). Piscataway, NJ: IEEE. Martorell, F., Cotofana, S., & Rubio, A. (2007). Fault tolerant structures for nanoscale gates. In Proc. IEEE-NANO (pp. 605-610). Piscataway, NJ: IEEE. Martorell, F., Cotofana, S., & Rubio, A. (2008). An analysis of internal parameter variations effects on nanoscaled gates. IEEE Transactions on Nanotechnology, 7, 24–33. doi:10.1109/ TNANO.2007.913429 Martorell, F., Rubio, A., & Cotofana, S. (2005). Analysis of the noise and parameter variationstolerance of the averaging cell. In Proc. Int. Work. on Design and Test of Defect-Tolerant Nanoscale Architectures (NANOARCH) (pp. 1–6). Piscataway, NJ: IEEE.

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Chapter 8

Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs Takuya Kaizawa Hokkaido University, Japan

Yukinori Ono NTT Corporation, Japan

Mingyu Jo Hokkaido University, Japan

Hiroshi Inokawa Shizuoka University, Japan

Masashi Arita Hokkaido University, Japan

Yasuo Takahashi Hokkaido University, Japan

Akira Fujiwara NTT Corporation, Japan

Jung-Bum Choi Chungbuk National University, Korea

Kenji Yamazaki NTT Corporation, Japan

ABSTRACT A highly functional Si nanodot array device that operates by means of single-electron effects was experimentally demonstrated. The device features many input gates, and many outputs can be attached. A nanodot array device with three input gates and two output terminals was fabricated on a silicon-oninsulator wafer using conventional Si MOS processes. Its feasibility was demonstrated by its operation as both a half adder and a full adder when the operation voltage was carefully selected.

INTRODUCTION Recent progress in large-scale integrated (LSI) circuitry has been achieved by using metal-oxide semiconductor (MOS) field-effect transistors (FETs). Although MOSFETs are small and enable

high performance, such as high current drivability, their integration into small chips is limited due to the huge power dissipation that results from high levels of integration (Chin & McAlister, 2005). In addition, their extremely small feature size (< 10 nm) results in various nanometer-scale fluctua-

DOI: 10.4018/978-1-60960-186-7.ch008 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

tions in, for example, the surface and/or interface roughness, the impurity distribution, and the line edge roughness (Fukutome et al., 2006; Roy et al., 2006; Skotnicki et al., 2005). Controlling these fluctuations is quite difficult, so scaling down Si MOSFETs is problematic. Single-electron devices (SEDs) are an attractive alternative for future LSI circuits because of their small size and very low power consumption. In contrast to MOSFETs, they can operate at low power because they can control the flow of a very small number of electrons (Grabert & Devoret, 1992; Kastner, 1993; Likharev, 1999; Ono et al., 2005; Takahashi et al., 2002; Wasshuber et al., 1998). However, this results in low current drivability, which generally makes it difficult to send the output signal to the next stage of the circuit. One way to overcome this is to use multiple-valued logic with cascode MOSFETs (Inokawa et al., 2001, 2002). This enables the device to output a voltage usable in conventional CMOS circuits. Many kinds of highly functional devices using multiple valued operations have been demonstrated (Degawa et al., 2004; Inokawa et al., 2003, 2004). The power consumption in the complicated circuits of these devices is reduced due to the use of multiple values. However, the reduction level is limited because the circuits also have to output high current to send the signal to the next logic stage. Ono et al (2002) proposed multibit adders in which the drain terminal of one single-electron transistor (SET) is directly connected to the source terminal of the next SET, and only the inputs and outputs are connected by wiring. This reduces the total wiring length in SED circuits although the capacitances of the source and drain terminals should be charged and then discharged on the basis of the signal transfer. In short, to achieve low power consumption, as many logic and/or arithmetic calculations as possible should be done in an SED network in which single-electron islands or SETs are directly connected to one another without wiring. A possible structure for this is the nanodot array. Moreover, with a nanodot

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array device, the nanodots can inherently couple with many input gates (Takahashi et al., 2002). Size fluctuation is another challenge, not only in MOSFETs but also in SEDs. It is particularly challenging for SETs because the island size for practical SEDs will likely be smaller than the gate length of MOSFETs. Consequently, systems that can tolerate size fluctuation are needed. A previously proposed flexible nanodot-array device acts as a logic-function selectable device (Kaizawa et al., 2006). We have now obtained higher functionality by increasing the number of outputs and inputs. In addition, as a feasibility study, we fabricated a small nanodot array device with three input gates and two output terminals on a silicon-on-insulator (SOI) wafer and experimentally demonstrated its elementary characteristics as a half-adder and as a full adder.

DEVICE CONCEPT Nanodot arrays should be the most effective structure for achieving low power consumption if we can implement high functionality. The larger the nanodot array, the more suitable it is for low power operation because the wiring is reduced more. Consequently, our ultimate goal is to build a very large nanodot array on which many input gates are attached so as to couple capacitively with many nanodots. It should also have many current output terminals because important, highly functional circuits, such as multibit adder circuits, generally need multiple outputs as well as multiple inputs. Figures 1(a) and (b) show one possible structure for a nanodot array that has many gates and terminals. As illustrated in the diagrams, we can connect each output terminal to a different nanodot of the array. Since each gate couples capacitively with many dots, the voltage applied to each gate changes the electrical potential of the underlying dots, which induces current oscillation as a function of the gate voltage. However, the size

Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

Figure 1. Schematic top view (a) and cross section (b) of nanodot array device consisting of many randomly distributed nanodots. Top gate, which overlays entire region of array, is not shown in top view for clarity. Application image of multiinput and multi-output nanodot array device (c). Some gate electrodes are used as control gates to define logic function of device. Logic function can be changed by using control-gate voltages read from memory.

fluctuation of nanodots makes it difficult to fabricate large SED circuits that operate as designed. To overcome this problem so that we can achieve highly functional, low-power nanodot arrays, we

use some input gates as control gates to define the required logic functions. Figure 1(c) shows a conceptual image of a device in which some gate electrodes are used as control gates. Since it is difficult to control the precise size and position of the dots and small gates, we cannot predict the actual characteristics of the device. However, if the nanodots are fabricated with a size close to the designed size, current oscillations in which the periods and phases are slightly distributed are achieved as a function of the input- and control-gate voltages. Since the oscillation period and phase change in accordance with the input- and control-gate voltages, we can identify a combination of control-gate voltages that produces a required logic function if the array size is appropriately large. After the voltages are identified, they should be stored into nonvolatile memory. Then another series of control gate voltages for another logic function should be identified and stored. Once several series of control-gate voltages corresponding to various logic functions are identified and stored, the device can be used as a multifunctional device with functions selected by using the stored control-gate voltages. To demonstrate the feasibility of this device, we fabricated a small nanodot array with a two-by-two array at the center. Figure 2 shows a schematic top view. It consisted of seven nanodots connected to each other by tunnel capacitors. It had three gates, two lower gates and one top gate, and three current terminals, one used as a drain and the other two as a source. It thus had two output current terminals. Since each one was connected to a different nanodot, the two outputs were almost independent of each other. The narrow lower gates were coupled capacitively to nanodots underneath, resulting in current oscillation as a function of the gate voltage. The top gate was coupled to all dots in the array through the space between the two lower gates, so the potential of all dots could be changed at once.

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Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

Figure 2. Schematic top view of fabricated nanodot array device. Top gate, which overlays entire region of array, is not shown for clarity.

DEVICE FABRICATION The nanodot array was fabricated on an SOI wafer by using the pattern-dependent oxidation (PADOX) process (Ono et al., 2000; Takahashi et al., 1995, 1996, 1999), which is compatible with the conventional CMOS process. Scanning electron microscope (SEM) images of the fabricated device are shown in Figure 3. The array of nanowires connecting the wide Si area shown in Figure 3(a) was formed on the SOI substrate by electron-beam lithography and dry etching. The initial wire width was ~50 nm, the length was 120

nm, and the height, which corresponds to the thickness of the SOI layer, was 25 nm. The PADOX process in dry oxygen at 1000ºC converted each Si wire into a Si nanodot with tunnel barriers at both ends. This resulted in an Si nanodot array in which the nanodots were connected to each other by tunnel barriers. After performing the PADOX process, we attached narrow lower gates made of 170-nm-thick P-doped polycrystalline Si on the wires by electron-beam lithography and electron cyclotron resonance etching (Figure 3(b)). The overlay accuracy of the electron-beam lithography was better than 10 nm. Then, a 50-nm-thick SiO2 interlayer was deposited, and a wide polycrystalline Si top gate was formed (Nishiguchi et al., 2006). The top gate covered the entire area shown in Figure 3(b). The PADOX process forms a nanodot in the middle of the nanowire with tunnel barriers at both ends. This means that tunnel barriers are formed at the intersections of wires. As a result, the device shown in Figure 3 should act as the nanodot array device model shown in Figure 2. A simple SET was also fabricated on the same SOI wafer to confirm that the PADOX process had been performed correctly. We fabricated a 44-nm-wide, 50-nm-long, and 25-nm-high onedimensional wire that should act as a SET with only one island (Takahashi et al., 1996). This

Figure 3. SEM images taken before formation of two lower gates, Vg1 and Vg2, (a), and after formation of input gates (b). Top gate covers entire area shown in figure (not shown here).

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Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

SET has a wide top gate but does not have any lower gates.

RESULTS AND DISCUSSION The electrical characteristics of the fabricated SEDs were measured at 8 K using a low-temperature probe station and an Agilent 4156C precision semiconductor parameter analyzer. To confirm fabrication of an SED by the PADOX process, we first measured a device consisting of only one wire and that acted as an SET. The measured drain current Id versus gate voltage Vg characteristics are shown in Figure 4(a). Current oscillations due to Coulomb blockade are clearly evident. Figure 4(b) shows a contour plot of the differential conductance (∂Id/∂Vd) as a function of drain voltage Vd and gate voltage Vg. Clear Coulomb diamonds are evident. From Figure 4(b), the total capacitance, gate capacitance, source capacitance, and drain capacitance were estimated to be 7.6, 3.2, 2.7, and 1.7 aF, respectively. These results clearly show that the fabricated device had a nanodot with a diameter of ~30 nm, which is consistent with the wire size after oxidation.

Two source currents, I1 and I2, of the nanodot array device shown in Figure 3 were measured as a function of top-gate voltage Vg at 8 K. Figure 5 shows the results for four combinations of the two lower-gate voltages, Vg1 and Vg2, as a parameter. We set a “low” input voltage of 0.2 V for Vg1 and 0.3 V for Vg2, and a “high” input voltage of 0.3 V for Vg1 and 0.5 V for Vg2. Due to the Coulomb blockade effect of multidot systems, complicated current oscillations were observed. The oscillation shape and phase drastically changed depending on the combination of the lower-gate voltages. We used two lower gates as input gates for a logic device and introduced a threshold current that determined the high and low output current levels. When we set the threshold current to 1.0 nA, the logic functions of I1 and I2 were determined as a function of top-gate voltage Vg, as shown in Figure 6. For example, I1 operated as a two-input AND gate in a Vg range of 6.75–7.20 V, operated as an XOR (exclusive OR) gate in a Vg range of 7.45–7.75 V, and changed to an XNOR (exclusive NOR) gate in a Vg range of 8.05–8.50 V. Likewise, I2 operated as an AND gate in a Vg range of 7.30–7.80 V and changed to an OR gate in a Vg range of 8.05–8.40 V. These functions are illustrated in Figure 6 as a function of the top-gate

Figure 4. Drain current Id versus gate voltage Vg characteristics at drain voltage of 2 mV (a), and a contour plot of the differential conductance (∂Id/∂Vd) as a function of drain voltage Vd and gate voltage Vg (b) of a SET fabricated by PADOX method. From the figure, gate capacitance Cg, source capacitance Cs, drain capacitance Cd, and total capacitance CΣ were calculated to be 3.2, 2.7, 1.7, and 7.6 aF, respectively.

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Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

Figure 5. Output currents I1 and I2 versus controlgate voltage characteristics of a nanodot-array device measured at drain voltage Vd of 1 mV for four combinations of Vg1 and Vg2. The “0” and “1” correspond to “low” and “high” input gate voltages.

voltage. These results indicate that we can use the top gate as a control gate for selecting the device function. When we set the top-gate (control-gate) voltage, Vg, between 7.45 and 7.75 V, the device acted as a half adder, as shown in Figure 6, because I1 operated as an XOR (sum) gate and I2 operated as an AND (carry) gate. A half adder is one of the most important circuits for arithmetic logic units and usually consists of many transistors. It is thus remarkable that the function was achieved with only one device. We can use the top gate as a 3rd input gate or as a carry input from the lower bit for a full adder. Table 1 shows the truth table for a device that operates as a full adder. As described above, when Vg was ~7.6 V, I1 operated as an XOR gate and I2 operated as an AND gate. However, when Vg is ~8.2 V, I1 operated as an XNOR gate and I2 operated as an OR gate. Thus, when we set “low” Vg (carry from the lower bit) to 7.6 V and “high” to 8.2 V, the device operated exactly

Figure 6. Two-input logic functions as a function of top-gate voltage Vg, deduced from I1 versus top-gate voltage characteristics (a) and from I2 versus top-gate voltage characteristics (b) when threshold current was set to 1.0 nA and the characteristics were as shown in Figure 5.

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Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

Table 1. Truth table for full adder device. The high input “1” for Vg1, Vg2, and Vg corresponds to 0.3, 0.5, and 8.2 V, respectively. The low input “0” for Vg1, Vg2, and Vg corresponds to 0.2, 0.3, and 7.6 V, respectively. Input voltage

Output current

Vg1 (A1)

Vg2 (B1)

Vg (Carry0)

I1 (Sum1)

I2 (Carry1)

0

0

0

0

0

1

0

0

1

0

0

1

0

1

0

1

1

0

0

1

0

0

1

1

0

1

0

1

0

1

0

1

1

0

1

1

1

1

1

1

like a full adder, as shown in Table 1. We have thus demonstrated that an array SET device with multiple inputs and outputs offers the possibility of further high performance although the operation characteristics reported here are simply an elementary demonstration. The current on/off ratio of this device seems to be too low to enable the output current level to be distinguished between “high” and “low”. In addition, the circuit uses current outputs, while conventional CMOS circuits use voltage outputs. It is difficult to obtain a large voltage output in the SED circuit itself because the drain voltage has to be set small enough to keep the Coulomb blockade condition. To overcome these two problems, we can use cascode MOSFETs, as proposed by Inokawa et al.(2002). By using a cascode circuit, we can switch the output voltage between drain voltage Vdd and 0 V. These results confirm the basic operation of a multi-input, multi-output selectable logic device. However, as shown in Figure 5, the voltage levels of the two input voltages for “high” and “low” are different. This is due to the relatively large asymmetry of the input gate capacitance and to the large dot size fluctuation caused by fluctuation during the lithographic fabrication. These properties make

the device unsuitable for practical application. They can be improved by developing and using a more sophisticated lithographic system.

CONCLUSION We described a multi-input, multi-output device in which a nanodot array plays an important role as a single-electron device. Nanodots can inherently couple multiple gates, so many current terminals can be attached to a nanodot, with each terminal connected to a different dot. Such a device should operate as a function-selectable logic device with high functionality. Its feasibility was investigated by fabricating a two-by-two Si nanodot array with three input gates (two lower and one top) and three current terminals on an SOI wafer by using the PADOX process. The device can operate as a two-output device because two of the three terminals can be used as current output terminals. When the top gate was used as a control gate and the two lower gates were used as input gates, the device operated as a half adder when a suitable control-gate voltage was applied. That is, one of the outputs was XOR, and the other was AND.

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In addition, the device operated as a full adder when the operation voltage was carefully selected.

ACKNOWLEDGMENT This work was partly supported by Grants-in-Aid from the Japan Society for the Promotion of Science (JSPS KAKENHI (18560640, 20035001) and by the Korean Ministry of Science and Technology through the Frontier 21 National Program for Tera-level Nano-Devices.

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Inokawa, H., Fujiwara, A., & Takahashi, Y. (2002). A Merged single-electron and metal-oxidesemiconductor transistor logic for interface and multiple-valued functions. Japanese Journal of Applied Physics, 41, 2566–2568. doi:10.1143/ JJAP.41.2566 Inokawa, H., Fujiwara, A., & Takahashi, Y. (2003). A multiple-valued logic and memory with combined single-electron and metal–oxide–semiconductor transistors. IEEE Transactions on Electron Devices, ED-50(2), 462–470. doi:10.1109/ TED.2002.808421 Inokawa, H., Takahashi, Y., Degawa, K., Aoki, T., & Higuchi, T. (2004). A simulation methodology for single-electron multiple-valued logics and its application to a latched parallel counter. IEICE Transactions of Electron. E (Norwalk, Conn.), 87-C, 1818–1826. Kaizawa, T., Oya, T., Arita, M., Takahashi, Y., & Choi, J.-B. (2006). Multifunctional device using nanodot array. Japanese Journal of Applied Physics, 45(6A), 5317–5321. doi:10.1143/ JJAP.45.5317 Kastner, M. A. (1993). Artificial atoms. Physics Today, 46, 24–31. doi:10.1063/1.881393 Likharev, K. K. (1999). Single-electron devices and their applications. Proceedings of the IEEE, 87(4), 606–632. doi:10.1109/5.752518 Nishiguchi, K., Fujiwara, A., Ono, Y., Inokawa, H., & Takahashi, Y. (2006). Room-temperature-operating data processing circuit based on single-electron transfer and detection with metal-oxide-semiconductor field-effect transistor technology. Applied Physics Letters, 76, 183101. doi:10.1063/1.2200475 Ono, Y., Fujiwara, A., Nishiguchi, K., Inokawa, H., & Takahashi, Y. (2005). Manipulation and detection of single electrons for future information processing. Journal of Applied Physics, 97, 031101. doi:10.1063/1.1843271

Full Adder Operation Based on Si Nanodot Array Device with Multiple Inputs and Outputs

Ono, Y., Inokawa, H., & Takahashi, Y. (2002). Binary adders of multigate single-electron transistors: specific design using pass-transistor logic. IEEE Transactions Nanotechnologies, 01, 93–99. doi:10.1109/TNANO.2002.804743

Takahashi, Y., Fujiwara, A., Nagase, M., Namatsu, H., Kurihara, K., Iwadate, K., & Murase, K. (1999). Silicon single-electron devices. International Journal of Electronics, 86, 605–639. doi:10.1080/002072199133283

Ono, Y., Takahashi, Y., Yamazaki, K., Nagase, M., Namatsu, H., Kurihara, K., & Murase, K. (2000). Fabrication method for IC-oriented Si single-electron transistors. IEEE Transactions on Electron Devices, ED-47(1), 147–153. doi:10.1109/16.817580

Takahashi, Y., Fujiwara, A., Yamazaki, K., Namatsu, H., Kurihara, K., & Murase, K. (2000). Multigate single-electron transistors and their application to an exclusive-OR gate. Applied Physics Letters, 77(5), 637–639. doi:10.1063/1.125843

Roy, G., Brown, A. R., Adamu-Lema, F., Roy, S., & Asenov, A. (2006). Simulation study of individual and combined sources of intrinsic parameter fluctuations in conventional nano-MOSFETs. IEEE Transactions on Electron Devices, 53, 3063–3070. doi:10.1109/TED.2006.885683 Skotnicki, T., Hutchby, J. A., King, T.-J., Wong, H.-S. P., & Boeuf, F. (2005). The end of CMOS scaling. IEEE Circuits and Devices Magazine, 21, 16–26. doi:10.1109/MCD.2005.1388765 Takahashi, Y., Fujiwara, A., Nagase, M., Namatsu, H., Kurihara, K., Iwadate, K., & Murase, K. (1999). Silicon single-electron devices. International Journal of Electronics, 86, 605–639. doi:10.1080/002072199133283

Takahashi, Y., Nagase, M., Namatsu, H., Kurihara, K., Iwadate, K., & Nakajima, Y. (1995). Fabrication technique for Si single-electron transistor operating at room temperature. Electronics Letters, 31(2), 136–137. doi:10.1049/el:19950082 Takahashi, Y., Namatsu, H., Kurihara, K., Iwadate, K., Nagase, M., & Murase, K. (1996). Size dependence of the characteristics of Si single-electron transistors on SIMOX substrates. IEEE Transactions on Electron Devices, ED-43(8), 213–1217. Takahashi, Y., Ono, Y., Fujiwara, A., & Inokawa, H. (2002). Silicon single-electron devices. Journal of Physics Condensed Matter, 14(39), R995– R1033. doi:10.1088/0953-8984/14/39/201 Wasshuber, C., Kosina, H., & Selberherr, S. (1998). A comparative study of single-electron memories. IEEE Transactions on Electron Devices, ED-45(11), 2365–2371. doi:10.1109/16.726659

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 58-69, copyright 2009 by IGI Publishing.

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Chapter 9

Investigation on Stochastic Resonance in Quantum Dot and its Summing Network Seiya Kasai Hokkaido University & Japan Science and Technology Agency, Japan

ABSTRACT Stochastic resonance behavior of single electrons in a quantum dot and its summing network is investigated theoretically. Dynamic behavior of the single electron in the system at finite temperature is analyzed using a master equation with a tunneling transition rate. The analytical model indicates that an input-output correlation has a peak as a function of temperature, which confirms the appearance of the stochastic resonance. The peak position and height depend on charging energy, tunnel resistance, and input signal frequency. It is also found that the correlation is enhanced by formation of a summing network integrating quantum dots in parallel. The present model quantitatively explains the stochastic resonance behaviors of the single electrons predicted by a circuit simulation (Oya, Asai, & Amemiya, 2007).

INTRODUCTION single electron devices and their integrated circuits are expected to play important roles in future ultra-small and ultra-low-power nanoelectronics. However, there are several problems preventing their practical use. The most serious one is fluctuation, such as thermal and threshold voltage fluctuations. Decreasing the size for increasing operation temperature as well as increasing inteDOI: 10.4018/978-1-60960-186-7.ch009

gration density, the device becomes very sensitive to various fluctuations. It is obvious that atomiclevel imperfection of the structure is inevitable in large-scale integrated circuits (LSIs) integrating over million devices. This means that fluctuation of device characteristics is also inevitable. At this stage, a key issue in the single electron devices and circuits is to find a way to increase robustness against fluctuation rather than to remove or suppress it. Recently, it has been pointed out that stochastic resonance (SR) serves this purpose. It is a unique phenomenon in which response to a

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Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

weak signal is enhanced by adding noise (Benzi, Sutera, & Vulpiani, 1981; Gammaitoni, Hänggi, Jung, & Marchesoni, 1998). It has been found to work in various biological systems (Douglass, Wilkens, Pantazelou, & Moss, 1993; Funke, K., Kerscher, N. J., & Wörgöter, 2007; Moss, Ward, & Sannita, 2004) and contribute to the robustness of the systems against fluctuation, even having molecular-level fine structures. The SR is also known to occur artificially in various electronic systems, such as Schmitt trigger circuits (Fauve & Heslot, 1983), pn-junction diodes (Jung & Wiesenfeld, 1997), Josephson junctions (Hibbs, Singsaas, Jacobs, Bulsara, Bekkedahl, & Moss, 1995), carbon nanotubes (Lee, Liu, Zhou, & Kosko, 2006), and nanowire field effect transistors (Kasai & Asai, 2008). Recently, Kagaya, Oya, Asai, & Amemiya (2005) and Oya, Asai, & Amemiya (2007) predicted the appearance of the SR in single electron systems and demonstrated by circuit simulation. A noteworthy indication is that the response becomes robust against temperature by forming a summing network and high correlation value is kept even when temperature far exceeds the charging energy. In addition, the system can work even with threshold variation under finite thermal fluctuation. These behaviors agree with “without tuning” nature of the stochastic resonance indicated by Collins, Chow, & Imhoff (1995). However, at present, the mechanism of the single electron SR as well as physical parameters controlling the phenomenon have not been understood yet. The purpose of this paper is to theoretically study the single electron SR in a quantum dot and its summing network. First, a system for the single electron SR is described and the behavior of the single electron is analyzed using a master equation with simple approximations. Then, the calculated results are presented and discussed. Comparison with simulation results reported by Oya, Asai, & Amemiya (2007) is also shown.

MODELING AND ANALYSIS Figure 1 shows schematics of a quantum dot and its summing network studied in this paper. The quantum dot in Figure 1(a) consists of a normal capacitor, Cn, and a tunnel capacitor, Cj. The network in Figure 1(b) integrates quantum dots in parallel. Each connects to with a common voltage input and a summing circuit. The numbers of electrons in the dots are summed up using the summing circuit and it is measured as output of the system. When the capacitors of the dot are quite small, tunneling of a single electron into the dot results in non-negligible increase of electrostatic potential. This blocks the tunneling of another electron into the dot, called Coulomb blockade. Charging energy, EC, is an energy to overcome the blockade, given by EC = q2/C, where C = Cn + Cj and q is the elemental charge. When positive voltage, V, is applied to the normal capacitance, potential of the dot decreases and the electron in the right-hand side of the tunnel capacitor is attracted to the dot. When V exceeds EC/q, the Figure 1. Schematics of (a) a quantum dot and (b) its summing network

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Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

Figure 2. Potential and number of electrons in the dot as a function of the applied voltage

electron can tunnel into the dot. Then, the potential of the dot increases by EC/q. Further increasing V, the system repeats the above process. Changes of the potential and the number of electrons in the dot are schematically shown in Figure 2. A total dot capacitance can be roughly estimated by 2 x 8ε R, with a dielectric constant ε and the size of junction R. The modern semiconductor technology can realize 10 nm-size structures. When R = 10 nm, the capacitance of 20 aF is obtained for a semiconductor dot. This results in the charging energy of 8 meV, which corresponds to 90 K in temperature. Thus, for room temperature operation of the single electron device, the dot size of a few nm or less is necessary. In this study, energy quantization due to the quantum mechanical effect is ignored for simplicity. It mainly affects on the threshold voltage in addition or removal of an electron. Usually the effect is considered an origin of the fluctuation of the threshold voltage. Here, it should be mentioned that such fluctuation should be canceled out when the SR takes place (Collins, Chow, & Imhoff, 1995). In the present system, input is V and output is the number of electrons in the dot, n(t). When V is smaller than EC/q, it is enough for the analysis to consider a limited voltage range as shown in

142

Figure 2 by un-hatched region, where n = 0 or 1. Then, the system has a double-well potential divided by a barrier of EC. This bistable potential is essential for appearance of the stochastic resonance. Here, it should be pointed out that there are two differences between the present and conventional SR systems; the output is measured by the number of electrons, not by the real-space position of the medium, and the transition rate is given by the tunneling rate, not by a Kramers rate (Gammaitoni, Hänggi, Jung, & Marchesoni, 1998). The single electron tunneling obeys a Poisson distribution, although a Boltzmann distribution is applied in general cases. In this study, the analysis carried out considering above conditions. A master equation describing the dynamic behavior of a single electron in the system is given by, ¶ p± (t ) ¶t

= ± Γ+ p− (t ) − Γ− p+ (t )  

(1)

where, p+(-)(t) is probability density of an electron into (out of) the dot, Γ+(-) is electron transition rate for tunneling into (out of) the dot. The transition rate of the single electron through a tunnel junction is given by the next equation, Γ± (V ) =

±qV − EC  ±qV − E  q 2RT C  1 − exp −  kT   1

(2)

where k is a Boltzmann constant, T is temperature, and RT is a tunnel resistance. In the present case, the random force is given by thermal fluctuation. The tunneling transition rate also decreases monotonically with increasing T. It changes slowly as compared with the Kramers rate. In order to solve Equation (1) analytically, Γ = -x/[1-exp(x)] is approximated by exp[-x•ln(e-1)] for x << 2 and

Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

Box 1.  E  E    1 1    C   C  − kT << EC 2  1 ±   qV  2 exp −  kT    kT EC   q RT   Γ± (V ) =    E    kT 1 qV   , a = EC 2 << kT  2 exp − C  1 ±  akT   akT  ln (e − 1) q RT 

(

)

(

x•exp(-x) for x >> 2, respectively. In addition, the first order of the expansion of exp(qV/kT) is considered, since the input voltage is small. Then, the next approximation formulas are obtained for the tunneling transition rate, see Box 1. To solve Equation (1), a sinusoidal input, V(t) = Vincos(Ωt), is assumed for simplicity. Then p±(t) can be obtained analytically by solving Equation (1) with Equation (3). Using p±(t) on the two-state system with n = 0 and 1, the ensemble of the number of electrons in the dot, , is given by the next equation, see Box 2, where ϕ = tan-1(Ω/2k0). From Equation (4), the output is found to exhibit a peak. The response of the system is measured by input-output correlation coefficient, C1, given by,

C1 =

(3)

)

V (t )nS (t ) −V (t ) ⋅ nS (t )

( )

V 2 (t ) − V (t )

2

(

)

nS 2 (t ) − nS (t )

2

,

(5)

where ns(t) is the sum of n(t) in all dots in the network. In order to compute n(t) from , time-dependent variance is added to the ensemble as in the next formula (Collins, Chow, Capela, & Imhoff, 1996), n (t ) ≈ n(t ) + K ⋅ kT x (t )

(6)

where ξ(t) is Gaussian white noise and K is a constant. From Equations (4) and (5), analytical formulas of the input-output correlation is derived Box 2.

n (t ) =

1 A + 2 2

1 2

 Ω  1 +    2k0 

cos (Ωt − f )

 1  E   EC 1   C  A =  −  qV , k0 = 2 exp −  (kT << EC / 2)  kT   q RT  kT EC  (4)   E   qV kT  C  , k0 = 2 exp − A = (EC / 2 << kT )   akT  akT  q RT

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Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

Box 3.  1 1  (kT << EC / 2)  2 2 2         1   K '  kT ⋅ EC Ω Ω  ⋅  1 +  1 +    1 +  2k   2k 0   N  kT − EC Vin    0    C 1 =   1 1 (EC / 2 << kT )  2 2 2           1 + aK '  kT  1 +  Ω   1 +  Ω    2k   2k    N Vin    0  0    

as follow, see Box 3, where K’ is a constant and K is put into it. Thus, only K’ is an empirical parameter for calculating C1.

RESULTS AND DISCUSSION Calculated curves for N = 1 using Equations (7) are shown in Figure 3. In this figure, Cn = Cj = 10 aF, RT = 1 MΩ, Ω = 100 MHz, and K’ = 2.5 are assumed. The charging energy is 8 meV and corresponding temperature is 90 K. Equation (7a) shows a sharp and narrow peak in low temperatures. On the other hand, Equation (7b) shows a wide peak and the tail prolonged over 300 K. The two curves are crossed around T = EC/2k = 46 K. An extrapolated curve from the two analytical curves is shown in Figure 3 by a dotted line, which appears a single peak. This result confirms the appearance of the stochastic resonance in the single electron system. The peak position is at 16 K, which is smaller than the charging energy of 90 K. From Figure 3, it is found that Equation (7a) determines peak height and position in the case of the assumed parameters. Equations (7) also show that the response of the system depends on not only the charging energy but also the tunnel resistance. The peak shifts to higher temperature with increasing RT. This behavior is understood by the suppression of the tunnel event in low

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(7a)

(7b)

temperature region due to a larger tunnel barrier of the tunnel junction. Figure 4 shows input-output correlation curves of the network system for various numbers of quantum dots, N. Increasing N, the peak becomes high and wide. When N = 500, the peak height exceeds 0.9. Tail of the peak in the high temperature region also increases and C1 = 0.37 is obtained even at room temperature. Therefore, formation of the summing network is effective for the single electron system to realize robustness against thermal fluctuation. To see the validity of the present model and analysis, peak position and height are compared with those from the circuit simulation (Oya, Asai, & Amemiya, 2007). The results are shown in Figure 3. Calculated input-output correlation as a function of temperature for N = 1

Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

Figure 4. Input-output correlation for various N

Figure 5. The simulation data could be quantitatively reproduced by the present model for both peak height and position, assuming K’ = 2.5. This result verifies the validity of the analysis. However, the tail of the peak in the high temperature region is lower than that from the simulation. The origin of such discrepancy has not been clarified yet. One of the possible reasons is in the approximation description of the time-variance of the output by Equation (6). The fluctuation of the single electron might be expressed by a maximallength sequence rather than the white noise. The difference in the input waveform between the Figure 5. Peak height and position of the correlation values from the simulation (Oya, Asai, & Amemiya, 2007) and the present analysis

simulation and the present analysis might also cause the discrepancy of the correlation. The simulation used a pulse train for the input signal, although the present analysis assumes a sinusoidal wave. It has been indicated that binarization of input and output signals tends to increase the input-output correlation in the stochastic resonance phenomenon (T. Asai, private communication, 2007). Thus, the differences pointed out above should result in the better response in the simulation than the present analysis. Figure 6 shows the peak height and position of the input-output correlation as a function of the charging energy, EC. With increasing EC, the peak position, Tpeak, shifts to higher temperature. Figure 6. Charging energy dependence of correlation peak height and position

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Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

It is almost proportional to the charging energy, that is Tpeak ≈ α EC/k. The data in Figure 6 indicates α = 0.15. When EC equals to the thermal energy at room temperature, Tpeak becomes 48 K. On the other hand, the peak height decreases with increasing the charging energy. There is a trade-off between peak height and position. However, the peak tail in the high temperature region is almost unchanged even when the charging energy changes. Equations (7) also show that the SR peak depends on the tunnel resistance, RT. With increasing RT, the peak height decreases and the position shifts to higher temperature. High tunnel resistance increases EC dependence of the peak position, resulting in large α. The response of the system also depends on the input frequency. The inputoutput correlation decreases quickly when the input frequency exceeds the characteristic frequency, f0 = π/k0. f0 is increased with decreasing the tunnel resistance. Above results show that the SR response of the present system can be controlled by changing EC and RT. Finally, the possibility of experimental observation of the single electron stochastic resonance is mentioned. At present, no experimental study on the single electron SR has been reported yet. However, real-time single-electron counting in a quantum dot has been recently demonstrated using a system integrating a high-sensitive charge detector on a chip, such as a single electron transistor (Bylander, Duty, & Delsing, 2005), a quantum point contact (Fujisawa, Hayashi, Tomita, & Hirayama, 2006; Gustavsson, Leturecq, Simovic, Schleser, Ihn, Studerus, Ensslin, Driscoll, & Gossard, 2006; Reilly, Marcus, Hanson, & Gossard, 2007), or a nanowire FET (Nishiguch, Fujiwara, Ono, Inokawa, & Takahashi, 2006). Wide bandwidth of 20 MHz has been achieved (Cassidy, Dzurak, Clark, Petersson, Farrer, Ritchie, & Smith, 2007). Equations (7) indicate that input frequency much lower than 100 MHz is also possible for observation of the phenomena. Thus, experimental demonstration of the single electron SR and the verification of the present model are now

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possible using such single electron counting system.

CONCLUSION The single electron stochastic resonance in a quantum dot and its summing network was theoretically investigated. The behavior of a single electron was analyzed analytically using a master equation with a tunneling transition rate. The obtained model showed that an inputoutput correlation had a peak as a function of temperature, which confirmed the appearance of the stochastic resonance. The peak position and height were found to depend on charging energy, tunnel resistance, and input signal frequency. The analysis also showed the correlation is enhanced by formation of a summing network integrating quantum dots. The present model quantitatively explained the stochastic resonance behaviors of single electrons predicted by circuit simulation (Oya, Asai, & Amemiya, 2007). The possibility of experimental observation of the phenomena was also mentioned.

ACKNOWLEDGMENT The author thanks Dr. Tetsuya Asai for his valuable discussions and comments. He also thanks Professor Takashi Fukui and Professor Tamotsu Hashizume for their continuous supports. This work is partly supported by Exploratory Research (#18651074) from MEXT, Japan.

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Investigation on Stochastic Resonance in Quantum Dot and its Summing Network

Bylander, J., Duty, T., & Delsing, P. (2005). Current measurement by real-time counting of single electrons. Nature, 434, 361–364. doi:10.1038/ nature03375

Gammaitoni, L., Hänggi, P., Jung, P., & Marchesoni, F. (1998). Stochastic resonance. Reviews of Modern Physics, 70, 223–287. doi:10.1103/ RevModPhys.70.223

Cassidy, M. C., Dzurak, A. S., Clark, R. G., Petersson, K. D., Farrer, I., Ritchie, D. A., & Smith C. G. (2007) Single shot charge detection using a radio-frequency quantum point contact. Applied Physics Letters, 91, 222104.1-3.

Gustavsson, S., Leturecq, R., Simovic, B., Schleser, R., Ihn, T., Studerus, P., Ensslin, K., Driscoll, D. C., & Gossard, A. C. (2006). Counting Statistics of Single Electron Transport in a Quantum Dot. Physical Review Letters, 96, 076605.1-4.

Collins, J. J., Chow, C. C., Capela, A. C., & Imhoff, T. T. (1996). Aperiodic stochastic resonance. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54, 5575–5584. doi:10.1103/PhysRevE.54.5575

Harry, J. D., Niemi, J. B., Priplata, A. A., & Collins, J. J. (2005). Balancing act. IEEE Spectrum, 42, 36–41. doi:10.1109/MSPEC.2005.1413729

Collins, J. J., Chow, C. C., & Imhoff, T. T. (1995). Stochastic resonance without tuning. Nature, 376, 236–238. doi:10.1038/376236a0 Douglass, J. K., Wilkens, L., Pantazelou, E., & Moss, F. (1993). Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature, 365, 337–340. doi:10.1038/365337a0 Duan, F., & Abbott, D. (2007). Binary modulated signal detection in a bistable receiver with stochastic resonance. Physica A, 376, 173–190. doi:10.1016/j.physa.2006.10.046 Fauve, S., & Heslot, F. (1983). Stochastic resonance in a bistable system. Physics Letters, 97A, 5–7. Fujisawa, T., Hayashi, T., Tomita, R., & Hirayama, Y. (2006). Bidirectional Counting of Single Electrons. Science, 312, 1634–1636. doi:10.1126/ science.1126788 Funke, K., Kerscher, N. J., & Wörgöter, F. (2007). Noise-improved signal detection in cat primary visual cortex via a well-balanced stochastic resonance-like procedure. The European Journal of Neuroscience, 26, 1322–1332. doi:10.1111/ j.1460-9568.2007.05735.x

Hibbs, A. D., Singsaas, A. L., Jacobs, E. W., Bulsara, A. R., Bekkedahl, J. J., & Moss, F. (1995). Stochastic resonance in a superconducting loop with a Josephson barrier. Journal of Applied Physics, 77, 2582–2590. doi:10.1063/1.358720 Jung, P., & Wiesenfeld, K. (1997). Too quiet to hear a whisper. Nature, 23, 291. doi:10.1038/385291a0 Kagaya, R., Oya, T., Asai, T., & Amemiya, Y. (2005, October) Stochastic resonance in an ensemble of single-electron neuromorphic devices and its application to competitive neural networks, Paper presented at Proceedings of the 2005 International Symposium on Nonlinear Theory and its Applications, Bruges, Belgium. Kasai, S. & Asai, T. (2008). Stochastic Resonance in Schottky Wrap Gate-controlled GaAs Nanowire Field-Effect Transistors and Their Networks. Applied Physics Express, 1, 083001.1-3. Lee, I., Liu, X., Zhou, C., & Kosko, B. (2006). Noise enhanced detection of subthreshold signals with carbon nanotubes. IEEE Transactions on Nanotechnology, 5, 613–627. doi:10.1109/ TNANO.2006.883476

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Moss, F., Ward, L. M., & Sannita, W. G. (2004). Stochastic resonance and sensory information processing: A tutorial and review of applications. Clinical Neurophysiology, 115, 267–281. doi:10.1016/j.clinph.2003.09.014 Nishiguch, K., Fujiwara, A, Ono, Y., Inokawa, H., & Takahashi, Y. (2006). Room-temperatureoperating data processing circuit based on singleelectron transfer and detection with metal-oxidesemiconductor field-effect transistor technology. Applied Physics Letters, 88, 183101.1-3. Oya, T., Asai, T., & Amemiya, Y. (2007). Stochastic resonance in an ensemble of single-electron neuromorphic devices and its application to competitive neural networks. Chaos, Solitons, and Fractals, 32, 855–861. doi:10.1016/j.chaos.2005.11.027

Reilly, D. J., Marcus, C. M., Hanson, M. P., & Gossard, A. C. (2007). Fast single-charge sensing with a rf quantum point contact. Applied Physics Letters, 91, 162101.1-3. Russell, D. F., Wilkens, L. A., & Moss, F. (1999). Use of behavioural stochastic resonance by paddle fish for feeding. Nature, 402, 291–294. doi:10.1038/46279 Simonotto, E., Riani, M., Seife, C., Roberts, M., Twitty, J., & Moss, F. (1997). Visual Perception of Stochastic Resonance. Physical Review Letters, 78, 1186–1189. doi:10.1103/PhysRevLett.78.1186

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 70-79, copyright 2009 by IGI Publishing.

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Chapter 10

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation Andrew Kilinga Kikombo Hokkaido University, Japan Tetsuya Asai Hokkaido University, Japan Takahide Oya Yokohama National University, Japan Alexandre Schmid Swiss Federal Institute of Technology (EPFL), Switzerland Yusuf Leblebici Swiss Federal Institute of Technology (EPFL), Switzerland Yoshihito Amemiya Hokkaido University, Japan

ABSTRACT We propose a bio-inspired circuit performing pulse-density modulation with single-electron devices. The proposed circuit consists of three single-electron neuronal units, receiving the same input and are connected to a common output. The output is inhibitorily fedback to the three neuronal circuits through a capacitive coupling. The circuit performance was evaluated through Monte-Carlo based computer simulations. We demonstrated that the proposed circuit possesses noise-shaping characteristics, where signal and noises are separated into low and high frequency bands respectively. This significantly improved the signal-to-noise ratio (SNR) by 4.34 dB in the coupled network, as compared to the uncoupled one. The noise-shaping properties are as a result of i) the inhibitory feedback between the output and the neuronal circuits, and ii) static noises (originating from device fabrication mismatches) and dynamic noises (as a result of thermally induced random tunneling events) introduced into the network. DOI: 10.4018/978-1-60960-186-7.ch010 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

INTRODUCTION For the past 3 decades, the scaling of semiconductor devices has been the primary driving force behind improving the performance of LSI processors and systems. The decreasing feature sizes of transistors have been accompanied by dramatic increase in speed and integration densities, which have in turn led to increased and diversified functionality. This trend has been viable mainly due to guaranteed reliability in the downscaled devices even with decreasing process technologies. Reliability corresponds to high yields per die, hence low production costs (high cost efficiency), giving the circuit designer the opportunity to create reliable integrated systems with improved processing speeds, and increased functionality. However, as the physical feature sizes approach the deep sub-micron regime, process variations and undesirable internal (and or external) noises associated with nano-scale properties pose critical concerns in the future of scaling and in system system design; they dramatically reduce the reliability of electronic devices on the edge of the nano-scales (Bowman, 2002; Constantinescu, 2003; Jose, 2003; Way & Taeho, 1999). This reduced reliability is even more conspicuous as electronic device sizes are further scaled down to the nano-meter regime (Calhoun, 2008; Orshansky, 2002; Stolk, 1998). Getting rid of these nano-scale characteristics would involve introducing error-detecting circuits within the system, which leads to advanced complexity, and design tradeoffs in using high integration capacities available to the circuit designer. Some design techniques offering possible ways to mitigate the impact of within-die variations have been explored (Marculescu & Talpes, 2005; Tiwari, 2007). Other works involving introducing error-detecting circuitry in electronic systems include architectures proposed by Milor (1989) and Chatterjee (1993). Unfortunately, these approaches offer only a short term solution. The uncertainty in coming up with a long-lasting solution to these

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challenges has paved the way into a new field of the so called emerging research nano devices, which effectively utilize nano-scale characteristics in their operation. Such devices are viewed as promising blocks for creating application-specific processors, and ultra low-power systems in coming generations of LSI platforms. Such devices would include single-electron devices (Grabert & Devoret, 1992; Nakajima, 1997). Single-electron devices inherently operate with extreme low power dissipation, and provide a high integration density per unit area. Thus, they are viewed as potential building blocks for low-power, parallel-based computational applications in future LSI platforms. However, one of the major problems facing single-electron devices is that they are potentially unreliable. Their low reliability originates from two factors: i) large variations in the features of fabricated devices, hence device characteristics, and ii) sensitivity to internal and external noises. Therefore, despite all the appealing features in utilizing nano-electronic devices in future electronic systems, we have to address and solve a fundamental question; how do we build reliable systems from error-prone building devices? Improvements in fabrication technology alone cannot accomodate such enormous device failures. Therefore in designing functional electronic devices in the deep sub-micron and post-silicon era, we need to keep in mind the fact that we have to build reliable systems with unreliable (ITRS, 2005), and error-prone devices (Nikolic, 2001; Schimid & Leblebici, 2004; Goser, 1997). Thus the need to address robustness and design systems with large enough signal-to-noise ratio is inevitable (Hamed, 1997). An innovative architectural approach to increasing reliability is to exploit the internal and external noises, and the heterogeneity originating from fabrication mismatches in designing new electronic systems. For example, if we look at how living organisms code and transmit signals in their systems, we find similarities between neurons

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

(the basic elements responsible for information processing in neuronal systems) and nano-meter sized electronic devices. Neurons are sensitive to noises, operate asynchronously because of differences in their structural properties, and have large time jitters—that is, they are imperfect and unreliable (Shadelen & Newsome, 1998; Shint, 1993; Softky, 1993)—-but nevertheless they carry out information processing effectively. Similarly, nano-electronic devices (for instance single-electron devices) are sensitive to external interferences and noises, and have diverse fabrication variations in feature sizes, resulting in heterogeneity in parameters and device characteristics. Thus in creating electronic systems with such imperfect units, obtaining hints from living organisms is evidently of much importance. Such electronic systems that mimic neurological systems are referred to as neuromorphic circuits (Douglas, 1995; Mead, 1998). A number of neuromorphic circuits that operate by utilizing noises and device fabrication mismatches have been proposed. They include neuromorphic CMOS circuits utilizing device fabrication mismatches and environmental noises (Utagawa, 2007), single-electron circuits employing thermally induced stochastic resonance (SR) (see Collins (2002) for details on SR) in signal transmission (Oya, 2007), and single-electron networks performing synchrony detection (Oya, 2006). This paper explores the possibility of creating novel circuit architectures with single-electron devices, by employing environmental (dynamic) noises, and static noises originating from fabrication mismatches. The circuit architecture is inspired by information coding mechanisms in biological neural networks that convert analog input signals into spike densities (digital-pulse streams) in the time domain. This operation is also referred to 1-bit analog-to-digital conversion, and is often implemented with ΔΣ modulators (Aziz, 1996; Schreier & Temes, 2004). Such converters exhibit noise-shaping properties (see Mayr & Schueffny (2005); Shin (2001-a); Shin (2001-b))

for details on neuronal noise-shaping), separating signal and noises into low and high frequency bands respectively. A theoretical investigation of noise-shaping in neural networks is elaborated by Mar (1999). In their work, they demonstrated that noise-shaping was improved by introducing an inhibitory coupling between noisy model integrate-and-fire neurons (IFNs). In addition, the authors note that the noise-shaping properties were improved due to heterogeneity and noises introduced into the network. Inspired by their work, we propose and investigate the performance of a single-electron pulse-density modulating circuit that exhibits noise-shaping properties. This paper is organised as follows. Firstly, a brief review of pulse-density modulation in neurons is presented. Secondly, implementation of integrate-and-fire neurons, together with fundamental operation of single-electron devices is illustrated. Thirdly, a model on how to realize pulse-density modulation employing excitatory and inhibitory mechanisms is explained. This is followed by the circuit structure implementing the model with single-electron oscillators. Fourthly, the performance of the proposed circuit is investigated. The paper is summarized by noting on a possible architecture that also employs noises in achieving improved signal-to-noise ratio in singleelectron circuits and nanowire transistor networks.

A SHORT REVIEW OF PULSEDENSITY MODULATION IN NEURONS A neuron aggregates inputs from other neurons connected through synapses. The aggregated charge raises the membrane potential until it reaches a threshold, where the neuron fires generating a spike. This spike corresponds to a binary output 1. After the firing event, the membrane potential is reset to a low value, and it increases again as the neuron accepts inputs from neighboring neurons (or input signals) to repeat the same

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A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

Figure 1. (a) Pulse density modulation in neurons: analog input is converted into a pulse train whose density is proportional to the net amplitude of the input signal. (b) Fundamental structure and operation of integrate-and-fire neurons (IFNs). The IFN receives input voltages through excitatory and inhibitory synapses, and produces pulses when the net input voltage exceeds the threshold. The output pulse density (firing rate) is proportional to the net input voltage.

cycle; producing a stream of one and zero pulse trains. The spike interval (density of spikes per unit time) is proportional to the the analog input voltage i.e. the level of analog input is coded into pulse density. Thus a neuron can be considered as a 1-bit A-D converter (Cheung & Taung, 1993; Hovin, 2002) operating in the temporal domain. Figure 1(a) shows a schematic representation of analog-to-digital conversion in neurons. The output pulse density is proportional to the amplitude of the input signal. The operation of neurons is often modeled with spiking neurons such as the integrate-and-fire neurons. Figure 1(b) illustrates the fundamental operation of an integrate-and-fire (IFN) neuron. The open circles (◦) and shaded circles (•) represent excitatory and inhibitory synapses, respectively. The IFN receives input signals (voltages) through the excitatory synapses (to raise its membrane voltage) and inhibitory synapses (which decrease the membrane voltage) from adjacent neurons, to produce a spike if the summed input voltage (∑ Viex − ∑Vjin) exceeds the threshold voltage. After the IFN fires, its membrane voltage is reset to a low value, and

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the integration action resumes. The output pulse density is proportional to the net input voltage.

SINGLE-ELECTRON INTEGRATE AND FIRE NEURON A single-electron oscillator (Averin & Likharev,1986; Grabert & Devoret, 1992; kikombo, 2008, Likharev & Zorin, 1985; Oya, 2005) is used to model the operation of an integrate-andfire neuron (IFN). A single-electron oscillator (Figure 2(a)) consists of a tunneling junction (capacitance = Cj) and a high resistance R connected in series at the node (•) and biased with a positive or a negative voltage Vd. It produces self-induced relaxation oscillations if the bias voltage is higher than the tunneling threshold (Vd > e/(2Cj)) (where e is the elementary charge and kB is the Boltzmann constant). The node voltage V1 increases as the capacitance Cj is charged through the series resistance (curve AB), until it reaches the tunneling threshold e/(2Cj), at which an electron tunnels from the ground to the nanodot

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

Figure 2. Single-electron tunneling (SET) oscillator: (a) circuit configuration and (b) waveform showing oscillation of node voltage V1, as capacitor Cj is charged through resistance R (from A to B) and reset by an electron tunneling from the ground to the node (voltage drop from B to C). This sudden drop in the node voltage (BC) corresponds to a pulse output.

which aggregates input voltages (or inputs from neighboring neurons) producing a pulse when its node voltage reaches the threshold voltage (Figure 2(b)). By feeding a sinusoidal input to a singleelectron oscillator, one can adjust the probability of electron tunneling in the circuit: the tunneling rate increases as the input voltage rises above the threshold and gradually decreases to zero as the input approaches and falls below the threshold value. In other words, a single-electron oscillator converts an analog input into digital pulses. A single-electron oscillator can thus be viewed as a PDM converter, that produces a spike train (or produces zero) if the input signal exceeds (or falls below) the threshold value.

CIRCUIT IMPLEMENTATION

across the tunneling junction, resetting the node voltage to −e/(2Cj). This abrupt change in node potential (from B to C) can be referred to as a firing event. The nanodot is recharged to repeat the same cycles. Therefore, a single-electron oscillator could be viewed as an integrate-and-fire neuron,

Figure 3 shows the model of the proposed circuit, consisting of three neuronal elements. The neurons receive the same analog input through excitatory synapses (◦) and produce digital pulses toward the global inhibitor Σ (Asai; 2003). The output is fed-back to the three elements through inhibitory synapses denoted by shaded circles (•) in the net-

Figure 3. Model of pulse-density modulation circuit employing excitatory and inhibitory mechanisms. A common input is fed to the three neurons through excitatory synapses (◦), while the output is fed back to the three neurons through inhibitory synapses (•).

