Neuroelectrodynamics: Understanding the Brain Language - Volume 74 Biomedical and Health Research

NEUROELECTRODYNAMICS

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Neuroelectrodynamics Understanding the Brain Language

Dorian Aur Barrow Neurological Institute, Phoenix, Arizona, USA

and

Mandar S. Jog University of Western Ontario, London, Ontario, Canada

Amsterdam • Berlin • Tokyo • Washington, DC

© 2010 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-60750-091-9 (print) ISBN 978-1-60750-473-3 (online) Library of Congress Control Number: 2009943663 doi: 10.3233/978-1-60750-473-3-i Publisher IOS Press BV Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the USA and Canada IOS Press, Inc. 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: [email protected]

LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Neuroelectrodynamics D. Aur and M.S. Jog IOS Press, 2010 © 2010 The authors and IOS Press. All rights reserved.

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Preface The brain is an incredibly powerful computing machine shaped by an evolutionary process that has developed over millions of years. The essence of brain function consists in how information is processed, transferred and stored. Understanding how our brain computes can explain why our abilities of seeing and moving in space are substantially more powerful as compared to any artificially built system. Without a clear model of how computation is performed in the brain, we are not able to understand the relationship between perception learning, memory and molecular changes at DNA, gene or protein level. Many models in computational neuroscience claim the existence of parallel, distributed processing system in the brain. Similarly many advances in the structural, microscopic and genetic fields have shed light on the molecular machinery within the brain. Additionally, we know that the brain processes and communicates information through complex electrochemical processes. Any theoretical and computational model must be able to bring all of these aspects together and build a generally true model of brain computation. However, this has not happened. From a neurophysiological point of view, current doctrine remains focused within a spike timing paradigm. This paradigm has limited capability for advancing the understanding of how the brain works. Additional hypotheses such spike stereotypy (a visual assumption), temporal coding (which does not explain information processing) and cable function (which cannot explain memory storage and simplifies a complex, dynamic axonal system into a transmission cable) are required, due to which the reach of the neural computational model is limited. Assumptions made in any theoretical construct that are unproven make the theory limited and thus inherently weak. Taken from this perspective any subsequent result and further hypotheses are invalid. These reasons were the beginning of our dissatisfaction with the state of the current trend in neuroscience and neural computation and the main reason for writing this book. Our view is that information processing in the brain transcends the usual notion of action potential occurrence with millisecond precision. The experimental evidence and the models presented in this book extensively demonstrate, for the first time to our knowledge, that spike timing theory considerably restricts the richness of how the brain carries out computations. The ability to record voltage extracellulary directly implies the existence of significant changes in local electric fields. Indeed action potentials themselves are shaped by the electric field and the presence, distribution and dynamics of charges that locally change molecular dynamics and electrical excitability of voltage-gated ion channels in active neurons. We therefore set out to put forward a new model from a ground-up set of universal principles on the basis established by the laws of physics. These laws such as those of classical mechanics, thermodynamics and quantum physics can be used as guiding principles based upon which a general theoretical construct of brain computational model can be built. Such model attempts to incorporate the neurobiology of the cells, the molecular machinery itself along with the electrical activity in neurons to explain experimental results and in addition predict organization of the system. In this book we show that spike timing models are special cases of a broader theoretical construct – NeuroElectroDynamics (NED) neuro- from neurons, electrofrom electric field and -dynamics meaning movement. The main idea of NED is that under the influence of electric fields, charges that interact perform computations and