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A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

work. Firing in any of the neurons in the network decreases the membrane potential of the other neurons, reducing the probability of their firing. The neuronal structure in Figure 3 is implemented with single-electron oscillators that receive the same analog input. The input induces electron tunneling in the single-electron oscillators, generating pulses toward the global inhibitor. The global inhibitor Σ sums the pulses to produce a train of spikes representing tunneling (firing) events in the three neurons. Figure 4 shows the circuit configuration. Each neuron in the network is implemented with a single electron oscillator. The global inhibitor is realized by numerically summing the firing events in the network. Inhibitory synapses are implemented by coupling capacitances (C) that decrease the node voltages of all the oscillators once a pulse is released at the output. Each neuron in the network receives the same input (V (t)) raising its node voltage. Whenever any of the three single-electron oscillators reaches its threshold voltage, it fires, releasing a pulse toward the global inhibitor. The global inhibitor, through the coupling capacitors C, subtracts a Figure 4. Single-electron circuit performing pulse-density modulation. The structure consists of three single-electron oscillators, and a global inhibitor Σ. The output is fed back to all the other oscillators through the capacitive coupling C.

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certain amount of voltage from the other oscillators, suppressing them from tunneling for a certain period of time. This contributes to the distribution of output pulses. In the absence of the global inhibitor, all the neurons would fire randomly and with almost the same timing, producing a Poissonlike distribution of inter-spike intervals (ISIs). Contrally, by introducing the global inhibitor, consecutive firing events in the network are suppressed, resulting in a Gaussian-like distribution of ISIs in the coupled network.

SIMULATION RESULTS As mentioned in the introduction, the noise-shaping properties of the network of model neurons were reportedly improved by introducing dynamic and static noises (Mar, 1999). In our circuit, this was realised as follows. As noted earlier, thermal noises lead to random electron tunneling in singleelectron devices. We therefore introduced dynamical noises by tuning the temperatures in both the coupled and the uncoupled networks. Static noises were introduced only in the coupled network, by varying the values of series resistances R. In the coupled network, all the series resistances were set to 44 MΩ, whereas in the coupled network, the mean value of the three resistances was 44 MΩ, and the variance was ±12.5%. The inhibitory coupling in the coupled network was implemented with a capacitive coupling of 4 aF. The temperature was set to 0.5 K in all simulations. The performance of both the coupled and the uncoupled circuits was investigated through Monte-Carlo based computer simulations. All the circuit units in both the coupled and the uncoupled networks were fed with a sinusoidal input V (t) = V0 +Asin(2πft), where amplitude A = 2.5 mV, frequency f = 100 MHz, and bias voltage V0 = 7.85 mV. Figure 5 shows the raster plots of the firings of the network elements. The top diagrams of (a) and (b) show the random pulses for each unit in

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

Figure 5. Raster plots for firing events for uncoupled (left diagrams) and coupled (right diagrams) networks. The top diagrams show firing events for each neuron, while the bottom diagrams show summed output spike train at the global inhibitor Σ. Firing events in the uncoupled network were random and almost consecutive, whereas firing timings in the coupled network were well distributed as a result of the inhibitory coupling inhibiting concurrent events.

the uncoupled and coupled networks, respectively. The bottom diagrams in (a) and (b) show the summed output (pulse train) for all the elements in the uncoupled and coupled networks, respectively. From the diagrams, we could observe that the firing timings in the uncoupled network were random and all the neurons fired with almost the same timing. In the coupled network, however, the firing of one of the neurons inhibited the others from firing, thus reducing the probability of consecutive firing in the network. In addition, the variance in the series resistances results in variations in the time constants of the network neurons. This reduced the probability of neurons attaining the firing threshold at the same time, and thus improved the distribution of firing intervals in the network. Consequently, these two factors resulted in well distributed firing timings in the network, leading to a Gaussian-like distribution of inter-spike intervals. Figure 6 shows the ISI distribution of firing events in the whole network. The histogram for the coupled network shows a Gaussian-like distribution with an inter-spike interval of 1.65 ns at the maximum number of firing counts. The his-

togram for the uncoupled network, in contrast, shows a Poisson-like distribution. We also investigated the effect of increasing the variance in the series resistances on the standard deviation of the Gaussian-like distribution. We found that the standard deviation increases as the variation decreases below or increases above 12.5%. As the variance decreases, the probability that multiple neurons in the network reach the threshold voltage at the same time increases. This shifts the ISI at the maximum firing rate toward zero, consequently leading to a larger standard deviation of the ISI distribution. The ISI distribution can, however, be tuned by adjusting the value of the inhibitory coupling capacitance C. As the coupling strength increases, the number of neurons reaching the threshold concurrently decreases drastically. In other words, the firing timings tend to distribute evenly, resulting to a sharper Gaussianlike distribution. However, increasing the coupling strength to a relatively high value, beyond an optimal value (of 4 aF in our simulations), leads to a winner-takes-all (Cohen & Grossberg, 1983; Kaski & Kohen, 1994) operation (where only one neuron in the network produces the highest spike

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Figure 6. Histogram of inter-spike intervals (ISIs) for coupled and uncoupled networks. The uncoupled network shows a Poisson-like distribution of ISIs where the firing events in the network elements are almost concurrent. The coupled network shows a Gaussian-like distribution, as a result of distributed firing events.

rate and inhibits all the others from firing). This would be undesirable, especially in a network of fault- and defect-prone elements, where increasing the probability of correct operation requires that all the elements play a substantial part in the network operation (i.e. a winners-share-all (Fukai & Tanaka, 1997) operation, where several neurons in the network survive). Thus obtaining an ideal operation of the network requires tuning the firing rates of individual neurons though the series resistances, and also tuning the summed firing rate of the network through the capacitive coupling to obtain a winners-share-all type function. Figure 7 shows the power spectra for the coupled and uncoupled networks. The power in both cases was calculated with 25 runs averaged with a square window. From the results we can confirm that the global inhibitory coupling and the heterogeneity in series resistances collectively helped reduce the noise level in the coupled network substantially. The signal-to-noise ratio in the uncoupled network was 22.96 dB, while that in the coupled network was 27.30 dB below the cutoff frequency of 1 GHz. The harmonic distortions in the results are due to (i) the intrinsic

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firing rates of the individual neurons in the network and (ii) non-linear feedback introduced to the network. These distortions degraded the SNR characteristics. They could be decreased by setting the input signal frequency to a value much lower than the firing frequencies of individual neurons in the network. Another way of increasing the Figure 7. Power spectra of coupled and uncoupled networks. The coupled network shows a reduced noise level in the lower frequencies (signal band), improving the SNR with 4.34 dB as compared to the uncoupled network.

A Neuromorphic Single-Electron Circuit for Noise-Shaping Pulse-Density Modulation

SNR without tuning the input frequency would be by filtering the output signals, to get rid of the higher frequencies. This is often realized with digital filters in the feedback loop of ΣΔ converters (Kim, 2007).

DISCUSSIONS AND CONCLUSION To provide a basis for designing electronic circuits with mismatch-prone single-electron devices, this paper proposed and investigated the performance of a bio-inspired 1-bit analog-to-digital converter. The circuit elements are coupled to each other through a global inhibitory coupling. Through Monte-Carlo based computer simulations, we demonstrated that the presence of static and dynamic noises, and the global inhibitory coupling introduced into the circuit play an important role in improving its noise-shaping properties. The signal-to-noise ratio improved by 4.34 dB in the coupled network as compared to the un-coupled one. In the present network we extensively investigated the effect of static noises as a result of variations in series resistances, and of the inhibitory coupling in the network to noise-shaping properties. Investigating the effect of dynamic noises at higher temperatures, would also give a guideline into actual circuit design with such noise sensitive devices. From the results of these investigations, we can deduce that the performance of the circuit would improve up to the optimal value of thermal noises, and then deteriorate drastically as randomly induced firing further increases. This is as a result of decreased effect of the inhibition strength which contributes to the Gaussian-like distribution as discussed in the simulation results. Also, choosing the optimal number of neurons to use in the network would play an important role in improving its performance. As the number of neurons increases, we would obtain better resiliency toward faults and defects in the network. This would however, come at the expense of

tuning the optimal inhibitory coupling strength to realize a winners-share-all operation. Before summarizing the paper, it’s worth noting on similar promising works in achieving robust electronic systems by utilizing noises in improving signal-to-noise ratio in electronic systems. This approach has been demonstrated with single-electron devices (Oya, 2007), and nanowire transistor networks (Kasai & Asai, 2008) by some of the authors of this paper. The architectures effectively employ stochastic resonance (SR) (Collins, 2002), and demonstrate a viable novel approach to realizing robust systems in noisy environments. Stochastic resonance is a phenomenon where weak signals can be retrieved from a noisy output (Gammaitoni, 1998; Simonotto, 1997) by applying an optimal amount of random noise. Oya (2007) proposed a single-electron neural network that utilizes SR in signal transmission in neural networks, and successfuly demonstrated that using SR indeed improved the temperature performance of the circuit. Kasai and Asai (2008) experimentally investigated the performance of nanowire transistors with variations in threshold voltages and operating in a noisy experimental setup. In both cases, the effects of SR were investigated by setting the input signal to a value lower than the tunneling (firing) threshold of the network elements. By applying noises, network elements with non-zero inputs were induced to tunnel—tunneling events synchronized with the input signal to a certain quantity of noises. The authors showed that the SNR in their circuits was enhanced through partially using noises. Such innovative approaches, in addition to the neuromorphic methodology described in this paper would be indispensable in addressing reliability issues in electronic circuitry with nanoelectronic devices. From the investigation results in this paper, we can conclude that by learning from biological systems: high levels of redundancy where information processing depends on many neurons operating in parallel, controlled signal transfer through excitatory and inhibitory

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synapses, and stochastic resonance mechanisms, we could get hints on how to design circuits that perform better even in noisy environments and (or) with failure-prone electronic devices.

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This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 80-92, copyright 2009 by IGI Publishing.

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Section 3

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Chapter 11

Simple Collision-Based Chemical Logic Gates with Adaptive Computing Rita Toth University of the West of England, UK Christopher Stone University of the West of England, UK Ben de Lacy Costello University of the West of England, UK Andrew Adamatzky University of the West of England, UK Larry Bull University of the West of England, UK

ABSTRACT We present a method that is capable of implementing information transfer without any rigidly controlled architecture using the light-sensitive Belousov-Zhabotinsky (BZ) reaction system. Chemical wave fragments are injected into a subexcitable area and their collisions result in annihilation, fusion or quasielastic interactions depending on their initial positions. The fragments of excitation both pre and post collision possess a considerable freedom of movement when compared to previous implementations of information transfer in chemical systems. We propose that the collision of such wave fragments can be controlled automatically through adaptive computing. By extension, forms of unconventional computing, i.e., massively parallel non-linear computers, can be realised by such an approach. In this study we present initial results from using a simple evolutionary algorithm to design Boolean logic gates within the BZ system. DOI: 10.4018/978-1-60960-186-7.ch011 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Simple Collision-Based Chemical Logic Gates with Adaptive Computing

INTRODUCTION Previous theoretical and experimental studies have shown that reaction-diffusion chemical systems are capable of information processing (Adamatzky, 2003; 2004a; Adamatzky, de Lacy Costello & Asai, 2005; Adamatzky & de Lacy Costello, 2007; de Lacy Costello & Adamatzky, 2005). “In the strict sense of the term, reaction-diffusion systems are systems involving constituents locally transformed into each other by chemical reactions and transported in space by diffusion” (Nicolis & De Wit, 2007). In our previous work, see overview in Adamatzky et al. (2005) we demonstrated that reaction-diffusion chemical systems are capable of implementing various kinds of computational procedures. Experimental prototypes of reaction diffusion processors have been used to solve a wide range of specialised computational problems, including image processing (Adamatzky, de Lacy Costello & Ratcliffe, 2002a; Rambidi, 2003), path planning (Adamatzky & de Lacy Costello, 2003a; Steinbock, Toth & Showalter, 1995), robot navigation (Adamatzky et al., 2004b), computational geometry (Adamatzky & de Lacy Costello, 2003b), counting (Gorecki, Yoshikawa & Igarashi, 2003) and implementing memory (Motoike & Yoshikawa, 2003). Logic gates were also constructed in excitable chemical systems (de Lacy Costello & Adamatzky, 2005; Sielewiesiuka & Gorecki, 2002; Toth & Showalter, 1995) and in simple inorganic precipitation reactions (Adamatzky & de Lacy Costello, 2002b). Several researchers have created prototype reaction diffusion processors. These implement logical computation mimicking a conventional hardware type approach with wires and gates in a fixed morphology (Adamatzky & de Lacy Costello, 2002b; Motoike & Adamatzky, 2005; Sielewiesiuka & Gorecki, 2002; Steinbock, Kettunen & Showalter, 1996; Toth & Showalter, 1995). In previous works we introduced the possibility of constructing gates using a dynamical architectureless approach based on collision-based computing. We demonstrated both in compu-

tational (Adamatzky, 2004a; Adamatzky & de Lacy Costello 2007) and experimental studies (De Lacy Costello & Adamatzky, 2005) using a subexcitable BZ system that under carefully controlled conditions compact wave fragments develop in the medium, they then travel for reasonably long distances when undisturbed and their collisions can be interpreted as the implementation of logical operations. In the current experimental approach we utilise small chemical wave fragments propagating in a weakly excitable BZ medium (defined as a level set in experiment just above the critical subexcitable threshold) and study their interactions when they are collided whilst travelling on different trajectories. These interactions, i.e. collisions, produce dynamic structures which can be interpreted in terms of a computation. When two or more wave fragments collide, they may fuse, annihilate, generate new wave fragments or change their trajectories. We use the photosensitive BZ system (Gáspár, Bazsa & Beck, 1983) with tris(2,2’-bipyridine)ruthenium(II) ion catalyst immobilised on silica gel and immersed in the catalyst-free BZ solution. The excitability of the system is controlled by illumination of the reaction medium. Previous studies showed that two excitability limits define a subexcitable range where small wave fragments can form but they don’t exist for any significant period of time (Karma, 1991; Mihaliuk, Shakurai, Chirila & Showalter, 2002; Zykov & Showalter, 2005). Close to the upper excitability limit unbounded planar wave fragments (critical fingers) propagate for a relatively long time (Karma, 1991; Zykov & Showalter, 2005), providing the opportunity to use them as signals in unconventional computing methods. Evolutionary Algorithms (EA) (e.g., DeJong, 2005) are being increasingly used in the design and analysis of complex systems. Example applications include data mining, time series analysis, scheduling, process control, robotics and electronic circuit design. Such techniques can be used for the design of computational resources

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in a way that offers substantial promise for application in non-linear reaction-diffusion media computing. This is because the algorithms are almost independent of the medium in which the computation occurs. This is important in order to achieve effective non-linear media computing since an EA does not need to directly manipulate the material to facilitate learning and the task itself can be defined in an unsupervised manner. In contrast, most traditional adaptive computing algorithms use techniques that require detailed knowledge of and control over the computing substrate involved. Harding and Miller (2008) have recently described the use of an EA to design a computational system using liquid crystal and Tour et al. (2002) constructed a number of logic gates from molecular switches using an EA. Previously, we have shown that it is possible to control a BZ system via an approach which uses coevolutionary computing to create heterogeneous Cellular Automata (Stone, Toth, Adamatzky, Bull & de Lacy Costello, 2007) and an evolutionary rule-based system (Budd et al., 2006).

MATERIALS AND METHODS Sodium bromate, sodium bromide, malonic acid, sulphuric acid, tris(bipyridyl) ruthenium (II) chloride, 27% sodium silicate solution stabilized in 4.9 M sodium hydroxide were purchased from Aldrich (U.K.) and used as received unless stated otherwise. To create the gels a stock solution of the sodium silicate solution was prepared by mixing 222 mL of the purchased sodium silicate solution with 57 mL of 2 M sulphuric acid and 187 mL of deionised water (Wang, Kádár, Jung, & Showalter, 1999). Ru(bpy)3SO4 was recrystallised from the chloride salt with sulphuric acid (Gao & Försterling, 1995). Solutions for making gels were prepared by mixing 2.5 mL of the acidified silicate solution with 0.6 mL of 0.025 M Ru(bpy)3SO4 and 0.65 mL of 1.0 M sulphuric acid solution. Using capillary action,

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portions of this solution were quickly transferred into a custom-designed 25 cm long 0.3 mm deep Perspex mould covered with microscope slides. The solutions were left for 3 hours to permit complete gellation. After gellation the adherence to the Perspex mould is negligible leaving a thin gel layer on the glass slide. After 3 hours the slides were carefully removed from the mould and the gels on the slides were washed in deionised water at least five times to remove by-products. The gels were 26 mm by 26 mm, with a wet thickness of approximately 300 µm. The gels were stored under water and rinsed just before use. The catalyst-free reaction mixture was freshly prepared in a 30 mL continuously-fed stirred tank reactor (CSTR), which involved the in situ synthesis of stoichiometric bromomalonic acid from malonic acid and bromine generated from the partial reduction of sodium bromate. This CSTR in turn continuously fed a thermostatted open reactor with fresh catalyst-free BZ solution in order to maintain a nonequilibrium state. The final composition of the catalyst-free reaction solution in the reactor was: 0.42 M sodium bromate, 0.19 M malonic acid, 0.64 M sulphuric acid and 0.11 M bromide. The residence time was 30 minutes. An InFocus Model LP820 Projector was used to illuminate the computer-controlled image. Images were captured using a Lumenera Infinity2 USB 2.0 scientific digital camera. The open reactor was surrounded by a water jacket thermostatted at 22 oC. Peristaltic pumps were used to pump the reaction solution into the reactor and remove the effluent. A diagrammatic representation of the experimental setup is shown in Figure 1. The spatially distributed excitable field on the surface of the gel was achieved by the projection of a light pattern generated by a computer. Three light intensity levels were used: 0.035 (level 0), 1.35 (level 1) and 3.5 mW cm-2 (level 2), representing excitable, weakly excitable and non-excitable domains, respectively. The pattern was projected onto the catalystloaded gel through a 455nm narrow bandpass

Simple Collision-Based Chemical Logic Gates with Adaptive Computing

Figure 1. A block diagram of the experimental setup where A: computer, B: projector, C: mirror, D: microscope slide with the catalyst-laden gel, E: thermostatted Petri dish, F: CSTR, G1 and G2: pumps, H: stock solutions, I: camera, J: effluent flow, K: thermostatted water bath.

interference filter and 100/100 mm focal length lens pair and mirror assembly. The size of the projected pattern was approximately 14 mm square. Every 10 seconds, the pattern was replaced with a uniform grey level of 3.5 mW cm-2 for 10 ms during which time an image of the BZ fragments on the gel was captured. The purpose of removing the grid pattern during this period was to allow activity on the gel to be more visible to the camera and assist in subsequent image processing of chemical activity. Captured images were processed to identify chemical wave activity. This was done by differencing successive images on a pixel by pixel basis to create a black and white thresholded image. The images were cropped to the grid location and the grid superimposed on the thresholded images to aid analysis of the results.

WEAKLY EXCITABLE BZ MEDIUM Small changes in light intensity can significantly change the excitability of a ruthenium-catalyzed BZ system. There are two excitability boundar-

ies. The first defines the excitability limit below which wave propagation is not possible (nonexcitable medium). The other, higher, boundary defines the excitability limit above which wave fragments expand and their free ends form spirals (excitable medium). Between these limits the medium is subexcitable and chemical waves cannot fully develop but at some stage shrink and finally disappear. Close to the upper excitability limit unbounded planar wave fragments (critical fingers) propagate for a relatively long time without curling into spirals. Apart from the excitability of the medium, the evolution of the wave fragments also depends on their size. Smaller fragments shrink as they propagate, while bigger ones expand and there is a critical fragment size between the two extremes where the fragment either shrinks or expands when perturbed. In our experiments wave fragments were periodically initiated from a black area (minimal light intensity), in which the medium is oscillatory. In order to produce consistent size of wave fragments the oscillatory area and the weakly excitable area were connected through an excitable (level 0) channel of width ~2 mm surrounded by inhibitory (level 2) light intensity (Figure 2). When the fragments reached the weakly excitable area, the channels were removed, the whole area was illuminated with light at level 1 and from that point the collision of the fragments was monitored. By changing the position of the channels we were able to study different types of collisions. In these experiments we used a weakly excitable medium and not a subexcitable medium because if a wave fragment entered a subexcitable area (light intensity: 1.4 mW cm-2) it rapidly shrank and finally disappeared. If we managed to get two wave fragments close enough that they were able to collide before disappearing, then the collision resulted in either annihilation or ephemeral daughter fragments. Conversely, lowering the light intensity to 1.3 mW cm-2 resulted in fragments that expanded and typically formed double spiral waves that are indicative of an excitable medium.

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Figure 2. Areas of different light intensity projected onto the BZ medium

Thus, to produce unbounded planar wave fragments with a relatively long lifetime we used a light intensity of 1.35 mW cm-2, which we define as the weakly excitable level. When a small wave fragment enters the weakly excitable area (level 1) it starts slowly expanding and after it has reached a critical size it typically splits into two fragments in the monitored area (Figure 3).

COLLISIONS To examine head-on collisions initiation sites were positioned at the two opposite sides of the weakly excitable area. Shifting the channels up and down enabled us to study a range of different collision types. When the channels were offset by

more than the channel width there was no direct interaction between opposing fragments. By this we infer that there was no direct collision of the waves and they simply passed each other. However, it can be seen from Figure 4(a) that when waves of this type pass relatively closely there is a small but noticeable effect on the trajectories of the fragments. Fragments may also be de-stabilised so the overall propagation efficiency is reduced. Although we only have limited qualitative evidence for this effect it is indicative of how control and manipulation of signals in reaction diffusion based processors need not even be collision-based but may indeed just be proximity based. When the offset of the channels was less than the channel width or they were directly opposite to each other, then the collision resulted in “reflection” and two

Figure 3. Wave fragment propagates through a weakly excitable medium (15s between images)

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daughter fragments propagated in a perpendicular direction to the parent fragments (Figure 4(b) and (c)). In Figure 4(b) the waves were still slightly offset resulting in a disproportionate collision whereby a larger daughter fragment moved to the north away from the collision site. The fragment moving south was smaller and was relatively short lived. This gives an example of how critical the spacing, offset and angle of collision is and how this can be utilised to produce a huge array of subtly different phenomena which can be interpreted in terms of signal manipulation and thus computation. For example in figure 4(c) the waves are directly opposite resulting

in a balanced collision giving rise to two equal sized daughter fragments. The type of collisionbased gate implemented here is of the generic type〈x,y〉→〈x AND y, x AND NOT y, NOT x AND y〉. This is discussed in greater depth in (De Lacy Costello & Adamatzky, 2005) and adopts a formalism discussed in (Adamatzky, 1998). In this gate design if one of the wave fragments is not present, the other fragment will continue on its original trajectory. If it is assumed that the presence of a wave fragment along trajectory x is representative of logical TRUTH and the absence of the fragment FALSE, then these unchanged trajectories represent the operation x AND NOT y

Figure 4. Reflections of fragments from 180° collisions. (a) No collision between wave- fragments initiated from the opposite sides of the square when they are spaced far from each other. (b) and (c) Shifting the fragments closer to each other made the fragments collide and reflect. The two daughter fragments travel perpendicularly to the original direction (5s between images). (d) Scheme of the collision gate implemented in 4(c).

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Figure 5. Fusion of fragments from 45° collision. The daughter fragment propagates diagonally (5s between images). (b) Scheme of the collision gate implemented.

and NOT x AND y, respectively. If both fragments are present then the trajectories of the daughter fragments represent the operation x AND y. This is analogous to the Fredkin-Toffoli interaction gate (Fredkin & Toffoli, 2007), however the BZ reaction just simulates but does not actually support conservativeness. This gate has two output trajectories representing the same logical variable x AND y, so this type of experimental gate can also be used in signal splitting. When two fragments approached each other at an angle of 45° the collision resulted in the fusion of the two fragments. The resulting daughter fragment propagated on a diagonal trajectory away from the collision site (Figure 5). In this general case the colliding fragments implement the gate〈x,y〉=〈x AND y, x AND NOT y, y AND NOT x〉as there are two input trajectories and three output trajectories. When two fragments approached each other at an angle of 90° the collision also resulted in the fusion of the fragments. When both of the fragments were initiated at the middle of two neighbouring sides the daughter fragments propagated diagonally (Figure 6(a)). However, by shifting one of the initiation sites along the side this influenced the propagation trajectory of

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the daughter fragment. When this initiation site was shifted towards the other fragment, the resulting daughter fragment tended to propagate in the direction of the unshifted fragment (which arrived Figure 6. 90° collision of two wave fragments initiated from the sides of the square. The fused daughter fragment propagates (a) diagonally when both fragments are initiated from the middle of the sides, (b) to the left when the initiation channels are close to each other and (c) upwards when the initiation channels are far from each other. (5s between images)

Simple Collision-Based Chemical Logic Gates with Adaptive Computing

from the middle of the square) (Figure 6(b)). However, the original trajectory of the unshifted fragment is “pushed” slightly upwards. When the initiation site was shifted away from the other fragment the resulting daughter fragment travelled in the general direction of the shifted fragment (north in Figure 6(c)). However, in this case the dominant fragment is “pulled” slightly towards the direction of the weaker fragment. These behaviours are in agreement with observations gained using a model BZ system (Adamatzky & de Lacy Costello, 2007). It should be noted that these “pulling” and “pushing” type interactions are not always simple and it is possible for both to be seen in the result of one collision sequence. For example, prior to the collision there is often an “inductive” effect whereby the colliding fragments (usually the dominant fragment) may be “pulled” towards the collision site but in the resulting collision the dominant fragment may then be “pushed” away from its original trajectory by collision with the weaker “control” fragment. Therefore, even though the generic type of collision is of a fusion type it is apparent that relatively fine control of the size, trajectory and velocity of the daughter fragments may be achieved by simple shifts of the fragments’ input trajectories. This is obviously of great use in developing more complex composite type gates, adders etc. which would require multiple sequential collisions and the use of daughter fragments in subsequent collisions. When the initiation sites were positioned diagonally at adjacent corners a 90° collision was observed resulting in fusion of the parent fragments. However, depending on the phase of the colliding fragments the daughter fragment changed its direction. It propagated upwards, in the direction of the opposite side of the square, when the parent fragments were “in phase” i.e., arrived at the middle of the square at the same time (Figure 7 (a)). When there was a phase difference between the parent fragments, the propagation direction of the daughter fragment was close to

Figure 7. 90° collision of two wave fragments initiated from the corners of the square. (a) The fused daughter fragment propagates upwards when the parent fragments get to the middle of the square at the same time. (b) When parent fragments get to the middle with a phase difference the daughter fragment propagates in the direction of the more advanced fragment. (c) When one of the fragments extinguishes just before the collision the surviving fragment changes its direction towards the area where the other fragment disappeared (pulling). (d) When the phase-shifted fragments start to expand before collision two daughter fragments form (5s between images).

the direction of the more advanced fragment. In this case the lagging fragment collided with the free end of the already expanded fragment and fused into it (Figure 7(b)). Even though one of the fragments was significantly weaker than the other it still exerted some influence over the dominant fragment’s trajectory and pushed it

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from its original diagonal trajectory. The smaller the phase difference between the two parent fragments, the greater the effect the second fragment had on the first and the more it pushed it upwards. Again, these observations can be used to exert a fine control over fragment collisions and thus aid the construction of more advanced collisionbased computers. Even when the fragments did not collide because one of them extinguished before reaching the collision site, the other fragment changed its direction of propagation (Figure 7(c)). This fragment tended to propagate towards the vacated area, where the other fragment had disappeared and the extinguished fragment thus exerted a “pulling” type interaction over the remaining fragment even without a collision. When the phase-shifted fragments started to expand before collision and the phase difference between them was such that they collided perpendicularly, two daughter fragments were formed. The lagging fragment split and half of it fused with the other fragment and propagated upwards while the other half propagated parallel with the bottom of the projected area (Figure 7(d)).

With a 135° collision one of the fragments propagated diagonally from a corner and the other from the middle of the opposite side. This collision resulted in a fusion type interaction, and again by shifting the phase of the initiation of the parent fragments we were able to subtly influence the trajectory of the daughter fragment. Once again the more advanced parent fragment dictated the direction of the daughter fragment (Figure 8(a)-(c)). These results are also in line with our previous theoretical studies.

Evolving Simple Logic Gates As noted above, we are interested in exploring the development of collision-based computing systems within this chemical computing approach. To begin examining the potential for the embodied coevolution of such structures in the continuous, non-linear 2D media described we designed a simple scheme to simulate a number of two-input Boolean logic gates. To evolve simple logic gates we adopt a similar approach to that of Harding and Miller (2007) in their work on evolving logic gates in liquid crystal. The spatially distributed excitable

Figure 8. 135° collision of two wave fragments results in fusion. Changing the phase of the initiation. ((a), (b) and (c)) of the parent fragments influences the direction of the propagation of the daughter fragment in a way that the more advanced parent fragment dictate the direction of the daughter fragment (5s between images).

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field on the surface of the gel was achieved by the projection of a 7-by-7 cell grid pattern comprised of cells with the three intensity levels (level 0,1,2) we used for the collision experiments. Chemical fragments are again generated at the bottom of the grid from a small oscillating area and travel along the channel (level 0) and up the sides of the grid approximately symmetrically (Figure 9(a)). Three sides of the projected grid are used to provide inputs to the system. A one-cell boundary around the edge of the grid is set by default to level 2. This light level acts as a barrier to prevent spurious entry of chemical fragments into the centre of the grid where computation is to occur. The boundary at the left and right sides of the grid is modulated to allow entry of fragments. Under program control the left and right sides of the grid are treated as being the two binary inputs to the system. There are thus four possible states – 00, 01, 10 and 11 – for which chemical activity must be supplied to the grid in order to test its operation implementing Boolean logic. For a ‘0’ input, light at the appropriate boundary remains at level 2 prohibiting fragments from passing from the initiation channel into the grid. For an input of value ‘1’ light level at the left and/or right side boundary of the grid is controlled by the genome used by the simple evolutionary algorithm. The genome is a binary string of length 10 bits, with 5 bits coding for the 5 active boundary cells along the left side of the grid and 5 for the right side. Depending upon the genome, the wlight level associated with each cell in the active boundary is given the level 0 or 2. A light level of 0 allows fragment activity to pass into the 5 by 5 cell centre area of the grid, which is illuminated with a uniform light level of 1. In this way the evolutionary algorithm is able to influence the spatial and temporal dynamics of chemical activity in the centre of the grid with the aim of inducing ‘useful’ activity in the form of computation. In addition to the two inputs controlled by the EA, a constant truth input is provided to the grid by means of a single cell in the centre of the bottom boundary that has a fixed light level of 0.

Figure 9. (a) Initiation pattern, (b) example input configuration representing input logic states and (c) collisions of fragments occurring during configuration (b). The black area (level 0) in (a) and (b) represents the excitable medium (where chemical waves can fully develop) whilst the white area (level 2) is non excitable (where the formation of waves is inhibited). The grey area (level 1) in the centre of the grid is weakly excitable where wave fragments just manage to propagate.

The EA used is a simple genetics-based hill climber: starting from an initial random solution, an offspring is evaluated and adopted as the parent if at least as fit as the original solution. This 171

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choice is motivated by the impracticality of testing multiple individuals in a population with a real (slow) chemical system such as used here. Bitwise mutation is the only variation operator, with a single gene being mutated after each complete cycle of four logical input presentations. The output of computations occurring in the centre of the grid is obtained by monitoring the 25 cells in the centre of the grid using the image processing techniques described above. Each output cell is tested 300s after the system has been presented with one of the four possible Boolean input configurations and is assigned the Boolean output value of 1 if the total excitation level in the cell is 20% or more and a value of 0 otherwise. The centre of the grid thus implements (potentially) 25 logic functions in parallel. To determine the fitness of each cell position with respect to a target logic function, each of the four possible input configurations is presented in turn and a fitness of 1 is assigned to the cell if it presents the correct output for that configuration. A complete logic function is discovered when a single cell presents all four correct outputs in a single learning episode. For this preliminary work we investigated the ability of the system to discover two-input AND and NAND gates. These gates are complementary in function and both were investigated to avoid the

possibility of results being biased by particular properties of the chemical system used.

RESULTS Figure 10(a) shows the discovery of correctly functioning AND gates obtained by running the system described above. For these experiments the goal was for the EA to design the input boundary to the grid in such a way that a single correctly functioning AND gate resulted in at least one of the cells in the central sub-excitable area of the grid. We were able to run a maximum of 30 input configurations (which required about 6 hours) in the experiments because after that time the excitability of the system changed due to the “desensitization” effect of high intensity light (Cassidy and Müller, 2006). Figure 10(b) shows similar results for design of single NAND gates. From these results it is evidently somewhat trivial for the EA to design grid boundary conditions such that single instances of correctly functioning two-input logic gates can be produced by collision-based computing. We therefore wanted to attempt a more testing problem, namely the simultaneous design of multiple concurrently functioning logic gates. Figure 11 shows that it is possible to design concurrently functioning

Figure 10. Number of input configurations necessary to discover the first (a) AND and (b) NAND gate. Gate is found when fitness reaches four. Fitness is an average of 3 runs. Error bars show minimum and maximum fitness.

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Figure 11. Number of input configurations necessary to discover the first two cells implementing concurrently functioning AND and NAND gates. Both gates are found when fitness reaches eight. Fitness is an average of 3 runs. Error bars show minimum and maximum fitness.

AND and NAND gates using the same technique with only a marginal increase in difficulty.

CONCLUSION We have reproduced in chemical experiments a large majority of the collision types observed in previous numerical simulations of the subexcitable BZ reaction (Adamatzky & de Lacy Costello, 2007). We have achieved the aim of implementing collision-based logical gates in a homogeneous architectureless chemical medium. In line with previous studies (Adamatzky, 2003; 2004a; Adamatzky & de Lacy Costello, 2007; de Lacy Costello & Adamatzky, 2005) we have demonstrated that the trajectories of travelling wave fragments can be dynamically changed, adjusted and tuned by colliding other control wave fragments with them. These collisions can be interpreted in terms of logic gates. We have showed that the outcome of any collision strongly depends on whether fragments collide front on, or one fragment hits the tail or edge of another fragment. We also affirmed that it is not necessary for two fragments to actually

collide for there to be an effect. We observed the case where a fragment had already collapsed but still exerts a strong influence over the trajectory of the remaining fragment. The main contribution of these results to the field of collision-based computing and unconventional computing methods in general is that we have proved it possible in experiment to design flexible computing architectures where the propagating signals are manipulated by each other. In this type of system it is not necessary to introduce any stationary impurities such as heterogeneous illumination into the excitable medium. This differentiates it from previous “pseudo-wired” or hardwired approaches to constructing gates/information transfer in excitable and sub-excitable systems. Excitable and oscillating chemical systems have previously been used to solve a number of simple computational tasks. However the experimental design of such systems has typically been non-trivial. In this paper we have presented initial results from a methodology by which to achieve the complex task of designing such systems — through the use of adaptive computing. We have shown that it is possible to use adaptive computing methods in order to control the behaviour of a light-sensitive BZ reaction and construct a number of logic gates.

ACKNOWLEDGMENT This work was supported under EPSRC Grant No. GR/T11029.

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Adamatzky, A. (2003). Collision based computing. Springer. Adamatzky, A. (2004a). Collision-based computing in Belousov Zhabotinsky medium. Chaos, Solitons, and Fractals, 21, 1259–1264. doi:10.1016/j. chaos.2003.12.068 Adamatzky, A., Arena, P., Basile, A., CarmonaGalan, R., de Lacy Costello, B., & Fortuna, L. (2004b). Reaction Diffusion navigation robot control: From chemical to VLSI analogic processors. IEEE Trans. Circ. Syst. I., 51, 926–938. doi:10.1109/TCSI.2004.827654 Adamatzky, A., & de Lacy Costello, B. (2002b). Experimental logic gates in a reaction diffusion medium: the XOR gate and beyond. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 66, 046112. doi:10.1103/PhysRevE.66.046112 Adamatzky, A., & de Lacy Costello, B. (2003a). Reaction diffusion path planning in a hybrid chemical and cellular automaton processor. Chaos, Solitons, and Fractals, 16, 727–736. doi:10.1016/ S0960-0779(02)00409-5 Adamatzky, A., & De Lacy Costello, B. (2003b). On some limitations of reaction diffusion computers in relation to the Voronoi diagram and its inversion. Physics Letters. [Part A], 309, 397–406. doi:10.1016/S0375-9601(03)00206-8 Adamatzky, A., & De Lacy Costello, B. (2007). Binary collisions between wave-fragments in a sub-excitable Belousov–Zhabotinsky medium. Chaos, Solitons, and Fractals, 34, 307–315. doi:10.1016/j.chaos.2006.03.095 Adamatzky, A., De Lacy Costello, B., & Asai, T. (2005). Reaction-Diffusion Computers. Elsevier. Adamatzky, A., de Lacy Costello, B., & Ratcliffe, N. (2002a). Experimental reaction diffusion preprocessor for shape recognition. Physics Letters. [Part A], 297, 344–352. doi:10.1016/S03759601(02)00289-X

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Harding, S., & Miller, J. (2008). (in press). Evolution in Materio: Exploiting the Physics of Materials for Computation. Int. J. Unconventional Computing. Harding, S. L., & Miller, J. F. (2007). Evolution in Materio: Evolving Logic Gates in Liquid Crystal. J. Unconventional Computing, 3(4), 243–257. Karma, A. (1991). Universal limit of spiral wave propagation in excitable media. Physical Review Letters, 66, 2274. doi:10.1103/PhysRevLett.66.2274 Mihaliuk, E., Sakurai, T., Chirila, F., & Showalter, K. (2002). Feedback stabilization of unstable propagating waves. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 65, 065602. doi:10.1103/PhysRevE.65.065602 Motoike, I. N., & Adamatzky, A. (2005). Three valued logic gates in reaction-diffusion excitable media. Chaos, Solitons, and Fractals, 24(1), 107–114. Motoike, I. N., & Yoshikawa, K. (2003). Information operations with multiple pulses on an excitable field. Chaos, Solitons, and Fractals, 17, 455–461. doi:10.1016/S0960-0779(02)00388-0 Nicolis, G., & De Wit, A. (2007). Reaction-diffusion systems. Scholarpedia, 2(9), 1475. doi:10.4249/ scholarpedia.1475 Rambidi, N. G. (2003). Chemical based computing and problems of high computational complexity: the reaction diffusion paradigm. In Seinko, T., Adamatzky, A., Rambidi, N., & Conran, M. (Eds.), Molecular Computing. The MIT press.

Steinbock, O., Toth, A., & Showalter, K. (1995). Navigating complex labyrinths: Optimal paths from chemical waves. Science, 267, 868–871. doi:10.1126/science.267.5199.868 Stone, C., Toth, R., Adamatzky, A., Bull, L., & De Lacy Costello, B. (2007). Towards the Coevolution of Cellular Automata Controllers for Chemical Computing with the B-Z Reaction. In D. Thierens et al. (Eds.), GECOO-2007: Proceedings of the Genetic and Evolutionary Computation Conference. ACM Press, (pp. 472-478). Toth, A., & Showalter, K. (1995). Logic gates in excitable media. The Journal of Chemical Physics, 103, 2058–2066. doi:10.1063/1.469732 Tour, J. M., Van Zandt, W. L., Husband, C. P., Husband, S. M., Wilson, L. S., Franzon, P. D., & Nackashi, D. P. (2002). Nanocell Logic Gates for Molecular Computing. IEEE Transactions on Nanotechnology, 1(2), 100–109. doi:10.1109/ TNANO.2002.804744 Wang, J., Kádár, S., Jung, P., & Showalter, K. (1999). Noise driven Avalanche behavior in subexcitable media. Physical Review Letters, 82, 855–858. doi:10.1103/PhysRevLett.82.855 Zykov, V. S., & Showalter, K. (2005). Wave front interaction model of stabilized propagating wave segments. Physical Review Letters, 94, 068302. doi:10.1103/PhysRevLett.94.068302

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 1-16, copyright 2009 by IGI Publishing

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Toward Biomolecular Computers Using ReactionDiffusion Dynamics Masahiko Hiratsuka Sendai National College of Technology, Japan Koichi Ito Tohoku University, Japan Takafumi Aoki Tohoku University, Japan Tatsuo Higuchi Tohoku Institute of Technology, Japan

ABSTRACT This article investigates a possibility of constructing massively parallel computing systems using molecular electronics technology. By employing the specificity of biological molecules, such as enzymes, new integrated circuit architectures that are free from interconnection problems could be constructed. To clarify the proposed concept, we present a functional model of an artificial catalyst device called an enzyme transistor. In this article, we develop artificial catalyst devices as basic building blocks for molecular computing integrated circuits, and explore the possibility of a new computing paradigm using reaction-diffusion dynamics induced by collective behavior of artificial catalyst devices.

INTRODUCTION The purpose of this article is to discuss the possibility of constructing massively parallel computing DOI: 10.4018/978-1-60960-186-7.ch012

architectures using molecular electronics technology. By employing the specificity of biological catalysts, such as enzymes, new circuit/system integration could be realized. Recently, we have presented a model of molecular computing using

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Toward Biomolecular Computers Using Reaction-Diffusion Dynamics

artificial catalyst networks (Aoki et al, 1998; Hiratsuka et al, 1999, 1999a). We have shown that artificial catalysts having activity control function, such as enzyme transistors (Aoki et al, 1998), can realize various analog/digital computing circuits as molecular reaction networks. Currently, we are interested in creating new information processing functions, which are essentially different from those of conventional electronic systems, using artificial catalyst devices (Hiratsuka et al, 1999, 1999a). An important feature of molecular information processing is its massive parallelism based on reaction-diffusion molecular dynamics in a continuous signal transfer/processing medium. Our current project aims

Figure 1. Enzyme transistor model. (a) Schematic illustration of an enzyme reaction regulated by an effector. (b) Model of an enzyme transistor. (c) Typical sigmoid characteristic for k1(C).

i.

To develop artificial catalyst devices as basic building blocks for molecular computing integrated circuits, and ii. To explore the possibility of a new computing paradigm using reaction-diffusion dynamics induced by collective behavior of artificial catalyst devices.

TOWARD WIRE-FREE SYSTEM INTEGRATION A Model of Enzyme Transistors An enzyme transistor presented here is a molecular device based on activity-controlled enzyme reactions. Figure 1(a) schematically illustrates an effector-controlled enzyme reaction. An enzyme E catalyzes the reaction that converts its substrate S into the corresponding product P. The catalytic activity of the enzyme is regulated by the specific effector C. We assume that the enzyme transistor is an artificial catalyst whose activity is controlled by some effector. Figure 1(b) shows the model of an enzyme transistor, where C, S, P, and J represent the concentrations of the effector, substrate, product, and the substrate flux (the rate of substrate sup-

ply), respectively. The catalytic activity of the enzyme transistor varies in response to the concentration of the effector C. We assume that the rate constant k1 of the enzymatic reaction, which is a measure of catalytic efficiency, is expressed as a function of the effector concentration C, and is denoted by k1(C). In general, k1(C) is assumed to have a sigmoid characteristic which has a steep slope near the operating-point concentration CO, as illustrated in Figure 1(c). (These types of characteristics are widely found in living systems.) By coupling multiple enzyme transistors, we can construct an artificial network of biochemical

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reactions. Papers (Aoki et al, 1998; Hiratsuka et al, 1999) describe model-based analysis of enzyme transistor circuits (artificial catalyst networks) and their possible functionality. The structure of this type of network is defined by the molecular selectivity of enzyme transistors. Each chemical substance corresponds to a specific node in the network, and each enzyme transistor reaction converting a specific substrate into a specific product corresponds to the edge connecting the two nodes. All the information in our system is coded into varieties of molecular agents, and is discriminated by the selectivity of enzyme transistors. Thus, no physical structure is required for device-device interaction, in contrast to the present solid-state circuits using metal interconnects.

Implementation Issues There are several possibilities for realizing the function of enzyme transistors. In the future, it might be possible to design and mass produce artificial protein catalysts which can serve as enzyme transistors. Tailoring such single-molecule devices, however, will require technological breakthroughs in protein engineering. One of the most practical approaches at this early stage may be bioelectronic implementation, where we can

employ the state-of-the-art biosensor technology for device implementation. Figure 2 shows the conceptual realization of an enzyme transistor based on electrical control of enzyme reactions. The detector electrode, an enzyme-based biosensor, responds to the effector concentration C and generates electric output. This signal controls the catalytic activity of the enzyme immobilized on the modulator electrode; this enzyme catalyzes the reaction to convert the substrate S to the product P. As a result, the function of enzyme transistors could be realized. The fundamental question now is how to control enzyme reactions electrically. Redox enzymes, a class of enzymes which catalyze reduction/oxidation reactions, are useful for this purpose since the dynamics of these enzymes is based on electron transfer among molecules and this electron flow can be controlled by electrode devices. The authors have demonstrated a possibility of electrical control of redox enzyme reactions (Aoki et al, 1998). In our experiments, the cofactor NAD+/NADH and the mediator hydroquinone/benzoquinone are used to shuttle electrons from the redox center of the enzyme, glucose dehydrogenase, to the surface of the platinum electrode.

Figure 2. Conceptual realization of an enzyme transistor based on electrical control of enzyme reactions

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Figure 3. Electrochemical implementation of molecular computing integrated circuits

An Experimental Model for Future Biomolecular Computing Using Integrated Circuits Figure 3 shows the idea of bio-electronic integrated circuits using enzyme transistors. The enzyme transistors are integrated on a substrate in the form of two-dimensional (2-D) array structure. The device surface is covered by a thin layer of buffer solution that serves as a massively parallel communication channel among enzyme transistors. Each enzyme transistor detects a specific effector molecule, and regulates a specific reaction in response to the effector concentration. If we now take the spatio-temporal development of concentration patterns into consideration, more complex and interesting phenomena, such as pattern formation and wave propagation, can be observed. An important principle underlying these phenomena is reaction-diffusion dynamics in which enzyme transistors act as programmable artificial catalysts for controlling reaction networks. An experimental prototype of a redox micoroarray has been proposed in (Hiratsuka et al, 2002, 2003). In (Hiratsuka et al, 2003) we presented first attempt to demonstrate the possibility of implementing artificial reaction-diffusion dynamical systems under the control of electrode devices. The redox microarray model can be regarded, in a sense, a simplified abstraction of a

future molecular processor, which may employ massively parallel reaction-diffusion dynamics for signal/pattern processing. Figure 4(a) shows Figure 4. Experimental system for a redox microarray. (a) Configuration of the electrodes: WE - working electrodes; CE - counter electrode; RE - reference electrode. (b) Redox cycling of a benzoquinone/hydroquinone couple.

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the layout of integrated microelectrodes in the redox micoroarray. Platinum (Pt) microelectrodes are used as artificial catalysts. On a glass substrate, 64 Pt microelectrodes are integrated in the form of a 2-D array structure. Every microelectrode can be accessed independently from the reverse side of the glass plate. The surface of the electrode array is covered with a liquid solution layer, where a benzoquinone/hydroquinone couple (Figure 4(b)) is used as reversible redox species. The reaction kinetic function of each electrode is implemented with software in an external controller, and is fully programmable. We have designed the reaction kinetic function that exhibits the same qualitative behavior as the excitable FitzHughNagumo (FHN) dynamics. Figure 5 shows the photo snapshots of wave propagation observed on the redox microarray with excitable dynamics. The proposed model could be extended to other reaction-diffusion dynamics having useful computational capabilities, and may provide a foundation for future massively parallel molecular computing devices. Figure 5. Photo snapshots of active wave propagation in the redox microarray. High concentration of hydroquinone is indicated by the dark brown color.

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TOWARD REACTION-DIFFUSION COMPUTING Artificial Reaction-Diffusion System Using Enzyme Transistors The above studies toward future molecular computing technology demonstrated the possibility of creating massively parallel computers that are free from interconnection problems. An important feature of the proposed molecular computation is its massive parallelism based on reaction-diffusion molecular dynamics in an active continuous signal transfer/processing medium. It is well known that reaction-diffusion dynamics plays a significant role in biological self-organization and adaptive behaviors (Murray, 1993). Even the simplest reaction-diffusion phenomena, such as Turing pattern formation and active wave propagation, can be used to solve complex computational tasks (Sienko et al, 2003; Adamatzky et al, 2005). Understanding the principle of such pattern formation phenomena is essential for designing bio-inspired molecular computing systems employing high functionality of reaction-diffusion dynamics. At this point, an important technical challenge may arise as to whether it is possible to synthesize artificial reaction-diffusion dynamical systems for specific applications. Our previous papers (Hiratsuka et al, 1999, 1999a) address this question and present a method of implementing specific reaction-diffusion dynamics using enzyme transistor circuits. To demonstrate a synthetic approach for designing reaction-diffusion computers, we have illustrated how typical reaction-diffusion mechanisms capable of generating spatio-temporal patterns can be realized by enzyme transistors. The enzyme transistor model can thus be used to implement an active continuous medium which exhibits useful pattern formation phenomena. It seems that the proposed method could be extended to other reaction-diffusion dynamics having useful computational capabilities. Further investigation

Toward Biomolecular Computers Using Reaction-Diffusion Dynamics

Figure 6. Shortest path search using an excitable DRDS. A snapshot of wavefronts represents an equidistant surface measured from the starting point. By tracing back the equidistant surfaces, the shortest path from the starting point (S) to any specified destinations (G1, G2, ..., G9) can be obtained.

is required to establish the method for designing reaction-diffusion computers.

REACTION-DIFFUSION DYNAMICS AND PARALLEL COMPUTING The framework of reaction-diffusion system may provide a technique for applying the principle of biological pattern formation phenomena to many engineering problems. For this purpose, we have proposed a Digital Reaction-Diffusion System (DRDS) --- a model of a discrete-time discretespace reaction-diffusion dynamical system (Ito et al, 2001, 2003, 2006).

Figure 6 shows the application of the DRDS to shortest path search, where the DRDS simulates an excitable reaction-diffusion dynamics (Ito et al, 2006). A traveling wave propagates through a 2-D map (specifying collision-free space and blocked space) as a boundary condition for the excitable reaction-diffusion dynamics. The shortest path from the starting point to any specified destinations can be found by tracking back the wavefronts of the traveling wave. The traveling wave in the excitable DRDS can be used to generate a Voronoi diagram, as shown in Figure 7. Figure 8 shows the other application of the DRDS to fingerprint restoration, where the DRDS simulates a model of Turing instability (Ito et al,

Figure 7. Generation of a Voronoi diagram using an excitable DRDS. A snapshot of wavefronts represents an equidistant surface measured from the 11 points. A Voronoi diagram of 11 points can be constructed when opposing wavefronts collide, as shown in the red lines.

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Figure 8. Reconstruction of a fingerprint pattern from a subsampled image. (a) Original image. (b) Subsampled image with subsampling rate 1/(5 x 5). (c)--(e) Restored images by using an adaptive DRDS.

2003). The use of Turing instability for fingerprint enhancement/restoration allows active processing of fingerprint images, including the generation of most likely local patterns that interpolates missing fingerprint textures. Thus, the use of DRDS makes possible to understand the mechanism of morphogenesis within the framework of multidimensional digital signal processing theory, and apply the mechanism of morphogenesis to various engineering problems.

CONCLUSION This article investigated a possibility of constructing massively parallel computing systems using molecular electronics technology. By employing the specificity of biological molecules, such as enzymes, new integrated circuit architectures that are free from interconnection problems could be constructed. A molecular device called the “enzyme transistor” was proposed as a building block of wire-free integrated circuits. The listed below are major results of our study. i.

A prototype enzyme transistor was successfully implemented using redox enzymes with electrode devices. Also, a possibility of creating an artificial reaction-diffusion system for wire-free computation was demonstrated using an array of integrated microelectrodes. ii. A new class of signal processing algorithms based on artificial reaction-diffusion dynamics was proposed. The related investigations

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provide useful spin-offs to some real applications, such as image restoration and optimal path planning.

REFERENCES Adamatzky, A., De Lacy Costello, B., & Asai, T. (2005). Reaction-diffusion computers. Amsterdam: Elsevier. Aoki, T., Hiratsuka, M., & Higuchi, T. (1998). Enzyme transistor circuits. IEE Proceedings. Circuits, Devices and Systems, 145(4), 264–270. doi:10.1049/ip-cds:19982008 Hiratsuka, M., Aoki, T., & Higuchi, T. (1999). Enzyme transistor circuits for reaction-diffusion computing. IEEE Transactions on Circuits and Systems -- I, 46(2), 294--303. Hiratsuka, M., Aoki, T., & Higuchi, T. (1999). Pattern formation in reaction-diffusion enzyme transistor circuits. IEICE Transactions on Fundamentals. E (Norwalk, Conn.), 82-A(9), 1809–1817. Hiratsuka, M., Aoki, T., Morimitsu, H., & Higuchi, T. (2002). Implementation of reaction-diffusion cellular automata. IEEE Transactions on Circuits and Systems -- I, 49(1), 10--16.

Toward Biomolecular Computers Using Reaction-Diffusion Dynamics

Hiratsuka, M., Aoki, T., Morimitsu, H., & Higuchi, T. (2003). Implementation of a redox microarray: an experimental model for future nanoscale biomolecular computing using integrated circuits. IEE Proceedings. Nanobiotechnology, 150(1), 9–14. doi:10.1049/ip-nbt:20030518 Ito, K., Aoki, T., & Higuchi, T. (2001). Digital reaction-diffusion system -- a foundation of bioinspired texture image processing. IEICE Transactions on Fundamentals. E (Norwalk, Conn.), 84-A(8), 1909–1918.

Ito, K., Hiratsuka, M., Aoki, T., & Higuchi, T. (2006). A shortest path search algorithm using an excitable digital reaction-diffusion system. IEICE Transactions on Fundamentals. E (Norwalk, Conn.), 89-A(3), 735–743. Murray, J. D. (1993). Mathematical biology. Berlin: Springer-Verlag. doi:10.1007/b98869 Sienko, T., Adamatzky, A., Rambidi, N., & Conrad, M. (Eds.). (2003). Molecular computing. Massachusetts: The MIT Press.

Ito, K., Aoki, T., & Higuchi, T. (2003). Fingerprint restoration using digital reaction-diffusion system and its evaluation. IEICE Transactions on Fundamentals. E (Norwalk, Conn.), 86-A(8), 1916–1924.

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 17-25, copyright 2009 by IGI Publishing.

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Chapter 13

Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions B. P. J. de Lacy Costello University of the West of England, UK J. Armstrong University of the West of England, UK I. Jahan University of the West of England, UK N. M. Ratcliffe University of the West of England, UK

ABSTRACT Under normal reaction conditions [AlCl3 0.28-0.34M and NaOH 2.5M A.Volford et al.] spontaneous spiral and circular travelling precipitate waves were observed. We constructed a phase diagram for the reaction and identified a large controllable region at lower aluminium chloride levels. We show that it is possible to selectively initiate travelling circular waves and other self-organised structures within this controllable region. In previous work initiation was undertaken before adding the outer electrolyte resulting in disorganised waves. However, marking the gel one minute after adding outer electrolyte resulted in cardioid waves. Increasing the time interval to two minutes caused a transition to single circular waves. If the gel is marked sequentially nested circular waves (target waves) are formed. These reactions were used to calculate simple and additively weighted Voronoi tessellations. The fine control of self-organisation in precipitation reactions is of interest for the synthesis of novel and functional materials.

INTRODUCTION When considering pattern formation in inorganic systems the majority of work spanning over 100 DOI: 10.4018/978-1-60960-186-7.ch013

years has been focussed on Liesegang type reactions (Liesegang, 1896). These are periodic patterns formed due to non-equilibrium crystallisation and precipitation. There have been many models proposed to explain the phenomenon but

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Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions

still none are able to predict robustly the position, spacing and morphology of the precipitate bands in the full range of chemical systems. For example reactions based on cobalt chloride and ammonium hydroxide exhibit chaotic dynamics due to redissolution of the chemical complex at the stratum coupled with continued band formation ahead of the reacting front. This phenomenon means the entire pattern migrates within the tube (Ghoul and Sultan, 2001). More recently there has been renewed interest in this class of chemical reactions and the control thereof in order to synthesise functional materials by utilising the inherent self-assembly mechanisms. Work has been carried out to apply electrical fields (Lagzi, 2002), magnetic fields (Sorensen and Madsen, 2000) and changing the conventional geometry of the reactor (Kuo, Lopez Cabarcos and Bansil, 1997). This work is still at an early stage of development and lacks a fundamental mechanistic understanding which would allow a quantum leap forward. Basically our understanding of crystallisation processes and molecular processes per se is very limited and to some extent limits the potential of this class of reaction. If better molecular computation and material synthesis via self-assembly is to be achieved then this knowledge gap must be eradicated. The recent discoveries of a series of very simple inorganic reactions exhibiting travelling wave phenomena serves to further highlight this issue (Hantz, 2000; Adamatzky, De Lacy Costello and Asai, 2005; Volford et al, 2008; De Lacy Costello, Ratcliffe and Hantz, 2004). In terms of modelling inorganic pattern forming reactions a significant amount of work has been directed towards the modelling of sea shell patterns. Shells consist of calcified material secreted by the mantle, they then increase their size by accretion of new material at the “growing edge” of the shell. They show a diverse range of complex pigmentation patterns which can be modelled via cellular automata models (Wolfram, 1994), neural models (Murray, 1993) and reaction dif-

fusion models (Meinhardt, 1995). Thus there is some driving force to study pattern formations in simple inorganic systems in order to gain a better understanding of pattern formation in natural systems. This is in addition to the need to understand the underlying mechanisms in order to exert high levels of control. Recently a new class of inorganic reactions were discovered based on gels of copper chloride reacted with sodium hydroxide (Hantz, 2000). These reactions exhibited a range of self-organised patterns only observed previously in more complex reactions such as the BZ reaction. This is highly significant given the very simple nature of the chemical reactants. Most striking was the formation of travelling cardioid and spiral precipitation waves. This phase of the reaction was further studied in (Adamatzky, De Lacy Costello and Asai, 2005) where the use of a different gel media and outer electrolyte (potassium hydroxide) enabled spiral wave evolution to be observed at much smaller length scales. More recently another inorganic system has been identified based on aluminium chloride gels reacted with sodium hydroxide (Volford et al, 2007). This system is particularly remarkable as unlike the systems mentioned above (Hantz, 2000; Adamatzky, De Lacy Costello and Asai, 2005) the wave evolution can be easily observed in real-time. In previous work we were able to exert control over another class of inorganic reaction in order to realise a useful computation (De Lacy Costello, Ratcliffe and Hantz, 2004). The reaction involved the addition of copper chloride to potassium ferricyanide immobilised in an agarose gel. At high concentrations of potassium ferricyanide circular expanding waves are spontaneously generated by heterogenities. Where these waves collide they form a natural tessellation of the plane similar to those obtained in crystallisation processes (multiplicatively weighted crystal growth Voronoi diagrams). By reducing the concentration of potassium ferricyanide then the point and time at which

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circular waves are initiated could be controlled by marking the gel with a very fine glass needle. By using this methodology user defined precipitation patterns could be created. These user defined precipitation patterns equate to a calculation of a Voronoi tessellation of the plane. In previous work (De Lacy Costello, 2008) involving the aluminium chloride sodium hydroxide reaction we adopted the same strategy and were able to show that initiation of waves was possible. However, marking the gel in this way produced complex travelling waves made up of multiple expanding fragmented waves. This led to highly disorganised patterns. In the current paper we report an extension of this approach where we first analysed the entire phase diagram for the reaction in order to identify the controllable region and boundary. Then via selection of the appropriate chemical conditions and by marking the gel at various time intervals after the start of the active phase we are able to show selection and transition between different self-organised waves. This paper details the first example of initiation of self-organised structures namely cardioid waves in chemical systems. The approach described provides an important step in the control of precipitation waves and brings closer the prospect of using the phenomena for computing and materials synthesis.

EXPERIMENTAL Natural Wave Evolution Experiments The reaction detailed here is adapted from (Volford et al., 2007). Agarose gel (Sigma Aldrich Chemical Co.) was dissolved in distilled water to produce a 1% solution. It should be noted that we found the same general behaviour using a range of gels including agar. This solution was heated to 70oC with constant stirring until a clear solution was obtained. The heat was removed and Aluminium chloride hexahydrate was added to the solution in

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order to obtain a final inner electrolyte concentration of between 0.28 and 0.34M. The solution was again stirred to ensure complete dissolution and then poured into a Petri dish of diameter 9cm – to a depth of 7mm. The gels were then left to set at room temperature for at least an hour prior to the reaction. To start the reaction 21 cm3 of a 2.5M solution of sodium hydroxide was added directly to the gel surface. The reactions and any pattern formation were then monitored in transmitted light using a flatbed scanner interfaced to a PC.

Controlled Wave Evolution Experiments Disorganised Complex Fronts As with our previous papers on controlling wave evolution in inorganic systems (De Lacy Costello, Ratcliffe and Hantz, 2004; De Lacy Costello, 2008) we reduced the concentration of the inner electrolyte. In order to obtain controlled user defined pattern formation the recipe detailed in the Natural wave evolution section was adapted to give a final AlCl3 concentration of 0.26M. If at this concentration the gel is reacted with 5M sodium hydroxide solution then limited natural wave evolution occurs. However, if the gel surface is marked carefully with a fused silica needle of diameter 0.25mm then reacted with 5M sodium hydroxide

Initiation of Single Circular Waves The concentrations were the same as those used in the Disorganised complex fronts section but instead of marking the gel prior to adding the outer electrolyte the gel was marked 2-5 minutes after the start of the reaction.

Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions

Initiation of Nested Circular Waves (Target Waves) Target waves could be initiated by marking the gel sequentially at time intervals according to the reaction kinetics. In this experiment we marked the gel at 5 minute intervals.

Initiation of Travelling Cardioid Waves Cardioid waves could be initiated using the reaction conditions stated in the Disorganised complex fronts section but marking the gel one minute after addition of the outer electrolyte solution. To show transition between different wave structures a gel was prepared and the reaction initiated by pouring on sodium hydroxide. The gel was them marked three times after 1 minute and this was repeated at 15 second intervals.

Results and Discussion Natural Wave Evolution in the Aluminium Chloride Sodium Hydroxide Reaction When the sodium hydroxide is added to the surface of the aluminium chloride gel layer it forms a diffusion front and in the wake of this front a precipitate (Al(OH)3) is formed. At the same time at some point behind the diffusing front the excess sodium hydroxide re-dissolves the precipitate via complex formation. The result is that a thin precipitate layer (approximately 0.1-0.2 mm in depth) is formed which moves through the reactor. A few minutes after the addition of the sodium hydroxide a number of self organised patterns can be observed within the moving precipitate layer (see Figure 1). At the lower inner electrolyte (“lower excitability”) concentration these mainly take the form of single expanding circular fronts. These are of the same generic type observed previously in inorganic systems in (De Lacy Costello, Ratcliffe and Hantz, 2004). However, in (De Lacy

Costello, Ratcliffe and Hantz, 2004) the evolution of the expanding waves can only be observed by viewing the petri dish from underneath. This is easily explained because in (De Lacy Costello, Ratcliffe and Hantz, 2004) which describes the reaction between potassium ferricyanide (immobilised in gel) and copper chloride only limited redissolution occurs. Therefore, as the diffusion front advances a layer of precipitate masks the reacting front. From studying (De Lacy Costello, Ratcliffe and Hantz, 2004) it is obvious that the original uniform precipitation front has split to form cone shaped regions which enclose unreacted zones of inner electrolyte. It was the subsequent collision and annihilation of these expanding precipitate cones that formed a generalised Voronoi diagram. This phenomena has been the focus of much research (Hantz, 2004; De Lacy Costello, 2008) in order to identify the materials which form the boundary of the cones (or wave boundaries in thin 2D systems). These boundaries effectively limit and even stop completely the diffusion driven reaction in certain zones of the reactor. Therefore,

Figure 1. Natural wave evolution in the aluminium chloride (0.294M) sodium hydroxide reaction (2.5M). Double spiral cardioid type waves, circular waves, single spiral waves and disorganised waves can all be observed.

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we postulate that what is viewed in the aluminium chloride sodium hydroxide reaction is actually identical but that the material forming the upper stratum of the cone shaped zones has been redissolved by excess sodium hydroxide. The result is that active waves can be seen to move within the precipitating region. At higher levels of the inner electrolyte concentration (>0.29M) double spiral waves are seen to evolve resulting in cardioid like patterns. These structures were first observed in [9] in the copper chloride, sodium hydroxide reaction. However, due to the reaction chemistry only the static phase of the reaction could be observed with ease. What is remarkable about the aluminium chloride, sodium hydroxide reaction is that because of the redissolution kinetics these complex waves can be viewed in real time. However, as with the copper chloride-sodium hydroxide reaction eventually the diffusion front ceases to move through the reactor and the reaction is effectively frozen – preserving stationary patterns of complex wave interactions. If the gel is thinner in the aluminium chloride reaction then excess sodium hydroxide re-dissolves all the structures and no permanent structure remains.

Complex Travelling Waves If the concentration of aluminium chloride is lowered to a level where natural structures are absent or minimised then pre-marking the gel results in wave evolution at specific points (see Figure 2). The concentration of aluminium chloride used was 0.26M and the concentration of sodium hydroxide 5M. It should be noted that the controllable region of the phase diagram with respect to the level of aluminium chloride changes depending on the concentration of sodium hydroxide used. Figure 2 shows that by pre-marking the gel at six points prior to initiation “controlled” evolution is observed after the reaction is started. However, the structures evolving from the marked points are highly complex structures. It can also be seen that

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the complex structures are unique despite being initiated by an “identical” method. Therefore, control is only exerted over wave initiation not form. These findings are in contrast to a previously studied system [12] where the same approach gave single expanding circular waves corresponding to each marked point. The explanation is that in the previous system complex wave evolution was not observed per se. Therefore, only circular waves were obtained in both the controlled and natural phases of the reaction. This means that the timing of the initiation is not critical in less complex systems. The evolution of complex waves of this type is not particularly useful in a computational context. Even for geometric calculations such as the computation of a Voronoi diagram where the outer wave corresponding to the original initiation could be used to partition reactor space it is obviFigure 2. Controlled initiation of six complex waves in the aluminium chloride (0.26M) and sodium hydroxide reaction (5M). The gel was marked once at each of six positions in a hexagonal pattern. After marking the gel the sodium hydroxide was poured onto the gel to initiate the precipitation front. Note that some natural structures still persist due to large heterogeneities in the system such as air bubbles and the interface region between the gel and the reactor wall.

Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions

ous that the complex nature of the initiation and subsequent interactions produces non-circular asymmetric wave structures that grow at different rates

Initiation of Single Circular Waves For simple geometric calculations it is useful to be able to initiate circular waves from point sources at user defined time intervals. We investigated the strategy of marking the gel at different times after the initiation of the primary reaction. We found that circular waves could be obtained by marking the gel two minutes after initiating the reaction. However, to be certain of obtaining circular waves across the whole controllable region of the phase diagram it was necessary to leave five minutes between the start of the reaction and marking the gel. We also found that the controllability of structures was dependent on the diameter of the fused silica needle –with the best results obtained with the smallest diameter of 0.25mm. Figure 3. shows the evolution of a reaction where a hexagonal array has been marked five minutes after pouring on the sodium hydroxide solution. It should be noted that the separation between the circular fronts marked on the gel was over 1cm. This was to show an absence of natural

structures across a large area of the reactor necessary for undertaking useful computation and synthesising functional materials. However, this gives the impression that the computational potential of these reactions is lower than in reality. With this class of chemical processor the more points in a given unit area (upto the theoretical maximum) the faster the calculation is completed. This is one of the inherent benefits of parallel processors of this type.

Initiation of Target Waves We found that if over the course of the reaction the same point at the centre of the circular wave was marked repeatedly then a series of waves could be initiated forming a target wave like structure (Figure 4). This is because as mentioned previously the circular wave is expanding due to its movement through the thickness of the gel. The area within an expanding circular wave is comprised of uniform “amorphous” precipitate. When the gel is marked again the reacting front in this locality is disrupted causing the formation of a differentiated product. The exact nature of this product has not been established but we postulate that it is simply a phase change to a more structured crystalline form of the precipitate. The dark area observed as

Figure 3. A controlled evolution of circular waves in the aluminium chloride (0.26M) and sodium hydroxide reaction (5M). The gel was marked after 5 minutes. Figure 3a. 25 minutes after initiation. Figure 3b. 75 minutes after initiation showing the formation of bisectors where the travelling circular fronts meet.

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Figure 4. Formation of target waves in the aluminium chloride (0.26M) sodium hydroxide (5M) reaction. The gel was marked twice at 5 minute intervals. Image was taken 35 minutes after initiation of the reaction.

the “wavefront” is simply a void at the boundary between the two phases. It appears dark because the reaction is placed on a black background for visualisation purposes. In Figure 3a and b the white boundary in advance of the dark front is the actual differentiated “boundary” material. The formation of target waves of this type may be useful in the synthesis of new materials. We have shown it is possible to harness and precisely control the inherent self-organised structures in this reaction.

Initiation of Cardioid like Double Spiral Waves Most natural structures evolving in the aluminium chloride sodium hydroxide reaction are either circular waves or double spiral waves. There is a definite point of transition in the phase diagram where predominantly circular wave evolution switches to predominantly double spiral wave evolution at given concentrations of aluminium chloride and sodium hydroxide. The formation of single spiral waves seems to be via the disruption of double spiral waves. This disruption tends to occur late in the evolution of the reaction. For

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double spiral waves to maintain stability the counter-rotating tips must meet within a specific time interval in order that they can undergo proportionate collision. This proportionate collision forms the new wave front in the central region. Any alteration in this timing will result in asymmetric double spirals and may eventually result in the annihilation of one of the tips. By marking the gel at short time intervals after the initiation we were able to initiate single wave fragments whose free ends curl to form double spiral cardioid waves (Figure 5). This process is not yet 100% reproducible even though figure 5 shows the initiation of three double spirals from three marks. Under identical reaction conditions when repeated in a larger single array using a one minute time interval only 60% of the marked points

Figure 5. The initiation of cardioid like double spiral waves in the aluminium chloride (0.26M) sodium hydroxide (5M) reaction. The reaction was marked three times at the following time intervals following intiation (from top to bottom 60 seconds, 75 seconds, 90 seconds, 105 seconds, 120 seconds). There are some natural waves at the edges of the reactor. The reaction is highly non-linear but a transition from double spiral waves to circular waves between 1 and 2 minutes can be observed.

Fine Control and Selection of Travelling Waves in Inorganic Pattern Forming Reactions

formed double spiral waves. The remainder formed expanding wave fragments which simply collided and annihilated to form “circular” waves. Therefore, this suggests what we observe as circular waves in the natural evolution of the reaction may be formed via this mechanism (and not from the formation of perfectly expanding circular waves). As the controlled phase of the reaction is carried out in a region of lower “excitability” then this may explain the difficulty in reproducibly forming double spiral waves. At lower excitability levels the spiral tips travel with a lower velocity and are therefore more prone to annihilation during the collision phase. Also the region above the controllable zone possesses mainly circular waves as natural structures indicating they will be favoured in the controllable zone. This work is the first example of the control and selection of self-organised complex structures in precipitating reactions. This phenomena may be useful in the synthesis of new functional materials. The fact that the phenomena is manifested in this type of simple chemical precipitating system means that potentially there are many reactions of this type where such control and selection could be exerted given specific experimental conditions.

GEOMETRIC COMPUTATIONS Construction of Simple Voronoi Diagram We marked the gel using a hexagonal array in order to initiate single circular waves. Where these waves meet they form bisectors and the bisectors constitute the Voronoi diagram of the original set of points (Figure 6). A Voronoi diagram of a collection of objects is a partitioning of the reactor space into cells such that each cell consists of the points closer to that object than any other. We have carried out a number of experiments on the construction of Voronoi type tessellations of the plane see overview in (Adamatzky, De Lacy Costello, Asai, 2005). However, these systems where drops of reagents are used are only capable of constructing generalised Voronoi diagrams and the accuracy of the calculation can be affected by drop size and morphology etc. In this set of experiments the Voronoi diagram is computed via a series of sub millimetre point sources and therefore, the accuracy of the calculation is far higher.

Construction of an Additively Weighted Voronoi Diagram We show that it is possible to exert a high level of control over the reactions evolution via construc-

Figure 6. Construction of a simple Voronoi diagram of a set of marked points in the aluminium chloride sodium hydroxide reaction.

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tion of an additively weighted Voronoi diagram. An additively weighted Voronoi diagram is constructed when points are initiated at different times but all expand at the same rate. Now where the expanding circles meet they will not form straight line bisectors but instead they will be hyperbolic line segments. In Figure 7 we show the construction of an additively weighted Voronoi diagram. In this construction one set of points was marked 5 minutes after initiation whereas a second set of interpenetrating points was marked 10 minutes after initiation.

CONCLUSION AND FUTURE WORK We have demonstrated that it is possible using simple techniques to exert a high level of control over the evolution of the Aluminium Chloride Sodium Hydroxide reaction. We were particularly successful in exerting control over the evolution of single circular waves. This enabled us to undertake a number of geometric calculations based Figure 7. An additively weighted Voronoi diagram constructed in the aluminium chloride sodium hydroxide reaction. Note that a bubble in the gel has caused a single wave to be initiated separate to the marked set of points. However, this does not disrupt the calculation.

around constructing Voronoi diagrams of a set of points. We were also able to construct additively weighted Voronoi diagrams as the reaction could be stimulated at different time intervals during its evolution. The same phenomena allowed us to construct a number of user defined target waves. Even more significantly we were able to select different modes of self-organised structures. For example we were able to initiate double spiral cardioid waves in preference to circular waves with a high degree of certainty. However, there is still more work to be done in this area. What the experiments do show is that with some improvement it may be possible to construct useful structures in precipitating media where the boundaries of travelling waves are functional highly crystalline materials. In the future we hope to show that wave initiation and dynamical evolution may be controlled by the application of external fields – for example an electrical field. We would also hope to show that it is possible to more accurately select for certain pattern types. For example the formation of large single spiral crystalline waves may be possible if selective annihilation of double spiral tips could be achieved. These reactions are very much observed on a macroscopic scale with relatively long time intervals. However, they are the result of co-operative and co-ordinated molecular processes which occur in parallel after the uniform initiation (which just involves pouring the outer electrolyte onto the inner electrolyte gel). There is no doubt that with better understanding of this type of molecular interaction there is no barrier to implementation at significantly smaller length scales.

ACKNOWLEDGMENT The authors wish to acknowledge the support of the EPSRC grant number EP/E016839/1 for the support of Ishrat Jahan.

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REFERENCES Adamatzky, A. B., & De Lacy Costello, T. (2005). Asai, Reaction Diffusion Computers. Elsevier. Al Ghoul, M., & Sultan, R. (2001). Front propagation in patterned precipitation. Simulation of a migrating Co (OH)2 Liesegang pattern. The Journal of Physical Chemistry A, 105, 8053–8058. doi:10.1021/jp011158o

Kuo, C. S., Lopez Cabarcos, E., & Bansil, R. (1997). Two dimensional pattern formation in reaction-diffusion systems. Influence of the geometry. Physica A, 239, 120–128. doi:10.1016/ S0378-4371(97)00050-2 Lagzi, I. (2002). Formation of Liesegang patterns in an electrical field. PCCP, 4, 1268–1270. Liesegang, R. E. (1896). Naturwiss Wocheaschr, 11, 353.

De Lacy Costello, B. P. J. (2008). (in press). Control of complex travelling waves in simple inorganic systems –the potential for computing. International Journal of Unconventional Computing.

Murray, J. D. (1993). Mathematical biology. Berlin: Springer. doi:10.1007/b98869

De Lacy Costello, B. P. J., Ratcliffe, N. M., & Hantz, P. (2004). Voronoi diagrams generated by regressing edges of precipitation fronts. The Journal of Chemical Physics, 120(5), 2413–2416. doi:10.1063/1.1635358

Skytte Sorensen, J., & Lundager Madsen, H. E. (2000). The influence of magnetism on the precipitation of calcium phosphate. Journal of Crystal Growth, 216, 399–406. doi:10.1016/ S0022-0248(00)00449-8

Hantz, P. (2000). Pattern Formation in the NaOH + CuCl2 Reaction. The Journal of Physical Chemistry B, 104, 4266–4272. doi:10.1021/jp992456c

Volford, A., Izsak, F., Ripszam, M., & Lagzi, I. (2007). Pattern Formation and self organisation in a simple precipitation system. Langmuir, 23(3), 961–964. doi:10.1021/la0623432

Meinhardt, H. (1995). The algorithmic beauty of Sea Shells. Berlin: Springer.

Wolfram, S. (1994). Cellular automata as models of complexity. Nature, 311, 419–422. doi:10.1038/311419a0

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 26-35, copyright 2009 by IGI Publishing.

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Dynamics of Particle-Based Reaction-Diffusion Computing:

Active vs. Passive, Attraction vs. Repulsion Jeff Jones University of the West of England, UK

ABSTRACT Reaction-diffusion computing utilises the complex auto-catalytic and diffusive interactions underlying self-organising systems for practical computing tasks – developing variants of classical logical computing devices, or direct spatial embodiments of problem representations and solutions. We investigate the concept of passive and active approaches to reaction-diffusion computing. Passive approaches use front propagation as a carrier signal for information transport and computation. Active approaches can both sense and modify the propagation of the underlying carrier signal. We also consider the differences in attraction and repulsion behaviour for both passive and active approaches. Using particle approximations of reaction-diffusion behaviour in chemical systems, and the plasmodium of Physarum polycephalum, we demonstrate the similarities and differences between the passive and active approaches using both attraction and repulsion behaviour. We provide examples of how the approaches can be used for complex spatially represented computational tasks. We note that the active approach results in second-order emergent behaviour, exhibiting complex quasi-physical properties such as apparent surface tension effects and network minimisation which may have utility in future physical implementations of reaction-diffusion computing devices.

INTRODUCTION The natural world (both purely physical and living) appears to take the assembly of complex structures in its stride - producing complex, dynamic, DOI: 10.4018/978-1-60960-186-7.ch014

functional and reliable structures in what would be considered very noisy and unpredictable environments. Such complex pattern forming behaviour exploits properties of self-organisation – the spontaneous formation of order when energy is input into a system. One of the most complex selforganisation mechanisms is reaction-diffusion

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Dynamics of Particle-Based Reaction-Diffusion Computing

(RD) patterning, where complex patterns emerge from almost homogeneous environments. Turing’s explanation of how complex patterning could emerge in an almost homogeneous two chemical system, which would previously have been expected to remain at equilibrium, laid the foundation for future exploration of spatial chemical patterning (Turing, 1952). Babloyantz suggested that the travelling wavefronts from chemical reactions using excitable media could be used to propagate information for spatial problems (Babloyantz & Sepulchre, 1989). The non-equilibrium dynamics of the chemical reactions that underlie excitable media have both an autocatalytic component and a diffusion component. The resulting wavefront propagates forwards at a constant velocity, does not dissipate (at least not significantly) over time and has a post-excitation refractory period so that the shape of the advancing front is made visible. Chemical waves are annihilated at boundaries and where fronts meet, so avoiding reflection and interference patterns. Steinbock, Toth and Showalter used wave propagation in the Belousov-Zhabotinsky (BZ) chemical reaction to traverse the shortest path through a maze (Steinbock, Toth, & Showalter, 1995). Although computationally efficient, since the waves perform a parallel search or route branches, the spatial requirements are substantial since a direct spatial encoding of the problem must be stored, as opposed to a graph encoding in conventional approaches. There are some disadvantages to the solution of the maze by chemical wave approaches. The first is that the propagation speed of the reaction is inherently slow. Steinbock et al. estimated that their wavefront moved at 2.41mm per minute (approximately 0.04 mms-1). In an attempt to overcome the speed limitation, Asai et al. developed hybrid analogue/digital silicon VLSI circuits that implemented wave propagation and were used to compute approximations of Voronoi diagrams (Asai, Costello, & Adamatzky, 2005). Karahaliloglu also implemented a hardware shortest path

device in a coupled cellular neural network on a CMOS substrate (Karahaliloglu, 2006). Ito et al. developed a software model of reaction-diffusion and used a separate backtracking algorithm to traverse the path backwards through the obstacle field to find the shortest path (Ito, Hiratsuka, Aoki, & Huguchi, 2006). Although reaction-diffusion front propagation provides a powerful means of parallel search, the difficulty in eliciting the results from the wavefront map necessitates a separate stage to ‘read out’ the results. This stage can be a separate software algorithm to traverse the front path, or hybrid systems where chemical processor result is sampled and recorded for further semi-automated processing. Adamatzky noted the difficulty in configuring and programming chemical processors and suggested a multi-layered approach where the output of one reactor could influence the behaviour of another (Adamatzky, 2005). As an example of a multi-layered approach Adamatzky et al. presented a software robot whose trajectory was influenced by reaction-diffusion processor fields via chemo-attraction (as a homing device) and chemo-repulsion (as an obstacle avoidance measure) (Adamatzky et al., 2004). Most experimental models of reaction-diffusion computing are based upon the reactiondiffusion processes seen in chemical systems, as described by Turing’s original description. The prototypes have in common the feature that they use the non-equilibrium kinetics of the chemical reactions to support the propagation of information. By using the propagation to encode data signals, and channelling the signal paths, useful computational information can be inferred by the timing of the arrival of signal pulses, such as distance or the presence or absence of logical values. It is also possible to obtain complex reaction-diffusion patterning without explicitly requiring chemical components. Oster noted that many different models of pattern formation (including classical RD) could be described in terms of systems which contained local autocatalytic

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activation and long range inhibition (Oster, 1988). Examples of natural systems exhibiting complex patterning which exploit non-chemical means of reaction-diffusion include stone sorting in patterned ground (Kessler & Werner, 2003), complex patterning in granular media (Ouyang & Swinney, 1991), and the spontaneous assembly of vascular networks (Serini et al., 2003). Bonabeau has shown that complex RD-type patterning can be generated in discrete multi-agent systems without reference to chemical concentration gradients (Bonabeau, 1997). The final result of self-organised pattern formation are spatial patterns which, although they may have functional properties (such as vascular networks), are usually apparently static. One organism, the true slime mould Physarum polycephalum, has been shown to exhibit complex behaviour which may be described as dynamic pattern formation. Physarum is a member of the Myxomycota phylum, a unicellular multinucleate organism whose single cell is very large and often visible to the naked eye. During its vegetative stage of growth, the plasmodium stage, the organism is usually visible to the naked eye, its amorphous cytoplasm usually lending it an amoeboid appearance. The plasmodium is a syncytium of nuclei within a cytoplasm comprised of a complex gel/ sol network. Local oscillations in the thickness of the membrane spontaneously appear with approximately 2 minutes duration (Takagi & Ueda, 2007). The spatial and temporal organisation of the oscillations has been shown to be extremely complex (Takamatsu, 2006) and affects the internal movement of sol through the network by assembly and disassembly of the local actin-myosin structures. The protoplasm moves backwards and forwards within the plasmodium in a characteristic manner known as shuttle-streaming. The plasmodium is able to sense local concentration gradients and the presence of nutrient gradients appears to alter the structure of external membrane areas. The softening of the outer membrane causes a flux of

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protoplasm towards the general direction of the gradient in response to internal pressure changes caused by the local thickness oscillations. The strong coupling between membrane contraction and streaming movement is caused by the incompressibility of the fluid requiring a constant volume – the weakening of the membrane provides an outlet for the pressure. When the plasmodium has located and engulfed nearby food sources, protoplasmic veins appear within the plasmodium, connecting the food sources. The purpose of the veins is to transport protoplasm amongst the distributed extremes of the organism. The effect is to both maximise foraging area (during the foraging stage) and to minimise nutrient transport distance by the formation of the protoplasmic veins, whilst at the same time maintaining a fault tolerant connectivity that is resilient to damage (Nakagaki, Kobayashi, Nishiura, & Ueda, 2004). The relative simplicity of the cell and the distributed nature of its control system make Physarum a suitable subject for research into distributed computation substrates. In recent years there has been a wealth of research into its computational abilities, prompted by Nakagaki et al. who reported the ability of Physarum to solve path planning problems (Nakagaki, Yamada, & Toth, 2000). Subsequent research has confirmed and broadened the range of abilities to spatial representations of various graph problems (Nakagaki, Yamada, & Hara, 2004), (Shirakawa, Adamatzky, Gunji, & Miyake, 2008), (Adamatzky, 2008a), combinatorial optimisation problems (Aono & Hara, 2007), construction of logic gates (Tsuda, Aono, & Gunji, 2004) and logical machines (Adamatzky, 2007), and as a means to achieve distributed robotic control (Tsuda, Zauner, & Gunji, 2007), robotic amoeboid movement (Ishiguro, Shimizu, & Kawakatsu, 2006), and robotic manipulation (Adamatzky & Jones, 2008). Adamatzky et al., have demonstrated that the computational properties of the propagation and minimising behaviour of Physarum can be considered as equivalent to those seen in chemical instantiations of reaction-

Dynamics of Particle-Based Reaction-Diffusion Computing

diffusion computing (Adamatzky, De Lacy Costello, & Shirakawa, 2008). In this paper we explore the notion of passive and active approaches to reaction-diffusion computing. We consider two almost identical software approaches, inspired by chemical prototypes of RD computation and physical instantiations of Physarum computers. Despite being inspired by very different physical substrates, both approaches can be placed within the framework of reaction-diffusion computation and thus both can be based upon the same sensory behaviour of simple agent particles in diffusive environments. The two approaches (active and passive) differ only in their interactions with the diffusive environment. We also consider chemo-attraction and chemo-repulsion dynamics for both approaches. We find that both approaches exhibit complex and desirable computational behaviour and conclude that the active approach yields particularly rich behaviour, due to the emergence of second-order quasi-physical behaviour. The remainder of the paper continues as follows: We introduce the underlying multi-agent particle framework by describing the particle morphology and algorithmic behaviour common to both approaches. We then present results from the passive approximation of chemical RD computing for image processing and path planning tasks using repulsion and attraction dynamics respectively. We then describe the active approach, describing the critical difference which separates passive from active. We explore the complex pattern formation behaviour seen in the active approach using both attraction and repulsion dynamics and demonstrate results pertaining to the formation and evolution of dynamic transport networks, network minimisation tasks, and the emergence and control of collective oscillatory behaviour and amoeboid movement. We conclude with a summary of both active and passive approaches for both attraction and repulsion dynamics and suggest possible advantages and disadvantages of each approach.

PARTICLE BASED APPROXIMATIONS OF REACTIONDIFFUSION COMPUTING The method used in both passive and active approaches is based upon emergent behaviour exhibited by a population of simple, reactive mobile particles. The particles are coupled to diffusive environments in which attractive and repulsive stimuli are represented by areas which propagate synthetic chemical gradients. The difference between the two approaches is simple: The passive approach only couples the sensory apparatus of the particles to the diffusing field – the particles’ resulting actions do not affect the properties of the field (collective particle actions are ‘written’ to a separate homologous structure). In the active approach the particles both sense and modify the diffusing field, providing a strong coupling between particle and environment interactions. Due to space constraints, only minimal details of the model are discussed, in favour of an overview of results from each method. For more detailed descriptions of the model and parameter settings used for synthetic chemical approximation and Physarum approximation see (Jones, 2008) and (Jones, 2010a) respectively. Due to the dynamical nature of the model output the reader is encouraged to refer to the video recordings available at: (http://uncomp.uwe.ac.uk/jeff/passive_active_attraction_repulsion.htm).

PARTICLE MORPHOLOGY AND BEHAVIOURAL ALGORITHM Both passive and active approaches to particle based RD computing use the same representation for the particles. Differences in the approaches will be indicated in the text. The behaviour of each particle is very simple indeed, amounting to a reflexive stimulus-response dynamic. Each particle is located on a discrete two-dimensional lattice. Each particle can be addressed by its unique

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position on the lattice and each also has an internal representation of its current angular orientation. To overcome the orthogonal limitations of the lattice each particle also has an internal floating point representation of its position and orientation within the lattice which can be rounded to an integer representation. The morphology of the particle and pseudocode demonstrating the sensory algorithm is shown in Figure 1. The particle behaviour is forward biased and its location in the lattice is denoted at position ‘C’. The particle samples chemo-attractant levels at the positions of its three forward offset sensors (FL, F, FR) at every scheduler step. The angles of the two outer sensors from the forward position can be specified by the Sensor Angle (SA) parameter. The sensor values are used as inputs to the behavioural algorithm. The outcome of the algorithm may be a change in angular orientation by the Rotation Angle (RA) parameter. The attraction or repulsion dynamics can be altered by setting the change in orientation preference, which can be either chemo-attractive (rotate towards the sensor with the strongest concentration) or chemo-repulsive (towards the lowest concentration). The pseudocode algorithm shown is for the chemo-attractive (attraction) response and the chemo-repulsive (repulsion) response is simply the opposite behaviour in response to chemo-attractant levels, in which case

Figure 1. Particle behavioural algorithm based on chemo-attraction and particle morphology

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the particle will attempt to move away from the stronger source. [Motor stage] - Attempt move forwards in current direction - If (moved forwards successfully) ╅╅╅╅╅ Deposit trail record at new location [Sensory stage] - Sample concentration gradient diffusion field values↜- if (F > FL) && (F > FR) ↜ ̀̀̀̀ - Stay facing same direction ╅╅╅╅╅ - Return - Else if (F < FL) && (F < FR) ╅╅╅╅╅ Rotate randomly left or right by RA - Else if (FL < FR) ╅╅╅╅╅ Rotate right by RA - Else if (FR < FL) ╅╅╅╅╅ Rotate left by RA - Else ╅╅╅╅ Continue facing same direction

At each iteration of the sensory algorithm, the particle attempts to move forwards one step in the current orientation. If the step is successful, a record of the movement is deposited. If the particle is not able to move forwards successfully (for example, if the new site is already occupied), the move is abandoned and a new orientation (from 0-360 degrees) is randomly selected. In the passive approach, the record is stored in a separate data structure and the diffusive field is not disturbed. In the active approach the record of movement is inserted directly into the diffusing field, directly altering the contents and propagation of the diffusion field.

Dynamics of Particle-Based Reaction-Diffusion Computing

PASSIVE PARTICLE APPROXIMATION OF REACTIONDIFFUSION COMPUTING The passive particle approximations of RD computing use a spatial representation of the problem definition and the problem solution. A full description of the method can be found in (Jones, 2008) and a brief overview of the method follows. The problem data is represented by a greyscale image which is projected onto a multi-layered two dimensional environment and is subject to diffusion according to the following algorithms: Data Projection: For each pixel, Ix,Iy in Image I: {↜Set diffusion map at position (Mx, My) to value in (Ix, Iy) * projection_ copy_weight↜} Data Diffusion: For each pixel Mx, My in diffusion map M in parallel: IF (Mx, My == WALL or BOUNDARY)

Absorb all chemoattractant at this location ; Else: Diffuse Mx, My: { Temp = Mx-1, My-1 + Mx, My-1, + Mx+1, My-1 + Mx-1, My + Mx+1, My + Mx-1, My+1 + Mx, My+1, + Mx+1, My+1 ; Temp /= 4 ; Mx, My = Temp – Mx, My ; Mx, My = Mx, My * diffusion_damping_ value ;}

The data in the projection layer diffuses out from the initial sites at every scheduler step and the projection data is refreshed every 5 scheduler steps to maintain the diffusion gradients. The refreshing of the projection data and the diffusion results in an outward propagation of the data pattern, as shown in Figure 2.

Figure 2. Illustration of the effect and visualisation of data projection and diffusion. Top row: Point source data, plot of diffusion gradient, actual diffusion gradient, enhanced gradient to reveal front propagation. Bottom row: Propagation of wavefronts from concave and convex initial sources at identical speeds to a planar front shape

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The propagation of the projected stimulus on the diffusion map is not easily visible due to the rapid dissipation of the signal away from the stimulus point. The cross-section plot illustrates the effect of removing the D.C. component of the wave (by clipping values below zero). Visualisation of the diffusion field was improved by employing a local dynamic scaling of the diffusion map which rendered the propagating fronts visible. The wavefront propagating outwards from the point source is approximately circular. The front propagation is independent of the stimulus orientation and propagation from concave and convex stimuli proceeds at identical rates, leading to planar waves. The particle population resides on a second layer with identical dimensions to the diffusive layer (Figure 3). Each particle in the population senses the synthetic chemo-attractant concentrations in the diffusion layer and orients its sensors and moves in response. The particle response is either positive chemo-attraction (towards concentration gradient) for path planning applications, or negative chemo-repulsion (away from the gradient) for skeletonisation and Voronoi diagram approximation. The collective and historical movement of the particles is recorded Figure 3. Multi-layered approach used in passive approximation of chemical reaction-diffusion computing

in another separate layer so that the result of the computation can be recorded. Note that although the particles sense the diffusing chemo-attractant levels their response does not affect the propagation of information in the diffusion layer. The behaviour can be said to be passive – the particles can be considered as floating mobile ‘markers’ to indicate features such as the path towards the strongest chemo-attractant source, or the locations where wavefronts are annihilated (as in chemical reaction-diffusion systems where a precipitate indicates the computational output). Different applications of the passive approach to particle based RD systems depend on the particle responses to the chemo-attractant gradients and sample results from skeletonisation, Voronoi diagram approximation and path planning are provided below.

IMAGE SKELETONISATION Image skeletonisation is a transformation to recover an efficient and minimal shape descriptor for a shape. For the skeletonisation task, the desired behaviour of the particles is to orient themselves away from the diffusing diffusion field (chemo-repulsive behaviour). The particles are also initialised so that they do not populate the areas outside the area of problem representation – a black silhouette of the image. If a particle moves outside this area it will immediately be moved to a randomly chosen unoccupied position inside the black area. Since only the silhouette areas can be occupied, care must be taken that the %p parameter (population size as a percentage of image area) accurately reflects the possible habitable area as opposed to the entire image landscape. A pseudocode description of the emergent pattern formation approach to skeletonisation follows: Initialise population: Place particles at random positions within the silhouette area.

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Figure 4. Utilising front propagation and chemo-repulsion for skeletonisation. Top Row: original source data, projected diffusion data, visualisation of wavefront, result of standard Matlab algorithm↜Bottom Row: Particle population snapshot, record of collective movement, trails after erosion, overlay of skeleton on source data

Figure 5. Particle approximation of chemical processor performing internal skeleton from planar shapes. Top Row: Digitised representation of problem, particle population distribution, emergent synthetic precipitate pattern. Middle Row: Comparison of emergent pattern formation skeleton (left) with chemical processor skeleton (right). (Chemical processor images courtesy of Adamatzky and Costello). Bottom Row: Enlargement of particle framework bisector shows accurate placement but thinning of line edges (left). The same area in the chemical processor shows a distortion of the bisector position closer to the convex point

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For each system step: { Copy weighted contents of image landscape to the diffusion field map Diffuse map: For each particle: { Sample local diffusion field levels Orient particle towards locally lowest of the frontal sensors Attempt to move forwards in current orientation If Moved forwards successfully: Deposit trail record at new location If new site is blocked: Jump to randomly chosen unoccupied area inside the silhouette } }

The repulsive field initiated at the shape border causes the particles to move away from the field. The furthest points away from the field are the interior sections of the image. The particle distribution shown in Figure 4 (second row, left) shows particle aggregation at the medial axis of the figure (the points where the wavefronts collided) and this is reflected in the strength of the emergent record of movement patterns. Compared to the classical skeletonisation algorithm, the emergent patterns show a more natural curvature and symmetry. Pruning the spurs (which are less evident in the particle approach when compared to the classical algorithm) is simply a matter of performing a standard uniform erosion operator (external environment selection pressure) to the movement record field as the system evolves. If the erosion value is just less than the global trail deposition rate (the mean amount of trail deposited per cell, per system step), the central skeleton trail persists whereas the spur artifacts are eroded away. The natural curvature of the lines reflects the convex circular shape of the propagating wavefronts and the probabilistic behaviour of the particle movements.

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APPROXIMATION OF CHEMICAL PROCESSOR SKELETONISATION A reaction-diffusion chemical processor was used for skeletonisation experiments by Adamatzky, De-Lacy Costello and Ratcliffe (Adamatzky, de Lacy Costello, & Ratcliffe, 2002). Masks of the source images impregnated with potassium iodide were placed on an agar dish containing a quasitwo dimensional layer of palladium chloride gel. Wavefronts were initiated at the border of the mask and a precipitate formed at all points except for where two wavefronts converged. The chemical processor was simulated using the framework by having the filter paper areas (white) acting as diffusion sources and the particle population as chemorepulsive markers. The results from the chemical processor and the emergent pattern formation approach are shown in Figure 5.

APPROXIMATING THE INVERSION OF THE CHEMICAL PROCESSOR PLANAR SKELETON By using the original greyscale trail data it is also possible to invert the skeletonisation of planar shapes. Figure 6 shows the emergent trail pattern generated by the simulation of Adamatzky’s chemical processor when used as a diffusion source. The particle population is repelled from the wavefronts emanating from the skeleton and migrate to positions which represent the central locations of the source bars in the original problem.

APPROXIMATION OF VORONOI DIAGRAM The same propagation of information method that was used to construct skeletons of shapes may be used to construct approximations of Voronoi diagrams. Instead of a front emanating from the border of an enclosed planar shape, a collection

Dynamics of Particle-Based Reaction-Diffusion Computing

Figure 6. Reconstruction of original planar shape positions from external greyscale skeleton. Top Row: Emergent synthetic precipitate pattern skeleton source image (inverted for clarity), initial stages of experiment showing particle migration from diffusion sources. Bottom Row: Final position of particles, Emergent trail pattern overlaid onto original image shows reconstruction of shape positions

of circular wavefronts are created from individual point sources. Since the waves propagate at a constant velocity, the areas at which the expanding waves collide (and are annihilated) correspond to the bisectors of the Voronoi partitions (Figure 7).

DYNAMIC RESPONSE TO CHANGING DATA CONFIGURATION The software approach to RD computing allows the possibility of computation in dynamically changing environments. The problem dataset can be changed by simply specifying a new pattern to

Figure 7. Construction of Voronoi diagram approximation using chemo-repulsion method. Top Row: Evolution of expanding diffusive wavefronts from point sources. Bottom Row: Diffusion source points, particle positions, approximation of Voronoi diagram at annihilation points

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be projected, replacing the previous pattern. The particle population is able to respond automatically to the change in configuration, although there is a short period of confusion when the old wavefront pattern competes with the emergence of the new front. By applying a simple erosion operator to the record of particle movement trails, the old record gradually fades to be replaced by the new result. Examples of the population response to a change in problem configuration can be seen in Figure 8 for both the skeletonisation task (a change in problem configuration) and the Voronoi diagram approximation task (an increase in the number of data point sources).

PATH PLANNING BY CHEMOATTRACTION RESPONSE For path planning problems the repulsive chemotaxis behaviour was changed to chemo-attraction behaviour. The particle this time orients itself towards the locally highest concentration of diffusing chemo-attractant. Initial experiments comprised a simple path choice between two channels. The image (Figure 9, top) contains a source for the diffusion (left) and the particles must be confined to start within a particular area of the path (white area on right). The light grey area represents the area where waves are free to propagate and the surrounding darker grey areas are obstacles at which all waves will be absorbed. When particles reach the diffusion source they are

Figure 8. Response to dynamic problem environments by the particle population. Top Row, left to right: Skeleton emerging from sitting dog image (left), particle population distribution, emergent trail pattern.↜Original landscape is then replaced with the (far right) image of a standing dog. Second Row, left to right: Dynamic reformation of the skeleton shown by particle positions. Emergent trail patterns show ‘confusion’ state where new record of standing dog is stored with the old sitting dog. Finally only the new stimulus remains. Bottom Two Rows: Increasing number of diffusion point sources from two to seven (one addition per panel) introduces new competing diffusion point sources whose competing wavefronts represent Voronoi bisectors

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Figure 9. Shortest path solution in a very simple path choice problem

immediately transported back to the initial start position area. Panel a) shows the partial propagation of the wavefront along the labyrinth. The wave is initiated at the source on the left side of the figure and splits when it encounters a junction, traversing both paths in parallel. Because the waves move at a constant velocity, the wavefront traversing the lower (shorter) path moves further along the labyrinth than the wave travelling the upper path. Before the wave has completed the labyrinth there is a period of initial confusion in the population (since there is no chemotactic path to follow), and random movement ensues with particles taking both the upper and lower path (panel b). Shortly afterwards the wave has completed its traversal (panel c) and the lower wave, arriving at the final junction first, travels the remaining distance to

the particle source. When the two separate wave fronts meet, the waves are apparently annihilated at the point of contact. The particles choose the shorter path (panel d) and the historical emergent trail record left behind by the particles (panel e) corresponds to the shortest path through the simple labyrinth. The path following behaviour can be extended to more complex examples. The wavefront is initiated at the exit and propagates backwards throughout the maze in parallel (Figure 10). When the wavefront reaches the particle start area, the particles can follow the path of strongest chemoattractant and identify the shortest path through the maze (Figure 11) by the record of the collective particle movement trails.

Figure 10. Initiation and propagation of wavefront through a maze

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Figure 11. Shortest path planning in more complex environments

DYNAMIC RESPONSE TO CHANGING PATH PLANNING ENVIRONMENTS The particle population is also able to dynamically track changes to the environment without having to restart the algorithm. The iterative nature of the framework ensures that the population is collectively able to maintain a record of the shortest path via the trail map. An illustration of the response timeline to changes in the environment is shown in Figure 12.

HARNESSING QUANTITATIVE PROPAGATION FOR PATH COST ASSIGNMENT One significant difference between the particle based approach and chemical implementations of RD computing is that in the particle approach, the wave propagation has quantitative as well as qualitative properties. In chemical approaches only the timing of the wave propagation is used for the computation, the wavefront which arrives first is the ‘winner’ and the waves are annihilated on contact. In the particle based approach the amount of diffusing chemo-attractant also has an

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influence on particle path choice. In labyrinths with identical path widths, this property is not taken into consideration and only the timing of propagation is utilised. The quantitative aspect of the front propagation can be harnessed and exploited to assign path costs, as illustrated in Figure 13. Figure 13 shows a labyrinth with two possible paths, the right (narrow) side of which is considerably shorter. As befitting the shorter length, the wavefront on the right side arrives first at the particle start location and would be expected to be traversed by the particles. The narrow width of the right side path, however, acts to constrict the amount of chemo-attractant flowing through that path and the annihilation point where the two waves met is shifted to the right hand side by the strong competition of the flow in the longer left channel (top row, circled). For this reason the particles follow the path of greater flow and choose the longer side. When the width of the right side channel is changed to match that of the left side, the greater flow in the right side shifts the annihilation point back to the left and the particles choose the shorter right side path (middle row, circled). When a simple erosion operator is used on the particle movement trail map the new path choice replaces the previous path choice over-time, in response to path width changes (bottom row,

Dynamics of Particle-Based Reaction-Diffusion Computing

Figure 12. Evolution of diffusion field and shortest path trail pattern in a dynamic environment. a) Initiation of wavefront from source, particle movement is random at this point (~100 system steps); b) Wavefront reaches particle positions, particles begin to follow concentration gradient (~430 steps); c) Wavefront completes traversal of entire maze (~700 steps); d) The first of the particles completes the path through the maze (~920 steps); e) The landscape configuration is replaced with a longer, more complex maze (~1100 steps); f) Population confusion as the new wavefront competes with the older collapsing front (~1500 steps); g) The new wavefront completes its traversal through the entire maze and particles follow the new path (although the path is longer, the older and shorter path is no longer available). The particle positions indicate the new path but remnants of the older path persist in the trail pattern (~2100 steps); h) The erosion selection pressure ensures only the new path persists and the old path is ‘forgotten’ (~4000 steps)

right). The width of the channels may therefore be used to assign additional costs to the problem routes, as with real life problem instances where, for example, a shorter path may be subject to more congestion.

ACTIVE PARTICLE APPROXIMATION OF REACTIONDIFFUSION COMPUTING The active approach to particle RD Physarum computing was introduced in (Jones, 2010a) as a means of exploring the bottom-up construction of emergent transport networks. It was found that the collective behaviour of the particle population mimicked the behaviour of the Physarum

plasmodium in a number of ways (including foraging, streaming movement, and network optimisation). A single particle represents a hypothetical particle of Physarum plasmodium gel/sol structure. When a particle moves, the movement can be said to represent the protoplasmic flux of sol. When a particle is not able to move it can be said to represent the immobile gel matrix. The active method differs from the passive approach by the fact that the record of particle response to stimulus is inserted into the diffusion field instead of being recorded into a separate map structure. Modifying the diffusion field significantly affects the properties of the field, both in terms of the spatial distribution of chemo-attractant and the strength of diffusion sources. The autocatalytic component of the reaction-diffusion system is

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Figure 13. Utilising the quantitative properties of front propagation to assign path costs

provided by the collective deposition by the particle population into the diffusion field and the attraction to their own deposition of chemoattractant (autocrine stimulation), and the long range inhibition is generated both by the diffusing field and the reduction in stimuli provided by the evacuation of particles from low stimulus areas of the environment. Since there is no requirement to visualise the shape and peaks of the propagating synthetic chemical wavefronts, a more simple scheme of diffusion, based on a mean filter kernel, is used to propagate the information outwards. The diffusion can be damped to restrict the area

of the propagation, and weighted to change the slope of the diffusion gradient.