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are capable to read, write and store information in their spatial distribution at molecular level within active neurons. In this domain, the universal physical laws from classical mechanics to thermodynamics and quantum mechanics are applied in order to give mathematical and theoretical support to charges as the operators of the brain computing machine. Experimental data is used where necessary to corroborate, support and extend these observations. This novel and substantial approach of the NeuroElectroDynamic model extends the previous approach to a different level where the physical aspect of information processing is fundamental and can be directly related to macromolecular changes, thermodynamic properties and intrinsic quantum phenomena. This new model represents the crucial paradigm shift which opens the door to a new subfield of neuroscience that bridges neural computational models and research in molecular computation. Any model requires certain assumptions and the proposed theory is no different. However, the assumptions that are made in this theory are those based upon already validated laws and physical principles that should and indeed must be transferable between the fields of physics and neuroscience. No simplifications to these laws are made in the NED construct. The very novelty and strength of the model is in such an application of basic physical principles which are used to explain how the brain computes. This makes the theoretical framework used to generate NED “general” and by default very amenable to expansion, with clear possibility for experimental verification. The book is organized into five large chapters. The first chapter outlines and provides an overview of the deficiencies of the current approaches to neural computation. The current development of neuronal computation based upon the assumption of spike stereotypy, known as spike timing theory is fraught with several gaps and controversies. Many of these issues are then discussed in detail in individual chapters along with a parallel development of the new theoretical architecture alluded to in this first chapter. The chapter is also a gentle introduction to laws and principles required to build a new model that addresses the issues raised in the chapter. This description highlights a systemic approach where universal laws should shape the nature of computations. Since the link between electrophysiological properties of neurons and theory of neural computation, is critical, Chapter 2 describes such experiments and their results. The analyses performed in the in-vivo behaving animals are presented with a very fresh approach to reflect the essential characteristics of neurons and neuronal ensembles and disqualify spike timing as a unique modality which can fully describe brain computations. Spike timing analyzed in local or larger areas does not seem to be rich enough and does not tell us the whole story. Fundamental processing is hidden, observables such the timing of spikes (firing rate, ISI), oscillations partially include information regarding these hidden processes. Importantly, new methods and techniques, designed to characterize neuronal activity well beyond the spike timing approach are revealed in this chapter. This new description provides the required link between computation and semantics analyzed during behavioral experiments. Chapter 3 highlights some important concepts which relate the theory of information with computation and electrophysiological properties of neurons. The presentation differentiates between popular view of ciphering (i.e. performing encryption and decryption) and fundamental mathematical principles of coding and decoding information. Previous theories of neural coding are discussed in this context

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and new concepts are presented. This novel view relates information processing and computation with cognitive models. Information flow within the nervous system is driven by electrical phenomena with full support from the biological and neurochemical infrastructure of the neurons. Chapter 4 is the heart of the presentation of the new model of computation and forms the raison d’être for the book. The chapter builds a systemic approach of brain computation in terms of interactions, dynamics of charges (NeuroElectroDynamics) from the nanoscale level to macroscale phenomena using a theoretical design, the computational cube, which can be regarded as a compartmental model. This novel model proves that complex, grained charge level of information processing is the subtle phenomenology behind the commonly observed spiking activity and temporal pattern occurrence. Such processes have implicit quantum mechanisms that can be demonstrated as intrinsic results of experimental data analysis. A unified mathematical view of computation expressed in terms of charge dynamics as successive coding and decoding phases as developed in the NED theory is able to characterize the activity within single cells or neuronal ensembles. It is clear that computation depends on intrinsic physical laws (the principle of least action, second law of thermodynamics) and anatomical substrate of neurons and is reflected in their electrophysiological properties. The concepts of computation based on the dynamics and interaction of charges, is no different and is the grounding upon which NED model has been developed. Additionally, the new approach is able to explain “coding and decoding” mechanisms, information retrieval, perception, memory formation, and other cognitive processes. We show as an example that temporal organization/temporal patterns can be obtained as a result of ‘learning’ starting with a charge movement model. The description of these state-of-the-art techniques of information processing (coding and decoding) are gradually developed in this book, using a simple mathematical formalism. This approach will allow readers from different backgrounds to implement and experiment new models and perform computer simulations of brain computation. There is no real equivalent for such development in the models that consider temporal aspects of spike occurrence (firing rate, ISI). The theoretical approach of NED builds a mathematically general model of brain computation that incorporates the current spike timing theory as a particular case. This ability of generalization is sorely missing from the current models of brain function. As the NED theory is accepted and developed in the future, it is likely that it can itself be incorporated within another even more generalized approach. Chapter 5 establishes and expounds the new NED model and several examples of its application are shown in order to reveal and characterize brain computations. These applications are only partially developed here in this book and will form the basis of our next work. Several processes including memory and cognition can be explained if spike timing models are replaced with the new systemic approach of dynamics and interaction of charges. The emergent topology of charges as a result of their intrinsic dynamics during action potentials and synaptic spikes are at the origin of distancebased semantics, a phenomenon previously experimentally evidenced in several brain recordings. NeuroElectroDynamics is a completely different model where the intrinsic computational processes are described by the dynamics and interaction of charges. Such a description very naturally includes and incorporates the internal neuronal molecular processes, the effect of neurotransmitters and genetic regulatory mechanisms. At fundamental molecular level, information carried by electric field and