ATTRACTION BEHAVIOUR EVOLUTION OF COMPLEX DYNAMIC TRANSPORT NETWORKS The results shown in Figure 14 indicate the simplest possible results from the active approach using chemo-attraction behaviour when the diffusion field is initially devoid of stimuli. The particle population is initialised at random positions and orientations within the environment

Figure 14. Emergence and evolution of complex transport networks in the active approach to particle reaction-diffusion

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which has periodic boundary conditions. The movement of particles results in mass deposition into the diffusion field. The subsequent diffusion of the field results in local areas of stronger chemo-attractant gradient which attract nearby particles. The initially random arrangement of particles begins to self-assemble into streams of movement. The streams merge and a network is formed, composed of bi-directional particle movement. The movement of individual particles within the network flows, however, mimics the shuttle streaming seen in propagating Physarum pseudopodia. The arrangement of the network shows paths composed of particles surrounding lacunae where no particles are present. The network undergoes a complex evolution where smaller lacunae are seen to shrink and close, and larger lacunae grow in size. At the default sensor angle (SA=22.5 degrees) and rotation angle (RA=45 degrees) the network evolution continues to reduce in complexity (the number of lacunae reduces) until bifurcations spontaneously appear in some of the network paths. The new sprouting paths cross the lacunae to merge with the network flow at a different area and the network evolution continues in this manner indefinitely. Not only do the networks form by self-assembly, but the networks appear to be self-balancing

during their evolution. Since the number of particles remains constant throughout a run of the simulation, an increase in particle flow in one area (for example in the sprouting of a new path) must result in a change of structure in another part of the network. Conversely, the closure of a cyclic area results in a temporary cluster of particles at the centre of the former cycle. This cluster of particles is redistributed automatically throughout the network as the evolution continues and the network path thickness remains evenly distributed. After the initial network formation and contraction, the network evolution becomes self-balancing as the network adapts to stabilise the number and mean size of the lacunae (Figure 15).

EMERGENCE OF QUASI-PHYSICAL PROPERTIES - APPARENT SURFACE TENSION AND NETWORK CONTRACTION When both sensor angle and rotation angle are both set to 45 degrees and boundary conditions are fixed, the bifurcation of network paths does not occur later in the network evolution and the network contracts as the lacunae close (Figure 16).

Figure 15. The dynamic network becomes self-balancing as the number and size of lacunae stabilises. Left: Number of lacunae in typical dynamic network over 5000 steps. Right: Mean size of lacunae over 5000 steps in same network

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Figure 16. Network condensation and contraction into a solid mass (top two rows) and plasmodial sheet (bottom row)

The length and area of the network is minimised by the contraction process. The contraction continues until the population forms a solid mass. For larger population sizes the contraction results in a deformable plasmodium-like sheet (Figure 16, bottom row). The plasmodial sheet maintains the minimal area in response to perturbation. When periodic boundary conditions are present, the contraction of the network cannot condense into a solid mass due to connectivity between the left and right side of the environment and between

the top and bottom of the environment. The contraction of the network does continue (with this constraint) until the network length is minimal. The network connectivity approximates a hexagonal structure and, when repeated (Figure 17, bottom right), a periodic hexagonal tiling is observed, satisfying the minimal network coverage of an area (Hales, 2001). Unlike the sprouting self-balancing networks which maintain their high connectivity rate by sprouting new paths, the contracting networks

Figure 17. Fixed boundary conditions results in network contraction to a hexagonal structure

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undergo a gradual reduction in the number of lacunae until the hexagonal pattern is reached (a single lacuna, Figure 18). The contraction of the dynamic network appears to exhibit second order, quasi-physical, properties resembling surface tension effects. The final hexagonal tiling is stable and self balancing (the structure is maintained despite the constant turnover of particle positions) and consists of the bi-directional flow of particles.

REPULSIVE DYNAMICS IN THE ACTIVE APPROACH – DISSIPATIVE PATTERN FORMATION The previous examples of network formation with the active approach used chemo-attraction behaviour, eliciting a quasi-physical behaviour within the network evolution. Chemo-repulsive behaviour may also be used. In this method the particles still deposit a record of their movement in the diffusion field but are repulsed by the concentration of material in the field – particles orient themselves away from the higher concentrations in the field. Complex collective patterns are also seen in the repulsion behaviour. Using SA and RA

values of 45 degrees the particles form regular self-organised hexagonal arrays whose size is dependent on the SO scale parameter (Figure 19, top row). Other parameter values resulted in the emergence of dynamical striped patterns (Figure 19, bottom row). There is no cohesion of the particle population when repulsive behaviour is used and the patterns are dissipative in their nature, strongly resembling classical Turing patterns. By smoothly varying the sensory parameters between zero and 180 degrees it is possible to obtain a parametric mapping for both attraction and repulsion behaviour in the active approach (Jones, 2010b). For each particle, at every scheduler step, the SA parameter is set according to the particle position across the lattice. The RA parameter is set according to the position down the lattice (see Figure 20). For attraction behaviour the mapping illustrates differences in cohesion of the collective ‘material’ composed of the particle interactions. The influence of the changes in cohesion results in different dynamical evolution of the parameter regions showing reticulated, labyrinthine and island-like patterning (Figure 20, left). It may be surmised that changes to the sensory parameters mimic physical changes to material properties (for example humidity or

Figure 18. Record of the decreasing number of lacunae in minimising networks. Number of lacunae in typical minimisation network over 30000 steps. Momentary spikes are caused by temporary erroneous detection of clusters as lacunae as the network closes

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Figure 19. Emergence of dissipative dynamical patterns when repulsive behaviour is used in the active particle approach. Darker regions indicate greater particle flux. Top Row: Hexagonal pattern - SA 45, RA 45, SO 23, %p 15 (6000 particles, 200x200 lattice). Bottom Row: Striped pattern - SA 112.5, RA 67.5, SO 15%p 15 (6000 particles, 200x200 lattice)

desiccation in the Physarum plasmodium). Changes in these properties thus affect patterning behaviour. As an example, in Physarum the patterning changes from reticulated patterns to dendritic patterns and to isolated island patterns when environmental conditions become less favourable (Takamatsu, 2009). In the case of repulsion behaviour (Figure 20, right) the mapping diagram illustrates the dissipative nature of the interactions – there are only a few regions which

yield persistent patterning behaviour, characterised by the hexagonal and striped regimes. Although both attraction and repulsion behaviour result in complex patterning there are definite differences concerning pattern type and persistence between the two. Attraction behaviour generates patterns with apparent physical properties of evolution whereas the repulsion behaviour shows a simpler patterning regime with more regular and periodic structures. Although the

Figure 20. The effect of adjusting SA (left to right) and RA (top to bottom) on pattern formation behaviour. Left: attraction behaviour. Right: repulsion behaviour

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repulsion behaviour generate simpler patterning it has been shown that these patterns can be more predictably regenerated when the pattern is damaged, reforming the original pattern (Jones, 2010b). The attraction behaviour can regenerate the general pattern type when damaged, but cannot exactly reproduce the original pattern.

EXPLOITING QUASI-PHYSICAL ATTRACTION BEHAVIOUR FOR NETWORK OPTIMISATION By inserting diffusion stimulus sources into the diffusion field, external nodes of attraction can be placed to affect the dynamics of network formation. As shown in Figure 21, the network condenses around the nodes as the attraction of the network flow to the stimulus nodes is greater than the attraction of the default network contraction. The distance between the nodes is minimised by the apparent tension effects on network evolution. For two nodes, the network simply straightens to form a straight line. For increasingly complex arrangements of nodes, the network stabilises into approximations of Steiner trees (the minimum

amount of network material required to connect all nodes in a network).

NETWORK MODIFICATION BY MODIFICATION OF STIMULUS WEIGHTS Modifying the strength of the stimulus node stimuli affects the condensation of the network around the nodes. Lower stimulus weights appear to reduce the influence of the nodes on network formation – the apparent ‘tension’ of the network is reduced. Increasing the stimulus weight appears to ‘tighten’ the network contraction at the nodes. Figure 22 illustrates an example where the network evolution around the stimulus nodes (first image) initially forms a cyclic area. By reducing node stimulus weights the cycle closes under the influence of the tension inherent in the network flow. Later, increasing the node stimulus strength appears to tighten the pattern, transforming the Steiner tree pattern into a spanning tree pattern (final image, the Steiner nodes are removed by the increase in tension).

Figure 21. Constraining network condensation results in network minimisation and approximation of Steiner trees

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Figure 22. Effects of reducing and increasing node stimuli weights on cyclic links and Steiner nodes

SYSTEMATIC DEFORMATION OF A ‘PLASMODIAL SHEET’ STRUCTURE The initial condensation of the network around the stimulus nodes is dependent on the initial (random) distribution of particles. When a large number of stimulus nodes are present, this can result in network formation and contraction that differs during each experimental run. These are manifested as cyclic regions in the network (which may or may not be desirable, depending on the requirements of network length vs. network fault tolerance), and subtle differences in the final contraction network pattern (Figure 23, top row, from three separate experimental runs). To ensure that the

final network pattern is reproducible over repeated runs it is possible to perform the contraction on a deformable plasmodial ‘sheet’ of particles. By initialising the environment with a large population of particles, a synthetic plasmodial sheet forms over the node stimuli, eliminating cyclic areas. By randomly removing particles at a predefined rate (for example a probability of removing each particle of 0.00025 per scheduler step) the plasmodial sheet shrinks and dynamically adapts to the reduction in size (Figure 23, middle row). Slow rates of particle removal are necessary to avoid tearing the plasmodial sheet – occurring when the population cannot adapt quickly enough to the changes in tension of the sheet. The adaptation

Figure 23. Use of plasmodial sheet shrinkage method provides identical network convergence and more regular results

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of the plasmodial sheet in response to population shrinkage results in more reliable convergence to identical network patterns (Figure 23, bottom row – three separate experimental runs). Variations in node stimulus concentration strongly and differing initialisation conditions significantly affect network evolution and the behaviour of the particle collective has been shown to reproduce the network connectivity of Physarum in the approximation of proximity graph construction (Adamatzky, 2008), (Jones, 2009).

EMERGENCE OF COLLECTIVE AMOEBOID MOVEMENT IN THE SYNTHETIC PLASMODIAL SHEET The cohesive properties of the synthetic plasmodial sheet of particles, mimics, to a certain extent, the large protoplasmic mass seen in Physarum plasmodia. The plasmodium of Physarum can exhibit amoeboid movement towards attractive stimuli (for example food or warmth gradients) and away from harmful stimuli (such as certain frequencies of visible light and noxious chemical stimuli). The movement is due to the extension of pseudopodia at the outer surface of the plasmodium into which protoplasmic sol surges, the actual underlying mechanism of which may be the assembly and disassembly of actin/myosin structures. The surging of sol through the gel/sol matrix results in strong inertial movement of the plasmodia. The exact control of the movement is still, to a large extent, unknown but is thought to be related to the spontaneous formation of distributed oscillators within the plasmodium. The fact that such a complex and distributed control mechanism can exist in a syncytium of a unicellular organism suggests possible mechanisms of distributed control of robotic systems. Unlike the Physarum plasmodium, the default behaviour of the synthetic particle based plasmodial sheet does not exhibit collective movement. This is due to the weak motor coupling between

the particles. Although the particles have strong sensory coupling (due to the sensors sensing chemoattractant concentration produced by other particles distant from the actual particle position), the motor coupling is weak. This is because an obstruction (such as an occupied cell) results in no forward movement and a loss of directional persistence (due to the subsequent random change in direction). Although this is sufficient to ensure cohesion of the collective and automatic adjustment of shape in response to environmental stimuli, it does not result in significant collective movement. The particle collective may be said to approximate soap film dynamics, or the evolution of lipid nanotube networks (Lobovkina, et al. 2008). By slightly modifying the particle behaviour to increase the motor coupling strength it was possible to generate complex emergent internal oscillations, amorphous plasmodium shape and collective amoeboid movement. To implement the strong motor coupling the particle behaviour was changed so as not to automatically select a new direction if the selected site was blocked. The particle instead halted at the current cell and incremented an internal positional vector until a space in front of the particle became unoccupied. This effectively results in particle being able to ‘push past’ each other in the event of a collision. Since there may be a momentary delay until a site becomes vacant, the result is a surging, inertial movement of the particle population. The effect mimics the inertial flux of sol in the gel/sol matrix of the Physarum plasmodium. The strength of the inertial effect can be modified by setting a parameter (pCD – probability of a change in direction, from 0-1) to provide a probability of resetting the particle’s internal position and selecting a random direction (the default behaviour of non-motor coupled particles). Low values of pCD result in stronger inertial movement and higher values weaken the motor coupling effect. The effect of the strong motor coupling can be seen in Figure 24. Instead of the uniform circular plasmodial sheet, the surging inertial flux causes

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Figure 24. Emergence of oscillations of particle flux in strong motor coupling and phase shifts in dumbbell configuration. Top Row: Oscillations of particle flux in plasmodial sheet with strong motor coupling (dark areas indicate greater flux). Bottom Row: Dumbbell chamber (left, measurement areas indicated by squares), and spontaneous phase switching of oscillating thickness in left (thin line) and right (thicker line) chambers

the synthetic plasmodium to become rough at the edges, and a complex pattern of distributed internal oscillations emerges, similar to those reported by (Takagi & Ueda, 2007). The varying brightness illustrates different regions of particle flux within the plasmodium. The oscillations correspond closely to the oscillating thickness of the plasmodium in Physarum and, when the plasmodium is contained within a dumbbell shaped chamber, show the characteristic switching of phase patterns from in-phase synchronisation to anti-phase oscillations, similar to those observed in experimental results with the organism (Takamatsu, Fujii, & Endo, 2000). The amorphous oscillating plasmodial sheet moves randomly about its environment. To be useful as a distributed means of robotic control a method of controlling the direction of movement must be found. One method is to use positive chemotaxis as an attractant to which the plasmodium moves. The results shown in Figure 25 indicate that the direction of movement can indeed be influenced by a chemotaxis gradient. A stimulus is introduced into the environment (at the

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position of the mouse cursor). The plasmodial sheet begins to extend a pseudopod process towards the diffusing stimulus source. A sequence of oscillations within the plasmodium emerges (waves appearing to move from the rear of the plasmodium towards the stimulus, similar to the effect observed by (Matsumoto, Takagi, & Nakagaki, 2008)) and the remainder of the plasmodium shifts towards the extending pseudopod before returning to the rough circular shape. The collective amoeboid movement may alternatively be guided by the presence of repulsive stimuli – instead of guiding the plasmodial sheet towards an attractive stimulus, the population moves away from a noxious stimulus. Two methods were employed to achieve this effect in attempts to mimic the response of Physarum plasmodium to exposure to visible light (where the organism attempts to move from exposed areas (Nakagaki, Yamada, & Ueda, 1999), (Nakagaki et al., 2007)). In the first method, the strong motor coupling was removed at areas of the environment which were directly exposed to light. The removal of motor coupling results in the ces-

Dynamics of Particle-Based Reaction-Diffusion Computing

Figure 25. Amoeboid movement by pseudopod extension and retraction towards chemotaxis stimuli. Food source stimuli indicated by mouse pointer location. Darker areas indicate greater particle flux

sation of oscillator formation in these areas. In the second method the motor coupling is not reduced at exposed areas but the chemoattractant flux (produced by mobile particles) was reduced in light exposed areas. Preliminary experimental results (Figure 26) indicate that both methods are effective at influencing the plasmodium to move away from the hazardous stimulus. The most visibly dramatic response was seen in the second method (reduction in flux) where the plasmodium quickly avoided areas exposed to light. The most realistic result

(albeit less dramatic) was demonstrated using the first method (cessation of motor coupling and oscillator formation at exposed areas). This result was more realistic because it more closely resembled the behaviour of Physarum – some plasmodium did remain on the exposed areas but the movement was weaker. It is entirely possible that a combination of both responses may be utilised in the real organism.

Figure 26. The oscillating plasmodial sheet moves away from exposure to simulated visible light stimuli. Top Row: Combining removal of motor coupling and flux reduction in exposed areas. Bottom Row: Light avoidance by flux reduction in exposed areas only. (Light gray mass represents plasmodial sheet, dark uniform rectangle indicates area exposed to light.)

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CONCLUSION: PASSIVE/ACTIVE AND ATTRACTION/REPULSION PARTICLE DYNAMICS Reaction-diffusion computing exploits mechanisms of self-organising pattern formation and propagation for useful computational tasks. The regularities of the patterning process (for example the uniform propagation speed of chemical wavefronts) provide a means of either encoding information (for example on the wavefront shape, or by the presence or absence of a signal), or indirectly inferring spatial properties of the system (differences in wave arrival time can be used to infer differences in distance). In this report we have presented results of particle approximations of reaction-diffusion computing, examining the effects of so-called active/passive approaches with both attraction and repulsion dynamics. Passive approaches are characterised by the weak sensory coupling of underlying reactiondiffusion mechanisms (such as wavefront propagation in an excitable medium) to the behaviour of an external system – in this instance a population of simple chemotaxis based particles sensing the concentration gradients of the propagating wavefronts. It should be noted that the computational functionality (specifically the parallel search by the wavefront propagation) is being performed in the propagation layer – the external system of particles is used to simply, and collectively, indicate the solution to the problem. We effectively obtain a computational free-ride from the reactiondiffusion behaviour, provided that the desired computational problems and that the solutions can be spatially represented (a non-trivial task in itself). Even this simple coupling can, however, provide complex results: The passive approach described in this chapter can cope with dynamically changing environments and, by utilising the quantitative properties of the propagation mechanism, there is the possibility for more complex problem representations, such as the assignment of costs to certain paths. When the difference

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between chemo-attraction and chemo-repulsion behaviour is considered, the weak particle/environment coupling of the passive approach provides differences only in the direction of particle movement. Repulsion behaviour results in efflux away from the diffusion source, resulting in either division of the plane (in the Voronoi diagram approximation) or migration towards the medial axis (in the skeletonisation example), as the particles try to avoid the propagating waves. Attraction behaviour results in an influx of agents towards the diffusion source, resulting in a directed path towards the sources of the waves when a single point source is used. Active approaches may be characterised by a strong coupling of the underlying reactiondiffusion mechanism to the behaviour of an external system – the particle behaviour disturbs the diffusion field. In the examples provided of synthetic approximations of Physarum computing there is both sensory and motor coupling of the particles to the diffusing concentration gradients. The modification of the concentration gradients by the particle population adds an autocatalytic component to the underlying diffusion mechanism. Unlike many purely physical examples of reactiondiffusion patterning, however, the patterns formed are mobile and dynamic only as long as there is energy input into the system (i.e. as long as the particles are mobile). In the case of chemoattraction behaviour the strong sensory coupling of the particles results in the emergence of complex network dynamics from the simple underlying particle interactions. Furthermore, these dynamics exhibit second-order, quasi-physical, emergent properties such as minimisation by surface tension which was used to provide simple examples of network minimisation tasks. By the addition of strong motor coupling to provide momentary interruptions to the normally smooth particle flux, the spontaneous emergence of spatially distributed oscillations can be observed. These oscillations may in turn be used to govern complex collective amoeboid movement.

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There are significant differences in patterning between chemo-attraction behaviour and chemorepulsion behaviour in the active approach. Attraction behaviour results in cohesive properties to the collective behaviour (for example, network formation and minimisation). The patterning formed by the cohesive behaviour can be tuned by adjusting the particles’ sensory parameters. Repulsion behaviour, on the other hand, does not exhibit cohesive behaviour between the agents. Instead, regular and periodic dissipative patterning processes (resembling classical Turing-type patterning) are observed. We may suggest that, in general, active approaches to reaction-diffusion computing provide richer computational properties than passive approaches. The complex adaptive evolution of the transport networks formed in the active approach, and the quasi-physical properties of their evolution, suggest that the active approach may be more computationally useful, and also more amenable to finding physical examples which could take the approach from the purely in-silico realm. The cost of these additional emergent properties borne from the active approach is the added complexity, which arises when motor coupling to the diffusive lattice is used. Careful tuning of sensory parameters is required to generate collective network behaviour which remains good connectivity and does not become isolated. Although we have dealt with global adjustment to sensory parameters in this report, it is certainly possible – perhaps highly likely – that living systems with collective behaviour (Physarum plasmodium, ant colonies, swarms etc.) may exhibit different regimes of local connectivity within different parts of the collective at the same time in response to the distributed nature of their environment. It is also the case that complex living collectives are exposed to both attraction stimuli (food, shelter, potential mates) and repulsive stimuli (unfavourable locations, hazards, threats etc.) within the same environment. Thus, it may be possible that the complex cohesive behaviour of the (attrac-

tion based) particle collective could be coupled to both attractive and repulsive external stimuli simultaneously. Preliminary experimentation (in preparation) using the active particle approach (with attraction behaviour), and simultaneous presentation of both attraction and repulsion stimuli, results in a further complex constraining of network evolution in response to the extremes of stimuli. To conclude, we have presented a multi-agent based particle collective, each member of which is comprised of very simple sensory and motor behaviours. This collective implements a simple approximation of reaction-diffusion dynamics and has been shown to exhibit complex emergent behaviour in response to both attraction and repulsion based stimuli. The particle collective was applied to a range of spatially represented computational problems. In the active approach (where both sensory and motor behaviour is coupled to the diffusive lattice) the dynamics and evolution of the collective was shown to be particularly rich and complex, forming a virtual material whose properties could be adjusted by sensory parameter adjustment. By a simple change to motor behaviour we have also been able to generate complex and emergent oscillatory dynamics which can be harnessed to provide distributed – and externally controllable – collective amoeboid movement which may be applicable as a means of implementing distributed robotics. The next step – moving from a virtual collective material to a real physical material with the same properties and behaviour - is certainly very challenging. But we hope that we have demonstrated some general approaches (active/passive diffusion interactions) and specific responses to stimuli (attraction/repulsion dynamics) which may be useful in the search for suitable candidates.

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ACKNOWLEDGMENT The work was partially supported by the Leverhulme Trust research grant F/00577/1 “Mould intelligence: biological amorphous robots”, led by Andrew Adamatzky.

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Chapter 15

Towards Arithmetical Chips in Sub-Excitable Media: Cellular Automaton Models Liang Zhang University of the West of England, UK Andrew Adamatzky University of the West of England, UK

ABSTRACT We discuss a theoretical design of an arithmetical chip built on an excitable medium substrate. The chip is simulated in a two-dimensional three-state cellular automaton with eight-cell neighborhoods. Every resting cell is excited if it has exactly two excited neighbors, the excited cells takes refractory state unconditionally. A transition from refractory back to resting state also happens irrelevantly to a state of the cell neighborhood. The design is based on principles of collision-based computing. Boolean logic values are encoded by traveling localizations, or particles. Logical gates are realized in collisions between the particles. Detailed blue prints of collision-based adders and multipliers presented in the article pave the way to future laboratory experimental prototypes of general-purpose chemical computers.

INTRODUCTION Excitable chemical media are amongst most promising ‘wet’ computing devices capable for solving a wide ranging of tasks from optimization, computational geometry, image processing and robot control. The excitable media can also implement functionally complete sets of logical gates thus qualifying as (logically) universal computing systems. See theoretical background and details of laboratory implementations of DOI: 10.4018/978-1-60960-186-7.ch015

reaction-diffusion computers in (Adamatzky, De Lacy Costello, Asai, 2005). So far all experimental laboratory prototypes of reaction- diffusion chemical computers are in fact specialized, or task oriented purposed processors. They are capable for solution of only one problem or family of problems. When logical universality is concerned the experimental implementations do usually deal with one or two logical gates, just as a matter of demonstration (Adamatzky, 2004; De Lacy Costello and Adamatzky, 2005; Adamatzky and De Lacy Costello, 2007; Toth et al, 2008). To move from abstract computational

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Towards Arithmetical Chips in Sub-Excitable Media

universality to general-purpose machine we must demonstrate how we can cascade simple logical gates in more complicated circuits able to execute sensible computational tasks. An implementation of arithmetical chip in reaction- diffusion medium would be enough to convince laymen that excitable chemical computing devices are not just a matter of curiosity but viable candidates for a role of future non-silicon computers. We envisage that novel arithmetic chip, to be built in an excitable medium, will be based on principles of collision-based computing. The proposed chips will be based on logical schemes of computation in Conway’s Game-of-Life (Berlekamp, Conway, Guy, 1982), Fredkin-Toffoli’s conservative logic (Fredkin and Toffoli, 1982) and Margolus’s physics of computation (Margolus, 1984). In collision-based computing (Adamatzky, 2003), quanta of information are represented by compact patterns traveling in an ‘empty’ space and performing computation by colliding with each other. The absence or presence of traveling patterns encodes values of Boolean logical variables. The trajectories of patterns approaching a collision site represent input variables, and the trajectories of the patterns ejected from a collision, and traveling away from the collision site, represent the results of logical operations, output variables. A sub-excitable Belousov-Zhabotinsky medium (Sedina-Nadal et al, 2001) is an ideal substrate to build collision-based arithmetical chips in chemical systems. In a normal, excitable, mode the Belousov-Zhabotinsky medium responds to local perturbations by forming target or spiral waves, which propagate in all possible directions away from perturbation site. In a sub-excitable mode, we observe generation of a localized excitations, or wave-fragments that preserve their shape and travel like dissipative solitons (Bode et al, 2002) in one pre-determined direction for a substantial amount of time. We have demonstrated (Adamatzky, 2004; De Lacy Costello and Adamatzky, 2005; Adamatzky and De Lacy Costello, 2007; Toth et al, 2008) that it

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is possible to implement logical gates by colliding excitation wave-fragments. In present article we use a cellular-automaton sub-excitable lattice (Adamatzky, 1995; Adamatzky, 1998) to simulate a sub-excitable chemical medium. We integrate together our previous results (Zhang and Adamatzky, 2008; Zhang and Adamatzky, 2008) on arithmetical operations in collision-based media.

CELLULAR-AUTOMATON MODEL OF SUB-EXCITABLE MEDIA We employ a two-dimensional three-state cellularautomaton model of an excitable medium — the 2+-medium, originally introduced in (Adamatzky, 1995; Adamatzky, 1998). The 2+-medium consists of an orthogonal array of finite automata, where every automaton, called a cell, takes three states: resting, excited ‘+’ and refractory ‘-‘ and updates its state depending on the states of its eight neighbors. All cells update their states simultaneously and in discrete time, using the same rule. A resting cell becomes excited if it has exactly two excited neighbors. The transitions from excited state to refractory state, and from refractory state to resting state are unconditional. Most common, and also minimal in size, localizations observed in 2+-medium, are particles traveling along horizontal and vertical directions of the cellular lattice (e.g. Figure 1a) and along diagonals of the lattice (e.g. Figure 1b). A 2+-particle consists of two excited cell-states and two refractory cell-states (Figure 1a). The particle can travel in four possible directions: East, West, North and South, and the particle conserves its configuration. A 3+-particle consists of three cells in excited states and three cells in refractory states (Figure 1b). The 3+-particle travels in the direction of North-East, South-East, North-West and South-West. Its topology changes in a cycle of four configurations. There is the only one stationary localization in 2+-medium. The stationary oscillator (Figure

Towards Arithmetical Chips in Sub-Excitable Media

Figure 1. Species of traveling localizations used to implement arithmetic circuits in 2+-medium (Adamatzky, 1995): (a) 2+-particle traveling East, (b) 3+-particle traveling North-East, (c) stationary oscillator. Excited states are shown by ‘+’, refractory ‘-’ and resting ‘.’.

1c) consists of four excited states and four refractory states. The oscillator’s configuration changes in a cycle of three configuration s and it remains at its original position all the time. We are employing these two species of traveling localizations and the oscillator in further designs. Presence of a particle represents value 1, Boolean Truth, and absence of the particle value 0, Boolean False. When the particles, produced by certain gates, are not used in further operations, we annihilate them by directly changing their excited states to refractory states (Figure 2).

ADDERS 1-Bit Half Adder A 1-bit half adder is a logical circuit that has two binary inputs A and B, and two binary outputs, a Sum value S and a Carry value C: S=A⊗B and C=A⋅B. The initial configuration of implementing the half adder in the 2+-medium is shown in Figure 3a. Two aforementioned inputs, A and B, are represented by a 2+-particle traveling South at the top of the graph and a 3+-particle traveling South-East in the middle. As stated before, the presence (or the absence) of these particles denotes the assignment of logical value ‘1’ (or

Figure 2. Garbage removal: examples of annihilation of a 2+-particle traveling East (a) and a 3+-particle traveling North-East (b), whose excited states are altered to refractory states. Both particles will disappear at the next time step.

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Figure 3. Space-time configurations of excitable cellular automaton implementing 1-bit half adder: (a) the initial configuration of input and control particles, both inputs A and B have value Truth in this example and represented by 2+-particle at the top, and 3+-particle in the middle of the lattice, while the control particles always hold Truth value; (b to e) trajectories of particles depending on different input values, produced as time lapsed superpositions of several consecutive configurations of the cellular automata, time-related positions of the particles are encoded by values of gray.

‘0’) to the input values A and B, therefore in this example, A = 1 and B = 1. Moreover, there are two additional control particles, a 2+-particle traveling East on the far left of the graph and a 3+-particle traveling North-East at the bottom. These two additional particles are always present regardless of the values of A and B, which means they always hold the value of logical ‘1’ or Truth. The 1-bit half adder in the 2+-medium occupies 18× 45 cells. All particles are injected in the system, i.e. start their travel in predetermined places at the same time (Figure 3a). The outputs are read at the right side of the lattice: the upper output represents Carry value and the bottom output represents Sum value. Trajectories of the particles constituting the 1-bit half adder are shown in Figure 3b to Figure 3e: when both input particles A and B hold logical value ‘0’ (Figure 3b), the control particles collide with and annihilate each other so that no particles reach the right side of the lattice and both outputs equal logical ‘0’; when

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only one of the input particles has logical value ‘1’ (Figure 3c and 3d), this input particle collide with and annihilate the control particle at the bottom, giving the 2+-particle on the far left a chance to travel all along to the right side of the lattice; while if both input particles are present (Figure 3e), they produce a new 2+-particle traveling East after their collision, making the Carry output having logical value ‘1’, and again, the control particles annihilate each other.

1-Bit Full Adder A 1-bit full adder has three binary inputs, addends A and B together with a Carry-in value Cin. Also the adder has two binary outputs, a Sum value S and a Carry-out value Cout: S=A⊕B⊕Cin=(Cin⊕A) ⊕B Cout=A⋅B+A⋅Cin+B⋅Cin=Cin⋅A+((Cin⊕A)⋅

Towards Arithmetical Chips in Sub-Excitable Media

Figure 4. The redirection gate: (a) the redirection gate itself (at time step t = 1) vanishes (t = 6) when there is no particle to redirect; (b) 2+-particle traveling East (t = 1) is redirected and heads South (t = 6).

The final form of the above equations shows that a 1-bit full adder can be realized by using two 1-bit half adders in the following way: 1. Two of the three inputs are input values of the first half adder (Cin and A, for example); 2. The third input (B) and the Sum value of the first half adder (Cin⊕ A) are input values of the second half adder; 3. The Sum value of the full adder is equivalent to that of the second half adder ((Cin⊕A) ⊕ B); 4. The Carry-out value of the full adder is equivalent to either the Carry value of the first half adder (Cin ⋅A) or that of the second half adder ((Cin ⊕ A)⋅ B). The 1-bit full adder in the 2+-medium is implemented exactly the same way described

above. In addition to the two half adders, there are some control elements used to complete the implementation. The redirection gate (Figure 4), consisting of two 2+-particles traveling in directions orthogonal to each other, functions as a gate that changes a 2+-particle’s traveling direction by 90 degrees. There are eight different instances of redirection gates in total depending on the traveling directions and relative positions of the consisting 2+-particles, and the redirection gate shown in Figure 4 is one of them. Figure 4a shows the redirection gate per se, which simply disappears when there is no 2+-particle to be redirected; whereas when collide with a 2+-particle traveling East (Figure 4b), a 2+-particle traveling South emerges after 5 time steps. A 1-bit full adder in the 2+-medium occupies 28× 59 cells (Figure 5) and it consists of two 1-bit half adders on the left, a redirection gate (RG1)

Figure 5. The configuration layout of a 1-bit full adder, composed of two 1-bit half adders and control elements. Different elements are injected in the system at different time steps.

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in the middle and a block of redirection gates on the right of the lattice. The block of redirection gates encompasses a stationary oscillator (OSC), fi9 redirection gates (RG2. .. RG6) and an additional 2+-particle traveling West within the range of RG6. The gate RG1 is used to redirect the Sum value of the first half adder to become an input value of the second half adder. The block of redirection gates is used to delay the Carry value of the first half adder by exactly 13 time steps, synchronizing it with the Carry value of the second half adder. Therefore there is eventually only one 2+-particle to represent the Carry-out value of the full adder. The stationary oscillator can perform other operations, we only use it for redirection purpose here. We use a stationary oscillator instead of another redirection gate because we want the 2+-particle to be redirected to travel one time-step faster. The block of redirection gates disappears if there is no particle to redirect and/ or delay. Note that different elements are injected in the system at different time steps when implementing the 1-bit full adder, contrary to the implementation of the 1-bit half adder, where all particles are injected at once. Here the first 1-bit half adder is added at time step t = 1, followed by the redirection gate RG1 and the stationary oscillator OSC at time step t = 28, and the second 1-bit half adder at time step t = 29. The rest of the redirection gates RG2 to RG6 are added at time steps t = 34, 37, 45, 48 and 56. The additional 2+-particle traveling

West is injected in the system at time step t = 57. The 1-bit full adder completes its computation and produces a result at time step t = 60. The input values are represented by particles as follows: • • •

Carry-in: 2+-particle traveling South in the first 1-bit half adder; A: 3+-particle traveling South-East in the first 1-bit half adder; B: 3+-particle traveling South-East in the second 1-bit half adder.

If we only implement a single isolate 1-bit full adder, the input values do not necessarily have to be assigned to the particles exactly the same way described above. The Carry-in value can also be represented by the 3+-particle traveling SouthEast in either of the half adder. However, when we use the 1-bit full adder as a module to implement multiple-bit adders, the above assignment is highly recommended, so that it is much easier to propagate the signal of Carry-out value of one full adder to become the Carry-in input value of the next full adder. The possible pathways of information flow when the 1-bit full adder computes are shown in Figure 6. Four flows of information, namely FL1 to FL4, are depicted. The FL1 appears only when both Carry-in=1 and A=1, thus the 2+-particle traveling South and the 3+-particle traveling South-East in the first 1-bit half adder collide

Figure 6. Flows of information during operating of 1-bit full adder

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and produce a 2+-particle traveling East, which is then redirected by the block of redirection gates six times (first the OSC, followed by the RG2, RG3, RG4, RG5 and RG6) and reach the right side of the adder. The FL2 appears when either Carry-in=1 or A=1, thus their sum equals 1 and the 2+-particle traveling East in the first 1-bit half adder would remain, be redirected by the redirection gate RG1 and become the 2+-particle traveling South in the second 1-bit half adder. The FL3 appears when the FL2 is appearing and B=1, thus the 2+-particle traveling South and the 3+-particle traveling South-East in the second 1-bit half adder collide and produce a 2+-particle traveling East, reaching the right side in the end. The FL4 appears when either the FL2 is appearing or B=1, thus the 2+-particle traveling East in the second 1-bit half adder would remain and reach the right side of the adder. Note that the FL1 and

FL3 cannot appear at the same time. However, if appears, both the FL1 and FL3 would reach the right side of the adder at the same time step t = 60. Trajectories of the particles constituting the adder are shown in Figure 7. Results of the computation are read at the right side of the adder: upper output represents the Carry-out value and the bottom output represents the Sum value.

Multiple-Bit Adders The 1-bit adders we have discussed so far are both modular, so that we can use them to produce multiple-bit adders in the 2+-medium as long as we deal with the carry bit, i.e. to propagate the carry information from one bit to its next higher significant bit. Here we show the realization of 2-bit adders and demonstrate one way to propagate the carry information with a redirection gate. Ad-

Figure 7. Trajectories of particles implementing 1-bit full adder

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ders of arbitrary-bit length can be implemented in the same way, provided all input values are correctly assigned to the input particles within each 1-bit adders. The 2-bit half adder occupies 38× 97 cells, consisting of a 1-bit half adder added at time step t = 1, a redirection gate at time step t = 28 and a 1-bit full adder at time step t = 37. The redirection gate is used to propagate the Carry value from the less significant bit to the next higher significant bit. The input values are represented as follows: • • • •

A0: 2+-particle traveling South in the 1-bit half adder; B0: 3+-particle traveling South-East in the 1-bit half adder; A1:3+-particle traveling South-East in the first 1-bit half adder of the 1-bit full adder; B1: 3+-particle traveling South-East in the second 1-bit half adder of the 1-bit full adder.

The 2-bit full adder occupies 45×109 cells, consisting of two 1-bit full adders activated in the system at time step t = 1 and t = 67, as well as a redirection gate at time step t = 60. The redirection gate is again used to propagate the carry information. The input values are represented as follows: •

•

•

•

•

Carry-in: 2+-particle traveling South in the first 1-bit half adder of the first 1-bit full adder; A0:3+-particle traveling South-East in the first 1-bit half adder of the first 1-bit full adder; B0: 3+-particle traveling South-East in the second 1-bit half adder of the first 1-bit full adder; A1:3+-particle traveling South-East in the first 1-bit half adder of the second 1-bit full adder; B1: 3+-particle traveling South-East in the second 1-bit half adder of the second 1-bit full adder.

Figure 8. Trajectories of particles implementing 2-bit adders

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Figure 9. Collision-based implementation of and gate: (a) initial configuration (within gray regions) with both inputs True, (b) 2+-particle produced by the gate when both operands are False is terminated at time step t = 2, (c) the termination operation affects the configuration produced by the gate when there is only one operand with value True, this does not influence the overall result of the computation, (d) the termination operation does not change the configuration produced by the gate when both operands are True.

Selected trajectories of the particles constituting the 2-bit adders are shown in Figure 8. Results of the computation are read at the right side of the adder, with the outputs from top to bottom representing the Sum value of the less significant bit, the Carry-out value and the Sum value of the higher significant bit respectively for both 2-bit adders.

MULTIPLIERS

collection) are always changed to refractory states at time step t = 2 no matter what initial values of the operands are. Figure 9c shows that the configuration produced by the gate when there is only one operand True at time step t = 2 is affected by the garbage-collection operation, whereas the overall result is not affected. The configuration produced by the gate when both operands are True is not affected by the garbage-collection operation at all (Figure 9d). Thus construction of collision-based and gate is completed.

AND Gate The initial configuration of the AND gate or the 1-bit multiplier at time step t = 1 consists of at most three 2+-particles (Figure 9a). Two of the particles traveling South and North (gray regions) represent two operands and one particle traveling East represents constant Truth. When both operands are True, collision of these three 2+-particles produce a new 2+-particle traveling East. This newly born particle indicates that output of the conjunction is True. When there is only one operand True, no excitation is generated in the result of collision, thus representing the output False. When both operands are False, the 2+-particle traveling East is unaffected, this garbage-particle is annihilated at time step t = 2 (Figure 9b). Note that the cells affected by the termination of particles (garbage

Figure 10. Design of the 2-bit multiplier

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Figure 11. Converting a 2+-particle traveling East to a 3+-particle traveling South-East. (a) Configuration at time step t = 1. The particle in the gray region is the one to be converted and the other one is the control particle. (b) A 3+-particle traveling South-East is produced at time step t = 5. (c) If there is no particle to be converted, the control particle will be annihilated at time step t = 4. (d) The termination operation affects the configuration at time step t=4 when there is a particle to be converted, but does not affect the final result.

2-Bit Multiplier 2-bit multiplier is implemented using four and gates and a 2-bit full adder, as shown in Figure 10. Boolean values of multiplicands A1A0 and B1B0 are distributed and fed to and gates. Output of the AND gate AG1: A0 and B0, is the lowest bit of the product C0. Outputs of other and gates

Figure 12. A 1-bit collision-based full adder used in designing of multipliers. Particles within the light gray regions are injected in the medium, as local perturbations, at different time steps and positions comparing to those constructed in Sect. 3.2. Particles within the dark gray regions are inputs of the adder, where the 2+-particle traveling South represents the Carry-in value and two 3+-particles traveling South-East represent addends. In this exemplar configuration, all three inputs take value Truth.

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(AG2. .. AG4) become addends of the 2-bit full adder, and output of the adder is comprised of the bits of the product C1, C2 and C3. The 2-bit full adder is constructed similarly to that discussed in previous sections, with a difference that we here adopt a modified construction of 1-bit full adder (Figure 12), where the second 1-bit half adder (the left-bottom gray region) as well as all particles except the stationary oscillator in the block of redirection gates (the right gray region) are added to the array six time steps later and six cells lower than those in the original construction. This leaves for newly introduced and gates enough time and space to produce addends for the 1-bit adder (0 or 1, depending on the absence or presence of a 3+-particle traveling South-East within the dark gray regions after conjunction operations take place). Since the AND gates have been already described in Sect. 4.1. The only design task left to complete is to convert the output of the and gate, i.e. 2+-particle traveling East, into the addend of the adder, i.e. 3+-particle traveling South-East. This is done by the gate shown in Figure 11a, where the 2+-particle within the gray region is the one to be converted and the particle traveling North is the control particle being constant truth. Using this gate at time step t = 1, we produce a 3+-particle traveling South-East at time step t = 5 (Figure 11b). However, if the output of the and gate is False, i.e.

Towards Arithmetical Chips in Sub-Excitable Media

Figure 13. Configuration layout of the 2-bit multiplier, formed by four and gates (AG1 ...AG4) and a 2-bit full adder. Two additional redirection gates are used to make outputs of the multiplier, which will be read at the right side of the lattice (four gray regions), in order of the lowest bit (C0) to the highest bit (C3) from the top to the bottom.

there is no 2+-particle in the gray region of Figure 11a, we would be left with an unwanted 2+-particle traveling North. To avoid this particle interfering with other particles, we terminate it at time step t = 4 by changing its excited states into refractory states (Figure 11c). Again, this termination operation affects the configuration when the output of the AND gate is True (Figure 11d), but it does not affect the overall result of the gate, because a 3+-particle traveling South-East is produced at time step t = 5 anyway. The 2-bit multiplier occupies a lattice of 82×149 cells (Figure 13). Four input bits A0, A1, B0 and B1 are distributed as shown in Figure 10 as operands of the four and gates (AG1. .. AG4). There 2+-particles traveling South represent bits from multiplicand A and the particles traveling North represent bits from multiplicand B. The AND gates AG1 and AG2 are added to the lattice at time step t = 1, and the gates AG3 at time step t = 35, and the gate AG4 at time step t = 110. The control particles, responsible for converting outputs of the and gates AG2,. . ., AG4 into

addends of the adders, are ‘injected’ in the lattice one time step after the gates are introduced. The two 1-bit full adders are added at time steps t = 6 and 81, respectively, and the redirection gate joining up the 1-bit adders is added at time step t = 71. In addition, two more redirection gates are added after the 2-bit adder at time steps t = 146 and 174, ensuring output bits of the multiplier in order of the lowest bit (C0) to the highest bit (C3) when they are read at the right side of the lattice from the top to the bottom. Exemplar trajectories of the particles constituting the 2-bit multiplier are shown in Figure 14. Figure 14a demonstrate the case when both A0 and b0 equal 1, the output of and gate AG1 which is also the lowest bit of the product C0 equals 1. In Figure 14b we see trajectories of particles for the situation when Sum values of the 2-bit adder, which are also two bits from the product C1 and C2 being True, and the connection between two 1-bit full adders as well as the Carry-out value of the 2-bit adder being redirected to become the highest bit of the product C3 can be seen in Figure 14c.

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Figure 14. Trajectories of particles implementing 2-bit multiplier. The configuration of trajectories is produced as time-lapsed superpositions of several consecutive configuration of the cellular automaton implementing the multiplier. Level of gray encodes time elapsed from the beginning of computation.

3-Bit Multiplier The 3-bit multiplier is implemented using nine and gates and two 4-bit full adders, as shown in Figure 15. Binary bits of the multiplicands A2A1A0 and B2B1B0 are distributed and are fed to the AND

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Figure 15. Design of the 3-bit multiplier

gates. Output of the AND gate AG1 — A0 and B0 — is the lowest bit of the product C0. Outputs of other and gates (AG2– AG9) become addends of the 4-bit full adders. The lowest bit of the Sum value from the first 4-bit adder is another bit of the product C1, the rest of bits of the Sum value from this 4-bit adder become input addends of the second 4-bit adder, whose Sum value constitutes the rest bits of the product, C2 to C6. The AND gate and conversion from output of the and gate to addend of the adder are implemented in previous sections and the 4-bit full adder is implemented using the way described in Sect. 3.3. Therefore the only thing we need to complete the construction of the 3-bit multiplier is to establish a connection between two 4-bit full adders. This is done by redirecting bits from Sum value of the first 4-bit adder (S1, S2 and S3 in Figure 15), which are all represented by 2+-particle traveling East, to addends of the second 4-bit adder, which are 3+-particles traveling South-East. This task is demonstrated in Figure 16. Let a first 1-bit full adder be added to the lattice at time step t = 1. The adder’s Sum value equals 1 and represented by 2+-particle traveling East. This particle is deflected North by a redirection gate added at time step t = 66, and then to the West by another redirection gate added at time step t = 82. After the deflections the particle col-

Towards Arithmetical Chips in Sub-Excitable Media

Figure 16. Trajectories of particles which establish connections between full adders. The 2+-particle traveling East, which represents Sum value of a first 1bit full adder, is converted to 3+-particle traveling South-East, which represents an addend of a second 1-bit full adder added to the lattice at the same position but 107 time steps later than the first 1-bit full adder.

lides with 2+-particle traveling South at time step t = 102. A 3+-particle is produced in the result of collision, t = 108, when a second 1-bit full adder is added to the lattice at the same position as the previous one. If the Sum value of the first adder

is 0, two redirection gates would disappear themselves and the 2+-particle traveling South would be terminated at time step t = 108. The termination operation does not affect the formation of the 3+-particle if the Sum value of the first adder equals 1. The 3-bit multiplier occupies a lattice of 143×250 cells, see Figure 17. Six input bits A0, A1, A2, B0, B1 and B2 are distributed as shown in Figure 15 as operands of the nine and gates (AG1 – AG9), which are added to the lattice at time steps t = 1, 1, 76, 35, 110, 185, 217, 292 and 367, respectively. The control particles responsible for converting outputs of the AND gates (AG2– AG9) to addends of the adders are added one time step after introduction of the gates. The eight 1-bit full adders are initiated on the lattice at time steps t = 6, 81, 156, 231, 188, 263, 338 and 421, respectively. The six redirection gates joining up the 1-bit adders are added at time step t = 71, 146, 221, 253, 328 and 403. Control particles responsible for converting Sum values of the first 4-bit full adder to addends

Figure 17. Configuration layout of the 3-bit multiplier, formed by nine and gates (AG1-AG9) and two 4-bit full adders or eight 1-bit full adders. Some of and gates and 1-bit full adders are added at the same position in the lattice but with a time difference of 107 time steps. All input values to the and gates are set to 1 in the example provided. Outputs of the multiplier are read at the right side of the lattice (six gray regions) in order of the lowest bit (C0) to the highest bit (C5) from top to bottom. Controlling particles responsible for linking 4-bit full adders are not shown in the picture because otherwise they would overlapping with the block of the redirection gates.

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Figure 18. Exemplar trajectories of particles implementing 3-bit multiplier

of the second one are added at the time steps 65, 81 and 101 later than the first adder, as described above. Product of the 3-bit multiplier is read at the right side of the lattice in order of the lowest bit (C0) to the highest bit (C5) from top to bottom. Sample trajectories of the particles constituting the 3-bit multiplier are shown in Figure 18, where connections of 1-bit full adders within and between the two 4-bit full adders can be seen. Using the constructs described above, we can implement multipliers of an arbitrary length. For any n-bit multiplier (n≥ 3), we will need n2 and gates and n- 1 of (n + 1)-bit full adders to complete the implementation, with the same mechanism used to construct the 3-bit multiplier.

DISCUSSION We represented basic designs of adders and multipliers which lay a theoretical ground for future designs of arithmetical circuits in experimental

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laboratory reaction-diffusion chemical media. We must admit that our designs have two – not dramatic but yet ‘annoying’ imperfections: cumbersome implementation of garbage collection and high reliance of the scheme functioning on timing of the signal generation or input. Regarding, garbage collection, some collisions between signals do produce auxiliary particles, which are not employed for further cascading of signals. These particles are removed by direct ‘extinguishing’: their excited components are switched to refractory states by external inputs. We are currently looking for a viable alternative of removing the ‘garbage’ particles. As to the timing, in the designs of full adders and multipliers we injected signals/particles in the computing medium at certain time steps. These time steps are calculated precisely. Perfect timing of particles generation is a necessary condition for robust functioning of our collision-based circuits. This weak point of the design decreases fault-tolerance of the excitable medium chips. Thus our next task would be to perfect designs

Towards Arithmetical Chips in Sub-Excitable Media

so all signal and control particles are injected in the medium at the same time or to modify the designs towards asynchronous implementations. And, of course, our ultimate goal is an experimental laboratory implementation of adders and multipliers in Belousov-Zhabotinsky excitable chemical medium. Preliminary steps have been already made in that direction. We demonstrated that it is possible, in principle, to realize basic logical gates and cascading of logical gates by colliding wave-fragments sub-excitable BelousovZhabotinsky medium (De Lacy Costello and Adamatzky, 2005), and cataloged all possible outcomes of binary collisions between wavefragments (Adamatzky and De Lacy Costello, 2007; Toth et al, 2008). The ‘only’ step left is to assemble all components together into a fullyfunctional arithmetic chip. This step requires detailed elaboration of the wave-fragment controlling procedures, including routing, elimination and generation of new wave-fragments.

Adamatzky, A. (2004). Collision-based computing in Belousov-Zhabotinsky medium. Chaos, Solitons, and Fractals, 21, 1259–1264. doi:10.1016/j. chaos.2003.12.068

ACKNOWLEDGMENT

De Lacy Costello, B., & Adamatzky, A. (2005). Experimental implementation of collision-based gates in Belousov-Zhabotinsky medium Chaos. Solitons & Fractals, 25, 535–544. doi:10.1016/j. chaos.2004.11.056

The authors wish to acknowledge the support of the EPSRC grant number EP/E016839/1 for the support of Liang Zhang.

REFERENCES Adamatzky, A. (1995). Controllable transmission of information in the excitable media: the 2+ medium. Advanced Materials for Optics and Electronics, 5, 145–155. doi:10.1002/amo.860050303 Adamatzky, A. (1998). Universal dynamical computation in multidimensional excitable lattices. International Journal of Theoretical Physics, 37, 3069–3108. doi:10.1023/A:1026604401265

Adamatzky, A., & De Lacy Costello, B. (2007). Binary collisions between wave-fragments in a subexcitable Belousov-Zhabotinsky medium Chaos. Solitons & Fractals, 34, 307–315. doi:10.1016/j. chaos.2006.03.095 Adamatzky, A., De Lacy Costello, B., & Asai, T. (2005). Reaction-diffusion computers (Elsevier, 2005). Berlekamp, E., Conway, J., & Guy, R. (1982). Winning Ways, 2, (Academic Press, 1982). Bode, M., Liehr, A. W., Schenk, C. P., & Purwins, H.-G. (2002). Interaction of dissipative solitons: Particle-Like behaviour of localized structures in a three-component reaction-diffusion system. Physica D. Nonlinear Phenomena, 161, 45–66. doi:10.1016/S0167-2789(01)00360-8

Fredkin, E., & Toffoli, T. (1982). Conservative logic Int. J. Theor. Phys., 21, 219–253. doi:10.1007/ BF01857727 Margolus, N. (1984). Physics–like models of computation. Physica D. Nonlinear Phenomena, 10, 81–95. doi:10.1016/0167-2789(84)90252-5 Sedina-Nadal, I., Mihaliuk, E., Wang, J., PerezMunuzuri, W., & Showalter, K. (2001). Wave propagation in subexcitable media with periodically modulated excitability. Physical Review Letters, 86, 1646. doi:10.1103/PhysRevLett.86.1646

Adamatzky, A. (2003). (Ed.) Collision-Based Computing (Springer).

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Towards Arithmetical Chips in Sub-Excitable Media

Toth, R., Stone, C., Adamatzky, A., de Lacy Costello, B., & Bull, L. (2008). (in press). Experimental validation of binary collisions between wave fragments in the photosensitive BelousovZhabotinsky reaction Chaos. Solitons & Fractals.

Zhang, L., & Adamatzky, A. (2008). (in press). Collision-based implementation of a two-bit adder in excitable cellular automaton Chaos. Solitons & Fractals. Zhang, L., & Adamatzky, A. (2008). Implementation of multipliers in collisions of mobile localizations in sub-excitable lattice. submitted.

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 63-81, copyright 2009 by IGI Publishing.

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Section 4

240

Chapter 16

Organization-Oriented Chemical Programming of Distributed Artifacts Naoki Matsumaru Friedrich Schiller University Jena, Germany Thomas Hinze Friedrich Schiller University Jena, Germany Peter Dittrich Friedrich Schiller University Jena, Germany

ABSTRACT The construction of molecular-scale machines requires novel paradigms for their programming. Here, we assume a scenario of distributed devices that process in-formation by chemical reactions and that communicate by exchanging molecules. Programming such a distributed system requires specifing reaction rules as well as exchange rules. Here, we present an approach that helps to guide the manual construction of distributed chemical programs. We show how chemical organization theory can assist a programmer in predicting the behavior of the program. The basic idea is that a computation should be understood as a movement between chemical organizations, which are closed and self-maintaining sets of molecular species. When sticking to that design principle, fine-tuning of kinetic laws becomes less important. We demonstrate the approach by a novel chemical program that solves the maximal independent set problem on a distributed system without any central control—a typical situation in adhoc networks. We show that the computational result, which emerges from many local reaction events, can be explained in terms of chemical organizations, which assures robustness and low sensitivity to the choice of kinetic parameters.

INTRODUCTION Nanotechnology and molecular computation are a great match since those share the same scale DOI: 10.4018/978-1-60960-186-7.ch016

medium: nanoscale molecules. Under the achievements of nanotechnology, lots of examples including logic gates using multiple nanotube transistors (Bachtold, Hadley, Nakanishi, & Dekkerdagger, 2001) have been reported. Wide varieties of

Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Organization-Oriented Chemical Programming of Distributed Artifacts

nanoparticle applications (Salata, 2004), for example, ultrasensitive biosensors (Wang, 2005) using gold nanoparticles coupled with enzymes (Willner, Basnar, & Willner, 2007), attribute to nanotechnological techniques of manipulating nano-scale objects. Synthesizing molecular machinery out of DNA molecules seems promising (Bath & Turberfield, 2007) even though the lack of stiffness of biomolecules in comparison with ‘dry’ nanotechnology materials has been argued (Merkle, 2000) to be the drawback. Despite the rapid development of nanotechnology, wet-lab experiments are commonly exercised to operate the boolean logics (de Silva & Uchiyama, 2007). DNA computing demonstrated by Adleman (1994) already addressed that limitation of imperative computation paradigms by utilizing particular operation modes of DNA molecules. Assuming DNA as data carrier, up to 1021 bytes can be saved and operated simultaneously within one liter of liquid providing a storage density of 1bit / nm 3 (Păun, Rozenberg, & Salomaa, 1998). One Joule allows up to1019 molecular operations on DNA (Pisanti, 1998). This highly parallelized operation on DNA strands with high data density is the key characteristics of the DNA computation approach. The significance, we note here, is that the computation model is in concert with computation medium. In the nano-scale world, molecules are regarded to constitute a medium, and chemical reactions play an important role in biological information processing principles (Küppers, 1990). Employing molecules and reaction rules as a metaphor, thus, novel computation paradigms have been explored (Banâtre & Mètayer, 1986; Păun, 2002; Banâtre, Fradet, & Redenac, 2004; Tschudin, 2003; Berry & Boudo, 1992). Essentially, those chemical computing models refer the elementary units as molecules, and the operations are described in the form of reactions among those molecules. Given the inputs of the computation as the initial configu-

ration of reaction vessels or reactors, the outputs emerge from local interactions in accordance with the given reaction rules (Banzhaf, Dittrich, & Rauhe, 1996). In these chemical computing models, programming corresponds to designing the reaction rules at the microscopic levels, and the desired computational result emerges at the macroscopic levels as a global systems’ state. The relation between those two levels is highly non-linear, and thus the question for effective programming techniques arises. It seems scarcely possible in this context to predict the macro behavior from the micro rules because of the parallel operations of the reaction rules that are possibly tangled in a complex manner. A common approach to this difficulty is to find a mapping from a known computation model like a Turing machine or a finite state automaton (Păun, 2002; Rothemund, 1996). Conrad (1989) argued that the conventional computers innately differ from natural molecular systems, such as brain or enzymes, and the natural molecular systems are (from today’s perspective) not highly programmable (Conrad, 1985). The conventional digital computers are designed to achieve high programmability by restricting the behaviors of computational entities, and the natural molecular systems operate to exploit the useful properties of the medium. In this paper, we present a programming technique utilizing a notion of chemical organization (Dittrich & Speroni di Fenizio, 2007). Our programming approach is specialized for chemical computing paradigms, and thus the computational medium in the nano-scale world is highly respected. The maximal independent set (MIS) problem (Luby, 1986) is employed for our case story. Dynamical behaviors of the chemical programs are simulated for the validation purpose. Finally, we also discuss that changing the analysis coverage is fruitful.

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Organization-Oriented Chemical Programming of Distributed Artifacts

APPROACHES TO CHEMICAL PROGRAMMING In general, there are several approaches to obtain a chemical program capable of solving a predefined computational problem. Here, we distinguish optimization versus construction. Optimization subsumes heuristically driven techniques: A more or less randomly chosen reaction network becomes successively improved e.g. by evolutionary methods (Ziegler & Banzhaf, 2001), learning inspired by neural networks, simulated annealing, or tabu search (Landweber, 2002). Within the optimization process, the reaction network topology as well as reaction parameters are fitted according to the desired behavior (Deckard & Sauro, 2004). Although pure optimization often generates solutions of astonishing efficiency, the absolute correctness of chemical programs is not guaranteed. In contrast, the roots of manual construction of reaction systems lie in engineering (Alon, 2006). This approach is based on some elementary computational units represented by predefined well-understood reaction network motifs (Aoki, Kameyama, & Higuchi, 1992). Equipped with specified interfaces, these motifs then can be combined towards interconnected networks of more complex functionality. Using construction principles like hierarchical modularization, malfunction of the final reaction network can be avoided (Papin, Reed, & Palsson, 2004).

CHEMICAL ORGANIZATION THEORY Inspired by Fontana and Buss (1994), a theory of chemical organizations has been developed by Dittrich and Speroni di Fenizio (2007). A chemical organization is defined as a set of molecular species that is closed and self-maintaining. These properties are only dependent on the stoichiometry of a reaction network. The theory operates on a relatively high abstraction level, sets of molecular species, and neglects, so far, quantitative aspects

242

such as concentration level. Yet, relations of the organizations to dynamical behaviors are proven. Supposing dynamics of reaction systems are given in the form of ordinary differential equations (ODE), the set of species in any fixed point with positive concentrations is an organization (Dittrich & Speroni di Fenizio, 2007). Thus, chemical organization theory allows predicting which species and reactions can persist in a long-term simulation of the reaction system (see Kaleta, Centler, Speroni di Fenizio, & Dittrich, 2008) for a practical application). This prediction does not only encompass stationary states where the concentration of no molecular species changes but also periodic attractors and long-term behaviors with an unbounded increase of concentrations (Peter, 2008). The definition of chemical organizations described in this section is adopted from (Dittrich & Speroni di Fenizio, 2007). Formally, a reaction network is a tuple M , R where M is a set of molecular species and R is a set of reaction rules among those species. One reaction rule r Î R is specified with the stoichiometric coefficients li,r ³ 0 and ri,r ³ 0 for i Î M , corresponding to the lefthand and right-hand side, respectively. Two map-

{

}

pings are defined: LHS(r) := i ∈ M li,r > 0

{

}

and RHS(r) := i ∈ M ri ,r > 0 , extracting the set of reactant species and product species, respectively, given a reaction rule r Î R . At this point, the closure property can be defined. A set A Î M is closed if, for all reaction rules that can happen in A , their products are also contained in A ; ∀r ∈ RA where

{

}

RA = r ∈ R LHS(r) ⊆ A

RHS(r) Í A .

This condition assures that reactions among molecules in a closed set cannot produce molecular species outside this set. Definitions of self-maintenance property involve the species’ production rates r = (ri ∈M ) = Sv described as the multiplication

Organization-Oriented Chemical Programming of Distributed Artifacts

of stoichiometric matrix S = (ri,r − li,r ) and flux vector (or kinetic laws) v = (v r∈R

) depending

on the current concentration vector x = (x i ∈M

)∈ℜ

M ≥0

. Without loss of generality,

we assume v r ³ 0 so that a reversible reaction needs two entries in v . Although the kinetic laws can be arbitrary functions, they are constrained by the underlying reaction network: Obviously, the flux v r of reaction r can only be positive, if all reactant species ( LHS(r) ) are present. It also makes sense to assume the reverse. Then we obtain the chemical ODE constraint: vr > 0 ⇔ for all i ∈ LHS(r),x i > 0 .

(1)

There is a couple of theoretical approaches that use this constraint, already formulated by (Feinberg & Horn, 1974), to relate the algebraic structure of the underlying reaction network to the dynamical behavior of the reaction system (e.g., Gatermann, Eiswirtha, & Senssea, 2005). Additionally, the flux v r should be zero if any of reactants is absent. Under these constraints, the self-maintaining set is capable of sustaining every element in the set. Set A Î M is self-maintaining if there exists a strictly positive flux vector such that all species in A are produced at a non-negative rate. That is, rA = (ri ∈A ) ≥ 0 . Equivalent to the chemical ODE constraints, the flux vector is constrained as follows: v r > 0 if r ∈ RA , v r = 0 if r ∉ RA . The self-maintenance property assures that all species consumed by reactions in A can be reproduced by some reaction pathways in the whole network of A . Using that notion of chemical organizations, the given reaction network is explored and is decomposed into hierarchical, overlapping sub-networks (organizations). That hierarchy is an overview of persisting sets of species in the reaction systems

after processed according to the reactions specified. When reaction processes are employed for computation, the hierarchical organizational structure can provide us with an overview of computation outputs. This view has motivated an organization-oriented chemical programming technique (Dittrich & Matsumaru, 2007), where the theory of chemical organizations serves as a tool for programming chemical reaction systems.

A CHEMICAL PROGRAM FOR THE MIS PROBLEM The MIS problem is, given an undirected graph, to find a maximal independent set (MIS) as illustrated in Figure 1. Let G = W,E be an undirected graph where W = {w1, , w N } is a set of N vertices and E ⊆ W ×W is a set of edges. An edge is represented by a pair (w p , wq ) ∈ E of vertices that are connected. The order of the pair is insignificant, that is, (w p , wq ) ≡ (wq , w p ) . A set of vertices I Í W is independent if no two vertices in the set are directly connected by an edge:

{

}

∀ w p , wq p ≠ q, w p ∈ I , wq ∈ I : (w p , wq ) ∉ E

An independent set is maximal if there is no larger independent set containing it. No vertex can be added to a MIS without violating its independence property. The MIS problem can be solved efficiently while finding the largest MIS (denoted as maximum independent set problem) is NP complete. The MIS problem is particularly interesting because the independence property only relies on local quality whether two vertices are connected or not, and the MIS is the global quality of the whole graph. In other words, the MIS emerges at the global level from the local-level attribute of two vertices. This problem has been tackled under the distributed computing environments without

243

Organization-Oriented Chemical Programming of Distributed Artifacts

Figure 1. Illustration of the maximal independent set (MIS) problem. (Left) The given undirected graph consists of four vertices. (Right) Schematic representation of solving the MIS problem. Adding a vertex w1 , w2 , or w 4 to the empty set, the independence property is satisfied, but these sets are not maximal because it is possible to add a vertex without violating the independence property. Adding vertex w 3 , on the other hand, results in a solution of the MIS problem. Another solution to this MIS problem is calculated by further adding vertices to the non-MIS sets. The set {w1, w2 } is not an independent set because there is an edge between those vertices. Consequently, there are three MISs for this problem instance. In passing, the solutions to the “maximum” independent set problem is {w1, w 4 } and {w 2 , w 4 } because of the largest size while the other MIS {w 3 } has the size of 1.

a central control or without a global outlook, and the following two predicates are suggested (Herman, 2003; Shukla, Rosenkrantz, & Ravi, 1995; Ikeda, Kamei, & Kakugawa, 2002): (I) If a neighboring vertex w j ( (wi , w j ) ∈ E ) of wi is included in the set I , then the vertex wi should be excluded from I . (II) If no neighboring vertex w j of wi is included in I , the vertex wi should be included in I . Formally: (I) ∃w j : (wi , v j ) ∈ E , w j ∈ I ⇒ wi ∉ I , (II) ∀w j : (wi , w j ) ∈ E , w j ∉ I ⇒ wi ∈ I .

(2)

(3)

These are repeatedly applied locally in each vertex, and the set becomes a MIS. Our chemical program is derived from those.(see Figure 2) Given a graphG , the chemical program for the MIS problem, MIS chemistry, is a reaction

244

network: M , R

MIS G

= 〈M NMIS, LNMIS ∪ T E MIS 〉

where M NMIS is a set of 4N molecular species:

{

}

M NMIS = s1i , s 0i , f 1i , f 0i i = 1, , N . T h e Figure 2. MIS chemistry. (Bottom) Reaction rules R as a combination of logic rules L and exchange rules T . (Top) Illustration of the reaction network for two vertices. See text for details.

Organization-Oriented Chemical Programming of Distributed Artifacts

first character in the species name s stands for self and f for foreign, and the second character is a binary number indicating the membership of the vertex in the set. The subscript is the corresponding vertex number. The species names11 , for instance, should be interpreted such that the vertex w1 is positively included in the set. Species f 01 represents that the vertex w1 is connected by an edge to a vertex that is excluded from or negatively included in the set. Next, we define a mapping DEC to decode the species combination into a set of vertices. Given a set of species A Í M NMIS

{

}

DEC (A) := wi ∈ W s1i ∈ A, s 0i ∉ A . Then,

our design goal of the MIS chemistry is that, when dynamical reaction systems are constructed based on that reaction network, only species combinations that are mapped to a MIS are more likely to be present and observable. That is, our MIS chemistry is designed such that those species combinations form chemical organizations. There are two sorts of reaction rules: logic rules LNMIS and exchange rules T E MIS . Elements in LNMIS are common for every vertex and play a role of logical computation:

{

LNMIS = s1i + s 0i → Ø , f 0i → Ø , f 1i → s 0i i = 1, N

}

(4)

Exchange rules are defined for each edge in E: T E MIS =

 (T

(wi ,w j )∈E

MIS i→ j

∪T

MIS j →i

) where

T i →MIS = {s1i → s1i + f 1j , s 0i → f 0 j } . j



(5)

Communications between vertices are established by exchanging molecules through undirected diffusion or directed transport (Abelson et al., 2000; Siehs & Mayer, 1999; Hiyama et al.,

2005). For example, withs1i → s1i + f 1j , vertex wi informs a neighbor vertex w j that wi is now included in the set. The bidirectional communication is assumed so that there are flows from w j to wi . The logic and exchange rules are, from a formal point of view, indistinguishable. We denote these reactions by the reactants. Three logic rules in LNMIS are r (s ) , r ( f 0) , and r ( f 1) , respeci

i

i

tively, for vertex wi . Exchange rules can be writ-

{

= r (s1) ten as T i →MIS j

i→j

, r (s 0)

i→j

}.

In a former work (Matsumaru, Lenser, Hinze, & Dittrich, 2007), we developed another chemical program for the same problem. The difference is that the former program involves an irrational reaction such that the number of reactants depends strictly on the number of neighbors. If a vertex has five neighbors, a fifth order reaction must be defined. Moreover, that reaction itself must be modified whenever the neighbor list is changed, and it becomes critical in a dynamically changing environment. These insufficiencies are overcome in this work. The order of the chemical reactions is restricted to at most two regardless of the graph topology. The neighboring vertices are presumed to be indistinguishable such that all neighbors are categorized as a foreign vertex. Even in the wireless network, maintaining a neighbor list is feasible with the help of neighbor discovery algorithms, but the improved chemical program does not require the list, saving energy and memory resources. Further distinction is that the former program is infeasible to analyze with local perspectives, introduced later.

ORGANIZATIONAL ANALYSIS OF CHEMICAL PROGRAM Organizational analysis is an algebraic analysis on a reaction network to explore which species combination is a chemical organization. We dem-

245

Organization-Oriented Chemical Programming of Distributed Artifacts

onstrate how the chemical organization theory can be applied, and simultaneously, our design intension is exposed. A particular graph structure has to be chosen. A simple graph consists of two

{w , w }, {(w , w )} , and there are two MISs: {w } and {w } . Given this graph, vertices: G2 =

1

2

1

1

2

2

the MIS chemistry 〈M 2

MIS

, L2MIS ∪ T E MIS 〉 con2

sists of 8 species and 6 logic rules inL2MIS . Exchange rules are dependent on the given set of edges E 2 = (w1, w2 ) :

{

}

{

T E MIS = T 1→MIS ∪ T 2→MIS = r (s1) 2 1

.

2

1→2

, r (s 0)

1→2

} ∪ {r (s1)

2→1

, r (s 0)

2→1

}

(6)

Applying the organization, this reaction network is decomposed into overlapping sub-networks of organizations. The result is a hierarchical organizational structure, as shown in Figure 3, composed of three organizations: {Ø } ,

{s1 , f 0 , s 0 , f 1 } , {s 0 , f 1 , s1 , f 0 } . The or1

1

2

2

1

1

2

2

ganizations of size 4 are decoded to the two MISs, respectively, and no organizations mapping to the non-MIS exist. Implication of this analysis is that Figure 3. Organizational structure within the reaction network for the MIS problem and corresponding MISs. We take a graph with two vertices linked together as a problem instance, and the MIS chemistry consists of 10 reactions as shown in Figure 2. There are three organizations within that reaction network. Two of those, except for the empty set, correspond to the two solutions to the MIS problem. Vertices with double circles specify the MIS.

it is possible to construct chemical reaction systems, based on that MIS chemistry, that behave in accordance with the solutions to the MIS problem. In the following, we briefly sketch how that hierarchy is extracted. Exact calculation of organizational structures can be referred in (Kaleta, Centler, Speroni di Fenizio & Dittrich, 2008). Our exploration for the chemical organizations starts from the smallest, that is, the empty set. The empty set is always the smallest organization unless there is an inflow, which produces molecules from none. Then, bigger sets are explored whether they satisfy the two criteria to be an organization. For instance, set {s11 } is not closed since an exchange rule produces a new molecular species which is not contained:

(

RHS r (s1)

1→2

) = {s1 , f 1 } ⊄ {s1 } . 1

Set

1

A = {f 01 } is closed but not self-maintaining. The production rate for the element in A can be w r i t t e n a s

( ) (

) (

rA = rf 0 = −v r(f 0) + v r(s 0) = −v r( f 0) 1 1 2→1 1

)

because v r(s 0) = 0 due to the chemical ODE 2 →1

(

constraint ( LHS r (s 0)

2→1

) ⊄ A ). The production

rate cannot be non-negative, and thus the set is not self-maintaining. Set O = {s11, f 01, s 02 , f 12 } is an organization. There are four applicable reaction rules:

{

RO = r (s1)

1→2

, r ( f 1) , r (s 0)

2→1

2

, r ( f 0)

1

}

.

Applying the mapping RHS to each, it is certain that no new molecular species, outside of O , will be produced by any of those reactions.

(

RHS r (s1)

1→2

) = {s1 , f 1 } ⊆ O , for instance. 1

2

After the closure, the self-maintenance property is checked. The production rate of the elements in O is rO = SO vR where: O

246

2

Organization-Oriented Chemical Programming of Distributed Artifacts

 profiles x (t ) ∈ X t0 ≤ t ≤ t1

{

(s11 ) 0 0 0 0    ( f 01 ) 0 0 1 −1  SO =  (s 02 ) 0 1 −1 0    ( f 02 ) 1 −1 0 0  ,  v = v  1 r (s 1) 1→ 2     v v =  2 r ( f 1)  2  vR =  . O v 3 = v r(s 0)  2 →1     v 4 = v r(f 0) 

(7)

1

By choosing the flux vector such that 0 < v 4 ≤ v 3 ≤ v2 ≤ v1 , all of the production rates c a n b e n o n - n e g a t i v e :

(

rO = 0 v 3 − v 4

v2 − v 3 v1 − v2

)

T

≥0

.

Since there exists a strictly positive flux vector satisfying rO ³ 0 , the set O is self-maintaining.

ORGANIZATION-ORIENTED DESIGN PRINCIPLES Our MIS chemistry presented in the previous section was, in the design phase, strongly influenced by the theory of chemical organizations. That programming technique was named organizationoriented chemical programming and described naively in (Dittrich & Matsumaru, 2007). Here, that description is elaborated in more detail, and our design process of the MIS chemistry is explained. There are seven principles, from which the first one (P1) describes a constraint on mainly coding schemes to insure the applicability of chemical organization theory. Considering P2-P6, a reaction network is designed. The basic idea is to arrange the network to conform the organizational structures to the desired ones. Then, kinetics and its parameters are specified for fine-tuning the computation as stated in P7. P1: There should be one organization for each output behavior class. Dynamical behaviors of reaction systems are time series of concentration

}

where X is the

systems’ state space. By some features, the dynamical behaviors are categorized in classes, and those behavior classes are interpreted as outputs of computation. This initial programming principle states that this output behavior classification should be arranged such that there should be one organization for each of that class. In other words, if there are two distinguishable behavior classes, then the corresponding organizations should be different from each other. In our MIS example, the feature is whether species s1 or s0 is present in the reactor at a certain time point. The time point is determined such that the reaction systems will reach to a steady state. Utilizing this classification feature, there are four behavior classes C = {cØ , cs 0 , cs 1, cs } depending on which species are present: neither (cØ ), s1 (cs1 ), s0 (cs 0 ), and both (cs ). Out of these, two can be outputs: C out = {cs 1, cs 0 } (either

s1 or s0 is exclusively present). An output behavior classcs1 , for example, is interpreted to include in the set. The other two behavior classes, cØ andcs , cannot be mapped to the output of the MIS problem, and they are denoted as uncompleted computations. Note that this classification applies to each vertex. To indicate a corresponding vertex, another subscript may be attached to the class name or the reactor name. This behavior classification or coding scheme is a simple way to fulfill this first principle. The output classescs1 , cs 0 are distinguishable at the abstraction level where chemical organization theory operates. The same feature can be used to distinguish organizations whether species s1 or s0 is included. In passing, there are approaches of chemical computation exploiting quantitative values of species concentration. Examples are enzymatic computation (Zauner & Conrad, 2001), where the concentration level (high or low) of reaction products are chosen for features to clas-

247

Organization-Oriented Chemical Programming of Distributed Artifacts

sify output behaviors. Another example is a chemical reaction system evolved by Deckard and Sauro (2004) to compute the square-root. The final concentration xT of a molecular species is the square root of the initial concentration x 0 ( xT = x 0 ). These coding schemes to map dynamical behaviors to computational output do not cooperate with this programming technique because of the violation of this principle. Quantitative aspects are indistinguishable from the organizational point of view. On the other hands, the classical DNA computing (Adleman, 1994) or a prime number artificial chemistry (Banzhaf et al., 1996) may be combined with chemical organization theory since the existing species characterize computational outputs. Adopting that coding scheme, we define reaction rules such that the resulting chemical program contains organizations that can be mapped to the solutions to the MIS problem. Namely, two organizations are desired in the two vertex example: {s11, f 01, s 02 , f 12 } and {s 01, f 11, s12, f 02 } . We first add a reaction r (s ) := s1 + s 0 → Ø , called

a cooperative decay, in order to eliminate organizations that are associated with a class of uncompleted computationscs . This reaction rule invalidates the self-maintenance property. For example, set {s11, s 01 } is not self-maintaining because of the negative production rate ofs11 : xs 1 = −v r(s ) where flux vr(s ) > 0 due to the 1

1

Furthermore, it is expected that the desired output set is contained in a self-maintaining set within that closure. The self-maintenance property of the set of molecular species indicates theoretical possibilities to sustain all the species in the dynamical reaction systems, so the desired output species may be sustained in the reaction system until the outcomes of the computation is observed. The ideal case is that the desired output is represented by a largest self-maintaining set within that closure. In case that there exists a larger self-maintaining set than the desired output set, the dynamics may settle above the desired one. This argument leads to P3. P3: The set of molecular species representing an input should generate the organization representing the desired output. To generate the organization from a set of species, by definition, the closure of the given set is taken at first. Then we remove species until we reach a largest self-maintaining set contained in the closure. This principle will be fulfilled on the following two conditions: the desired output is contained within the closure of the input (P2 is fulfilled), and the largest self-maintaining set contained in the closure corresponds to the desired output.

1

chemical ODE constraint. One species cannot be produced at a non-negative rate, and thus that set is not self-maintaining. P2: The set of molecular species (and the organization) representing a result should be in the closure of the species representing the initial input. The closure denotes a set of molecular species that is generated by adding all possible reaction products until no more new species can be produced. This principle assures that there is

248

a reaction path from the initial input configuration to the desired output species. Otherwise, the desired output will not appear as a result of the computation.

The largest self-maintaining set within a closure is not always unique in general although it is uniquely generated in a specific class of reaction networks, called semi-consistent (Dittrich & Speroni di Fenizio, 2007). In chemical computing, the uniqueness is not required. It can be even beneficial, on the contrary. In our example, the initial input configuration is represented by the self species for each vertex. Following P2, we created a reaction path from

Organization-Oriented Chemical Programming of Distributed Artifacts

s11 to s02 in accordance with the first predicate

(I) in Equation (2): r (s1)

:= s11 → s11 + f 12

1→2

and r ( f 1) := f 12 → s 02 . The first exchange rule 2

is catalytic due to P3. Otherwise, one of the target organizations {s11, f 01, s 02 , f 12 } would not be self-maintaining because production rate of s11 would be always negative: xs 1 = −v r s − v r s 1 ( )1

1

( )1→ 2

where v r(s ) = 0 and v r(s 1) > 0 . Species s12 1 1→ 2 may also generate s01 : r (s1) and r ( f 1) . 2→1

1

P4: Eliminate organizations not representing a desired output. Since each organization potentially includes fixed points, the reaction system’s dynamics may converge to one of the organizations. Hence, it makes sense to eliminate organizations not representing an output in order to avoid premature termination of a computation or even false computational outputs. This can be achieved by destroying either its closure property or its self-maintenance. Organization to be eliminated here is {s 01, f 01, s 02 , f 02 } because this set represents an

undesiredoutput.Reaction r (s 0)

or r (s 0)

2→1

1→2

:= s 01 → f 02

:= s 02 → f 01 contributes to destroy

its self-maintenance property while keeping the closure property. Production rate of s01 is always negative: xs 0 = −v r s + v r f 1 − vr s 0 where 1

( )1

( )1

vr(s ) = vr(f 1) = 0 and vr(s 0) 1

1

( )1→2

> 0 . This produc-

1→ 2

tion rate can be positive only when the set contains f 11 , applying positive constraint on flux v r(f 1) > 0 . Thus, that set is not an organization 1

any more. P5: An output organization should have no organization below. The dynamics of the reaction system that moves from one organization O1 to another O2 below (i.e.,O2 É O1 ) is called a

downward movement. This dynamical move can be theoretically prevented by the self-maintenance property with a right kinetics. Practically speaking, this move may still occur spontaneously due to, e.g., stochastic effects. Following this principle, a downward movement can be restricted. So far, the chemical program consists of eight reactions and contains eight organizations. Below the target organizations, there are four organizations: {Ø } , {f 01 } , {f 02 } and {f 01, f 02 } . Three of those (except the empty set) are not organizations any more when we added outflows r ( f 0) := f 0 → Ø . Additionally, the following sets are not self-maintaining, either: a n d {s11, f 01, s 02, f 12, f 02 }

{s 0 , f 1 , f 0 , s1 , f 0 } . Production rate of f 0 , 1

1

1

2

2

1

for example, is x f 0 = −v r( f 0) + v r(s 0) so that 1 1 2→ 1 s02 must coexist to apply positive constraint on the addition term. We now reach to the reaction network containing only three organizations as shown in Figure 3. It may be better to eliminate the empty set, but it is not possible in this case. The empty set is always self-maintaining, so the closure property must be tackled. An inflow generating molecules from none is considered. However, any of such reactions also destroy the closure property of one of the target organizations. Thus, the empty set as the organization cannot be eliminated. In other words, only empty set is allowed to connect those target organizations because there is no common element. P6: Assure, if possible, stoichiometrically the stability of an output organization. Instead of eliminating organizations below the desired output as in the previous principle P5, the downward movement can be ruled out by purely stoichiometric argument. It may be possible to design the reaction network such that the organization rep-

249

Organization-Oriented Chemical Programming of Distributed Artifacts

resenting the desired output is stable for any kinetic law. As a simple example consider the system R = {a → b, b → a } , which has two organiza-

tions: {Ø } and {a, b } . Due to mass-conservation,

the system can never move spontaneously from the organization with two species to the empty one. In the MIS example, this principle is implicitly conformed.

{w } . 3

within the reaction network, the correspondence between the organizations and MISs is, again, confirmed. Concentration dynamics of species s1i and s 0i for vertex wi are expressed as follows: d s1i  dt

P7: Use kinetic laws for fine tuning. The kinetic law determines the systems’ behavior within an organization and the transition dynamics between organizations. One of rationales for the right kinetics is to assure that the dynamical reaction systems are stable in the output organizations, restricting mainly the downward movement. Finding the right kinetic laws is in general a non-trivial task. However, the existence of such laws is ensured by chemical organization theory to a certain extend, and we have seen that following principles P1-P6 simplify this tasks significantly. Classical dynamical systems theory is certainly reliable for this task, and it is even possible to derive at least in some cases rigorously dynamical stability from network structure (Clarke, 1980; Feinberg & Horn, 1974). Another point of consideration is a trade-off between that stability and the speed of computation since chemical reaction systems may compute by moving amongst organizations.

d s 0i  dt

(8)

   = k2  f 1i  − k1 s1i  s 0i  −  ∑ k3 s 0i     (wi ,w j )∈E  (9)

k

k

1 2 reactions s1i + s 0i   → Ø and f 1i   → s 0i , respectively, and both are set to 0.1. Exchange k3 → f 0 j with a kinetic constant rule s 0i  

k 3 = 0.1 is considered for every neighboring

vertex w j with (wi , w j ) ∈ E . Species s1i is not

flowed out to the neighbor vertices because of the catalytic nature of the exchange rule: k4 s1i   → s1i + f 1j , k 4 = 0.1 . Concentration dynamics of species f 1i and f 0i for vertex wi are given: d  f 1i 

DYNAMICAL SIMULATION

250

= −k1 s1i  s 0i 

Where k1 and k2 are kinetic constants for

dt

Based on the MIS chemistry described above, a dynamical reaction system in the form of ordinary differential equation (ODE) systems can be constructed, assuming mass action kinetics for each reaction. In this section, we present dynamical behaviors of the MIS chemistry for the validation purpose. As a concrete problem instance, we utilize a graph shown in Figure 1, and the graph embraces three MISs: {w1, w 4 } , {w2 , w 4 } , and

Analyzing the organizational structure

d  f 0i  dt

   = −k2  f 1i  +  ∑ k4 s1j   ,     (wi ,w j )∈E    = −k5  f 0i  +  ∑ k 3  (wi ,w j )∈E

 s 0   .  j   

(10)

(11)

Both equations are composed of one term for an outflow and multiple terms for inflows from neighboring vertices. Kinetic constant k5 for k

5 → Ø is set to 0.1. outflow f 0i  

Organization-Oriented Chemical Programming of Distributed Artifacts

The equation system listed in Equations (8) – (11) is a reaction system based on our MIS chemistry. However, we introduced a modification on the dynamics of s1 concentration: d s1i  dt

0.1 = −k1 s1i  s 0i  − k6 s1i  + s 0  + 10  i  (12)

Species s1i is incorporated with an inflow that is inhibited by speciess 0i , expressed by the last term of Equation (12). The production rate of s1i is low when s 0i exists in a high concentration. This modification implements automatic regeneration of s1i species when s 0i is absent. Looking at the reaction network, there is no production rule of speciess1i . This becomes problematic when a vertex is desired to alter dynamically the membership state from negative (s0 ) to positive (s1 ). This may happen when, for example, all vertices are dynamically disconnected. In order to cope with that condition, the inflow is added to compensate the disappearance ofs1i . The inhibitory effect from s 0i is necessary to deactivate the inflow by the presence ofs 0i . Outflow term −k6 s1i  ( k6 = 0.001 ) is also added in order to avoid explosive increase of s1i concentration ( k

6 s1i   → Ø ).

Aproblem instance on the graph G = W4 ,E 4 as shown in Figure 1 is considered: V4 = {w1 , w2 , w 3 , w 4 } ,

E4 =

{(w , v ), (w , w ), (w , w ), (w , w )} . 1

2

1

3

2

3

3

4

archy of organizations within that reaction network is shown in Figure 4, and the correspondence between the organizations and MISs is confirmed. Three largest organizations are decoded to the three solutions of the MIS problem, and there is no organization associating with an incorrect solution. The analysis also indicates possibilities of the uncompleted computations denoted in the figure as Org 1, Org 2, Org 3, and the empty set. When the system is “caught” in one of these organizations, the outcome of the computation is inconclusive. This happens when there are not enough molecules, especially s1 in the system. This situation was circumvented in the ODE system by introducing an inflow of s1 species. Figure 5 shows a dynamical behavior of a MIS chemistry implemented as an ODE system listed in Equations (9) – (12). The reaction system is stochastically simulated using Copasi (Sahle et al., 2006) and compartmentalized (Amos, 2004) in order to emulate distributed system settings, in which each compartment equals a vertex. The compartment size is set to 20 ml. Initially, every compartment is set empty, and we disconnected

Figure 4. Global analysis of the MIS chemistry for the four-vertex topology as shown in Figure 1. The Hasse-diagram on the left panel shows all of seven organizations within the MIS chemistry M ,R

MIS 4

, and contents of each organization

is listed on the right panel. Three largest organizations are decoded to MISs, desired outputs of the computation. The other four organizations below cannot lead to a computational result, representing uncompleted computations.

(13) The MIS chemistry M , R

MIS

now par-

ticularly consists of 16 species and 28 reactions. There are 12 logic rules and 16 exchange rules since four edges connect four vertices. The hier-

251

Organization-Oriented Chemical Programming of Distributed Artifacts

Figure 5. Dynamical behavior of MIS chemistry. The graph structure is given at the top, and the dashed lines specify deactivated connections. The computational result from the reaction system is depicted as vertices with double circles. Initially, every connection is deactivated. Only s1 species for every vertex has a positive concentration, and this state is decoded to the set of four vertices. At t = 5000 , all of connections are activated. In vertex w1 and w 4 , s1 continues to exist with a high concentration. Vertex w2 and w 3 is populated by s0 species, and s1 species is vanished. The output of the chemical compu-

tation is {w1, w 4 } . In the whole simulation run, the chemical reaction system behaved in order to maintain the global MIS property of the underlying graph. Auxiliary species, f 1 and f 0 , are also present accordingly. In w1 , only f 0 species is with a positive concentration because that vertex is linked only with vertices excluded from the set. Vertex w2 contains both f 1 and f 0 species because of the connections to w1 and w 3 , respectively.

all of vertices so that no exchange of molecules is possible. Each compartment comes to populate with » 200 molecules of s1 species (10 #/ ml ) due to the appended inflows. Att = 5000 , all of four edges are activated. That steady state of each compartment is perturbed, and the ODE system settles on a new steady state after » 500

252

steps. Decoding those states, the computational output of the reaction system is the set of all four vertices in the first half of the run. In the second half, the output becomes {w1, w 4 } . Those are in accordance with the solutions of the MIS problem. We next studied the behaviors when the graph topology changes dynamically, and the result is

Organization-Oriented Chemical Programming of Distributed Artifacts

illustrated in Figure 6. At the beginning (t = 30000 ), every edge of four is deactivated, and all vertices are disconnected in the time range of 30000 : 35000 . Then every 5000 time steps, the graph topology is altered. The disconnected graph is reconnected in three steps: edge (w1, w 3 )

is activated att = 35000 , vertex w2 is connected by activating two edges att = 40000 , and edge (w3 , w4 ) is activated att = 45000 . In the next step, vertex w 2 is temporary disconnected by deactivating the two edges att = 50000 . In the end ( 55000 : 60000 ), all four edges become

activated. According to this scenario, we simulated the MIS chemistry in the form of the compartmentalized ODE system, and the dynamical behaviors are shown at the bottom of Figure 6. That concentration graph only shows that of s1 for each vertex. We made sure that s1 species concentration became high only when s0 concentration was (very) low, and vice versa. Observing thes1

concentration, thus, is enough to determine the membership state of the vertex. The decision whether a vertex is included in the set or not is utterly local, depending on the reaction process within the corresponding compartment. Nevertheless the maximal independence as global property is maintained all the time.

ORGANIZATIONAL ANALYSIS FOR DISTRIBUTED SYSTEM Our MIS chemistry is designed to consist of 4N molecular species, where N is the number of the vertices in the given graph. In fact, only 4 distinctive species are multiplied for each vertex. For example, speciess11 ands12 are identical but distinguished because of the location difference, either in vertex w1 or w2 . This global view is not scalable with respect to N , has much redundancy, and is dependent on a specific problem instance. Here, we show an alternative, a local perspective.

Figure 6. Exhibiting a robustness of the MIS chemistry against dynamical changes of the underlying graph. Every 5000 time step, the graph structure is changed by de-/activating connections between vertices as shown on the top. Accordingly, exchange rules in T MIS are affected. The computational output decoded from the reaction system state is represented as double circles of the vertices. The dynamical behaviors of the reaction system in terms of species concentrations are shown at the bottom. Even though the underlying graph structure is changed dynamically, the chemical reaction system has computed to maintain the global MIS property all the time. See text for details.

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Organization-Oriented Chemical Programming of Distributed Artifacts

It is significant for network analysis to determine the spatial regions to cover because of the effects on network structures. An enzymatic reaction, which is mediated by a certain molecular species, is only defined when that specific enzyme species is present in the coverage. The catalysts may be isolated by membrane or immobilized to a specific location. As argued in (Speroni di Fenizio & Dittrich, 2007), the organizational analysis can also provide valuable insights into the spatial structure of the system and information about the best spatial scale to consider. With respect to distributed computing systems, the physical structure provides two apparent perspectives: local focusing on each computational unit and global considering the whole system. Taking the local perspective, the MIS chemistry is adjusted so that only those distinctive MIS = {s1, s 0, f 1, f 0} species constitute the set M local . Three logic reactions among those are there. The exchange rules are modeled as an outflow and an influx because the neighboring vertices are out of the scope, identified as an environment. The

MIS MIS MIS set of reaction rules isRlocal = Llocal ∪ T local where: MIS Llocal = {s 0 + s1 → Ø , f 0 → Ø , f 1 → s 0} , MIS T local = {s 0 → Ø } . (14)

Note that there is no outflow of s1 because no exchange rule consumes it. Influxes are appended depending on the environmental conditions as shown in Figure 7: no neighbors (without any inflows), no neighbors are included in the set (append influx of f 0 ), all neighbors are included in the set (append influx of f 1 ), and some neighbors are included and some are not (append both influxes of f 1 and f 0 ). Analyzing the organizational structure in all cases, there are organizations with the desired species combination, and no organizations violating MIS property exist. Two organizations are nonessential: {Ø } and {f 0} , and those elucidate the possibilities of uncompleted computations. In fact, our dynamical mechanism of the inhibited s1 influx is to move the system from those organizations up to the “solution” ones.

Figure 7. Organizational analysis of the MIS chemistry from local perspectives. (Top) Reaction networks or sets of reactions. (Bottom) The organizational structure for each network. The organizational analysis focuses locally on each vertex. Since neighboring vertices are considered as in the environment, exchange rules are modeled as outflows and inflows. There are four cases because, for the MIS problem, these are the environmental conditions any vertices possibly encounter. The base reaction network with no inflow (leftmost) represents a vertex has no neighbors. The inflow of f 0 is appended when vertices with the negative membership in the set become neighbors. Otherwise, the inflow of f 1 is added to the base case. The rightmost case represents that the memberships of neighboring vertices are both positive and negative. Components of the organizations agree to the desired behavior of the vertex under any conditions. This analysis result is independent of the underlying graph structure.

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Organization-Oriented Chemical Programming of Distributed Artifacts

Since chemical organizations indicate the species combinations with a positive concentration in fixed points, each vertex will be controlled by the MIS chemistry in accordance with the independence property. If there exists a neighbor vertex containing in the set, the self vertex must be excluded. The self vertex shall be included in the set only when no neighbors are in the set or there are no neighbors at all. This analysis with the local perspective is independent of N and therefore scalable.

CONCLUSION AND OUTLOOK We have presented a theoretical principles for designing chemical computing systems, namely organization-oriented chemical programming. Our techniques were exemplified on the maximal independent set (MIS) problem, which could be regarded as a prototype of differentiation and morphogenesis. Applying our approach has led to a solution chemistry, and robustness against dynamical changes of the underling graph structure has been also demonstrated using simulations. The solution chemistry was analyzed using chemical organization theory additionally with a different spatial scale, focusing locally on each computation unit. From this local perspective, exchange of molecules between devices is modeled as inflows and outflows. The global perspective is, on the other hand, realized by attaching spatial coordinates to the species name in order to distinguish molecular species based on residing locations. The insight gained from the local and global analysis allowed us to understand how the system copes with qualitative perturbations and with a changing underlying graph structure. Consequently, we add a stochastic inflow of a particular species, which should initiate upward movements in the space of organizations. This modification led to a robust system that can adapt to a dynamically changing graph structure.

Molecules and chemical reactions are computation mediums in the nano-scale world, and the molecular computing metaphor is proposed as the models of computation for nano-scale devices. Communication between devices is assumed as the exchange of molecules through directed diffusion or transportation along communication links. This sort of communication paradigm has been provoked and investigated for biological, nanoscale devices, instead of communications with electrons or electromagnetic waves, because of the power and size limitations (Moore, Enomoto, Nakano, Okaie, & Suda, 2007). The traditional communication methods, e.g., by radio links are also suggested for nano-scale machines (Demoustier, Minoux, Le Baillif, Charles, & Ziaei, 2008). In that case, a molecule will be exchanged via virtual data packages transmitted by electromagnetic waves. Our focus here was to construct and design chemical programs from scratch. It will be practical when this manual construction method is combined with optimization. One method supplies building blocks or network motifs for the other. We have seen benefits when an artificial optimization process is guided with chemical organization theory (Lenser, Matsumaru, Hinze, & Dittrich, 2008). Instead of manufacturing chemical programs, different methods have been conceived such that reaction systems in nature are studied and explored for desired behaviors because natural systems are furnished with numerous functional motifs. For this, we call, explorative approach, the organizational analysis will be useful in order to detect systematically a particular one from that rich repository.

ACKNOWLEDGMENT The authors would like to thank our colleagues of our group for their support and useful advice. We are very grateful to Gerd Grünert and Christoph Kaleta for their help with the software develop-

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Organization-Oriented Chemical Programming of Distributed Artifacts

ment and Elisabeth zu Erbach-Schönberg for reviewing nanoparticle research activities. We acknowledge financial supports by the German Research Foundation (DFG) Grant DI 852/4 and by the European Union, NEST-project ESIGNET no 12789.

Banzhaf, W., Dittrich, P., & Rauhe, H. (1996). Emergent computation by catalytic reactions. Nanotechnology, 7(1), 307–314. doi:10.1088/09574484/7/4/001

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Rothemund, P. W. K. (1996). A dna and restriction enzyme implementation of turing machines. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 27(12), 75–113. Sahle, S., Gauges, R., Pahle, J., Simus, N., Kummer, U., Hoops, S., et al. (2006). Simulation of biochemical networks using copasi - a complex pathway simulator. In Proceedings of the 2006 Winter Simulation Conference (pp. 1698-1706). Salata, O. V. (2004). Applications of nanoparticles in biology and medicine. Journal of Nanobiotechnology, 2(1), 3. doi:10.1186/1477-3155-2-3 Shukla, S. K., Rosenkrantz, D. J., & Ravi, S. S. (1995). Observations on self-stabilizing graph algorithms for anonymous networks. In Proceedings of the Second Workshop on Self-Stabilizing Systems (pp. 7.1-7.15).

Tschudin, C. (2003). Fraglets - a metabolic execution model for communication protocols. Paper presented at AINS’03. Wang, J. (2005). Nanomaterial-based amplified transduction of biomolecular interactions. Small, 1(11), 1036–1043. doi:10.1002/smll.200500214 Willner, I., Basnar, B., & Willner, B. (2007, Jan). Nanoparticle-enzyme hybrid systems for nano-biotechnology. The FEBS Journal, 274(2), 302–309. doi:10.1111/j.1742-4658.2006.05602.x Zauner, K.-P., & Conrad, M. (2001). Enzymatic computing. Biotechnology Progress, 17(3), 553–559. doi:10.1021/bp010004n Ziegler, J., & Banzhaf, W. (2001). Evolving control metabolisms for a robot. Artificial Life, 7(2), 171–190. doi:10.1162/106454601753138998

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 1-19, copyright 2009 by IGI Publishing.

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Chapter 17

Dominant Spin Relaxation Mechanisms in Organic Semiconductor Alq3 Sridhar Patibandla Virginia Commonwealth University, USA Bhargava Kanchibotla Virginia Commonwealth University, USA Sandipan Pramanik University of Alberta, Canada Supriyo Bandyopadhyay Virginia Commonwealth University, USA Marc Cahay University of Cincinnati, USA

ABSTRACT We have measured the longitudinal (T1) and transverse (T2) spin relaxation times in the organic semiconductor tris(8-hydroxyquinolinolato aluminum) - also known as Alq3 - at different temperatures and under different electric fields driving current. These measurements shed some light on the spin relaxation mechanisms in the organic. The two most likely mechanisms affecting T1 are hyperfine interactions between carrier and nuclear spins, and the Elliott-Yafet mode. On the other hand, the dominant mechanism affecting T2 of delocalized electrons in Alq3 remains uncertain, but for localized electrons (bound to defect or impurity sites), the dominant mechanism is most likely spin-phonon coupling.

INTRODUCTION Recent interest in the field of “spintronics” stems primarily from the desire to use the spin degree DOI: 10.4018/978-1-60960-186-7.ch017

of freedom of a single electron, or a collection of electrons, to store, process, detect and communicate information. Digital information (in the form of binary bits 0 and 1) is encoded in the spin polarization of electron(s), then “processed” using spin-spin or spin-orbit interactions, subsequently

Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Dominant Spin Relaxation Mechanisms

communicated over long distances using spin waves or spin chains, and finally sensed using techniques that are able to measure spin polarizations of single or multiple electrons. A well known embodiment of this idea is the Single Spin Logic (SSL) paradigm (Bandyopadhyay, et al., 1994) where a single electron’s spin polarization is rendered “bistable” by placing it in a static magnetic field. The polarization can then point either parallel to the direction of the magnetic field, or anti-parallel to it, since only these two polarizations are allowed eigenstates of the Hamiltonian describing the electron. By engineering the exchange interactions between nearest neighbor spins, it is possible to implement (classical) digital Boolean logic gates and combinational logic circuits for universal computation (Bandyopadhyay, et al., 1994, Bandyopadhyay, 2005, Agarwal and Bandyopadhyay, 2007). These circuits have the advantage of being extremely energy efficient and amenable to high levels of integration, which results in enhanced computational prowess (Cahay and Bandyopadhyay, 2009). Evidently, the most important concern in such approaches is preserving the fidelity of the data that is being processed. The processed information must remain intact during the entire computational cycle, which can happen only if spin does not flip spontaneously while computation takes place. Coupling of an electron’s spin with the environment (phonons, magnons, etc.) can randomly flip the spin, leading to errors in the computation. The probability of such an error occurring during one computational cycle is p = 1 −e

−T Ts

,

(1)

where T is the duration of a computational cycle (typically the period of the clock that drives the computation) and Ts is the spin relaxation time. In order to make the error probability as low as

260

possible, we will have to make Ts as long as possible and/or T as short as possible. There are two distinct types of spin relaxation time Ts that matter. To understand them, consider the fact that an electron’s spin is a quantum mechanical entity and can exist in a state that is a coherent superposition of two mutually antiparallel polarizations, which we will label as “up” and “down”. An arbitrary spin can therefore be written as spin = a ↑ + b ↓ 2

2

a + b =1

,

(2)

where ­ denotes the “up” polarization and ¯ denotes the “down” polarization. The coefficients a and b are complex quantities. Because of this property, an electron’s spin is able to represent a quantum bit (or “qubit”) which is a coherent superposition of the classical binary bits 0 and 1. If we encode the classical bit 0 in the up-polarization and the classical bit 1 in the down-polarization, then a spin can represent the qubit qubit = a 0 + b 1 2

2

a + b =1

.

(3)

It is now obvious that the phase relationship between the coefficients a and b is vital since quantum mechanical information is contained in this relationship and plays a critical role in quantum computation. We can define a density matrix as:  2  a a *b  r  =  , 2    *  ab b  

(4)

Dominant Spin Relaxation Mechanisms

where the diagonal terms are real quantities and the off-diagonal terms are generally complex quantities [the asterisk denotes complex conjugate]. When a spin “relaxes”, both a and b have to change with time (see Equation (2)), so that both the diagonal and off-diagonal terms decay temporally. However, they can decay with very different rates. The rate at which the diagonal terms decay is expressed as 1 T1 and the rate at which the off-diagonal terms decay is expressed as1 T2 , where T1 is called the longitudinal relaxation time and T2 the transverse relaxation time. In quantum computing applications, it is important that both T1 and T2 be long so that the error probability is minimal. However, since the latter is usually much shorter than the former, it is more important to ensure that T2 is long. In classical computing applications, there is no issue of phase relationships and we only need to ensure that a bit does not flip randomly. In other words, we only require T1 to be long. In general, the classical bit error probability will be given by Equation (2) with Ts = T1 and the qubit error probability will be given by Equation (2) with Ts = T1 or T2, whichever is shorter.

ORGANIC SPINTRONICS A great deal of effort is currently being expended to find materials where both T1 and T2 are very long. Longitudinal spin relaxation times (T1) of the order of 1 second have been reported in inorganic semiconductor quantum dots, but only at very low temperatures of a few millikelvins (Amasha, et al., 2008). In contrast, nanostructures of the organic semiconductor tris(8-hydroxyquinolinolato aluminum), or Alq3, have exhibited T1 times that could exceed 1 second at the much more practical temperature of 100 K (Pramanik, et al., 2007) which makes organics the preferred platform for spin based classical computing. With T1 = 1 second and T = 10-9 seconds (for a 1 GHz clock),

the classical bit error probability according to Equation (2) will be 10-9, which may be sufficiently low to allow large scale fault-tolerant classical computing. Therefore, Alq3 is a promising material for spin-based classical computing above liquid nitrogen temperature (77 K). It also exhibits a T2 time of ~ 30 nanoseconds at room temperature, which may be long enough to allow fault tolerant quantum computing (Kanchibotla, et al., 2008). With a 1 GHz clock, the error probability associated with a qubit dephasing randomly is then ~ 3% according to Equation (2). This is low enough to permit fault tolerant quantum computing with the aid of error correction codes (Knill, 2005). Additionally, Alq3 has very special spin-dependent optical properties that are not found in inorganic semiconductors which make it particularly attractive for quantum computing applications since these optical properties can be harnessed for easy qubit read out (Kanchibotla, et al., 2008). Therefore, organic semiconductors may supersede their inorganic counterparts in spin-based computing applications – both classical and quantum mechanical. This motivates much of the interest in the emerging new field of “organic spintronics”.

SPIN RELAXATION MECHANISMS IN ORGANICS The Longitudinal Spin Relaxation Mechanisms In order to understand why organics have relatively long spin relaxation times, and to seek ways of making them even longer, we need to identify the physical mechanisms that relax spins in these materials. This is not an easy task since there is considerable controversy about what mechanisms are predominant. In almost all semiconductors, including the organic variety, the four primary longitudinal spin relaxation mechanisms that limit T1 are:

261

Dominant Spin Relaxation Mechanisms

1. 2. 3. 4.