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charges can be selectively transferred and can perform intricate operations on regulatory mechanisms at the gene level which are required to build appropriate proteins. Importantly, NeuroElectroDynamics shows that computational properties of neurons are built and change at the quantum level. The subtle changes that occur at a molecular level determined by the interactions of charges and electric fields can be related with qualitative information processing. Most important to the concept of human behavior, it is possible to relate these interactions to the emergent property of semantics of behavior, which has been elusive to date except in the realms of psychology and psychophysics. As presented this multimodal aspect of information transfer and processing in the brain incorporates and indeed requires interactions with intracellular neuronal and nonneuronal complex molecular regulatory mechanisms from gene selection/expression, DNA computations to membrane properties that spatially shape the dynamics and interaction of charges. Spike timing including spike patterns can be regarded as an epiphenomenon or a result of these hidden processes. The often observed heterogeneity in activation as response to a given stimulus reflects strongly processes that occur at molecular level within selected neurons. Therefore, NeuroElectroDynamics provides the structure upon which one can bridge molecular mechanisms and the spike timing framework. This systemic view unifies neural computation and may provide fresh guidance to research in artificial intelligence using newly developed models of computation. Instead of revolving around spike timing code the new model highlights several different levels where information is processed, emphasizes and indeed necessitates within the theory the role of fine sensitive structure of cells, the role of ion channels, proteins, genes and electric charges in information processing. Additionally, this chapter alludes to the role of afferent brain rhythms including sleep in learning and memory storage. Also, specific neuropathological situations where the brain is diseased such as in schizophrenia or Parkinson are briefly developed in the context of NED. The analysis of brain processes in terms of dynamic changes and resulting electric perturbations is fundamentally necessary to allow us to consider a true brain-computer interface or to perform what we feel is true neuromodulation. The future of these concepts and how we should achieve them are also discussed here. We explain why currently built artificial systems based on ultra-simplified neurons using spike timing paradigms are inconsistent with brain computations and their future is limited by this reductionist approach. Built on principles supported by established pillars of physical laws in describing brain computation NeuroElectroDynamics is a step forward, required to develop the next generation of artificial intelligent systems and understand the mind in computational terms. We anticipate that the development in nanotechnology will allow this progress to be made soon. The NED model highlights the importance of charge dynamics, electric interactions and electric field in information communication at a nanoscale level of protein synthesis. Traditional molecular biological approaches are unable to link the electrophysiological components of neural function to protein and genes and indeed most importantly with information models. Although NED does not provide a complete solution, we feel that it at least gives a path along which further research could be carried out in a way that emphasizes the full relationship between genes, proteins and complex regulatory mechanisms of information flow.

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We started this journey more than five years ago by demonstrating that spikes are not stereotyped events by computing spike directivity and providing electrical images of recorded spikes. Since the fundamentals of spike timing theory of computation in the brain are based on the assumptions of ‘spike stereotypy’, representation of axons as cables and others, the resulting temporal coding theory and its conclusions are restrictive, limited and possibly inaccurate. Indeed, they are certainly incomplete in terms of explaining the complexity of function and behavior that we see in the real world. After we began our effort of writing this book in February 2008, terms that we have defined such as – spike timing code as “a myth” – have already been borrowed from our proposal to publishers, by other experts in the field. We are delighted to see that some distinguished members of the academic community are beginning to incorporate these ideas into their own work. We are fully aware of how slow progress is in science and how hard it is to induce a paradigm change (see Planck, 1949; Kuhn, 1962). We hope that this book will appeal greatly to many students. The models hinted at in chapter 5 can serve as research pathways for both experimental and theoretical development. Those that study physics and mathematics will know that ideas put forward by giants in their fields such as those of Schrödinger and Sir Roger Penrose can indeed be applied in neuroscience. New tools and mathematical concepts will need to be developed and applied. The students that are interested in systems neuroscience can revel in the fact that complex systems dynamics can be made to work and deeply explain experimental data. To the neurophysiologist there are now clear testable conjectures that are predicted and will result in a new line of experiments well beyond the current spike timing measurements. To the computer science and AI group, a true and detailed framework for brain function will allow much better algorithmic development of computations that can be translated into applications. From philosophical, psychological or psychoanalytical point of view, mind and brain are inseparable and this model of computation will appear as a naturally significant change that resonates strongly with previous views and new concepts of the science of the mind. Finally, to biologists that are struggling to find connections between the myriads of proteins, genes, their properties and behavior, NeuroElectroDynamics can be way of making sense of their findings in a comprehensive way. The list of scientists that pioneered this work of in-depth investigation of intricate phenomena in brain is long. We can mention here all the Nobel laureates cited for their contribution in this field along other names such as Wilder Penfield, Mircea Steriade, Alwyn Scott, Roger Penrose, Stuart Hameroff and others that did not receive this honorary recognition but made significant observations in analyzing how the brain works. By contrast there is an even larger list of scientists that have simplified their observations to the temporal domain by building spike timing models, emphasizing their role and creating the current fashion in computational neuroscience. We hope that for a new generation of students, researchers and scientists an important legacy stands in novel concepts initiated and developed in this book. We anticipate that their further development would establish a new field of research to emerge that will determine a paradigm shift required to significantly change current insights in computational neuroscience and connected fields which finally will provide a better understanding of the human brain and mind.