Elliott-Yafet mode D’yakonov-Perel’ mode Bir-Aronov-Pikus mode Hyperfine interactions of electron spins with nuclear spins

The Elliott-Yafet mechanism (Elliott, 1954) arises from the fact that in the presence of spin-orbit interaction, electron eigenstates will not be spin eigenstates. This makes the eigenspinors momentum-dependent, which means that an electron’s spin orientation will depend on its momentum or velocity. Therefore, whenever a scattering event changes an electron’s momentum, it is likely to change the spin orientation as well. This is the basis of Elliott-Yafet spin relaxation. Since this mechanism is associated with momentum relaxing scattering events, the spin relaxation rate is proportional to the momentum relaxation rate and hence inversely proportional to that part of carrier mobility which is determined by scattering. The D’yakonov-Perel’mechanism (D’yakonov and Perel’, 1971) arises from spin-orbit interaction as well. This interaction acts like an effective magnetic field whose strength is proportional to the electron’s velocity (see, for example, Bandyopadhyay and Cahay, 2008). An electron’s spin will precess about this field, but since scattering will cause different electrons in an ensemble to have different velocities, they will experience different fields and therefore precess about different axes with different angular frequencies. As a result, the average spin (or the spin polarization) of the ensemble will decay with time. Obviously, the D’yakonov-Perel’ mechanism will be suppressed if the electron velocities are small so that the effective magnetic field (which is proportional to the velocity) is also small. As a result, the D’yakonov-Perel’ relaxation rate is proportional to average carrier velocity and hence carrier mobility1 (D’yakonov and Perel’, 1971) unlike the Elliott-Yafet mechanism. Consequently, materials with low carrier mobility will experience weak D’yakonov-Perel’ spin relaxation. Organ-

262

ics are expected to be relatively immune to the D’yakonov-Perel’ mechanism since they have low mobilities. More importantly, transport in organics takes place via hopping and the hopping duration is unlikely to be much longer than ~ 1 picosecond at reasonable temperatures (Ke, et al., (2006)). This time is too short for the spin to complete one precession about the effective magnetic field (caused by weak spin-orbit interaction) while it is in motion. The angular frequency of precession is given by the Larmor expression Ω=

g mB Beff 

,

(5)

where g is the Landé g-factor of the organic (g = 2 for Alq3), μB is the Bohr magneton and Beff is the effective magnetic field due to spin-orbit interaction. In order to complete a precession in less than 1 picosecond, Beff must exceed 5.5 Tesla, which is completely impractical, particularly in organics which have weak spin-orbit interaction. Therefore, an electron cannot even complete a single precession while it is in motion during a hop. Consequently, it will not experience D’yakonov-Perel’ spin relaxation. In other words, the D’yokonov-Perel’ mechanism is not likely to be dominant in organics. The Bir-Aronov-Pikus mechanism (Bir, et al., 1976) arises from exchange interaction between an electron and a hole. If the hole’s spin flips for any reason, it may make the electron’s spin flip as well. This mechanism is usually important in bipolar transport where both electrons and holes are present and carry current. In unipolar spin transport, this mechanism plays almost no role. The hyperfine interaction (Abragam, 1961) with nuclear spins has its origin in the fact that nuclear spins give rise to an effective magnetic field. This field interacts with electron spins and may cause the latter to randomize (or relax) with time. However, hyperfine interactions are not

Dominant Spin Relaxation Mechanisms

expected to be strong in most organics used in spintronics, particularly π-conjugated molecules like Alq3. There, the interaction is primarily due to the nuclear spins of the isotopes 1H (naturally abundant), 13C (not naturally abundant) and 14N (naturally abundant). In π-conjugated molecules, the wavefunctions of delocalized carriers have practically no overlap with the C and H atoms, since the wavefunctions consist of pz orbitals whose nodal plane coincides with the molecular plane (Naber, et al., 2007). In Alq3, the delocalized wavefunctions may have some overlap with the C atoms (Sanvito and Rocha, 2006), but the latter’s naturally abundant isotope is 12C has no net nuclear spin. Therefore, hyperfine interaction should be vanishingly small in Alq3.

Which Mechanism is Dominant? Among the four mechanisms discussed above, the two that are most likely dominant in organics and determine the T1 time are the Elliott-Yafet mode and the hyperfine interactions with nuclear spins. Both are weak; the latter because of weak hyperfine interaction and the former because of weak spin-orbit interaction. Consequently, one expects Alq3 to have an exceptionally long T1 time since spin will relax slowly in this material.

The Transverse Spin Relaxation Mechanisms In the case of T2 time, the considerations are slightly different. This time depends on how fast an electron’s spin loses phase memory. Phonons and magnons will be particularly effective in reducing the T2 time since they are time-dependent perturbations that are notorious phase breaking agents. Therefore, the dominant mechanisms affecting T1 and T2 may be very different, and these two times can differ by several orders of magnitude. The most likely dephasing mechanisms that are operative in organics and determine T2 are:

1. hyperfine interactions of electrons with nuclear spins 2. spin-phonon and/or spin-magnon coupling 3. spin-orbit interaction 4. spin-spin interaction One or more of these mechanisms may dominate in different temperature ranges and under different situations.

SPIN RELAXATION TIMES IN ALQ3 AT DIFFERENT TEMPERATURES In an effort to determine the dominant spin relaxation mechanism in Alq3, we have carried out a series of experiments to measure both T1 and T2 as a function of temperature. These experiments shed light on the likely dominant mechanisms that determine T1 and T2 in Alq3.

Measurement of the T1 Time We measured the T1 time in (1) nanowires and (2) thin films of Alq3 using spin valve structures. Measurements of the magnetoresistance of these structures allow us to determine the spin relaxation length (or spin diffusion length), from which the spin relaxation time can be extracted if we known the carrier mobility. This is an indirect method of measuring the spin relaxation time and is necessitated by the fact that time resolved optical measurements that are routinely used to determine the spin relaxation time in inorganic direct gap semiconductors are not applicable for organics. Organics do not posses the same selection rules as inorganics and therefore do not lend themselves to time resolved optical methods for determining the spin relaxation time. As a result, we must make steady-state transport measurements to determine the spin relaxation length and relate that to the spin relaxation time in order to determine the latter.

263

Dominant Spin Relaxation Mechanisms

Spin relaxation lengths are best measured using “spin valves”. A spin valve structure is a tri-layered device where the two outer layers are ferromagnetic and the middle layer is the material in which the spin relaxation length is to be measured. The middle layer is termed the spacer layer. Although exotic techniques such as muon spin rotation have been employed to measure spin diffusion lengths in organic spin valves (Drew, et al., 2009), the more usual technique is to measure the magnetoresistance and extract the spin diffusion length from this measurement. This is described below In a spin valve, one of the ferromagnetic layers injects spin polarized electrons into the spacer, and these electrons are driven to the other ferromagnetic layer by an electric field in the spacer. If the spins that arrive at the second layer are parallel to the majority spins in that ferromagnet, then they transmit and current flows. Otherwise, if they are not parallel, the electrons are reflected by the second ferromagnet and current is blocked. Therefore, the resistance of the device depends on whether the two ferromagnets are magnetized in the same (parallel) direction or opposite (anti-parallel) directions. We will call these two resistances RP and RAP (for “parallel” and “anti-parallel”). As long as both ferromagnets have the same sign of spin polarization at the Fermi energy, i.e. the majority spins of one are also majority spins in the other, we will find that RAP > RP . If the spin polarizations have opposite

enter the spacer layer from the first ferromagnet by tunneling through the Schottky barrier at the ferromagnet/spacer interface with spin polarization P1. Typically, the Schottky barrier is thin enough to not affect the spins. The injected electrons then drift and diffuse through the paramagnet with exponentially decaying spin polarization, so that when they arrive at the interface with the second ferromagnet, the surviving spin polariza-

sign, then RAP < RP . For example, if the ferromagnets are Co and Ni, which have the same sign of spin polarization at the Fermi energy, RAP > RP . If the ferromagnets are Co and LSMO, which have opposite signs of spin polarization, RAP < RP .

where ∆R = RAP − RP . The above equation was used to determine the spin relaxation length in Alq3 in many reported experiments [Pramanik, et al., 2007, Xiong, et al., 2004]. It is easy to see from the above equation that if P1 and P2 have the same sign, ΔR is positive; otherwise, it is negative. In order to measure the ratio DR RP , a spin

Let the spin polarization (n↑ − n↓ ) (n↑ + n↓ )

of carriers at the Fermi level [ ns is the concentration of carriers with polarization σ] in the two ferromagnetic contacts be P1 and P2. Electrons

264

-L L

s , where L is the distance traveled tion is Pe 1 (which is approximately the thickness of the spacer layer) and Ls is the spin diffusion length (or relaxation length). The arriving electrons then tunnel through the Schottky barrier at the second interface into the second ferromagnet where the spin polarization is P2. Julliére has shown that (Julliére, 1975)

1 RP − 1 RAP 1 RP + 1 RAP

 −L   Ls  P , = Pe  1  2  

(6)

which yields that ∆R = RP

2P1P2e

−L Ls

1 − P1P2e

−L Ls

,

(7)

valve device is placed in a magnetic field and the field is scanned. The two ferromagnetic contacts become magnetized in anti-parallel directions

Dominant Spin Relaxation Mechanisms

when the field is between the coercive fields of the two ferromagnets. The resistance measured in this field interval is RAP , while at any other field, the measured resistance will be RP . Our spin valves have cobalt and nickel (or nickel-iron) as the ferromagnetic contacts. They both have the same sign of spin polarization at the Fermi energy. Hence, we will expect to see peaks in the magnetoresistance between the coercive fields of cobalt and nickel since ∆R > 0 . This is exactly what we observed in our experiments (Pramanik, et al., 2007). From the measured peak heights DR , we can estimate the spin relaxation length Ls using Equation (7). From the measured relaxation length Ls(E,T) at any temperature T and electric field E, we can deduce the longitudinal spin relaxation time T1 [= ts ] using the relation (Saikin, 2005) 1

Ls (E ,T ) = −

eE 2kT

2

e E   1  + +   2kT  D(E ,T )τs (E ,T )  



1

= −

eE 2kT

2

e E   q  + +   2kT  kT µ (E ,T ) τs (E ,T )  

(8)

where k is the Boltzmann constant, q is the electronic charge, and D is the spin diffusion coefficient which we assume to be the same as the particle diffusion coefficient. This assumption may not always hold (Pramanik, et al., 2008), particularly if the main spin relaxing mechanism is D’yakonov-Perel’, but since it is never the dominant mechanism is organics, this assumption is approximately valid. We can then apply the Einstein relation to relate the diffusion coefficient to mobility to arrive at the last equation. At low electric fields E <<

kT , Equation (8) simpliq µ(E ,T )τs (E ,T )

fies to Ls (E ,T ) » yields

kT µ(E ,T )τs (E ,T ) , which q

τs (E ,T )

= T1(E ,T )

=

qL2s (E ,T )

kT µ(E ,T ) . (9) low E

At E >> f

i

high

low E

electric

fields

kT , Equation (8) simpliq µ(E ,T )τ(E ,T ) e s t o

Ls (E ,T ) = µ (E ,T ) E τs (E ,T ) = µ (E ,T ) ET1 (E ,T )

, which yields T1 (E ,T )

high E

=

Ls (E ,T )

E m (E ,T )



(10)

In any given electric field, whether we use Equation (9) or (10) to determine T1 from the measured spin diffusion length has to be determined iteratively. We can start with either equation and then verify if the extracted T1 satisfied the condition for applying that equation. If it did, we accept the value. Otherwise, we resort to the other equation. In organics, transport usually takes place by hopping and the carrier mobility is a strong function of both temperature and the electric field inducing current flow. The mobility is expressed as the product of a field-independent part and a field-dependent part (Ke, et al., 2006):  β E     Λ(T )  qd 2 (E ,T )   exp  β E   µ(E ,T ) = exp − tanh      kT  τ β E  kT   kT  0     field −independent

field −dependent

(11)

where b is the field activation constant, Λ (T ) is the activation energy for hopping that depends linearly on temperature as Λ (T ) = AT − B (Ke, et al., 2006), d(E,T) is the average hopping distance at temperature T in an electric field E and 1 t 0 is the hopping frequency in the limit of infinite temperature in an electric field E. The electric

265

Dominant Spin Relaxation Mechanisms

field in the organic can be found from the relation I R E = dc P , where L is the length of the organic L layer (ascertained from cross section transmission electron microscopy) and Idc is the constant current driven through the organic by the equipment which measures the magnetoresistance. Note that when b E << kT , Equation (11) reduces to   A (E ,T ) β E   Λ(T )  qd (E ,T )   exp  β E  = µ(E ,T ) = exp − exp     τ 0kT kT  kT   kT   kT  2

(12)

 Λ(T )  qd 2 (E ,T ) . where A (E ,T ) = exp −  kT  t0   The quantity d(E,T) depends weakly on electric field and also weakly on temperature if the major scattering mechanisms encountered by the carriers during a hop are elastic. Furthermore, we willassumethatintheexpression Λ (T ) = AT − B , AT >> B at all temperatures of interest, so that  Λ(T )   is a temperature indethe quantity exp −  kT    pendent constant. Therefore, the quantity A (E ,T )

is roughly independent of both electric field and temperature. Equation (12) has the same form as given by Chen et al. (Chen, et al., 1999), namely β E   , which immediµ(E ,T ) = µ0 (T ) exp    kT  ately yields that A = kT m0 (T ) . Chen et al. re-

ported m0 (T ) at room temperature to be between

10-11 and 10-13 m2/V-sec. In nanowires, the mobility could degrade by two orders of magnitude because of increased scattering from charged surface states. These nanowires have a very high surface charge density of 1013 cm-2 (Pokalyakin, et al., 2005). Therefore, in our samples, A = 0 . 0 2 6 × 1 . 6 1 × 1 0 -19× 1 0 -13 C - m 2/ s e c 0.026×1.61×10-19×10-15 C-m2/sec, i.e. A is between

266

4.18×10-34 and 4.18×10-36 C-m2/sec, which surprisingly straddles the value of 1.9×10-35 C-m2/sec estimated in Ke, at al. (Ke, et al., 2006). Therefore, this range of A is very reasonable. Combining Equations (9) – (12), we can write T1 (E ,T )

T1 (E ,T )

low E

high E

= lim E →0

= E

qL2 (E ,T ) qL2s (E ,T ) ≈ s     A b E A  b E  kT tanh   exp    b E  kT   kT  Ls (E ,T )

 b E      exp  b E  tanh      kT  b E  kT  A

(13)

The T1 Time in Organic Nanowires We fabricated both “nanowire” and “thin film” spin-valve structures and measured their magnetoresistances at temperatures between 1.9 K and 100 K. The nanowire structures consisted of three layers of Co, Alq3 and Ni sequentially deposited within 50 nm diameter pores of anodic alumina films produced by anodization of Al in oxalic acid. Their fabrication was described in (Pramanik, et al., 2007). The resulting structures were nanowire spin valves with diameters of 50 nm. The magnetoresistance of these nanowires were measured at different temperatures between 1.9 K and 100 K. The measurements were carried out with a constant current source delivering 1 μA. From the measured resistance, we could find the voltage drop over the sample using Ohm’s law and then divided this drop by the organic layer thickness to find the average electric field E in the organic. The value of E in our samples was roughly 460 V/cm. From our measurement of the ratio ΔR/R, we extracted the spin diffusion length Ls at different temperatures using Equation (7). We found that it varies slightly with temperature between 3 and 6 nm in this temperature range. In order to extract the T1 time at different temperatures using Equation (13), we first deter-

Dominant Spin Relaxation Mechanisms

Figure 1. (left panel) Spin diffusion length as a function of temperature, and (right panel) maximum and minimum T1 times as a function of temperature measured in a nanowire sample. The two orders of magnitude difference between the maximum and minimum times is due to two orders of magnitude uncertainty in the mobility

mined that we needed to use the low field formula for nearly the entire temperature range. We also needed to know the value of the field emission constant β. This value depends sensitively on the conditions under which the organic is deposited and can vary widely. The theoretical value given by Ke, et al. (Ke, et al., 2006) is ~ 5×10-5 eV-(m/V)1/2 for thin films. However, since our nanowires are of much poorer quality, we assume that our β is degraded by two orders of magnitude so that our β is 5×10-7 eV-(m/V)1/2. Because of such ad-hoc assumptions, the extracted values of T1 should be viewed as order estimates, rather than accurate values. Using the low field formula in Equation (13), we extracted the T1 time from the measured spin diffusion length Ls. This time turned out to be relatively temperature-independent in the range 1.9 K – 100 K. The magnitude of T1 was found to lie between 10 millisecond and 1 seconds. The spread in the magnitude by 2 orders (10 ms – 1 s) is entirely due to 2 orders of magnitude uncertainty in the reported value of the mobility m0 (T) (Chen, et al., 1999) and not due to any large uncertainty or inaccuracy in the measurement.

In Figure 1, we plot the spin diffusion length Ls and the T1 time as a function of temperature for a typical nanowire sample. The measured T1 time initially increases slightly with temperature (up to 15 K), then decreases slightly. This behavior is somewhat sample-dependent (some samples do not exhibit this non-monotonic trend). From this temperature dependence, we can infer the likely spin relaxation mechanisms. Note that the upper limit of the T1 time is about 1 second in the entire temperature range of 1.9 K – 100 K. This makes Alq3 unique in that it exhibits an exceptionally long T1 time above liquid nitrogen temperature (77 K). That makes it eminently suitable as a host for (classical) spin bits.

THE T1 TIME IN ORGANIC THIN FILMS We repeated the measurements in thin film organics (as opposed to nanowires) by fabricating spin valve structures where the Alq3 layer was a thin film. The samples were fabricated in a cross

267

Dominant Spin Relaxation Mechanisms

Figure 2. A cross bar spin valve structure

bar pattern on oxidized Si wafers with a 100 nm thick insulating silicon dioxide on the surface. The schematic of this structure is shown in Figure 2. It consists of a spin valve where the paramagnetic spacer layer is Alq3 and the two ferromagnets are Co and NiFe. A 6 Å tunnel barrier of Al2O3 is interposed between Co and Alq3 to facilitate spin injection from Co into the organic (Rashba, 2000). NiFe was chosen as the second ferromagnetic layer since it is deposited on top of Alq3 and does not diffuse too much into the Alq3 layer to cause electrical shorts (Santos, et al., 2008). The entire structure is sandwiched between two Au electrodes at the top and the bottom, which are at 900 with each other. The bottom Au electrode is fabricated using a shadow mask followed by lift off. The first ferromagnetic material Co is then evaporated through a window using thermal evaporation at a chamber pressure of 10-7 Torr, followed by 2 Å of Al to prevent oxidation of Co. During evaporation, the thicknesses of the films were monitored in situ by a crystal oscillator and a shutter on top of the evaporation source prevented any deposition after the intended thickness was achieved. The Al2O3 tunnel barrier was grown using an atomic layer deposition system. Trimethyl aluminum (TMA) and H2O were used as precursors and the deposition was performed at 1250C with long purge cycles of about 60 seconds. Low deposition temperatures and long purge

268

cycles were used to avoid hard baking of the photo resist used in the fabrication. The organic layer Alq3, the second ferromagnetic material NiFe, and the top contact were then evaporated using a system consisting of dual thermal and e-beam evaporators equipped with a planetary rotation system to change the position of the substrate in the vacuum chamber. This system is capable of evaporating eight different materials without breaking the vacuum. We fabricated two different sets of samples with two tunnel barrier thicknesses. Each set consisted of about 200 spin valve structures. The composition of set #1 was Co(10nm)/Al2O3(6Å)/Alq3(20nm)/ Ni80Fe20(10nm)Ti(5nm), and that of set # 2 was Co(10nm)/Al2O3(170A)/Alq3(20nm)/ Ni80Fe20(10nm)Ti(5nm). The energy band diagram of the device under an electrical bias is shown schematically in Figure 3. The area of each device was 100 μm × 100µm. Fig. 4 shows the angled view optical images of the fabricated cross bar spin valves. The magnetoresistance traces of the thin film spin valves yielded the spin diffusion length Ls at various temperatures from Equation (7). We then applied Equation (13) to find the T1 time, assuming the bulk value of μ0 = 2×10-7 - 2×10-9 cm2/Vsec reported by Chen, et al. (Chen, et al., 1999) which yielded the value of A to be 4.18×10-32 and 4.18×10-34 C-m2/sec. The electric field in the or-

Dominant Spin Relaxation Mechanisms

Figure 3. The idealized energy band diagram of the spin valve structure under an applied electrical bias

ganic layer was E =

I dc RP

= 1.76 kV/cm (neL glecting any drop over the ultrathin Al2O3). Even this field is low enough to apply the lowE formula in Equation (13). We used the same value of β as before and extracted the T1 time as a function of temperature from the measured spin diffusion length. Both the length and the time are plotted as functions of temperature in Figure 5. In thin films samples, the spin diffusion length decreases monotonically with increasing temperature and then saturates, while the T1 time is relatively temperature independent. This temperature dependence allows us to infer the dominant longitudinal spin relaxation mechanism in thin films samples, which we discuss later.

MEASUREMENT OF THE T2 TIME We had mentioned earlier that the T1 time determines the bit error probability in classical computing, while the T2 time will determine the same probability in quantum computing. Knill has

shown that fault-tolerant quantum computing becomes possible (with quantum error correction codes) if the qubit error probability remains below ~ 3% (Knill, 2005). According to Equation (1), this will require thatT2 ³ 33T , where T is the time needed to complete a quantum bit operation. Let us estimate a reasonable value for T in the context of spin based quantum computing utilizing organic semiconductor hosts. Qubit operations are unitary transformations resulting in coherent spin rotations. Such rotations are accomplished by exploiting Rabi oscillations (Rabi, et al., 1954) where a microwave field irradiates a spin bath in which spin degeneracy has been lifted with a static magnetic field. If the microwave photon energy is resonant with the Zeeman spin splitting energy due to the static magnetic field, then a microwave photon is absorbed coherently to cause a spin rotation by an angle q given by θ=

g µB Bac τ 

,

(14)

where Bac is the amplitude of the ac magnetic field in the microwave radiation, and τ is the duration for which the spin bath is irradiated. This is the standard technique for qubit rotation. The time taken to complete a qubit rotation by the angle θ is τ, which will be the value of T. In order to estimate a reasonable value for τ, we will assume that the ac magnetic field amplitude is 500 Gauss, which is available in electron spin resonance spectrometer cavities. Next, we need to have a value for the g-factor. Organics often have multiple g-factors; for example, Alq3 exhibits two g-factors of 2 and 4, associated with delocalized electrons and localized electrons bound to defect or impurity sites (Grecu, et al., 2005). Delocalized electrons have longer (one order of magnitude longer) T2 times than localized electrons (Kanchibitla, et al., 2008) and therefore are preferred as qubits hosts. If quantum operations are

269

Dominant Spin Relaxation Mechanisms

Figure 4. Optical image of the cross bar spin valves

performed on delocalized electrons, then in order to carry out a complete spin flip operation coherently (i.e. q = 1800 ), the duration t = T = 0.36 nanoseconds. Since we needT2 ³ 33T , the minimum value of T2 required for fault-tolerant quantum computing will be 33×0.36 nsec = 12 nsec. Consequently, Alq3 will serve as a suitable

platform for spin-based quantum computing if the T2 time of delocalized electrons exceeds ~ 12 nsec. Unfortunately, it is very difficult to measure the single particle T2 time in any material since it requires complicated spin echo sequences. What is much easier to measure is the ensemble averaged T2 time (known as the T2* time) which can

Figure 5. (left) Spin diffusion length as a function of temperature and (right) maximum T1 time as a function of temperature in thin film samples

270

Dominant Spin Relaxation Mechanisms

be ascertained from the linewidth of electron spin resonance (ESR) spectra. The T2* time can be orders of magnitude shorter than the T2 time (de Sousa and Das Sarma, 2003). We measured the T2* time in bulk Alq3 samples as well as mesoscopic samples containing one or two molecules (Kanchibotla, et al., 2008). The latter were fabricated by trapping Alq3 molecules in nanovoids with sizes of 1-2 nm, following the prescription of Huang, et al. (Huang, et al., 2005). Since the size of the Alq3 molecule is 0.8 nm, each nanovoid can contain no more than 2 molecules. We made the following observations: 1. The T2* time associated with delocalized spins is nearly temperature independent in the entire temperature range 4 K – 300 K. 2. Its magnitude is of the order of 3 nanoseconds. 3. The T2* time of delocalized spins in bulk samples is slightly shorter (by about 16%) than in mesoscopic samples. 4. The T2* time associated with localized spins is temperature-dependent, albeit not strongly. 5. Its magnitude drops off from about 0.34 nanoseconds to 0.18 nanoseconds in the temperature range 4 K – 300 K. 6. The T2* time of localized spins is 2.5 times longer in mesocopic samples than in bulk samples. From observations 1 and 2, we can conclude that the single particle T2 time of delocalized spins in Alq3 is most likely longer than the required 12 nanoseconds (for fault-tolerant quantum computing) at room temperature since the single particle T2 time is always considerably longer (often by more than 1 order of magnitude) than the ensemble averagedT2* time. Therefore, Alq3 is a suitable platform for spin based quantum computing.

A Unique Qubit Readout Scheme in Alq3 Although the T2 time in Alq3 is sufficiently long for fault-tolerant quantum computing at room temperature, it is by no means the longest T2 time demonstrated at that temperature in solids. Electron spins associated with nitrogen vacancies in diamond can have T2 times exceeding 1 millisecond (Balasubramanian, et al., 2009), which is at least 5 orders of magnitude longer than in Alq3. However, what makes Alq3 unique is its organic character which lends itself to unique selection rules governing electron-hole recombination. These rules can be exploited to implement a very simple and elegant scheme for single-shot qubit read-out, which has no analog in inorganic semiconductors (or diamond). We describe this scheme in the ensuing paragraphs. Alq3 is an optically active organic semiconductor that has been the staple of organic light emitting diodes. Only singlet excitons in this organic are radiative while triplet excitons are “dark”. That means if the recombining electron and hole in an exciton have anti-parallel spins, then they recombine radiatively to emit a photon, but if they have parallel spins, then they recombine non-radiatively to emit a phonon. This property can be exploited to formulate an elegant scheme for qubit read out. When a qubit is “read”, it collapses to a classical bit which can be either 0 or 1, i.e. either parallel or anti-parallel to a spin quantization axes. We will label these two spin orientations as “up” and “down”. In order to determine if a target spin that encoded a qubit collapsed to the “up” or “down” state, we will inject a single hole into the organic from a p-type ferromagnet such as GaMnAs. The spin polarization of the hole will be known apriori since it will point in the direction of the ferromagnet’s magnetization (assuming that the injected spin is the majority spin in the ferromagnet). We will label such a hole’s polarization as “up”. This hole will then form an exciton with the target electron spin.

271

Dominant Spin Relaxation Mechanisms

If the exciton recombines to produce a photon, then we will know that the target spin’s polarization is anti-parallel to that of the injected hole, so that it is “down”. If no photon is emitted, then we will know that the target spin is “up”. Thus, by merely observing whether a photon is emitted following the hole injection, we can determine the polarization of the target spin. This is a very simple and yet elegant scheme of qubit read-out which makes optically active organic semiconductors like Alq3 unique in quantum computing applications.

Possible Phonon Bottleneck Effect in Organics Observation 4 tells us that the primary dephasing mechanism for localized spins in Alq3 is most likely spin-phonon interaction since the dephasing time is temperature-dependent. If that is true, then observation 6 tells us that spin-phonon interaction is suppressed in mesoscopic samples compared to bulk samples because the mesoscopic samples exhibit a T2* time that is longer by a factor of ~ 2.5. Such suppression is suggestive of a novel phonon bottleneck effect, which, to our knowledge, has not been observed hitherto in organics. The traditional phonon bottleneck effect reported in inorganic quantum dots is a consequence of electron confinement (Benisty, et al., 1991). Threedimensional confinement in quantum dots discretizes the electron energy completely so that any electron transition between two energy states can take place only if the energy separation between the states matches the energy of optical phonons available in the material. If the energy separation (determined by geometry, size and material of the quantum dots) does not match available optical phonon energies, then the transitions are blocked since energy conservation cannot be satisfied. This is the traditional phonon bottleneck effect that suppresses electron-(optical) phonon interaction (Benisty, et al., 1991).

272

In addition to optical phonon induced transitions, there are also acoustic phonon induced transitions between electron states which are usually not blocked by electron confinement. Unlike optical phonons, acoustic phonons can have arbitrary energies as long as the acoustic modes form a continuum. Therefore, acoustic phonons can induce transitions between any two energy states in a quantum dot regardless of their energy separation. As a result, electron confinement may suppress optical phonon mediated transitions, but it does not suppress acoustic phonon mediated transitions. However, if the quantum dot confines acoustic phonons as well, then the acoustic modes are discretized and only phonons of certain energies are allowed within the dot. If these energies do not match the energy separation between electron states, then acoustic phonon induced transitions are also suppressed, which exacerbates the phonon bottleneck effect. Thus, there are two distinct contributions to the phonon bottleneck effect: (1) electron confinement, and (2) acoustic phonon confinement. Any one of them will cause only partial suppression of transitions, while the two together can cause more complete suppression. It is acoustic phonon confinement that can suppress spin-phonon coupling in organic molecules when they are placed within nanometer sized cavities. When such a cavity is placed in a magnetic field, the spin degeneracy of each molecular state is lifted by the Zeeman effect. Acoustic phonons can subsequently couple two split spin levels and induce transitions between them, which will cause spin dephasing. For such transitions to take place, energy conservation will mandate that g µB Bdc = ωph ,

(15)

where wph is the phonon energy and g mB Bdc is the Zeeman splitting energy in the magnetic field Bdc. In our ESR experiments, g mB Bdc was ~ 38

Dominant Spin Relaxation Mechanisms

μeV, so that any coupling between the resolved spin states must be through acoustic phonons that have low energies (optical phonons will have energies well in excess of 38 μeV in most materials). If confinement in the cavity discretizes the acoustic modes and makes acoustic phonons of energy 38 μeV unavailable within the cavity, then spin-phonon coupling and the resulting spin dephasing will be suppressed. This may explain why theT2* time measured in molecules housed in 1-2 nm sized cavity is 2.5 times longer than in bulk samples. Confirming the existence of such a phonon bottleneck effect (which is obviously different from the traditional one introduced by Benisty and co-workers (Benisty, et al., 1991)) will require demonstrating progressive suppression of dephasing with shrinking cavity dimensions, but this is not experimentally accessible.

IDENTIFYING THE DOMINANT SPIN RELAXATION MECHANISM IN ALQ3 Earlier, we had surmised that the Elliott-Yafet mechanism and hyperfine interactions with nuclear spins are the two likely mechanisms that play the dominant role in longitudinal spin relaxation. As far as transverse spin relaxation is concerned, spin-orbit interaction, hyperfine interactions, spin-phonon coupling, spin-magnon coupling, spin-spin interaction, etc. are all likely mechanisms and identifying the dominant one among them is challenging. In the rest of this paper, we will attempt to identify the dominant longitudinal and transverse spin relaxation mechanisms based on available data and information.

Identifying the Dominant Longitudinal Spin Relaxation Mechanism The two likely culprits that limit the T1 time in Alq3 are hyperfine interactions and the Elliott-

Yafet mode. Both these mechanisms are weak in Alq3, which is why the T1 time is so long (up to ~ 1 sec) even at 100 K. The data in Figure 1 indicate that the T1 time increases slightly with temperature up to 15 or 20 K in the nanowire samples and then decreases very slightly and saturates to a constant value. In thin film samples, it is nearly temperature independent up to 100 K. These behaviors are consistent with both hyperfine interactions and the Elliott-Yafet mechanisms. In hyperfine interaction, spin relaxation occurs due to the carrier spins interacting with the magnetic field caused by nuclear spins. Since the latter has no significant temperature dependence, we expect the T1 time to be relatively temperature independent. This is basically what we see for both nanowires and thin films. The slight monotonic behavior seen in Figure 1 is within the error margin and is therefore ignored. The Elliott-Yafet mechanism should also produce a nearly temperature independent T1 in our samples. This mechanism is associated with momentum relaxing scattering events and the Elliott-Yafet spin relaxation time is proportional to the momentum relaxation time (Yafet, 1963). Therefore, T1 will have the same temperature dependence as the momentum relaxation time (Yafet, 1963; Monod and Beuneu, 1979). If the latter is determined primarily by Coulomb scattering with interface states (as in our nanowires) instead of phonon scattering, then it will be relatively temperature-independent. In that case, the spin relaxation time (or T1) should be relatively temperature-independent as well. However, if the momentum relaxation time is determined by scattering mechanisms which have pronounced temperature dependences, then clearly the momentum relaxation time – and hence the T1 spin relaxation time – will exhibit strong temperature dependence. While our own experiments show weak temperature dependence of the T1 time measured under low electric fields, experiments by other groups have often showed a strong temperature dependence of the spin dif-

273

Dominant Spin Relaxation Mechanisms

fusion length (Xiong, et al., 2004; Drew, et al., 2009) measured under low electric fields. From Equation (13), we deduce that this can happen only if T1 decreased with increasing temperature in those experiments. Such a decrease can never be consistent with hyperfine interactions, but is consistent with the Elliott-Yafet mechanism. Therefore, available data on temperature dependence of the T1 time favor the Elliott-Yafet mechanism over hyperfine interactions. In order to probe this issue further, one should study the electric field dependence of the spin diffusion length. Bobbert et al., (Bobbert, et al. 2009) have carried out a theoretical simulation of spin transport in an organic and concluded that if hyperfine interactions are the primary route for spin relaxation, then the spin diffusion length should increase almost exponentially with the driving electric field. This is exactly what Equation (13) predicts as well since in the case of hyperfine interaction, T1 should be approximately independent of electric field. In contrast, if the Elliott-Yafet mechanism is the primary channel for relaxation, then the spin diffusion length should decrease with increasing electric field since the latter will tend to increase the spin-orbit interaction strength via the Rashba effect. Thus, the electric field dependence of the spin diffusion length may be able to differentiate between hyperfine interactions and the Elliott-Yafet mechanism. The spin diffusion length in spin valves of Alq3 were measured as a function of electric field (at a fixed temperature of 1.9 K) (Pramanik, et al., 2006). This experiment measured the spin valve peaks (and hence effectively the spin diffusion length) in nanowire organic spin valves as a function of the electric current through the spin valve. From the current I, one can deduce the voltage V over the organic using Ohm’s law (V=IR) where R is the measured resistance of the spin-valve (since the organic is much more resistive than the metallic ferromagnetic contacts of the spin valve, the resistance R is almost entirely due to the organic and the contacts have negligible contribution).

274

Knowing the thickness L of the organic layer, one can then find the average electric field E over the organic from the relation E = V/L. The data obtained (Pramanik, et al., 2006) showed clearly that the spin diffusion length actually decreases with increasing electric field, instead of increasing exponentially with it. Clearly, this is inconsistent with hyperfine interaction. This data alone can eliminate hyperfine interaction as the dominant spin relaxing mechanism in Alq3. Consequently, the Elliott-Yafet mechanism is most likely the dominant spin relaxing mechanism which determines the T1 time in organics. This issue has been discussed in (Bandyopadhyay, 2010).

Identifying the Dominant Transverse Spin Relaxation Mechanism The likely mechanisms that limit T2* in Alq3 are hyperfine interactions, spin-phonon and/or spinmagnon coupling, spin-spin interactions and spin-orbit interaction. Hyperfine interaction is often thought to be a significant contributor to spin dephasing of electrons in Alq3 and limits the T2* time to (Sheng, et al., 2006) T2* »

 , g µB αH

(16)

where aH is the hyperfine splitting constant. This dephasing however is suppressed if an external magnetic field of flux density B > aH is applied to the sample since it tends to pin the spins (Sheng, et al., 2006). The value of aH is typically a few mTesla (Lewis and Singer, 1965), and since all our experiments to measure T2 times are carried out at higher magnetic fields (> 300 mTesla) (Kanchibotla, et al., 2008), it is unlikely that hyperfine interactions would have caused much spin dephasing in our experiments.

Dominant Spin Relaxation Mechanisms

Our measurement of the T2* time as a function of temperature in both bulk samples and mesoscopic samples containing 1-2 Alq3 molecules showed that the T2* time of delocalized spins in mesoscopic samples increases slightly (~ 16%) with temperature up to 30 K and then saturates. In bulk samples, it increases even more slightly (~ 8%) with temperature up to 50 K and then saturates (Kanchibotla, et al. 2008). These increases are too slight to conclude definitively that hyperfine interactions are dominant at low temperatures, although that possibility cannot be ruled out. Over the entire temperature range of 4 K – 300 K, the T2* time varies by less than 20% for mesoscopic samples and about 8% for bulk samples. This relative temperature independence is equally consistent with a number of mechanisms such as spin-orbit interaction, spin-spin interaction (Sanvito, 2007), etc. The only mechanism that can be excluded is spin-phonon coupling since that would be strongly temperature-dependent. Thus, it is impossible to identify the dominant dephasing mechanism of delocalized electrons from the temperature dependence alone. In contrast, the T2* time of localized spins showed a much stronger temperature dependence in both mesoscopic and bulk samples, where this time decreased considerably with increasing temperature. This is strongly indicative of spinphonon coupling being the dominant cause of spin dephasing. Therefore, although we are unable to identify the dominant dephasing mechanism for delocalized spins, we can be relatively certain that the dominant dephasing mechanism for localized spins is spin-phonon coupling.

CONCLUSION Our measurements of the longitudinal (T1) and transverse (T2) spin relaxation times in Alq3 samples (nanowires, thin films, bulk powder, and mesoscopic samples) as a function of temperature have shed some light on the dominant spin relax-

ation mechanism in this organic semiconductor. We draw the following conclusions: 1) The dominant longitudinal spin relaxation mechanism is most likely either hyperfine interactions or the Elliott-Yafet mode. Only a measurement of the electric field dependence of the spin diffusion length can allow us to further differentiate between the two and identify the dominant mode. 2) The primary transverse spin relaxation mechanism (dephasing mechanism) of delocalized electrons is uncertain. The suspects are hyperfine interaction, spin-spin interactions and spin-orbit interaction. Spin-phonon coupling is excluded. 3) The dominant transverse spin relaxation mechanism (dephasing mechanism) of localized electrons is spin-phonon coupling. These results enhance our knowledge and understanding of spin transport in organics.

ACKNOWLEDGEMENT The work at Virginia Commonwealth University is supported by the US National Science Foundation under grant CCF-0726373.

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Ke, L., Chua, S. J., Han, R. C. C., Ting, L. T., & Vijila, C. (2006). Brownian Motion Field Dependent Mobility Theory of Hopping Transport Process. Journal of Applied Physics, 99, 114512. doi:10.1063/1.2201852 Knill, E. (2005). Quantum Computing with Realistically Noisy Devices. Nature (London), 434, 39. This paper showed that fault tolerant quantum computing is possible if the qubit error probability remains below ~ 3%. Since then, there have been further developments in error correction protocols and the most recent result claims that errors can be corrected as long as they occur with a probability less than ~ 47% [H. Buhrmann, R. Cleve, M. Laurent, M. Linden, A. Schrijver and F. Unger, “New Limits on Fault-tolerant Quantum Computation”, Proc. 47th Annual IEEE Symp. Found. Comp. Sci., 411, (2006)]. However, the resources and overhead required to correct errors occurring so frequently can be overwhelming. Lewis, I. C., & Singer, L. S. (1965). Electron Spin Resonance of Radical Cations Produced by the Oxidation of Aromatic Hydrocarbons with SbCl5. The Journal of Chemical Physics, 43, 2712. doi:10.1063/1.1697200 Monod, P., & Beuneu, F. (1979). Conduction Electron Spin Flip by Phonons in Metals: Analysis of Experimental Data. Physical Review B: Condensed Matter and Materials Physics, 19, 911. doi:10.1103/PhysRevB.19.911 Naber, W. J. M., Faez, S., & van der Wiel, W. G. (2007). Organic Spintronics. Journal of Physics. D, Applied Physics, 40, R205. doi:10.1088/00223727/40/12/R01 Pokalyakin, V. (2005). Proposed Model for Bistability in Nanowire Non-volatile Memory. Journal of Applied Physics, 97, 124306. doi:10.1063/1.1937477

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ENDNOTE 1



Or, more precisely, the part of carrier mobility that is determined by scattering.

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Chapter 18

The Synthesis of Stochastic Circuits for Nanoscale Computation Weikang Qian University of Minnesota, USA John Backes University of Minnesota, USA Marc D. Riedel University of Minnesota, USA

ABSTRACT Emerging technologies for nanoscale computation such as self-assembled nanowire arrays present specific challenges for logic synthesis. On the one hand, they provide an unprecedented density of bits with a high degree of parallelism. On the other hand, they are characterized by high defect rates. Also they often exhibit inherent randomness in the interconnects due to the stochastic nature of self-assembly. We describe a general method for synthesizing logic that exploits both the parallelism and the random effects. Our approach is based on stochastic computation with parallel bit streams. Circuits are synthesized through functional decomposition with symbolic data structures called multiplicative binary moment diagrams. Synthesis produces designs with randomized parallel components—and operations and multiplexing—that are readily implemented in nanowire crossbar arrays. Synthesis results for benchmarks circuits show that our technique maps circuit designs onto nanowire arrays effectively.

1 INTRODUCTION As the semiconductor industry contemplates the end of Moore’s Law, there has been considerable interest in novel materials and devices (IRTS, 2006). Technologies such as molecular switches

and carbon nanowire arrays offer a path to scaling beyond the limits of conventional CMOS (FENA, 2006). Most such technologies are in the exploratory phases, still years or decades from the point when they will be actualized. Accordingly, the development of software tools and techniques for logic synthesis remains speculative.

DOI: 10.4018/978-1-60960-186-7.ch018 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Synthesis of Stochastic Circuits for Nanoscale Computation

And yet, for some types of new technologies, we can identify broad traits that will likely impinge upon synthesis. For instance, nanowire arrays are stochastically self-assembled in tightly-pitched bundles. Accordingly, they exhibit the following (DeHon, 2005):

Figure 1. nanowire crossbar with random connections.

1. A high degree of parallelism. 2. Minimal control during assembly. 3. Inherent randomness in the interconnect schemes. 4. High defect rates. Existing strategies for synthesizing logic for nanowire arrays are based on routing schemes similar to those used for field-programmable gate arrays (FGPAs) (DeHon, 2005). These rely on probing the circuit and programming interconnects after fabrication. We describe a general method for synthesizing logic that exploits both the parallelism and the random effects of the self-assembly, obviating the need for such post-fabrication configuration. Our approach is based on stochastic computation with parallel bit streams. Circuits are synthesized through functional decomposition with symbolic data structures called multiplicative binary moment diagrams. Synthesis produces designs with randomized parallel components -- AND operations and multiplexing -- operating on the stochastic bit streams. These components are readily implemented in nanowire crossbar arrays. We present synthesis results for benchmarks circuits illustrating the method. The results show that our technique is effective in implementing designs with nanowire arrays, with a measured tradeoff between the degree of redundancy and the accuracy of the computation.

2 CIRCUIT MODEL Our discussion of synthesis is framed in terms of a conceptual model for nanowire arrays. (In the

280

later part of the paper, we justify this model with implementation details.) A nanowire crossbar is illustrated in Figure 1. The connections between horizontal and vertical wires are random. However, we assume that these connections are nearly one-to-one, that is to say, nearly every horizontal wire connects to exactly one vertical wire, and vice-versa. This is a specific attribute of types of nanowire arrays, controlled during self-assembly (DeHon, 2005).

2.1 Parallel Stochastic Bit Streams Our synthesis method implements digital computation in the form of parallel stochastic bit streams. We refer to a collection of parallel nanowires as a bundle. The width of a bundle is the number wires. Its current weight is the number of logical 1’s on its wires. The signal that it carries is a real value between zero and one corresponding to the fractional weight: for a bundle of N wires, if k of the wires are 1, then the signal is k / N . Let P (X = 1) denote the probability that any given wire in bundle X carries a 1.

2.2 Shuffling Devices We implements computation with two basic nanowire constructs: shuffled ANDs and Bundleplexers. We describe these only in conceptual terms

The Synthesis of Stochastic Circuits for Nanoscale Computation

Figure 2. A shuffled AND element, for bundles of width 3.

here; implementation details are postponed until the later part of the paper.

2.2.1 Shuffled AND A shuffled AND has two bundles of N wires as inputs and a bundle of N wires as the output. Each wire in the output bundle is actually the output of an AND gate, which takes one input from the first input bundle and the other from the second. The choice of which inputs are fed into which AND gate is random. Figure 2 shows a simple shuffled AND with N = 3 . Suppose that the signal carried by the first input bundle A is a , that carried by the second input bundle B is b , and that carried by the output bundle C is c . Provided that the bits in the first and second input bundles are independent, for large N we can assume that c = P (C = 1)

(1)

= P (A = 1 and B = 1)

(2)

= P (A = 1) × P (B = 1)

(3)

= a × b.

(4)

We see that a shuffled AND in effect performs the multiplication of the signals carried by the two input bundles.

2.2.2 Bundleplexer A bundleplexer has two bundles of N wires as its inputs and a bundle of N wires as its output. It is tagged with a fixed selecting ratio, 0 < s < 1 . The output bundle is composed of a randomly selected choice of sN bits from the first input bundle and (1 - s )N bits from the second. The choice is not ordered: rather, a random shuffling occurs. Figure 3 shows a bundleplexer with N = 4 and s = 3 / 4 . The output bundle has three wires from input bundle A and one wire from input bundle B . Suppose that the signal carried by the first input bundle A is a , that carried by the second input bundle B is b and that carried by the output bundle C is c . For large N , we can assume that c = P (C = 1)

(5)

= sP (A = 1) + (1 − s )P (B = 1)

(6)

= sa + (1 − s )b.

(7)

281

The Synthesis of Stochastic Circuits for Nanoscale Computation

Figure 3. A bundleplexer with N = 4 and s = 3 / 4 .

We see that a bundleplexer in effect performs a scaled addition on the signals carried by the two input bundles.

2.3 Stochastic Circuits Our synthesis method produces a circuit design that operates on the fractional-weighted values carried by bundles of wires. Our approach is analogous to the formulation of a real-valued polynomial representation of a circuit, with arithmetic multiplication and addition. (In fact, we perform synthesis with symbolic data structures called binary moment diagrams.) For example, consider a circuit with the Boolean truth table shown in the top-right in Figure 4. Its output y can be represented as y = a + b − 2ab. Evaluating this polynomial for all Boolean values of a and b gives the correct Boolean output y . We use shuffled ANDs for multiplication and bundleplexing for addition. For a circuit with m inputs and n outputs, we have m input bundles and n output bundles (each bundle consisting of N parallel wires). For computation, all the wires in each input bundle are set to the corresponding Boolean input value (so all the wires in each bundle are set to 0 or to

282

1). With bundleplexing, wires are randomly selected from separate bundles. As a result, the internal bundles carry stochastic bit streams with fractional weightings. We assume that the output of the circuit is directly usable in a fractional-weighted form. For instance, in sensor applications, an analog voltage discriminating circuit might be used to transform an output bundle of bits into a Boolean value. We assume direct quantization: an output signal greater than or equal to 0.5 corresponds to logical 1; less than this corresponds to 0.

Figure 4. An example of the formulation of a stochastic circuit.

The Synthesis of Stochastic Circuits for Nanoscale Computation

Figure 5. A simple circuit.

Figure 4 illustrates the formulation. Bundles of width N = 4 are used. The truth table shown in the bottom-right gives the fractional weight on the output bundle Y . For inputs A = 1 and B = 0 , we have Y = 3 / 4 , which corresponds to logical 1. For A = 1 and B = 1 , we have Y = 1 / 4 , which corresponds to logical 0. Thus, the stochastic circuit implements the same Boolean function as that shown in the top-right truth table.

ranges. Figure 6 shows an example of a *BMD for the circuit in Figure 5, implementing the function: y = ((x 1 ∧ x 2 ) ∧ (x 3 ∨ x 4 )) ∨ (x 3 ∧ x 4 ). Figure 6. The *BMD for the circuit in Figure 5.

3 SYNTHESIS OF STOCHASTIC CIRCUITS Our synthesis procedure begins with the specification of a combinational circuit, say in the form of a netlist, and produces a stochastic design consisting of shuffled AND elements and bundleplexer elements. Synthesis is performed through functional decomposition with a variant of binary decision diagrams called multiplicative binary moment diagrams (*BMDs) (Bryant & Chen, 1995).

3.1 Multiplicative Binary Moment Diagrams Like binary decision diagrams, *BMDs are a graphic representation of functions over Boolean variables; however, they can have non-Boolean

283

The Synthesis of Stochastic Circuits for Nanoscale Computation

There is a total ordering of the variables in a *BMD. Also, each non-terminal vertex has outgoing edges to two children. Three salient features of *BMDs are: 1. There can be more than two terminal vertices and each terminal vertex can have a number other than 0 or 1. (Terminal vertices are shown as square boxes at the leaves of the tree.) 2. Each edge has an associated weight which can either be an integer or a real value. (The weights are shown in square boxes written directly on the edges; an edge without a box is assumed to have weight 1; the weights of edges that connect to terminal vertices are simply the weights of the terminal vertices themselves.) Note that the edge pointing to the root can also have a weight. (For example, see Figure 7(b).) The function represented by

Figure 7. The positive and negative *BMDs for the *BMD in Figure 6. (a) Positive *BMD; (b) Negative *BMD.

a *BMD is the product of the root weight and the function at the root vertex. 3. The left edge from each vertex indicates the case where the function is independent of the vertex variable; the right edge indicates the case where the function depends linearly on that variable. (In diagrams, we sometimes exchange “left” and “right” to get prettier graphs; when we do so, we indicate this by annotating the edges with L and R .) Thus, the function f at a vertex with variable x is f = wL fL + wR fR x ,

(8)

where w L is the weight of the left edge and wR is the weight of the right edge; fL is the function at the vertex pointed to by the left edge and fR is the function at the vertex pointed to by the right edge. We obtain the function for the *BMD through such recursive decomposition. In order to construct a *BMD, we begin with base functions corresponding to constants and individual variables, and then we build more complex functions by combining these. Given a circuit, we obtain a *BMD for its Boolean function by expressing it in terms of addition and multiplication operations. For the *BMD in Figure 6, the function is f = x 1x 2x 3 + x 1x 2x 4 + x 3x 4 − 2x 1x 2x 3x 4 .

(9)

In general, for a circuit with multiple outputs, there are separate *BMDs for each output. However, in the data structure, significant portions of different *BMDs can often be shared (Bryant & Chen, 1995).

284

The Synthesis of Stochastic Circuits for Nanoscale Computation

3.2 Decomposing a *BMD into Positive and Negative *BMDs After obtaining a *BMD for a circuit, the next step in our procedure is to decompose it into two *BMDs, both with non-negative-weighted edges, such that the function of the original *BMD equals that of the first *BMD minus that of the second. We call the first *BMD the positive *BMD and the second the negative *BMD. Figure 7 shows the positive and negative *BMDs for the *BMD in Figure 6. The function of the positive *BMD is x 1x 2x 3 + x 1x 2x 4 + x 3x 4 and the function of the negative *BMD is 2x 1x 2x 3x 4 . The procedure for this decomposition is given as pseudo-code in Figure 8.

In the pseudo-code, we represent a *BMD as a weighted pair of the form (w, v ) , where v designates the root vertex and w is the root edge weight. (This pair also refers to the function represented by the *BMD.) A vertex v = L denotes a terminal leaf. The function Var( v ) returns the variable of vertex v . The function Left( v ) returns the left pair of v : (wL , vL ) , where wL is the weight of the left edge of v and vL is the left child of v . Similarly, the function Right(v ) returns the right pair of v : (w R , vR ) , where wR is the weight of the right edge of v and vR is the right child of v . The function MakeBranch( x , (w L , vL ) , (wR , vR ) ) constructs a new *BMD. It returns a pair (w, v ) designating a *BMD, such that

Figure 8. Procedure for decomposing a *BMD into positive and negative *BMDs.

285

The Synthesis of Stochastic Circuits for Nanoscale Computation

wf (v ) = wL f (vL ) + wR f (vR )x . Here, f (v ) denotes the function of vertex v . The function ApplyWeight( w ¢ , (w, v ) ) multiplies the function of the pair (w, v ) by a constant w ¢ and returns the resulting pair. The functions MakeBranch and ApplyWeight are described in (Bryant & Chen, 1995). The function PosNegBMD in Figure 8 takes a pair (w, v ) representing a *BMD as an input argument and returns two pairs (w P , vP ) and (wN , vN ) representing the positive and negative *BMDs, respectively. It first obtains the positive and negative *BMDs without considering the weight w . In the non-trivial case, i.e., when v is not a terminal vertex, it recursively calls PosNegBMD to obtain the positive and negative *BMDs of the *BMDs designated by the left and right pairs of v . Then, the procedure calls the function MakeBranch to construct a positive *BMD based on the two positive *BMDs of the left and right pairs of v . Similarly, it constructs a negative *BMD based on the two negative *BMDs of the left and right pairs of v . Finally, it calls the function WeightChange to apply the weight w to the previously obtained positive and negative *BMDs. If w ³ 0 , then WeightChange just calls the function ApplyWeight to multiply both the positive and negative *BMDs by the weight w . Otherwise, the positive *BMD is taken to be the previously obtained negative *BMD multiplied by -w ; the negative *BMD is taken to be the previously obtained positive *BMD multiplied by -w .

3.3 Transforming a *BMD into a Unit-Weight *BMD In order to build a stochastic circuit, we need to transform *BMDs with integer edge weights into *BMDs of a special form, which we call unitweight: a *BMD is unit-weight if the absolute values of the edge weights of each non-terminal vertex sum to 1.

286

Figure 9 gives the unit-weight *BMD corresponding to the positive *BMD in Figure 7(a). The unit-weight *BMD corresponding to the negative *BMD in Figure 7(b) is just itself. The procedure for this transformation is given as pseudo-code in Figure 10. Assume that the original *BMD has variables x 1, x 2 , , x n and that they are ordered as x n < x n -1 <  < x 1 . (Here the root vertex has

Figure 9. The unit-weight *BMD corresponding to the *BMD in Figure 7(a). The numbers in parentheses gives the FuncWeight of the corresponding vertices.

The Synthesis of Stochastic Circuits for Nanoscale Computation

Figure 10. Procedure for transforming a *BMD into a unit-weight *BMD.

variable x n .) Each vertex in the unit-weight *BMD has three data members recording the weights: 1. LeftWeight: The edge weight of its left branch. 2. RightWeight: The edge weight of its right branch. 3. FuncWeight: The weight used to keep the function at that vertex unchanged. The function at a vertex in the unit-weight *BMD multiplied by its FuncWeight equals the function at the corresponding vertex in the original *BMD. In Figure 9, we show the FuncWeight of each vertex in parentheses. For example, the FuncWeight for vertex m4 equals 2 and the function

1 1 x 1x 2 + x 3 . The function for 2 2 the corresponding vertex n 4 in Figure 7(a) is

for m 4 is f4′ =

f4 = x 1x 2 + x 3 = 2 f4′ . In the initialization, we set the FuncWeight for each terminal vertex to the weight of that vertex. Then the procedure modifies the edge weights for all the vertices with variable x 1 ; then for all the vertices with x 2 ; and so on through to x n . For each vertex v in the unit-weight *BMD, let its LeftWeight and the LeftWeight of the corresponding vertex in the original *BMD be wUL and wOL , respectively. Let its RightWeight and the RightWeight of the corresponding vertex in the original *BMD be wUR and wOR , respectively. Let its FuncWeight be wUF .

287

The Synthesis of Stochastic Circuits for Nanoscale Computation

Denote the FuncWeight of its left child and its right child as wFL and wFR , respectively. We have the following equations to determine wUL , wUR and wUF : wUF =| wFL ⋅ wOL | + | w FR ⋅ wOR |,

wUL =

wUR =

wFL × wOL wUF wFR × wOR wUF

,

.

Finally, we set the root edge weight wU of the unit-weight *BMD to wU = wO × wF (root ),

that implements the function at that vertex. (Here, when we say “a bundle implements a function’’, we mean that the signal carried by the bundle equals the function output for all input combinations.) The procedure is as follows. We first set n bundles of inputs such that their signals are equal to the Boolean inputs x 1, x 2 , , x n of the original circuit. Also, we provide an input bundle with all bits equal to 1 (equivalent to a constant logical value of 1). Next, we build bundles implementing the functions of vertices in the unit-weight *BMD with variable x 1 ; then bundles implementing the functions of vertices with variable x 2 ; and so on through to those vertices with variable x n . For the circuit corresponding to a non-terminal vertex vk with variable x i , suppose that the functions of its left child vertex and right child vertex are fL and fR , respectively, and that LeftWeight and RightWeight are w L and w R , respectively.

where wO is the root edge weight for the original *BMD and wF (root ) is the FuncWeight for the root vertex. If the original *BMD has integer edge weights, then the edge weights of the unit-weight *BMD built by the procedure MakeUnitWeightBMD are all rational numbers. The root edge weight wU is an integer.

3.4 Transforming a Unit-Weight *BMD into a Stochastic Circuit In our method, we transform both the positive and negative *BMDs into unit-weight *BMDs. (We refer to these as UnitPosBMD and UnitNegBMD, respectively.) Both of these have non-negative edge weights. Given a unit-weight *BMD with non-negative edge weights, we can transform it directly into a stochastic circuit. For each vertex in the *BMD, we build a stochastic circuit with an output bundle

288

We h a v e

0 £ wL £ 1 ,

0 £ wR £ 1

and

wL + wR = 1 . According to Equation (8), the function of vk is f (vk ) = wL fL + wR fR x i .

(10)

At this point, since we are building the circuit according to the order of the vertex indices, we have already constructed a bundle sL implementing fL and a bundle sR implementing fR . To build the bundle implementing f (vk ) , we first build a shuffled AND on the input bundles sR and x i . Call the result of the shuffled AND sC and its signal fC . Since P (x i = 1) = 0 or 1 , we have P ( fR = a, x i = b) = P ( fR = a ) ⋅ P (x i = b), ∀a, b ∈ {0,1},

The Synthesis of Stochastic Circuits for Nanoscale Computation

Figure 11. A circuit fragment illustrating the computation of the function of a vertex vk .

which means that the bits in the bundle sR and the primary input bundle for x i are independent. Thus, according to Equation (4), we have fC = fR x i . If wL ¹ 0 , then we build a bundleplexer with inputs sL and sC . We set the selecting ratio of this bundleplexer to be s = wL with respect to sL . Thus, according to Equation (7), the output bundle of the bundleplexer implements the function wL fL + (1 − wL )fC = wL fL + wR fR x i ,

and so implements the function of vertex vk . A circuit fragment illustrating these steps is shown in Figure 11. In the circuit, we use a gate called “SAND’’ to denote a shuffled AND operation and a gate called “BUX’’ to denote bundleplexing. (We denote bundles by crossing a single wire with a slash and writing the number of wires, N , next to it.) The number on a bundleplexer denotes its selecting ratio with respect to the input bundle that is bubbled. The same conventions are also used in Figure 12. If wL = 0 , then wR = 1 and Equation (10) simplifies to f (vk ) = fR x i . Thus, the output bundle sC of the shuffled AND implements the function of the vertex vk .

Figure 12. The stochastic circuit obtained from the UnitPosBMD in Figure 9 and the UnitNegBMD in Figure 7(b). The number on the counter indicates the amount that it increments or decrements the count for each 1 on the corresponding bundle. The output of the counter implements the function of the original *BMD in Figure 6.

289

The Synthesis of Stochastic Circuits for Nanoscale Computation

Since both UnitPosBMD and UnitNegBMD are unit-weight *BMDs with non-negative edge weights, we can build two stochastic circuits implementing the functions of the root vertices of these *BMDs. Finally, we connect the output bundles of the two circuits to an analog counter. The 1’s in the output bundle of the circuit for UnitPosBMD will increment the counter by wUP , while the 1’s in the output bundle of the circuit for UnitNegBMD will decrement it by wUN , where wUP and wUN are the root edge weights of UnitPosBMD and UnitNegBMD, respectively. We call the increment and decrement coefficients of the counter the scaling factors. For the UnitPosBMD shown in Figure 9 and the UnitNegBMD shown in Figure 7(b), we obtain the stochastic circuit shown in Figure 12. The output bundle of BUX1 implements the function of vertex m 4 and the output bundle of BUX2 implements the function of vertex m5 , the root vertex of UnitPosBMD. The output bundle of SAND3 implements the function of the root vertex of UnitNegBMD. Finally, we connect the output bundle of BUX2 and the output bundle of SAND3 to a counter. The output of the counter implements the function of the original *BMD in Figure 6. Thus, the circuit implements the same logic as the circuit in Figure 5.

3.5 Summary of Synthesis Procedure In summary, our procedure for synthesizing a stochastic circuit consists of the following five steps: 1. Build a *BMD for each output of the circuit. 2. Decompose each *BMD into positive and negative *BMDs. 3. Transform these into unit-weight *BMDs. 4. Transform the unit-weight *BMDs into stochastic designs with shuffled ANDs and bundleplexers.

290

5. Realize the outputs with cumulative increment and decrement operations on the outputs.

4 EXPERIMENTAL RESULTS We chose 15 small benchmark circuits from the IWLS ‘93 set to test our synthesis technique. (For sequential circuits in this group, we have extracted the combinational part.) Table 1 shows some statistics of the original benchmark circuits and the corresponding stochastic circuits. Column “#Devices in Orig. Ckt.’’ gives the number of devices in the original circuit and column “#Devices in Stoch. Ckt.’’ gives the number of devices in the stochastic circuit. Here the devices are the shuffled AND elements and bundleplexers used in the design. The column ``Ratio’’ gives the ratio of the number of devices in the stochastic circuit to the number of devices in the original circuit. We see that that this ratio is on average one and a half. Given that the width of the bundles is finite, the outputs of the stochastic circuit might be erroneous. We analyze the error ratio defined as the number of outputs that return an incorrect value. For example, assume that for a given combination of inputs, the outputs of the stochastic circuit are  os = (o1, o2 , o3 , o4 ) = (0.78,1.01, -0.02, 0.16)  and that the correct values are o = (1,1,1, 0). After discriminating, we get a Boolean output    os ' = (1,1, 0, 0) . Comparing os ' with o , we find that 1 out of 4 bits is incorrect, so the error ratio is 25%. Of course, with larger bundle widths the error ratio will be lower. In our experiments, we do an average across a number of input combinations with the following rule: if the number of inputs is less than or equal to 5, then we run through all the input combinations; otherwise, we randomly select 25 = 32 input combinations and run experiments on them. Considering the inherent randomness in the circuit construction, we also run 20 trials for

The Synthesis of Stochastic Circuits for Nanoscale Computation

Table 1. Synthesis results for selected IWLS ‘93 benchmark circuits #Devices

#Devices

Ratio

in Stoch.

(col. 5 /

Circuit

#Inputs

#Outputs

in Orig. Ckt.

Ckt.

col. 4)

C17

5

2

14

26

1.86

b1

3

4

18

18

1.00

majority

5

1

18

23

1.28

lion

4

3

19

30

1.58

daio

5

4

26

29

1.12

mc

5

7

36

47

1.31

cm138a

6

8

43

104

2.42

bbtas

5

5

44

74

1.68

cm42a

4

10

49

61

1.24

tcon

17

16

58

73

1.26

beecount

6

7

62

108

1.74

decod

5

16

69

194

2.81

sqrt8ml

8

4

74

87

1.18

sqrt8

8

4

79

87

1.10

c8

28

18

184

272

1.48

Average

each input combination and average the results. (In our simulations, the randomness of the construction is generated by the standard C function rand().) We find that the width of the bundles needed to obtain an error ratio below a given threshold is linearly proportional to the maximal scaling factor of all the counters. Define a as the ratio of the width of the bundles to the maximal scaling factor. We run experiments to see how the error ratio changes with increasing a . We set a to five different values: 5, 10, 20, 50, 100. The result for each circuit is shown in Table 2. The error ratio is shown in the form of percentages. The smaller the error ratio, the better the result. We also give the maximal scaling factor for each circuit. The width of the bundles in each circuit is the maximal scaling factor multiplied by a . From Table 2, we see that:

1.54

1. With a increasing, the error ratio decreases. 2. For all the circuits, the error ratio is below 4% when a = 10 . 3. For most of the circuits, the error ratio is below 1% when a = 20 . 4. When a = 100 , the error ratio is almost 0. Some applications are characterized by a tolerance for less than perfectly accurate computation. For example, in image processing applications, a small error in a processed image will be masked by the limits of the display device and by the limits of human vision. For such applications, a non-zero error ratio is acceptable. Suppose that we choose 1% as our error ratio threshold. Then we obtain a » 20 . Given that the maximal scaling factor is around 10, on average, the width of the bundles in the stochastic circuit will be roughly 200.

291

The Synthesis of Stochastic Circuits for Nanoscale Computation

Table 2. Error percentages vs. a , the ratio of the width of the bundles to the maximal scaling factor. Max.

a : Width of Bundles over

Scaling

Max. Scaling Factor

Factor

5

10

20

50

100

C17

4

8.36

3.13

1.02

0.00

0.00

b1

3

5.63

1.72

0.00

0.16

0.00

majority

9

4.69

1.88

0.94

0.31

0.00

lion

4

4.27

1.56

0.31

0.00

0.00

daio

6

4.53

2.19

0.70

0.04

0.00

mc

6

3.97

2.12

0.42

0.07

0.00

cm138a

8

0.55

0.51

0.22

0.02

0.00

bbtas

7

5.84

1.91

0.78

0.09

0.00

cm42a

4

0.91

0.56

0.03

0.03

0.00

tcon

2

1.50

0.23

0.01

0.00

0.00

beecount

14

4.20

3.35

1.14

0.29

0.05

decod

16

4.81

1.90

0.72

0.11

0.05

sqrt8ml

24

3.56

1.76

0.82

0.39

0.04

sqrt8

24

6.60

1.52

0.86

0.12

0.12

c8

6

5.93

3.09

1.03

0.12

0.01

Average

9.13

4.36

1.83

0.60

0.12

0.02

Circuit

5 IMPLEMENTATION OF STOCHASTIC ELEMENTS WITH NANOWIRE CROSSBAR ARRAYS General features of nanowire technology are illustrated in Figure 13. The connections between horizontal and vertical wires are FET-like junctions with nearly a one-to-one ratio, i.e., there is nearly always one FET-like junction per horizontal nanowire. This is a specific attribute of nanowire arrays, controlled through doping during selfassembly (DeHon, 2005). When high or low voltages are applied to input nanowires, the FET-like junctions that cross these develop a high or low impedance, respectively.

292

Figure 13. The nanowire crossbar architecture.

The Synthesis of Stochastic Circuits for Nanoscale Computation

Because the doping regions for the junctions are randomly placed across the crossbar, the connections are random. We exploit this randomness to implement the shuffled AND and bundleplexing constructs.

5.1 Shuffled AND In order to implement a shuffled AND on two input bundles, four crossbars are required. Two invert the signals on the input bundles. Two more invert the results and compute the AND of pairs of randomly shuffled signals from each bundle. This is illustrated in Figure 14. Consider the third wire from the bottom. It produces the AND of a 0 and b1 . To see this, note that the horizontal wire with input a 0 runs through a FET-like junction that inverts the value on the first vertical nanowire from the left. Similarly the input b1 gets inverted on the second vertical nanowire from the right. These vertical nanowires are tied together by FET-like junctions on the horizontal nanowire that produces the output. This effectively computes

the complement of the OR of the inverted values, so the AND of a 0 and b1 .

5.2 Bundleplexing In order to implement the bundleplexing operation on two input bundles, we set a different density of doping for the FET-like regions on the corresponding crossbars. The density dictates that a certain ratio of the output stream is affected by one input stream and the rest affected by the other input stream. The implementation is composed of three crossbars. Two select wires from the input bundles and invert the values. A third inverts the values a second time, producing the output. This is illustrated in Figure 15, which shows 3 a bundleplexer with a selecting ratio of . We 4 Figure 15. The nanowire crossbar architecture implementing a bundleplexer.

Figure 14. The nanowire crossbar architecture implementing a shuffled AND.

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The Synthesis of Stochastic Circuits for Nanoscale Computation

dope the first three vertical wires in the upper-most crossbar and the right-most vertical wire in the middle crossbar. This effectively chooses three bits from bundle A and one bit from bundle B and inverts these. The lower-most crossbar inverts these choices a second time. This gives the requisite output values: a randomly shuffled selection of bits from the two input bundles, with a ratio of 3 . 4

6 DISCUSSION & FUTURE DIRECTIONS The trials with benchmarks in experimental results section show that our technique produces circuits with tunable characteristics: with small bundle widths, the circuits require relatively little area yet compute somewhat inaccurately; with larger bundle widths, the circuits consume more area yet compute more accurately. With sufficiently wide bundles, the computation is perfectly accurate (i.e., no errors occur in the outputs). For many applications, such as control circuitry, perfect accuracy is a requisite. However, for other applications, such as image processing and telemetry, the tolerance for errors might be quite high. Stochastic circuits are particularly applicable in these domains. Although not the focus of this paper, defect and fault-tolerance provide the impetus for our work. Indeed, with parallel stochastic bit streams, the random shuffles need not be perfect. There can be errors in the shuffling ANDs and bundleplexing: bits can be flipped or duplicated. With sufficiently wide streams, quantization at the output will map the resulting fractional weights to the correct Boolean values. We are working to analyze and optimize fault and defect tolerance with stochastic implementations.

Also, in future work, we will tailor the synthesis of stochastic circuits to particular forms of nanowire technology, such as hybrid Nano/CMOS architectures (Strukov & Likharev, 2005; Snider & Williams, 2007).

REFERENCES Asgar, Z., Kodakara, S., & Lilja, D. (2005). Faulttolerant image processing using stochastic logic (Tech. Rep.). Retrieved from http://www.zasgar. net/zain/publications/publications.php Bryant, R., & Chen, Y. (1995). Verification of arithmetic circuits with binary moment diagrams. In Proceedings of the 32nd Design Automation Conference (DAC ’95), San Francisco (pp.535541). DeHon, A. (2005). Nanowire-based programmable architectures. ACM Journal on Emerging Technologies in Computing Systems, 1(2), 109–162. doi:10.1145/1084748.1084750 FENA. (2006). Mission statement. Retrieved from http://www.fena.org International Technology Roadmap for Semiconductors. (2006). ITRS 2006 update. Retrieved from http://www.itrs.net/Links/2006Update/200 6UpdateFinal.htm Snider, G., & Williams, R. (2007). Nano/CMOS architectures using a field-programmable nanowire interconnect. Nanotechnology, 18, 1–11. Strukov, D., & Likharev, K. (2005). CMOL FPGA: A reconfigurable architecture for hybrid digital circuits with two-terminal nanodevices. Nanotechnology, 16, 888–900. doi:10.1088/09574484/16/6/045

This work was previously published in International Journal of Nanotechnology and Molecular Computation, edited by Bruce MacLennan, pp 39-57, copyright 2009 by IGI Publishing.

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Chapter 19

Random Dynamical Network Automata for Nanoelectronics: A Robustness and Learning Perspective1 Christof Teuscher Portland State University, USA Natali Gulbahce Northeastern University, USA Thimo Rohlf Genopole, France Alireza Goudarzi Portland State University, USA

ABSTRACT It is generally expected that future and emerging nanoscale computing devices will be built in a bottomup way from vast numbers of simple, densely arranged components that exhibit high failure rates, are relatively slow, and connected in an unstructured way. Other than that, there is little to no consensus on what type of technology and computing architecture holds most promises to go far beyond today’s topdown engineered silicon devices. Highly structured crossbar-like and cellular automata architectures have been proposed as possible alternatives to the von Neumann computing architecture, which is not generally well suited for emerging, massively parallel and fine-grained nanoscale electronics. While the top-down engineered semi-conducting technology favors regular and locally interconnected structures, emerging bottom-up self-assembled devices tend to have to be unstructured and heterogeneous because of the current lack of precise control over these processes. In this paper, we survey and assess two types of random dynamical networks, namely Random Boolean Networks (RBNs) and Random Threshold Networks (RTNs), as candidates for alternative computing architectures and models for future nanoscale DOI: 10.4018/978-1-60960-186-7.ch019 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Random Dynamical Network Automata for Nanoelectronics

information processing devices. In a high-level approach that is based on previous work, we illustrate that they have the potential to offer superior properties over highly structured crossbar- or mesh-like cellular automata architectures, such as an inherent and scale-invariant robustness, more efficient communication capabilities, manufacturing benefits for bottom-up self-assembled devices, and the ability to learn and solve tasks successfully. We also show that RBNs can learn and generalize. Our investigation is driven by the need for alternative computing and manufacturing paradigms to mitigate some of the challenges traditional approaches face.

INTRODUCTION AND MOTIVATION The advent of multicore architectures and the slowdown of the processor’s operating frequency increase are signs that CMOS miniaturization is increasingly hitting fundamental physical limits. A key question the industry faces is how computing architectures will evolve as we reach these fundamental limits. What type of device and architecture will guarantee a sustainable and continuous progress in the information and communication technology for the next 10-20 years? A likely possibility within the realm of CMOS technology is that the integration density will cease to increase at some point, instead, only the number of components, i.e, the number of transistors, will continue to increase, which will lead to chips with a higher area. This trend can be observed with current multi-core architectures, and is expected to continue. Besides the unsolved challenge of programming large multi-core systems, the trend has implications on the interconnect architecture, the power consumption and heat dissipation, and the reliability. Another possibility lies in the smooth and incremental transition to hybrid systems, which combine traditional silicon technology with new nano- and molecular-scale components, such as carbon nanotubes and nanowires. Yet another possibility is to go beyond silicon-based technology and to radically change the computing and manufacturing paradigms, by using for example bottom-up self-assembled devices. Self-assembling nanowires (Ferry, 2008) or carbon nanotube electronics (Avouris, Chen, & Perebeinos, 2007)

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are promising candidates, although none of them has resulted in electronics that is able to compete with traditional CMOS so far. What seems clear is that the current way with build computers and the way we algorithmically solve problems with them may need to be fundamentally revisited. The goal of this paper is to explore such a radical new approach and to evaluate its potential. While the top-down engineered CMOS technology favors regular and locally interconnected structures, future bottom-up self-assembled devices tend to have irregular structures because of the current lack of precise control over these processes. We therefore hypothesize that, compared to current CMOS technology, which allowed engineers to implement a logic-based computing architecture with extreme precision and reliability, future and emerging computing architectures will be increasingly driven by manufacturing constraints and particularities. Independent of the emerging device and fabrication technologies, we assume in this paper that future nanoscale devices will be built from (1) vast numbers of densely arranged devices that (2) are arranged and interconnected in some unstructured way and that (3) exhibit high failure rates. We take this working hypothesis for granted in this paper and address it from a perspective that focuses on the interconnect topology. This is justified by the fact that the importance of interconnects on electronic chips has outrun the importance of transistors as a dominant factor of performance (Meindl, 2003; Ho, Mai, & Horowitz, 2001; Davis et al., 2001). The reasons for that are twofold: (1) the transistor switching

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speed for traditional silicon is much faster than the average wire delays and (2) the required chip area for interconnects has increased. In (Zhirnov, Cavin, Leeming, & Galatsis, 2008), Zhirnov et al. explored integrated digital Cellular Automata (CA) architectures—which are highly regular structures with local interconnects (see Section)—as an alternative paradigm to the von Neumann computer architecture for future and emerging information processing devices. They argue that CAs are well suited for semiconductor technology, which “[…] favors the realization of regular, locally connected structures […].” Here, we are interested to explore and assess a more general class of discrete dynamical systems, namely Random Boolean Networks (RBNs) and Random Threshold Networks (RTNs), for emerging technologies. We will focus on RBNs, but RTNs are included in this paper because they offer an alternative paradigm to Boolean logic, which offers significant potential for new and efficient devices. We are interested to provide answers to the following questions: • • • • •

Do RBNs and RTNs offer benefits over CA-architectures? If yes, what are they? How does the interconnect complexity compare between RBNs/RTNs and CAs? Does any of these architectures allow to solve certain problems more efficiently? Is any of these architectures inherently more robust to simple errors? Can CMOS and beyond-CMOS devices provide a benefit for the fabrication of any of these architectures?

We will argue and illustrate that—at least from a theoretical perspective—random dynamical networks offer superior properties over classical regular CA-based architectures, such as inherent robustness as the system scales up, more efficient information communication capabilities, and manufacturing benefits for bottom-up fabricated

devices, which motivates this investigation. We will present recent results on the dynamic behavior and robustness of such random dynamical networks while also including manufacturing issues in this mini-assessment. To answer the above questions, we will present and extend recent results on the complex dynamical behavior of discrete random dynamical networks (Rohlf, Gulbahce, & Teuscher, 2007), their ability to solve problems (Mesot & Teuscher, 2005; Teuscher, Gulbahce, & Rohlf, 2007), novel interconnect paradigms (Teuscher, 2007; Teuscher & Hansson, 2008), damage spreading in spatial and small-world networks (Lu & Teuscher, 2009), and self-assembled interconnect models (Teuscher et al., 2009).

RANDOM DYNAMICAL NETWORKS Random Boolean Networks A Random Boolean Network (RBN) (Kauffman, 1968; Kauffman, 1984; Kauffman, 1993) is a discrete dynamical system composed of N nodes, also called automata, elements or cells. Each automaton is a Boolean variable with two possible states: {0,1}, and the dynamics is such that F : {0, 1}N  {0, 1}N ,

(1)

where, F = ( f1,...fi ,...fN ) , and each fi is represented by a look-up table of inputs randomly chosen from the set of N nodes. Initially, K i neighbors and a look-table are assigned to each node at random. Note K i (i.e., the fan-in) can refer to the exact or to the average number of incoming connections per node. A node state sit Î {0, 1} is updated using its corresponding Boolean function:

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si t +1 = fi (x xt , x xt ,..., x xt ) i1

i2

Ki

(2)

These Boolean functions are commonly represented by lookup-tables (LUTs), which associate a 1-bit output (the node’s future state) to each possible K-bit input configuration. The table’s out-column is called the rule of the node. Note that even though the LUTs of a RBN map well on an FPGA or other memory-based architectures, the random interconnect in general does not. We randomly initialize the states of the nodes (initial condition of the RBN). The nodes are updated synchronously using their corresponding Boolean functions. Other updating schemes exist, see for example (Gershenson, 2003) for an overview. Synchronous random Boolean networks as introduced by Kauffman are commonly called NK networks or models. Figure 1 shows a possible NK random Boolean network representation (N=8, K=3).

consists of N randomly interconnected binary sites (spins) with states si = ±1. For each site i, its state at time t+1 is a function of the inputs it receives from other spins at time t: s(t + 1) = sgn( fi (t ))

(3)

with N

fi (t ) = ∑ cij s j (t ) + h

(4)

j =1

The N network sites are updated synchronously. In the following, the threshold parameter h is set to zero. The interaction weights cij take discrete values cij=+1 or  1 with equal probability. If i does not receive signals from j, one has cij=0.

Random Threshold Networks Random Threshold Networks (RTNs) are another type of discrete dynamical systems. An RTN Figure 1. Illustration of a random Boolean network with N=8 nodes and K=3 inputs per node (selfconnections are allowed). The node rules are commonly represented by lookup-tables (LUTs), which associate a 1-bit output (the node’s future state) to each possible K-bit input configuration. The table’s out-column is commonly called the rule of the node

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CELLULAR AUTOMATA ARCHITECTURES

DAMAGE SPREADING AND CRITICALITY

Cellular automata (CA) (Wolfram, 1984) were originally conceived by Ulam and von Neumann (Neumann, 1966) in the 1940s to provide a formal framework for investigating the behavior of complex, extended systems. CAs are a special case of the more general class of random dynamical networks, in which space and time are discrete. A CA usually consists of a D-dimensional regular lattice of N lattice sites, commonly called nodes, cells, elements, or automata. Each cell i can be in one of a finite number of S possible states and further consists of a transition function (also called rule), which maps the neighboring states to the set of cell states. CAs are called uniform if all cells contain the same rule, otherwise they are nonuniform. Each cell takes as input the states of the cells within some finite local neighborhood. Here, we only consider non-uniform, two-dimensional (D=2), folded, and binary CAs (S=2) with a radius-1 von Neumann neighborhood, where each cell is connected to each of its four immediate neighbors only. Figure 2 illustrates such an CA. The Boolean functions in each node must therefore define 24=16 possible input combinations. To be able to compare CAs with RBNs, we do not consider self-connections.

Random Boolean and Threshold Networks As we have seen above, RBNs and their complex dynamic behavior are essentially characterized by the average number of incoming links (fan-in) per node (e.g., Figure 1 shows a K=3 network with 3 incoming links per node). It turns out that in the thermodynamic limit, i.e., N → ∞ , RBNs exhibit a dynamical order-disorder transition at a sparse critical connectivity (Derrida & Pomeau, 1986) (i.e., where each node receives on average two incoming connections from two randomly chosen other nodes), which partitions their operating space into 3 different regimes: (1), subcritical, where K < Kc , (2) complex, where K = Kc , and (3) supercritical, where K > Kc . In the sub-critical regime, the network dynamics are too “rigid” and the information processing capabilities are thus hindered, whereas in the supercritical regime, their behavior becomes chaotic. The complex regime is also commonly called the “edge of chaos,” because it represents the network connectivity where information processing is “optimal” and where a small number of stable attractors exist.

Figure 2. Illustration of a binary, 2D, folded cellular automaton with N=16 cells. Each node is connected to its four immediate neighbors (von Neumann neighborhood)

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Similar observations were made for sparsely connected random threshold (neural) networks (RTN) (Rohlf & Bornholdt, 2002) for Kc = 1.849 . For a finite system size N, the dynamics of both systems converge to periodic attractors after a finite number of updates. At Kc , the phase space structure in terms of attractor periods (Albert & Barabási, 2000), the number of different attractors (Samuelsson & Troein, 2003) and the distribution of basins of attraction (Bastolla & Parisi, 1998) is complex, showing many properties reminiscent of biological networks (Kauffman, 1993).

RESULTS In (Rohlf et al., 2007) we have systematically studied and compared damage spreading (i.e., how a perturbed node-state influences the rest of the network nodes over time) at the sparse percolation (SP) limit for random Boolean and threshold networks with perturbations. In the SP limit, the damage induced in a network (i.e., by changing the state of a node) does not scale with system size. Obviously, this limit is relevant to

information and damage propagation in many technological and natural networks, such as the Internet, disease spreading in populations, failure propagation in power grids, and networks-onchips. We measure the damage spreading by the following methodology: the state of one randomly chosen node is changed. The damage is then measured as the Hamming distance between a damaged and undamaged network instance after a large number of T system updates. For an electronic system, for example, we can compare such a damage event with a single event upset caused by electromagnetic radiation. We have shown that there is a characteristic average connectivity K sRBN = 1.875 for RBNs and K sRTN = 1.729 for RTNs, where the damage spreading of a single one-bit perturbation of a network node remains constant as the system size N scales up. Figure 3 illustrates this newly discovered point for RBNs and RTNs. For more details, see (Rohlf et al., 2007).

Figure 3. Average Hamming distance (damage) d after 200 system updates, averaged over 10,000 randomly generated networks for each value of K , with 100 different random initial conditions and one-bit perturbed neighbor configurations for each network. For both RBN and RTN, all curves for different N approximately intersect in a characteristic point Ks

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DISCUSSION Both and are highly relevant for nano-scale electronics for the following reason: assuming we can build massive numbers of N simple logic gates that implement a random Boolean function, the above findings tell us that on average, every gate should be connected somewhere close to both and in order to (1) guarantee optimal robustness against failures for any system size and (2) optimal information processing at the “edge of chaos.” We are also hypothesizing that natural systems, such as the brain or genetic regulatory networks, may have evolved towards these characteristic connectivities. This remains, however, to be proved and is part of ongoing research.

lular automata (e.g., as pictured in Figure 2), we have adopted the following methodology: (1) for a desired average number of links per cell K for a given CA size of N cells, the total number of links in the automaton is given by L=N K ; (2) we then randomly choose L possible connections on the regular CA-grid with uniform probability and establish the links. Damage is induced in the same way as for RBNs and RTNs: the state of one (or several) randomly chosen node(s) is changed. The damage is measured as the Hamming distance between a damaged and undamaged CA instance after a large number of T system updates, in our case T=200.

Cellular Automata

RESULTS

We have used the same approach as described above to measure the damage spreading in cellular automata. In order to vary the average number K of incoming links per cell in a cel-

Figures 4, 5, and 6 show the average damage of both RBNs and CAs for different system sizes and for a damage size of 1 and 10 respectively. We have left out RTNs for this analysis. As one can see, both the RBN and the CA average dam-

Figure 4. Average Hamming distance (damage) d after 200 system updates, averaged over 100 randomly generated networks for each value of K , with 100 different random initial conditions and a damage size of 1 node for each network. See text for discussion

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Figure 5. Average Hamming distance (damage) d after 200 system updates, averaged over 100 randomly generated networks for each value of K , with 100 different random initial conditions and a damage size of 10 nodes for each network. See text for discussion

age for different N approximately intersect in the characteristic point K sRBN = 1.875 . This point is less pronounced for the larger damage sizes (Figures 5 and 6). The reasons for this are finite size

effects because the damage becomes too big compared to the system size. This can be seen in Figure 6 for the RBN N=100 curve, which represents an outlier, i.e., does not intersect well with

Figure 6. Average Hamming distance (damage) d after 200 system updates, averaged over 100 randomly generated networks for each value of K , with 100 different random initial conditions and a damage size of 20 nodes for each network. See text for discussion

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the other curves. The RBN curves confirm what was already shown above in Figure 3, and are merely plotted here for comparison with the CA architectures and their system sizes imposed by square lattices. Interestingly, the CAs show different damage propagation behavior for different system sizes and connectivities. First, we observe that the average damage for one-bit damage events (Figure 4) is independent of the system size N for up to approximately K =2.5 average incoming connections per cell. This behavior disappears completely for large damage sizes (Figure 6). Second, Figure 4 shows that all curves intersect at K sRBN = K sCA = 1.875 . Third, Figure 6 suggest that for larger damage sizes, KsCA disappears for CAs. Fourth, the average damage for larger damage events, i.e., 10 and 20 in our examples, converges to the same final values for both RBNs and CAs as K approaches 4.

DISCUSSION We hypothesize that the particular behavior can be explained by the percolation limit of the cellular automata. Da Silva et al. (Silva, Hanson, & Roux, 1989) found that the link probability at the percolation limit is approximately p ∼ 0.6, which means that the average connectivity at the percolation limit in our CA topology with a maximum of 4 neighbors is given by k =4p=2.4. This value corresponds to the experimentally observed value where the damage spreading suddenly becomes dependent of the system size. Because of the local CA connectivity, there are lots of disconnected components below the percolation limit. Below this limit, the damage spreading is thus very slow and limited by the disconnected components, reason why it is essentially independent of system size. Above the percolation limit, the CA suddenly becomes connected and damage

spreading becomes therefore dependent on the system size. For larger damage events, such as 10 or 20, damage becomes more dependent on system size even below the percolation limit because there is a higher probability that damage is induced in several disconnected components at the same time. More recently, Lu & Teuscher (Lu & Teuscher, 2009) have also investigated the damage spreading in spatial random networks with extreme local connections and with small-world topologies. They found that spatially local connections change the scaling of the relevant component at very low connectivities (K ≪ 1), that there is no critical connectivity of stability Ks for locally connected networks, and that the critical connectivity Ks changes to lower values compared to random networks for small-world networks. In summary: for single-node damage events, CAs offer system-size independent damage spreading for up to about K = 2.4 (which corresponds to the percolation limit), however, this particular behavior disappears for larger damage events. We conclude that in the general case, CAs do not possess a characteristic connectivity Ks, where damage spreading is independent of the system size N. Such a connectivity, however, exists for both RBNs and RTNs, which makes them particularly suitable as a computing model in an environment with high error probabilities or systems with low system component reliabilities. Examples are logical gates based on bio-molecular components (Benenson et al., 2001), where high failure rates can be expected.

COMPLEX NETWORKS AND WIRING COSTS Most real networks, such as brain networks (Sporns, Chialvo, Kaiser, & Hilgtag, 2004; Eguéluz, Chialvo, Cecchi, Baliki, & Apkarian, 2005), electronic circuits (Cancho, Janssen, & Sole,

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2001), the Internet, and social networks share the so-called small-world (SW) property (Watts & Strogatz, 1998). Compared to purely locally and regularly interconnected networks (such as for example the CA interconnect of Figure 2), small-world networks have a very short average distance (measured as the number of edges to traverse) between any pair of nodes, which makes them particularly interesting for efficient communication. The classical Watts-Strogatz small-world network (Watts & Strogatz, 1998) is built from a regular lattice with only nearest neighbor connections. Every link is then rewired with a rewiring probability p to a randomly chosen node. Thus, by varying p, one can obtain a fully regular (p=0) and a fully random (p=1) network topology. The rewiring procedure establishes “shortcuts” in the network, which significantly lower the average distance (i.e., the number of edges to traverse) between any pair of nodes. In the original model, the length distribution of the shortcuts is uniform since a node is chosen randomly. If the rewiring of the connections is done proportional to a power law, l -a , where l is the wire length, then we obtain a small-world power-law network. The exponent a affects the network’s communication characteristics (Kozma, Hastings, & Korniss, 2005) and navigability (Kleinberg, 2000a), which is better than in the uniformly generated smallworld network. One can think of other distanceproportional distributions for the rewiring, such as for example a Gaussian distribution, which has been found between certain layers of the rat’s neocortical pyramidal neurons (Hellwig, 2000). In a real network, it is fair to assume that local connections have a lower cost (in terms of the associated wire-delay and the area required) than long-distance connections. Physically realizing small-world networks with uniformly distributed long-distance connections is thus not realistic and distance, i.e., the wiring cost, needs to be taken into account, a perspective that recently gained

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increasing attention in complex networks community (Petermann & Rios, 2006). On the other hand, a network’s topology also directly affects how efficient problems can be solved. Teuscher (Teuscher, 2007; Teuscher et al., 2009) has pragmatically and experimentally investigated important design trade-offs and properties of an irregular, abstract, yet physically plausible 3D small-world interconnect fabric that is inspired by modern network-on-chip paradigms. The results confirm that (1) computation in irregular assemblies is a promising and disruptive computing paradigm for self-assembled nano-scale electronics and (2) that 3D small-world interconnect fabrics with a power-law decaying distribution of shortcut lengths are physically plausible and have major advantages over local 2D and 3D regular topologies, such as CA interconnects. More recently, Teuscher et al. (Teuscher et al., 2009) proposed two simple growth models for non-classical nanowire interconnects with the goal to both the cost and the communication of such networks. They have also used an evolutionary algorithm to evolve optimal network topologies under given cost and communication constraints.

DISCUSSION There is a trade-off between (1) the physical realizability and (2) the communication characteristics for a network topology. A locally and regularly interconnected topology, such as that of a CA, is in general easy to build (especially for top-down engineered CMOS technology) and only involves minimal wire and area cost (as for example shown by Zhirnov et al. (V. Zhirnov et al., 2008)), but it offers poor global communication characteristics and scales-up poorly with system size. On the other hand, a random topology, such as that of RBNs or RTNs, scales-up well and has a very short-average path length, but it is not physically plausible because it involves costly long-distance connections established independently of the

Random Dynamical Network Automata for Nanoelectronics

Euclidean distance between the nodes. The RBN and RTN topologies we consider here as thus extremes, such as CA topologies. The ideal topology lies in between: small-world topologies with a distance-dependent distribution of the connectivity. Such topologies are located in a unique spot in the design space and also offer two other highly relevant properties (Kleinberg, 2000b; Teuscher, 2007): (1) efficient navigability and thus potentially efficient routing, and (2) robustness against random link removals. For these reasons, we can conclude that small-world graphs are the most promising interconnects for future massive scale devices. This outcome is further supported by the quantitative analysis presented in (Teuscher at al., 2009): unstructured network-on-chip topologies obtained by the growth models show specific wire-length distributions that are beneficial for the communication and minimize the wiring cost. These outcomes seem contradictory with Bilardi & Preparata (Bilardi & Preparata, 1995) argument that future architectures will naturally be driven toward regular and locally interconnected architectures. However, our findings have a different starting point, namely a set of computing nodes randomly distributed in space, which is motivated by the way we imagine such a system would be fabricated. Given that physical arrangement, we then ask the question of what the optimal interconnect is.

SOLVING TASKS WITH RANDOM BOOLEAN NETWORKS In their pioneering work, Carnevali & Patarnello (Carnevali & Patarnello, 1989) used two global stochastic optimization techniques, Simulated Annealing (SA) and Genetic Algorithms (GA), to train feedforward random Boolean networks to solve computational tasks. Paternello and Carnevali showed that global stochastic optimization is able to train networks for simple memorization and generalization tasks. Teuscher et al. (Teuscher et

al., 2007) used the same techniques to train random Boolean networks and showed that they can successfully solve simple computational tasks as well. Their preliminary results also hinted at the effect of the critical connectivity on the training and generalization performance. In this section we first replicate the above-mentioned results and then analyze the dynamics of learning and generalization in RBNs. In particular, we investigate how the network connectivity influences these dynamics.

EXPERIMENTAL SETUP For our purpose, we use genetic algorithms to train the networks for solving the bitwise AND, the even-odd, and the mapping task. The bitwise AND task consists of calculating the bitwise logical AND of two l-bit binary numbers. The networks thus have a 2 × l-bit inputs and an l-bit output. For the even-odd task, we have an l-bit binary input and a one-bit output. The output is one if the number of 1s in the input is an odd number, and 0 otherwise. The mapping task consists of an l-bit input and an l-bit output. The task consists of producing the same number of 1s in the output as there are in the input, but without considering the order of the bits. In order to perform these tasks with RBNs, we add additional input and output nodes to the regular network nodes. These nodes are connected randomly to other nodes in the network. While the input nodes only provide the input signals to the network, the output nodes may also compute a Boolean function. In order to use GAs, we encode the network into a bit-stream that consists of both the network’s adjacency matrix and the Boolean transfer functions for each node. The genetic operators consist of the standard mutation and one-point crossover operators that are applied to the genotypes in the network population. We further define a fitness function f and a generalization function g. For an

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input space N of size n and an input sample M of size m we write: EM =

1 m

∑

j ∈M

d ( j ), f = 1 − E M

where d(j) is the Hamming distance between the network output for the j-th input in the random sample from the input space and the expected network output for that input. Similarly, we write: EN =

1 n

∑

i ∈n

d (i ), g = 1 − E N

where d(i) is the Hamming distance between the network output for the i-th input from the entire input space and the expected network output for that input. The simple genetic algorithm we use to train the network is as following: 1. Create a random initial population of S networks. 2. Evaluate the performance of the networks on a random sample of the input space. 3. Apply the genetic operators to obtain a new population. 4. Continue with steps 2 and 3 until at least one of the networks achieves a perfect fitness or after Gmax generations are reached. While optimizing feedforward networks, we have to make sure that the mutation and crossover operators do not violate the feedforward topology of the network. Also, there is a subtle difference between feedforward RBNs and general (recurrent) RBNs: since recurrent RBNs can have feedback loops and thus have memory, we have to take into account the dynamical properties of the network and run them long enough to ensure they fall into an attractor. The network size N and

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the connectivity K determine the length of the transients, however, due to ambiguous outputs on cyclic attractors, we further calculate the majority activation of the output nodes over a number of time steps equal to the size of the network. Carnevali & Patarnello (Carnevali & Patarnello, 1989) introduced the notion of learning probability as a way of describing the learning and generalization capability of their feedforward random networks. They defined the learning probability as the probability of the training process yielding a network with perfect generalization, given that the training achieves perfect fitness on a sample of the input space. To calculate this measure, we run the training process r times and store both the f and g values.

RESULTS Figure 7 and 8 show the learning probability of feedforward RBNs on the even-odd and the bitwise AND task for K = 2 networks. As one can see, as the size of the input space increases, the training process requires a smaller number of training examples to achieve a perfect learning probability. We tune the GA parameters, such as the variation rate and the maximum number of generations experimentally, depending on how quickly we achieve perfect fitness on average. These result directly confirm Carnevali & Patarnello (Carnevali & Patarnello, 1989) experiments. Next, we create the same learning probability curves for RBNs to show that they too can solve these simple tasks. Figure 9 illustrates the results for the even-odd task. Similar to feedforward networks, we observe that as the problem size increases, a lower proportion of training samples is required to achieve high learning probabilities. On the other hand, we observe that it is more difficult to get the network to learn the tasks. The network reaches a learning probability of 1 only when we use the entire input space during the

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Figure 7. The learning probability of feedforward networks on the bitwise AND task on different problem sizes. As the number of input bits increases, the learning process requires a smaller fraction of the input space during the training to achieve a perfect learning score. N = 50, K = 2.0, Gmax = 3000, initial population size = 50, crossover rate = 0.6, mutation rate = 0.3. The GA was repeated for 700 times

Figure 8. The learning probability of feedforward networks on the even-odd task for various input sizes. As the number of input bits increases, the learning process requires a smaller fraction of the input space during the training to achieve a perfect learning score. For I = 3, some of the networks can solve a significant number of patters without training because the task is too easy. N = 50, K = 2.0, Gmax = 3000, initial population size = 50, crossover rate = 0.6, mutation rate = 0.3. The GA was repeated for 700 times

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Figure 9. (left) The learning probability of RBNs for the even-odd task for various input sizes. N = 20, = 2, Gmax = 500. To calculate the likelihood we repeated the GA 500 times. (right) The learning probability of RBNs for the mapping task for various input sizes. N = 40, = 2, Gmax = 1000. We calculated the likelihoods over 260 runs. For both the mapping and even-odd tasks we used an initial population size of 50, a crossover rate of 0, and mutation rate of 0.7 for the GA

training phase. The average node inputs K in this experiment was 2.0, so all networks were in the critical regime. To see how the three connectivity regimes (i.e., subcritical, critical, supercritical) affect the learning probability, we generated network populations of various K to solve the even-odd task with five input bits.

Figure 10 (left) shows the learning probability of RBNs with different average Ks for the even-odd task for five inputs (i.e., the input space has 32 patterns). As one can see, networks with supercritical connectivity ( K = 3.0) achieve a lower learning probability than the networks at the “edge of chaos” ( K = 2.0 and K = 2.5)

Figure 10. (left) Learning probability of RBNs with different average K for even-odd task with five inputs. (right) Training probability of RBNs with different K for even-odd task with five inputs. N = 20, Gmax = 500, initial population size = 50, crossover rate = 0, mutation rate = 0.7. We calculated the likelihood over 400 runs

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and at subcritical connectivities ( K = 1.0, 1.5, 1.875, 1.95). However, networks with subcritical connectivity show a interesting jump from zero to perfect learning probability when the training sample size reaches 0.6. For K = 2.0, 2.5, and 3.0, the network achieves some level of learning probability with smaller sample sizes (Figure 10 (left)). Figure 10 (right) shows the training probability for the even-odd task. We see that subcritical K networks make the training very difficult. As the input sample size increases, the amount of information in the samples also does and subcritical networks do not have the flexibility to learn the task easily. On the other hand, networks in the complex and chaotic regime can easily learn the task. Moreover, although the critical K = 2 networks have more difficulty in learning the samples, they can generalize better, and thus have a higher learning probability than the networks with supercritical connectivity, which simply memorize the patterns they have learned.

DISCUSSION We have shown that both feedforward and general (recurrent) RBNs can solve simple tasks and are able to generalize. While critical networks generalize better then subcritical and supercritical networks, the solution is harder to find. Our results show that there is great potential in training such networks more complex tasks and that the control of statistical properties of the network topology during the learning phase is an important parameter of generalization success.

MANUFACTURING ISSUES As Chen et al. (Chen et al., 2003) state, “[i]n order to realize functional nano-electronic circuits,

researchers need to solve three problems: invent a nanoscale device that switches an electric current on and off; build a nanoscale circuit that controllably links very large numbers of these devices with each other and with external systems in order to perform memory and/or logic functions; and design an architecture that allows the circuits to communicate with other systems and operate independently on their lower-level details.” While we can currently build switching devices in various technologies besides CMOS (see (V. V. Zhirnov, Hutchby, Bourianoff, & Brewer, 2006; Bourianoff, Brewer, Cavin, Hutchby, & Zhirnov, 2008; Hutchby, Cavin, Zhirnov, Brewer, & Bourianoff, 2008) for an overview), one of the remaining challenges is to assemble and interconnect these switching devices (or logic functions) to larger systems, and ultimately to design a computing architecture that allows to perform reliable computations. As mentioned before, there is little consensus in the research community on what type of technology and computing architecture holds most promises for the future. The motivation for investigating randomly assembled interconnects and computing architectures can be summarized by the following observations: •

•

•

•

Long-range and global connections are costly (in terms of wire delay and of the chip area used) and limit system performance (Ho et al., 2001); It is unclear whether a precisely regular and homogeneous arrangement of components is needed and possible on a multibillion-component or even Avogadro-scale assembly of nano-scale components (Tour et al., 2002) “[s]elf-assembly makes it relatively easy to form a random array of wires with randomly attached switches” (V. V. Zhirnov & Herr, 2001); and building a perfect system is very hard and expensive

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We have hypothesized in (Teuscher & Hansson, 2008) and (Teuscher, 2007) that bottom-up self-assembled electronics based on conductive nanowires or nanotubes can lead to the random interconnect topologies we are interested in, however, several questions remain open and are part of a 3-year interdisciplinary research project at Los Alamos National Laboratory (LANL). Our approach consists in using a hybrid assembly (as others explore as well, e.g., (Ferry, 2008)), where the functional building blocks will still be traditional silicon in a first step, while the interconnect is made up from self-assembled nanowires. Nanowires can be grown in various ways using diverse materials, such as metals and semiconductors. We have chosen a novel way to grow conductive nanowires, which Wang et al. (Wang, Li, Akhadov, & Jia, 2007) at LANL have pioneered and demonstrated: Ag nanowires can be fabricated on top of conducting polyaniline polymer membranes via a spontaneous electrodeless deposition (self-assembly) method. We hypothesize that this will allow to densely interconnect silicon components in a simple and cheap way with specific distance-dependent wire-length distributions. We believe that this approach will ultimately allow us to easily and cheaply fabricate RBN-like computing architectures. Random threshold networks, on the other hand, could be rather straightforwardly and efficiently implemented with resonant tunneling diode (RTD) logic circuits (see e.g., (Pettenghi, Avedillo, & Quintana, 2008)), and represent a very interesting alternative to conventional Boolean logic gates. The reported results in this paper on random threshold networks can thus directly be applied to the implementation of such devices. There has been a significant body of research in the area of threshold logic in the past (see e.g., (Muroga, 1971)), but to the best of our knowledge, random threshold networks have not been considered as computing models for future and emerging computing machines.

310

CONCLUSION The central claim of this paper is that locally interconnected computing architectures, such as cellular automata (CA), are in general not appropriate models for large-scale and general-purpose computations. We have supported this claim with recent theoretical results on the complex dynamical behavior of discrete random dynamical networks, their robustness to damage events as the system scales up, their ability to efficiently solve tasks, and their improved transport characteristics due to the short average path length. In a nutshell, the arguments why we believe that CAs are not promising architectures for future informationprocessing devices, are as following: •

•

•

•

Their local interconnect topology is not small-world and has thus worse global transport characteristics (than small-world or random graphs), which directly affects the effectiveness of how general-purpose algorithmic tasks can be solved. In terms of a complex dynamical system, they operate in the supercritical regime () with the widely used von Neumann neighborhood, which makes them sensitive to initial conditions; They do not generally have a characteristic connectivity Ks, where damage spreading is independent of system size, which makes a system inherently robust; and It is unclear whether a precisely regular and homogeneous arrangement of components is possible at the scale of future information processing devices.

We have assessed RBNs and RTNs as alternative models, however, as we have seen above, they come at a serious cost: the uniform probability to establish connections with any node in the system independent of the Euclidean distance between them is not physically plausible and too expensive in terms of wiring cost. The ultimate intercon-

Random Dynamical Network Automata for Nanoelectronics

nect topology is small-world and has a distancedependent distribution of the wires (Teuscher, 2007; Teuscher & Hansson, 2008; Petermann & Rios, 2006). It was also been confirmed recently (Lu & Teuscher, 2009) that a critical connectivity Ks exists for such small-world networks as well.

OPEN QUESTIONS AND UNADDRESSED ISSUES Naturally, there are a number of open questions and issues that we have not addressed because they are beyond the scope of this paper. In particular, an irregular topology with random logical functions makes the mapping of a given digital circuit much harder, if not impossible in certain cases. On the other hand, a regular interconnect topology clearly makes the mapping task easier. We believe, however, that automated design tools can address this challenge. After all, computation in random assemblies is not completely new and has been more or less successfully tried by others, e.g., (Patwardhan, Dwyer, Lebeck, & Sorin, 2006; Lawson & Wolpert, 2006, Tour et al., 2002), however in different contexts and with a different perspective in mind than we have presented here. We have deliberately not focused on any particular application in this paper because our results are independent of the application. However, it is noteworthy that locally interconnected CAs have been proven to outperform other general purpose architecture on very specific applications. A good example are cellular neural networks (CNNs) (Chua & Roska, 2002), which, e.g., allow to perform certain image processing tasks orders of magnitude faster than any other machine. Further, it is unknown at this point how exactly our findings fit into the interconnect predictions made by Rent’s rule, however, the rule may not be applicable to our non-traditional circuits since it is based on empirical results. Further research on this is planned.

Last but not least, we would like to mention that, although we have only considered 2D arrangements and interconnects here for simplicity, the future is clearly 3D (e.g., see (Pavlidis & Friedman, 2007)). The main reason is that the average wire length in 3D is shorter than in 2D interconnects.

OUTLOOK We believe that computation in random self-assemblies of simple components and interconnections is a highly appealing paradigm, both from the perspective of fabrication as well as performance and robustness. Future work will focus on (1) the manufacturing issues, (2) appropriate design methodologies, (3) addressing the mapping issues, and (4) more realistic models, which will allow to better assess the performance and cost, and (5) specific applications.

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ENDNOTE 1



This paper is a significant extension of work first presented at the Nanoarch 2008 workshop, Anaheim, CA, USA, Jun 12-13, 2008, and subsequently published in (Teuscher et al., 2009b). This present chapter contains a new section on “Solving Tasks with Random Boolean Networks.”

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About the Contributors

Bruce MacLennan has a BS in mathematics (with honors, 1972) from Florida State University, and an MS (1974) and PhD (1975) in computer science from Purdue University. He was a Senior Software Engineer with Intel Corporation (1975–9), after which he joined the Computer Science faculty of the Naval Postgraduate School (Monterey, CA) as Assistant Professor (1979–83), Associate Professor (1983–7), and Acting Chair (1984–5). Since 1987 he has been a member of the Electrical Engineering and Computer Science faculty of the University of Tennessee, Knoxville. MacLennan’s research includes the application of molecular computing to nanostructure synthesis and control and the development of novel models of computation intended to better exploit physical processes for computation. Prof. MacLennan has more than 60 refereed journal articles and book chapters and has published two books. He has made more than 60 invited or refereed presentations. *** Susumu Adachi received his M.S. in Engineering from Hiroshima University in 1995, and his Ph.D in Computer Science from Kobe University in 2001. His research interests are in cellular automata and complexity. Andrew Adamatzky is a Professor in Unconventional Computing, University of the West of England, Bristol. He does research in chemical, biological and physical computing devices, parallel computing, biological and distributed robotics, theory of computation, distributed intelligence and mathematical psychology. Martyn Amos received his B.Sc. degree in Computer Science from Coventry University, UK in 1993, and his Ph.D. in DNA Computation from the University of Warwick, UK in 1997. He then held a Leverhulme Special Research Fellowship, before taking up Lectureships in Bioinformatics, first at the University of Liverpool, UK and then at the University of Exeter, UK. He is currently a Senior Lecturer and Leader of the Novel Computation Group in the Department of Computing and Mathematics, Manchester Metropolitan University, UK. He is broadly interested in the development and analysis of computational methods based on natural or self-organizing systems, with particular reference to molecular and cellular computing and population-based approaches.

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About the Contributors

Takafumi Aoki is a Professor of the Graduate School of Information Sciences at Tohoku University. His research interests include theoretical aspects of computation, VLSI systems, multiple-valued logic, digital signal processing, image sensing, computer vision, biometric authentication, and secure embedded systems. Masashi Arita received B.S. in materials science in 1980 from Hiroshima University, Hiroshima, Japan and PhD in solid state physics in 1987 from ETH Zurich, Zurich, Switzerland. After joining Ciba-Geigy Japan in 1987, he worked on crystallography and electrophotography of organic substances. Since joining Nagoya University, Nagoya, Japan, in 1990 in 1997, he has been in research field of crystallography and physical properties of metallic nano particles and ultra thin films. Since 1997, he has been an associate professor at Graduate School of Engineering as well as Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. Dr. Arita is a member of the Japan Society of Applied Physics, the Japan Institute of Metals, the Physical Society of Japan, and the Magnetic Society of Japan. James Armstrong is studying chemistry at Warwick University and worked at UWE where he carried out experiments to map the phase diagram of the aluminium chloride sodium hydroxide reaction. Tetsuya Asai is an Associate Professor in the Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan, and is a Visiting Fellow of the Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol, UK. He received the B.E. and M.E. degrees in electrical engineering from Tokai University, Kanazawa, Japan, in 1993 and 1996, respectively, and the Dr. Eng. degree in electrical and electronic engineering from Toyohashi University of Technology, Aichi, Japan, in 1999. His research interests concentrate around developing nature-inspired integrated circuits and their computational applications. Current topics that he is involved with include; intelligent image sensors that incorporate biological visual systems or cellular automata in the chip, neuro-chips that implement neural elements (neurons, synapses, etc.) and neuromorphic networks, and reaction-diffusion chips that imitate vital chemical systems. John Backes received his Bachelor’s Degree in Computer Engineering from the University of Minnesota in May 2009. During his undergrad, John was actively involved in research, and is remaining at the University of Minnesota to pursue a PhD in Electrical Engineering. John’s main research interests include logic synthesis for cyclic topologies and SAT-Based algorithms for logic synthesis. Anirban Bandyopadhyay is a scientist in the Advanced Nano Characterization Center, in the National Institute for Materials Science (NIMS), Tsukuba, Japan. He received his PhD from the Indian Association for the Cultivation of Science (IACS), Kolkata, India on experimental condensed matter physics. He has been working on multilevel organic molecular switches for the last ten years, for the possible applications as molecular neurons. His past work ranged from inventing 2 D nano brain, single molecule neuron like multi-level switch, building world’s smallest neural network, designing introducing the concept of an adaptive molecule etc. Currently, he is working on the realization of a 3 D intelligent molecular bio-processor for details see www.anirbanlab.co.nr.

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About the Contributors

Supriyo Bandyopadhyay is a Professor of Electrical and Computer Engineering, and Professor of Physics at Virginia Commonwealth University, Richmond, USA, where he directs the Quantum Device Laboratory engaged in research in various aspects of nanotechnology. Prior to joining Virginia Commonwealth University, he had served as a visiting faculty member at Purdue University, West Lafayette, Indiana and as faculty member at University of Notre Dame and University of Nebraska-Lincoln. Prof. Bandyopadhyay has authored and co-authored over 300 research publications and given more than 80 invited talks and colloquia across four continents. He serves as the Chair of the Institute of Electrical and Electronics Engineers (IEEE) Technical Committee on Spintronics (Nanotechnology Council) as well as the Technical Committee on Compound Semiconductor Devices and Circuits (Electron Device Society). He is an IEEE Electron Device Society Distinguished Lecturer and served as a Vice President of the IEEE Nanotechnology Council. Prof. Bandyopadhyay is a Fellow of the Institute of Electrical and Electronics Engineers, the American Physical Society, the Institute of Physics (UK), the Electrochemical Society and the American Association for the Advancement of Science. Larry Bull is Professor of Artificial Intelligence within the Department of Computer Science at UWE and his research interests include natural and artificial complex adaptive systems. He is the founding Editor-in-Chief of the Springer journal Evolutionary Intelligence. Prof. Bull heads the Artificial Intelligence Group at UWE which carries out research in a number of areas such as data mining, adaptive control, complex systems, and optimization, with a strong emphasis on evolutionary computation. Marc Cahay is a professor in the Department of Electrical and Computer Engineering at the University of Cincinnati. His current research interests include modeling of nanoscale devices, spintronics, experimental investigation of mesoscopic systems, vacuum micro and nano electronics and organic light emitting diodes. He has published over 110 journal articles in these areas. He is an IEEE Electron Device Society Distinguished Lecturer and is currently a Vice President of the IEEE Nanotechnology Council. With Supriyo Bandyopadhyay, he has co-authored a textbook on an Introduction to Spintronics (CRC Press, Boca Raton 2008). He is a Fellow of the Electrochemical Society and IEEE. Ben de Lacy Costello is a Senior Research fellow in the Centre for Analytical, Material and Sensor Sciences at the University of the West of England. He is also a member of the unconventional computing group at UWE. His interests include synthesis and characterisation of chemical sensors, the study and control of non-linear chemical reactions. Peter Dittrich is junior group leader of the Bio Systems Analysis Group at the Friedrich-SchillerUniversity Jena, Department of Mathematics and Computer Science, and member of the Jena Centre for Bioinformatics. He is interested in modelling, simulation, and theory of complex biological and social systems, and especially in their reflexive information processing capabilities. Daisuke Fujita is Coordinating Director of Key Nanotechnologies Field, Managing Director of Advanced Nano-Characterization Center (ANCC) and Quantum Beam Center (QBC), and Principal Investigator of World Premier International Center for Materials Nano-Architectonics (WPI-MANA) of National Institute for Materials Science (NIMS), Japan. His research field is advanced nanometer-scale characterization and analysis on metals, semiconductors, superconductors, nanoclusters, and molecules in the extreme fields such as low-temperature, high-temperature, high-magnetic field, ultra-high vacuum,

350

About the Contributors

stress-strain field, and so on. Novel quantum phenomena and functionalities of artificial nanostructured materials observable at extreme environments are his main research targets. Development of novel nanostructured materials such as size-controlled metallic nanoclusters, one-dimensional nanowires, and two-dimensional nanosheets is also his research interests. The major investigation tools of his research are various types of nanoprobe technologies, such as scanning tunneling microscopy and spectroscopy, tunnel-electron-induced luminescence microscopy, and atomic force microscopy. Akira Fujiwara was born in Japan on March 9, 1967. He received the B.S., M.S., and Ph.D. degrees in applied physics from The University of Tokyo, Japan, in 1989, 1991, and 1994, respectively. In 1994, he joined LSI Laboratories, Nippon Telegraph and Telephone (NTT) Corporation, Kanagawa, Japan. He moved to the Basic Research Laboratories (BRL) in 1996. Since 1994, he has been engaged in research on silicon nanostructures and their application to single-electron devices. He was a guest researcher at National Institute of Standards and Technology (NIST), Gaithersburg, USA during 2003-2004. Since 2006, He is a group leader of Nanodevcies Research Group, NTT BRL. Since 2007, He is a Distinguished Technical Member, NTT BRL. He is a member of the Japan Society of Applied Physics and IEEE. Max H. Garzon is a professor of computer science, is affiliated with the bioinformatics program and is a member of ACM, IEEE and Sigma Xi. His research interest focus on the area of interactive computation, particularly DNA computing, Bioinformatics, and Human-Computer Interaction. His main goal is to develop better understanding of methods to encode and process information in ensembles of organic molecules such as DNA, RNA and protein. He is the co/author of over 150 books, research articles and publications in foundations of computing and biological information processing. Subrata Ghosh was born in 1978 in Howrah, India. He received his bachelor’s degree in 2000 from Burdwan University, India and Post graduated from the same University in 2002. He received his Ph.D. degree in the year 2010 from Jadavpur University and IACS, India for the Synthesis of Bioactive Natural Products. During his Ph.D he explored the total synthesis of allelopathic sesquiterpenoid compounds. In 2009 he joined the ANCC group in NIMS, Japan as a postdoctoral fellow. His current research interests include theoretical simulation of artificially intelligent molecular nano brain and practical synthesis of supramolecular architectures for showing in-vivo and in-vitro application of nano brain in human cell as a anti-cancer and anti-aging drug and synthesis of conjugated dendritic nano brain for gene therapy. Alireza Goudarzi received his B.Sc. in Information Technology and Computer Studies at Eastern Mediterranean University (Cyprus) in 2007. He is currently a dual degree graduate student in Computer Science and Systems Science department at Portland State University. His research interests include distributed autonomous computing, unconventional and biologically inspired computing, artificial intelligence and machine learning, and information processing on unorganized machines. His most recent works involve investigating the computational power of random automata networks and designing optimal controllers using Dual Heuristic Programming (DHP) at the North West Computational Intelligence Lab (NWCIL.) Natali Gulbahce is currently a postdoctoral research fellow at Dana Farber Cancer Institute at Harvard Medical school and Northeastern University with Albert-Laszlo Barabasi. She obtained her Ph.D. from Clark University studying statistical physics of phase transitions and nucleation. At the Center for

351

About the Contributors

Nonlinear Studies at Los Alamos National Laboratory she became increasingly interested in complex systems and networks. Her current research involves the modeling of large scale biological networks and their emerging global properties. The aim of her research is to increase the understanding of the effects of perturbations on cellular networks and the pathways to complex diseases. Tatsuo Higuchi is a Professor at Tohoku Institute of Technology and an Emeritus Professor at Tohoku University. His general research interests include the design of 1-D and multi-D digital filters, linear time-varying system theory, fractals and chaos in digital signal processing, VLSI computing structures for signal and image processing, multiple-valued ICs, multiwave opto-electronic ICs, and biomolecular computing. Thomas Hinze studied computer science with emphasis on computationally hard problems and graduated with diploma degree in 1997. After a three-year doctoral scholarship funded by the Studienstifftung des Deutschen Volkes, he received a PhD at the Dresden University of Technology for a thesis entitled “Universal Models and Selected Algorithms in DNA-Based Computing” in 2002. Since 2006, he is a research associate at the Friedrich-Schiller-University Jena, initially Bio Systems Analysis Group and currently Department of Bioinformatics. His research interests include models, methods, and concepts of biologically inspired computing, especially membrane systems. He became a member of the European Molecular Computing Consortium. Masahiko Hiratsuka received the B.E. degree in electronic engineering, and the M.S. and Ph.D. degrees in information sciences from Tohoku University, Sendai, Japan, in 1995, 1997 and 2000, respectively. He is currently an Associate Professor at Sendai National College of Technology, Sendai, Japan. From 1998 to 2000, he was a Research Fellow of the Japan Society for the Promotion of Science. His research interests include biomolecular computing and bioelectronics. Dr. Hiratsuka received the Outstanding Transactions Paper Award from the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan in 1997, the IEICE Inose Award in 1997, and the IEE Mountbatten Premium Award in 1999. Hiroshi Inokawa received B.S., M.S., and Ph.D. degrees in electrical engineering from Kyoto University in 1980, 1982 and 1985, respectively. In 1985, he joined Atsugi Electrical Communications Laboratories, NTT, Kanagawa, Japan. Since then, he has been engaged in the research and development of scaled-down MOS devices including those with reduced parasitic effects, and advance isolation schemes. In the years from 2000 to 2005, he was with NTT Basic Research Laboratories, NTT, Kanagawa, Japan, where he researched on silicon nanodevices, particularly single-electron devices, and their applications. Since 2006, he has been a professor at The Research Institute of Electronics, Shizuoka University, Hamamatsu, Japan. Dr. Inokawa is a member of the Japan Society of Applied Physics, the Institute of Electronics, Information and Communication Engineers of Japan, and IEEE. Teijiro Isokawa received his B.E. degree (Electronic Engineering), M.E. degree (Electronic Engineering), and D.E. degree (Doctor of Engineering) in 1996, 1999, and 2004, respectively, from Himeji Institute of Technology, Japan. He is currently an Associate Professor of the Division of Computer Engineering, Graduate School of Engineering, University of Hyogo, Japan. His research interests include nanocomputing, hypercomplex-valued neural networks, and cognitive models in visual systems.

352

About the Contributors

Koichi Ito is an Assistant Professor of the Graduate School of Information Sciences at Tohoku University. His research interests include signal and image processing, and biometric authentication. Ishrat Jahan is a PhD student sponsored by the EPSRC. Her PhD involves the control of chemical excitation waves towards tangible outcomes in computation. Mingyu Jo received B.S. in electrinics in 2008, respectively from Hokkaido University, Sapporo, Japan. Since 2008, he has been working toward the MS degree in the Laboratory of Nanoscience and Materials, the Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. His research focuses on application of Si nanodot array device. Mr. Jo is a member of the Japan Society of Applied Physics. Jeff Jones is a research associate at the Centre for Unconventional Computing, University of the West of England, Bristol, UK. He is researching particle based approximations of transport networks, inspired by the true slime mould Physarum polycephalum for applications in amorphous robotics. His research interests include multi-agent systems, emergent computation, pattern formation processes, cellular automata, spatially represented computation, unconventional models of computation, computational imaging, and computational models of vision. Takuya Kaizawa was born in 1981 in Japan. He received the B.S. and the M.S. degree from Hokkaido University, Sapporo, Japan, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree at the Laboratory of Nanoscience of Materials, the Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. His research interests include application of Si nanodot array single-electron devices. Mr. Kaizawa is a member of the Japan Society of Applied Physics. Bhargava Kanchibotla received MSc (Electronic Sciences) degree from University of Hyderabad, Hyderabad, India in 2001, MS (Physics) degree from Texas Tech University, Lubbock, TX, USA in 2005 and PhD (electrical and computer engineering) degree from Virginia Commonwealth University,Virginia, USA in 2009. Since then he has been with Intel Corporation, Hillsboro, Oregon as Process TD Engineer. Seiya Kasai is an associate professor in the Graduate School of Information Science and Technology and Research Center for Integrated Quantum Electronics (RCIQE), Hokkaido University. He is also a researcher of “Materials and Processes for Innovative Next-Generation Devices” research area of Basic Research Programs PRESTO, Japan Science and Technology Agency. His current research interests include III-V compound semiconductor-based quantum devices, functional nanodevices, nanodevice integration process technology, and implementation of novel logic architectures. He is a member of IEEE, the Japan Society of Applied Physics (JSAP) and the Institute of Electronics, Information and Communication Engineers (IEICE), Japan. Andrew Kilinga Kikombo received the B.E. degree in electronics engineering from the University of Electro-Communications Tokyo, Japan, in 2005 and the M.S. in electrical engineering from Hokkaido University, Japan, in 2007. He is currently working toward the PhD degree in Electrical engineering at Hokkaido University, Japan. His research interests include modeling, design and simulation of bioinspired single-electron circuits. Mr. Kikombo is a member of the Japan Society of Applied Physics.

353

About the Contributors

Yusuf Leblebici received his B.Sc. and M.Sc. degrees in electrical engineering from Istanbul Technical University, in 1984 and in 1986, respectively, and his Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign (UIUC) in 1990. Between 1991 and 2001, he worked as a faculty member at UIUC, at Istanbul Technical University, and at Worcester Polytechnic Institute (WPI). In 2000-2001, he also served as the Microelectronics Program Coordinator at Sabanci University. Since 2002, Dr. Leblebici has been a Chair Professor at the Swiss Federal Institute of Technology in Lausanne (EPFL), and director of Microelectronic Systems Laboratory. His research interests include design of high-speed CMOS digital and mixed-signal integrated circuits, computeraided design of VLSI systems, intelligent sensor interfaces, modeling and simulation of semiconductor devices, and VLSI reliability analysis. He is the coauthor of 4 textbooks, namely, Hot-Carrier Reliability of MOS VLSI Circuits (Kluwer Academic Publishers, 1993), CMOS Digital Integrated Circuits: Analysis and Design (McGraw Hill, 1st Edition 1996, 2nd Edition 1998, 3rd Edition 2002), CMOS Multichannel Single-Chip Receivers for Multi-Gigabit Optical Data Communications (Springer, 2007) and Fundamentals of High Frequency CMOS Analog Integrated Circuits (Cambridge University Press, 2009), as well as more than 200 articles published in various journals and conferences. He has served as an Associate Editor of IEEE Transactions on Circuits and Systems (II), and IEEE Transactions on Very Large Scale Integrated (VLSI) Systems. He has also served as the general co-chair of the 2006 European Solid-State Circuits Conference, and the 2006 European Solid State Device Research Conference (ESSCIRC/ESSDERC). He is a Fellow of IEEE and has been elected as Distinguished Lecturer of the IEEE Circuits and Systems Society for 2010-2011. Jia Lee received his PhD degree from Hiroshima University in 2001. From 2001 to 2006, he was a Post-Doc fellow at the National Institute of Information and Communications Technology (NiCT), Kobe, Japan. From 2006 to 2008, he was a CREST researcher of Japan Science and Technology Agency (JST) at Tsukuba University. Since 2008, he has been in research and development at Celartem Technology Inc. His research interests include image processing, pattern recognition, asynchronous systems, cellular automata, and formal language theory. Naoki Matsumaru had initially exposed himself to the studies of Computer Science and Engineering at University of Aizu in Fukushima, Japan, and graduated with Bachelor degree. To advance his study, he travelled to USA and pursued a chance to work with Michael Conrad and his colleagues, investigating on molecular computing. Their philosophy molded the basis of author’s present view on research activities. The author received Master degree from Wayne State University in Detroit, MI. His current position is a PhD student in Bio Systems Analysis Group at the Friedrich-Schiller-University Jena, Germany, and his work has been focusing on programming methodology of chemical computing. His scholarship is sponsored by German Research Foundation under the special keyword, Organic Computing, exploring a realization of artificial systems with useful “life-like” properties. His interests include an aspect of molecular computing to elucidate biological information processing principles. Takashi Morie received the B.S. and M.S. degrees in physics from Osaka University, Osaka, Japan, and the Dr.Eng. degree from Hokkaido University, Sapporo, Japan, in 1979, 1981 and 1996, respectively. From 1981 to 1997, he was a member of the Research Staff at Nippon Telgraph and Telephone Corporation (NTT). From 1997 to 2002, he was an associate professor of the department of electrical engineering, Hiroshima University, Higashi-Hiroshima, Japan. Since 2002 he has been a professor of Graduate School

354

About the Contributors

of Life Science and Systems Engineering, Kyushu Institute of Technology, Kitakyushu, Japan. His main interest is in the area of VLSI implementation of neural networks, mixed/merged analog-digital circuits, and new functional devices. Dr. Morie is a member of IEEE, the Institute of Electronics, Information and Communication Engineers (IEICE), the Japan Society of Applied Physics (JSAP) and the Japanese Neural Network Society (JNNS). Koji Nakajima received his B.E. M.E. and Dr. Eng. from Tohoku University, Sendai, Japan, in 1972, 1975, and 1978, respectively. Since 1978, he has been working at the Research Institute of Electrical Communication, Tohoku University, except for a ten month period in 1983 when he was a Visiting Assistant Research Engineer at the University of California, Berkeley. He is a professor at the Research Institute of Electrical Communication, Tohoku University, and is currently engaged in the study of VLSI implementation of neural network, and Josephson junction devices for digital applications. Dr. Nakajima is a member of the Japan Society of Applied Physics, the Institute of Electrical Engineers of Japan, and Japanese Neural Network Society. He received the Excellent Paper Award from the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan in 2000. Andrew J. Neel is an adjunct professor of computer science, a member of both IEEE and ACM, and affiliated with the University of Memphis. His research interests include DNA computing, operating systems, and security with a particular interests in network security. His current interests include (1) advancing DNA computers as semantic storage with capacities and speeds that far exceed conventional storage and (2) developing standards for quantifying the security of large networks. The effort is aimed at providing means to boost public confidence in businesses and ultimately creating effective defenses against increasing Internet Crime. Yukinori Ono received the B. Eng. degree in 1986, the M. Sci. degree in 1988, and the Dr. Eng. degree in 1996, all from Waseda University, Tokyo, Japan. Since joining Nippon Telegraph and Telephone Corporation (NTT) in 1988, he has been engaged in the research on physics and technologies of SiO2/ Si interfaces. From November 1996 till November 1997, he was a visiting scientist at Massachusetts Institute of Technology. His current work is the research on physics and technology of Si nanodevices, including single-electron devices for LSI applications. He is now a Senior Research Engineer at NTT Basic Research Laboratories, Atsugi, Kanagawa, Japan. Dr. Ono is a member of the Japan Society of Applied Physics. Takahide Oya received the B.E., M.E., and Dr. Eng. degrees in Electrical Engineering from Hokkaido University, Sapporo, Japan, in 2002, 2004, and 2006, respectively. He is now a lecturer with the Division of Electrical and Computer Engineering, Yokohama National University, Japan. His research interests are in development of novel nanodevices using carbon nanotubes and designing intelligent electronic circuits with single-electron devices. Ranjit Pati is an associate professor in the department of physics, Michigan Technological University, USA. He received his PhD from University of Albany, in the year 1998 in theoretical condensed matter physics. For years his research interest has been transport in the single molecular junction, electrical and magnetic properties of organic materials, for the possible applications in the nanoelectronic and spintronic devices. His current research interest is negative differential resistance occurring in single molecules.

355

About the Contributors

Sridhar Patibandla received his MSc degree from the University of Hyderabad, Hyderabad, India in 2001 and his MS in Physics from Texas Tech University, Lubbock, TX in 2004. He then joined Virginia Commonwealth University where he worked as a research assistant in the Quantum Device Laboratory on spin transport experiments in nanostructures and received his PhD in Electrical and Computer Engineering in 2008. Since then, he has been with Intel Corporation, Hillsboro, Oregon. Ferdinand Peper received his M.S. and Ph.D. degrees in computer science from Delft University of Technology, the Netherlands, in 1985 and 1989 respectively. Currently he is Senior Researcher at the Nano ICT Group at National Institute of Information and Communications Technology (NiCT), Kobe, Japan, and Visiting Professor at the University of Hyogo, Japan. His research interests include nanocomputing, cellular automata, and neural networks. Vinhthuy Phan is an assistant professor of Computer Science and is affiliated with the Bioinformatics Program and the W. Harry Feinstone Center for Genomics Research at the University of Memphis. His research interest is in the field of DNA computing and Bioinformatics. His current focus is on computational methods aimed to enhance the usage and improve the understanding of microarray technologies. These methods and approaches cover both traditional usage of microarray to detect gene expression as well as alternative usage of microarrays such as using them to sequence genomes or build in vitro memory devices Sandipan Pramanik received his BE degree in Electrical Engineering from Bengal Engineering and Science University (India) in 2001. He received MS (2003) and PhD (2006) degrees, both in Electrical and Computer Engineering, from Virginia Commonwealth University, Richmond, Virginia, USA. He spent six months as a post-doctoral associate at Virginia Commonwealth University before joining the Department of Electrical and Computer Engineering at the University of Alberta in August 2007 as an Assistant Professor. His research involves several areas in nanotechnology such as fabrication of novel nanostructures, spin polarized transport and spin coherence in nanoscale devices and modeling spin based devices. He has published numerous articles and book chapters in these areas and has given invited presentations at various international conferences. He is the recipient of Leadership and Services award (2006, 2007 at Virginia Commonwealth University), Phi Kappa Phi graduate student scholarship (2006) and the Disruptive Technology Challenge award (TRLabs, 2009). Weikang Qian received his B.S. degree in automation from Tsinghua University, Beijing, China, in 2006. He is currently a Ph.D. student in the Department of Electrical and Computer Engineering at the University of Minnesota, Twin Cities. During summer 2008, he worked as an intern at IBM T.J. Watson Research Center on verification of sequential logic circuit. His research interests include logic synthesis for VLSI circuits and new synthesis methodology for emerging technologies. Norman Ratcliffe is the Director of the Centre for Analytical, Material and Sensor Sciences at UWE. His interests include the synthesis of smart materials and sensors particularly for applications in disease diagnosis.

356

About the Contributors

Marc Riedel has been an Assistant Professor of Electrical and Computer Engineering at the University of Minnesota since 2006. He is also a member of the Graduate Faculty in Biomedical Informatics and Computational Biology. He has held positions at Marconi Canada, CAE Electronics, Toshiba, and Fujitsu Research Labs. He received his Ph.D. and his M.Sc. in Electrical Engineering at Caltech and his B.Eng. in Electrical Engineering with a Minor in Mathematics at McGill University. His Ph.D. dissertation titled “Cyclic Combinational Circuits” received the Charles H. Wilts Prize for the best doctoral research in Electrical Engineering at Caltech. His paper “The Synthesis of Cyclic Combinational Circuits” received the Best Paper Award at the Design Automation Conference. He is a recipient of the NSF CAREER Award. Thimo Rohlf received his PhD in Theoretical Physics at Kiel University (Germany) in 2004. He worked as a postdoctoral fellow at the Max-Planck-Institute for Mathematics in the Sciences, Leipzig (Germany) from 2004 to 2005 and at the Santa Fe Institute (Santa Fe, NM) from 2005 to 2008. Currently, he holds a position as a researcher in the joint CNRS-MPG progam on Systems Biology at the Epigenomics Project in Evry (Paris, France). His research interests cover a wide range of topics from complexity science, complex networks (dynamics and evolution), cellular automata and distributed computation, pattern formation and morphogenesis, as well as models of the structure, dynamics and evolution of gene regulatory systems. Satyajit Sahu was born in Odisha, India, in 1981. He received his undergraduate and masters degree from Berhampur University, India in the year 2001 and 2003 respectively. He received his PhD on Organic Photodetector from Indian Association for the Cultivation of Science, India in 2008. During his PhD he also worked on Organic LED and memory-switching devices. In 2008 he joined as a Postdoctoral Fellow in National Institute for Materials Science, Japan. His current research interest is to understand the electronic and computational properties of microtubules, single molecule multilevel-switching to realize molecular neurons and understanding the communication of nano-brain with human cell using microtubules. Shigeo Sato received B.E., M.E., and Ph.D. degrees from Tohoku University, Sendai, Japan, in 1989, 1991, and 1994, respectively. He was a postdoctoral fellow of the Japan Society for the Promotion of Science (JSPS) in 1995, and joined the Research Institute of Electrical Communication, Tohoku University in 1996. From 2000 to 2003, he served concurrently as a PRESTO researcher of the Japan Science and Technology Agency (JST). His research interests are in neural networks and quantum computing. He is a member of the Japan Society of Applied Physics and the Institute of Electrical Engineers of Japan. Alexandre Schmid received the M.S. degree in Microengineering and the Ph.D. degree in Electrical Engineering from the Swiss Federal Institute of Technology (EPFL) in 1994 and 2000, respectively. He has been with the EPFL since 1994, working at the Integrated Systems Laboratory as a research and teaching assistant, and at the Electronics Laboratories as a post-doctoral fellow. He joined the Microelectronic Systems Laboratory in 2002 as a Senior Research Associate, where he has been conducting research in the fields of non-conventional signal processing hardware, nanoelectronic reliability, bioelectronic and brain-machine interfaces. Dr. Schmid has published over 70 peer-reviewed journal and conference papers. He has served in the conference committee of The International Conference on

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About the Contributors

Nano-Networks since 2006, as technical program chair in 2008, and general chair in 2009. Dr. Schmid is an Associate Editor of the IEICE ELEX. Dr. Schmid is also teaching at the Microengineering and Electrical Engineering Departments/Sections of EPFL. Dennis Shasha is a Professor of Computer Science at the Courant Institute of New York University where he works with biologists on pattern discovery for microarrays, combinatorial design, and network inference; with physicists, musicians, and financial people on algorithms for time series; and on database applications in untrusted environments. Other areas of interest include database tuning as well as tree and graph matching. Because he likes to type, he has written six books of puzzles, a biography about great computer scientists, and technical books about database tuning, biological pattern recognition and time series. He has co-authored fifty journal papers, sixty conference papers, and eleven patents. For fun, he writes the puzzle column for Scientific American. Milos Stanisavljevic received the M.S. degree in electrical engineering from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia, in 2004, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in 2009. During 2004, he was an Analog Design and Layout Engineer for Elsys Design, Belgrade/Texas Instruments, Nice. In the end of 2004, he joined Microelectronic Systems Laboratory, EPFL, as a Research Assistant. During 2006, he was with International Business Machines Corporation (IBM) Research, Zurich, for six months, where he was involved in the project related to reliability emulation in the state-of-the-art nanoscale CMOS technology. He is currently engaged in the field of reliability and fault-tolerant design of nanometer-scale systems. His current research interests include mixed-signal gate and system level design, reliability evaluation, and optimization. Dr. Stanisavljevic received a Scholarship for Students with Extraordinary Results Awarded by the Serbian Ministry of Education from 1996 to 2004. Christopher Stone is a computer scientist interested in machine learning. He was awarded a PhD in 2005 and since then has worked in the areas of unconventional computing and discrete dynamical systems. Yasuo Takahashi received B.S., M.S., and Ph. D in electronics in 1977, 1979, and 1982, respectively, from Tohoku University, Sendai, Japan. Since joining NTT in 1982, he has been engaged in research on the physics and chemistry of semiconductor surfaces and interfaces, and the quantum physics of Si nanostructures and their electronic device applications, particularly single electronics. Recent his work includes research on many kinds of nanostructures such as ferromagnetic metals and resistance switching materials. Since 2004, he has been a professor at the Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. Dr. Takahashi is a member of the Japan Society of Applied Physics, the Institute of Electrical Engineers of Japan, and IEEE. Christof Teuscher currently holds an assistant professor position in the Department of Electrical and Computer Engineering (ECE) and in the Department of Computer Science at Portland State University (PSU). He also holds an Adjunct Assistant Professor appointment in Computer Science at the University of New Mexico (UNM). Dr. Teuscher obtained his M.Sc. and Ph.D. degree in computer science from the Swiss Federal Institute of Technology in Lausanne (EPFL) in 2000 and 2004 respectively. His main

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About the Contributors

research interests include emerging computing architectures and paradigms, biologically-inspired computing, complex & adaptive systems, and cognitive science. Teuscher has received several prestigious awards and fellowships. Rita Toth obtained PhD in the Nonlinear Chemical Dynamics Group at the University of Debrecen, in Hungary in 2002. Her research involved experimental studies of pattern formation in reaction-diffusion (universal dispersion relation of target patterns in excitable Belousov-Zhabotinsky (BZ) media; a method for initiation of chemical waves in the BZ reaction by using laser) and reaction-diffusion-convection systems (DIFICI and FDO). She developed new approaches to enable desired behaviour from networks of non-linear media using the light sensitive BZ reaction: chemical computing through coevolution and collision based computation in the subexcitable BZ reaction. Spiral formation was also observed in a heterogeneous BZ reaction. Degeneration of multiple spirals resulted in interacting spirals qualitatively resembling of tachycardia and fibrillation. Kenji Yamazaki was born in Japan in 1964. He received the B.S. degree in 1986, the M.S. degree in 1988, and the Ph.D. degree in 2004, all from Tohoku University, Sendai, Japan. In 1988, he joined LSI Laboratories, Nippon Telegraph and Telephone Corporation (NTT), where he studied scanning probe microscopy and its application to semiconductor process evaluation. In 1996, he moved to NTT Basic Research Laboratories, Kanagawa, Japan, where he was engaged in research on electron beam lithography for high-resolution and high-precision nanopatterning. He is now researching three-dimensional nanofabrication using electron beam lithography and its application to nanoelectromechanical systems. Dr. Yamazaki is a member of the Japan Society of Applied Physics and the Physical Society of Japan. Liang Zhang is a PhD student at the University of the West of England, Bristol. His research interests include chemical computing, collision-based computing, cellular automata and theory of computation. He receives a fully funded EPSRC studentship for his PhD research.

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360

Index

Symbols 3D space 6

A actin-myosin structures 196 adder 131, 132, 136, 137, 138, 223, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238 aluminium chloride 184, 185, 186, 187, 188, 189, 190, 191, 192 amoeboid 196, 197, 215, 216, 218, 219, 221 AND gate 101, 106, 107, 172, 231, 232, 233, 234, 235, 281 anti-parallel 260, 264, 271, 272 arithmetical chip 223, 224 artificial catalyst device 176 artificial catalyst networks 177, 178 Artificial Intelligence 39 artificial neural networks (ANNs) 101, 105, 111 associative memory 76, 77 associative processing 75, 76, 77, 78, 85, 97 associative processor 76, 77, 78, 79 asynchronous 29, 30, 31, 34, 35, 36, 37, 38, 39, 41, 42 asynchronous cellular automata 29, 34, 35, 36, 37, 41 asynchronous timing 28, 34 atom-scale components 1

B background-charge effects 75 basin of attraction (BA) 56, 57, 67

Belousov-Zhabotinsky (BZ) 162, 163, 164, 165, 166, 168, 169, 173, 195, 224, 237, 238 benchmark circuits 290, 291 bidirectional communication 245 bio-inspired circuit 149 biological catalysts 176 biological molecules 176, 182 biotechnology 16 bit-comparators (BCs) 77, 78, 79, 81, 82, 83, 84, 85, 86, 91 Boolean decision 1 Boolean logic gates 162, 170, 260, 310 bottlenecks 43 bottom-up 295, 296, 297, 310 Built-In Self-Test (BIST) 31 bundleplexer 281, 282, 283, 289, 293 bundleplexing 282, 289, 293, 294 BZ medium 163, 166 BZ reaction 185 BZ system 162, 163, 164, 165, 169

C capacitive input SET (C-SET) 117, 119, 120, 121 carbon nanotube (CNT) 60 cascode circuit 137 cellular arrays 117 cellular automata (CA) 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 72, 73, 295, 296, 297, 299, 301, 303, 304, 305, 310, 311, 312, 314 cellular automata models 185

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Index

cellular automaton 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 45, 66, 223, 226, 234, 238 cellular neural network (CNN) 33, 40, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 72, 117, 195, 311 central control unit (CCU) 44, 46, 47, 48, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67 chemical fragments 171 chemical organization 240, 241, 242, 243, 245, 246, 247, 248, 250, 255 chemical organization theory 240, 242, 246, 247, 248, 250, 255 chemical program 240, 241, 242, 244, 245, 248, 249, 255 chemical programming 243, 247, 255, 256, 257 chemical reactants 185 chemical systems 162, 163, 172, 173, 185, 186 chemo-attraction 195, 197, 198, 200, 204, 208, 211, 218, 219 chemo-repulsion 195, 197, 200, 201, 203, 218, 219 chemo-repulsive behaviour 200 chemorepulsive markers 202 chemotaxis 204, 216, 217, 218 circular waves 184, 186, 187, 188, 189, 190, 191, 192 CMOS fabrication process 114 collision-based adders 223 collision-based computing 163, 170, 172, 173, 223, 224 collision-based gate 167 complementary single electron transistors (CSETs) 101, 102, 104, 105 complex manner 241 complex waves 188 computing paradigm 176, 177, 304 configuration of SETs (CS) 82 continuously-fed stirred tank reactor (CSTR) 164, 165 control of self-organisation 184 Coulomb blockade 100, 101, 102, 113, 116, 117, 128, 129, 130, 135, 137, 141

Coulomb diamonds 135 Coulomb repulsion effect 84, 85, 88 crystallisation processes 185 cyclic voltammogram (CV) 48, 53, 54, 55 cytoplasm 196

D damage propagation 300, 303 daughter fragment 165, 167, 168, 169, 170 defect tolerance 29, 37, 41, 294 dendrimer 44, 52, 53, 54, 55, 56, 58, 60, 61, 62, 67 design principle 240 detector electrode 178 diffusion field 198, 200, 202, 207, 208, 209, 213, 218 digital logic 75 Digital Reaction-Diffusion System (DRDS) 181, 182 DNA 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 DNA computing 241, 248 DNA fingerprinting 19 DNA hash pooling 15, 19, 24 DNA molecules 241 DNA self-assembly 1 double spiral waves 188, 190, 191 double stranded DNA (dsDNA) 3 dynamic transport networks 197

E edge weight 285, 287, 288 electric fields 259, 265, 273, 274 electroencephalography (ECG) 67 electronic design automation (EDA) 114 electron transition rate 142 Elliott-Yafet mechanism 262, 273, 274 Elliott-Yafet mode 259, 262, 263, 273, 275 Environmental Gene Tags (EGTs) 17 enzyme reactions 177, 178 enzyme transistor 176, 177, 178, 179, 180, 182 erosion operator 202, 204, 206 error-correcting codes 1, 3, 5 error-preventing codes 1 error ratio 290, 291

361

Index

evolutionary algorithm (EA) 162, 163, 164, 171, 172, 304 evolutionary distance 17 excess electrons 108, 109 excitability levels 191 excitability limit 163, 165 excitable dynamics 180 excitable media 195, 223, 224, 237 excitable medium 165, 166, 171, 173, 223, 224, 236 excitatory PSP (EPSP) 92 exclusive-NOR (XNOR) 78, 81, 82 exclusive-OR (XOR) 78 execution units (EU) 44

F fabrication methods 28, 118 fault-free 121, 126 fault patterns 122 fault-tolerance 28, 37, 120, 124, 128, 129 fault tolerance capability 126 fault-tolerant 261, 269, 270, 271 fault-tolerant DNA computing 1 fault-tolerant realizations 126 fault-tolerant techniques 114, 115, 120, 121, 127, 128 FET-like junctions 292, 293 FET-like regions 293 field-effect transistors (FETs) 131 field-programmable gate arrays (FGPAs) 280, 298 FitzHugh-Nagumo (FHN) 180 four-layer architecture 119, 120, 128 full adder (FA) 126, 127, 131, 132, 136, 137, 138, 226, 227, 228, 229, 230, 232, 233, 234, 235

G Genetic Algorithms (GA) 305, 306, 307, 308 giant enzyme complexes 64 Gibbs energy 1, 3, 4, 5, 8, 10, 11, 12 Gibbs Energy 1, 4, 14 graph topology 245, 252, 253

362

H half adder 131, 136, 137, 225, 226, 227, 228, 229, 230, 232 Hamiltonian path 1, 6 Hamiltonian Path Problem (HPP) 1 hash function 19 hashing 15, 19 hash pooling 15, 19, 20, 22, 24 hybridization 1, 2, 3, 4, 5, 6, 14 hybrid systems 296

I information processing systems 75, 76 information transfer 162, 173 information transport 194 inhibitory PSP (IPSP) 92 inorganic systems 184, 185, 186, 187, 193 input frequency 146 input-output correlation 140, 143, 144, 145, 146 in silico 4, 5, 6, 7, 13, 15, 16, 17, 19, 21, 22, 23 integrate-and-fire neuron (IFN) 151, 152, 153 inter-spike intervals (ISIs) 154, 155, 156 in vitro 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 20

K Kaleidoscope of Life 29 kinetic laws 240, 243, 250

L large-scale integrated (LSI) circuitry 131, 132, 140, 150 learning probability 306, 307, 308, 309 Left-Hand-Side (LHS) 29, 30, 31, 35 logical gates 223, 224, 237 logic gates 162, 164, 170, 172, 173, 174, 175, 240, 260 logic synthesis 279 lookup-tables (LUTs) 298

M magnetic resonance imaging (MRI) 67 MATLAB 120, 121, 122, 129

Index

maximal independent set (MIS) 241, 243, 244, 245, 246, 247, 248, 250, 251, 252, 253, 254, 255 MEtaGenome ANalyzer (MEGAN) 18, 26 metagenomics 15, 16, 17, 18, 24, 26, 27 metal-oxide semiconductor field-effect transistors (MOSFETs) 131, 132, 137, 138, 139 metal-oxide semiconductor (MOS) 131 Microbiology 16, 25, 26, 27 microscopic levels 241 microtubule assisted protein (MAP2) 67 MIS chemistry 244, 245, 246, 247, 250, 251, 252, 253, 254, 255 modulus 19 molecular cellular neural network (m-CNN) 44 molecular computing 176, 177, 179, 180, 255 molecular electronics technology 176, 182 molecular information processing 177 molecular interaction 192 molecular machines 43, 64, 65, 71 molecular neurons 47, 48, 50, 52, 53, 59, 63, 68 molecular-scale machines 240 molecular species 240, 242, 244, 246, 248, 253, 254, 255 Monte-Carlo based computer simulations 149, 154, 157 Monte Carlo simulation 100, 103 Moore neighborhood 29, 35 multi-agent systems 196 multienzyme complexes 64 multi-input device 133, 137 multinucleate organism 196 multi-output device 137 multiplicative binary 279, 280, 283 multiplicative binary moment diagrams (*BMDs) 283, 284, 285, 286, 287, 288, 289, 290 multipliers 223, 232, 236, 237, 238 Myxomycota phylum 196

N NAND Boolean gate 122, 128 NAND gates 172, 173 nano-brain 43, 44, 45, 46, 47, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

nanocomputers 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 44 nano-devices 115, 120 nanodot 83, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 131, 132, 133, 134, 135, 136, 137, 138, 152, 153 nanodot array 83, 84, 94, 95, 96 nanodot array device 131, 132, 133, 134, 135 nano-electronic devices 150, 151, 157 nano-electronic systems 114, 118 nanoelectronic systems 115 nano-factory 44, 60, 64, 65, 71 nano-meter 150, 151 nanometer 115, 116, 118, 129, 131, 159, 272 nanoparticle applications 241 nanoparticles 241, 258 nanoscale 241, 255, 295, 296, 309, 312 nano-scale characteristics 150 nanoscale computation 279 nanoscale computing 295 nanoscale electronics 295 nanoscale molecules 240 nano-surgeon 44, 64, 65 nanotechnologies 115 nanotechnology 28, 240, 241, 257 nanowire 259, 263, 266, 267, 273, 274, 275, 277, 279, 280, 292, 293, 294, 296, 310 nanowire arrays 279, 280, 292 natural curvature 202 natural systems 185 negative *BMD 285, 286 negative differential resistance (NDR) 48, 52 network structures 254 Neumann computer 47, 297 Neumann neighborhood 29, 31, 35, 299, 310 neural network 100, 101, 105, 106, 107, 108, 112, 113 neuro-cognition 67 neuromorphic 151, 157, 159 neuromorphic circuits 151 neuromorphic CMOS circuits 151 neuronal circuits 149 neuron circuit 107, 111 noise-shaping 149, 151, 154, 157 noncrosshybridizing (nxh) 3, 7, 9, 10 non-silicon computers 224

363

Index

O ordinary differential equations (ODE) 242, 243, 246, 248, 250, 251, 252, 253 organics 259, 261, 262, 263, 265, 267, 272, 274, 275 organic semiconductor tris 259, 261 organic spintronics 261 Organizational analysis 245, 254 organization-oriented chemical programming 243, 247, 255, 257 oscillator (OSC) 228, 229

P parallel components 279, 280 particle approximations 194, 199, 218 pattern-dependent oxidation (PADOX) 134, 135, 137 pattern formation 184, 185, 186, 193 PCR Selection (PCRS) 3, 4, 7, 8, 9, 10, 12 phonon bottleneck effect 272, 273 Physarum 194, 196, 197, 207, 209, 212, 215, 216, 217, 218, 219, 220, 221, 222 Physarum plasmodium 207, 212, 215, 216, 219, 221 Physarum polycephalum 194, 196, 221, 222 plasmodium 194, 196, 207, 210, 212, 215, 216, 217, 219, 220, 221, 222 platinum electrode 178 Platinum (Pt) microelectrodes 180 polarization 259, 260, 262, 264, 265, 271, 272 Polymerase Chain Reaction (PCR) 3, 4, 6, 7, 8, 9, 10, 12, 13, 17, 26 positive *BMD 285, 286 post-evolution period 47 post-synaptic potential (PSP) 92, 93, 95, 96, 97 probability density functions (PDFs) 120, 122, 123, 124, 125, 126 protoplasm 196 pulse-density 149, 151, 153, 154 pulse-rate coding 91 pulse-width modulation (PWM) 91, 99 pyrosequencing 18

Q quantum bit (qubit) 260, 261, 269, 271, 272, 277 364

quantum computing 261, 269, 270, 271, 272, 277 quantum dot 140, 141, 144, 146 Quantum-dot Cellular Automaton (QCA) 32 quasi-physical properties 194, 219

R radio frequency superconducting quantum interference device (rf-SQUID) 108 random access memory (RAM) 48, 54, 61 Random Boolean Networks (RBNs) 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 308, 309, 310 random dynamical networks 295, 297, 299, 310, 313 random flies 38, 41 random number generator (RNG) 101, 104, 105, 107, 111, 113 Random Threshold Networks (RTNs) 295, 297, 298, 300, 301, 303, 304, 305, 310 reaction-diffusion computers 180, 181 reaction-diffusion computing 194, 195, 196, 197, 199, 200, 203, 206, 218, 219 reaction-diffusion dynamics 176, 177, 179, 180, 181, 182 reaction- diffusion medium 224 reaction-diffusion molecular dynamics 177, 180 reaction-diffusion phenomena 180 reaction-diffusion (RD) 194, 195, 196, 197, 199, 200, 202, 203, 206, 207, 208, 218, 219, 220, 223, 224 reaction-diffusion systems 163 relative error 126 relaxation time 259, 260, 261, 263, 265, 273, 275 resonant tunneling diode (RTD) 48, 310, 313 Restriction fragment length polymorphisms (RFLPs) 19 ribosomal RNA (rRNA) 16, 17, 19, 23, 26, 27 Right-Hand-Side (RHS) 29, 31 robustness 296, 297, 301, 305, 310, 311 Rose Bengal (RB) 51, 58 Rotation Angle (RA) 198, 209, 211, 212

Index

S scanning electron microscope (SEM) 134 scanning tunneling microscope (STM) 49, 50, 51, 53, 54, 62, 63 Schottky barrier 264 self-assembled nanowire arrays 279 self-assembly 1, 2, 12, 28, 310 self balancing 211 self-organisation 194 self-organised 196, 211 Self-Timed Cellular Automata (STCA) 36, 37 self vertex 255 semiconductor devices 150 Sensor Angle (SA) 198, 209, 211, 212 SET devices 81, 117, 122 shuffled AND 281, 283, 288, 289, 290, 293 signal-to-noise ratio (SNR) 149, 156, 157 silicon-on-insulator (SOI) 131, 132, 134, 137 SIMON 120, 121, 129, 130 Simulated Annealing (SA) 305 single circular waves 184, 191, 192 single electron 131, 132, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 157, 158, 159, 259, 260 single electron analog circuit 106 single electron box (SEB) 108 single-electron circuit 75, 78, 92 single-electron devices (SEDs) 75, 76, 78, 83, 97, 100, 101, 102, 111, 132, 133, 135, 137, 140, 149, 150, 151, 154, 157 single-electron operation 76 single-electron oscillator 152, 153 single-electron transistor (SET) 102, 103, 104, 105, 106, 107, 108, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 126, 128, 129, 130, 132, 134, 135, 137, 114 single electron turnstile 101, 108, 109, 111 Single Spin Logic (SSL) 260 single spiral waves 187, 190 skeletonisation 200, 201, 202, 204, 218 small-world (SW) 297, 303, 304, 305, 310, 311, 313, 314 smart glue 1, 2 sparse percolation (SP) 300 spherical assembly 47, 52, 55, 57, 59, 62, 65

spike densities 151 Spike Response Model (SRM) 92, 93, 94 spiking neuron 75, 76, 91, 92, 93, 97, 152, 160 spin-dependent 261 spin diffusion 263, 264, 265, 266, 267, 268, 269, 273, 274, 275, 276 spin-orbit 259, 262, 263, 273, 274, 275, 277 spin-phonon 259, 263, 272, 273, 274, 275 spin relaxation 259, 260, 261, 262, 263, 264, 265, 267, 269, 273, 274, 275, 276 spin-spin 259, 263, 273, 274, 275 spin transport 262, 274, 275 spintronics 259, 261, 263 spin valve 263, 264, 265, 266, 267, 268, 269, 270, 274, 276, 277 state-of-the-art biosensor technology 178 stimulus-response dynamic 197 stochastic associative processing 76, 77, 78, 85 stochastic circuits 290, 294 stochastic computation 279, 280 stochastic neural network 100, 107, 108 stochastic operation 75, 76, 82, 97, 100 stochastic property 101, 111 Stochastic resonance behavior 140 stochastic resonance (SR) 88, 100, 101, 108, 109, 110, 111, 112, 140, 141, 142, 144, 145, 146, 147, 148, 151, 157, 158 summing circuit 141 summing network 140, 141, 144, 146 synchronous 30, 33, 34, 35, 36, 37, 38

T target waves 184, 190, 192 temporal evolution function 47 terminal vertex 284, 286, 287, 288 thermal evaporation 268 thin film 266, 267, 268, 270, 273 three-dimensional (3-D) 84, 86 top-down 295, 296, 304 transport networks 197, 207, 208, 209 traveling salesman problem (TSP) 1 travelling circular waves 184 travelling wavefronts 195 travelling waves 186, 192, 193 Trimethyl aluminum (TMA) 268 triple modular redundancy 118, 124

365

Index

tunneling 149, 152, 153, 154, 157, 158 tunneling junction 116, 120, 152, 153 tunneling rate 142 two-dimensional (2-D) 179, 180, 181

U uniform probability 301, 310 UnitNegBMD 288, 289, 290 UnitPosBMD 288, 289, 290 unit-weight *BMDs 288, 290

366

V virtual test tube (VTT) 5, 6 Voronoi diagram 187, 188, 191, 192, 195, 200, 202, 203, 204, 218, 222

W wave-fragments 224, 237 wavefront 190 white noise 143, 145 winner-take-all (WTA) 77, 78, 82, 83 word-comparators (WCs) 77, 78, 79, 80, 81, 82, 83, 84, 85