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Acknowledgements We have succeeded in publishing this book in this environment where every model that does not display the fashion of spike timing is highly marginalized in the mainstream publications. We have to thank those people that had the vision and made this publication possible especially to reviewers and publisher Mark Eligh from IOS Press. Our foremost acknowledgement is to Shrikrishna Jog for carefully correcting the manuscript for valuable insights and useful comments. Also, the authors would like to express many thanks to Dr. Radu Balan, Dr. Neil Lambert and Liviu Aur for several suggestions to improve this book. Finally, we would like to thank our parents and families especially Monica, Teodora and our daughters, Alexandra, Rachna and Mihika for their patience. The Authors

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A Note on the Illustrations The online edition of this book has a free supplement with all the available colour illustrations of this publication. Please visit www.booksonline.iospress.nl and navigate to Book Series Biomedical and Health Research (Volume 74) or use the doi:10.3233/978-1-60750-473-3-i.

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CONTENTS Preface ........................................................................................................................ ... v A Note on the Illustrations .......................................................................................... xi CHAPTER 1. Understanding the Brain Language.................................................... 1 1.1.1. The importance of asking the right questions........................................... 1 1.1.2. Does neuroscience ask the right questions?.............................................. 1 1.1.3. Cognition, Computation and Dynamical Systems.................................... 2 1.2. Myths, Controversies and Current Challenges ............................................. 4 1.2.1. The inadequacies of temporal coding....................................................... 4 1.2.2. The cable myth ......................................................................................... 5 1.2.3. The dilemma of synapses ......................................................................... 6 1.2.4. The Hebbian approach of connectivity..................................................... 6 1.2.5. The myth of neurotransmitters ................................................................. 7 1.2.6. The myth of molecular biology ................................................................ 7 1.3. Spike Timing – Current Challenges ............................................................... 8 1.3.1. Is the neuron an integrator of incoming inputs – Temporal integration or coincidence detection? ....................................................... 8 1.3.2. Firing rate or ISI? Which technique is better to quantify neuronal activity? .................................................................................................... 8 1.3.3. Do irregular spikes convey more or less information than the regular ones?......................................................................................................... 8 1.3.4. Are synchrony and oscillations important? .............................................. 9 1.3.5. Across-fiber pattern, grandmother cell, or labeled-line.......................... 10 1.3.6. Optimal efficiency or a high metabolic cost of information transmission? .......................................................................................... 11 1.4. Issues of Brain Computation......................................................................... 11 1.4.1. Brain Computations – Classical or Quantum?........................................ 11 1.4.2. Brain computation and AI ...................................................................... 12 1.4.3. Is computation in the brain closed to a reversible process?.................... 12 1.4.4. Is ‘noise’ useful for brain computations? ............................................... 12 1.4.5. What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? ....................................... 12 1.5. How can Information be Represented in the Brain..................................... 13 1.5.1. Is postsynaptic activity required for dendritic development? ................. 13 1.5.2. How and where are memories stored?.................................................... 13 1.5.3. The key of the topographic map ............................................................. 13 1.6. Interdisciplinary Formalism ......................................................................... 14 1.6.1. Sets and functions................................................................................... 14 1.6.2. Dynamical Systems ................................................................................ 15 1.6.3. The laws of thermodynamics.................................................................. 17 1.6.4. Conclusion.............................................................................................. 18 CHAPTER 2. Imaging Spikes – Challenges for Brain Computations ................... 25 2.1. Methods, Models and Techniques................................................................. 25 2.1.1. Tetrodes.................................................................................................. 27 2.1.2. The Necessity for a New Model ............................................................. 28 2.1.3. The Model of Charges in Movement...................................................... 28

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2.1.4. The Triangulation technique................................................................... 30 2.1.5. Charges and Electric potential................................................................ 30 2.1.6. The Projective Field: Spatial Directivity of Charges.............................. 31 2.1.7. What is “Spike Directivity”? .................................................................. 33 2.1.8. Triangulation and Independent Component Analysis Together ............. 34 2.2. From Spikes to Behavior ............................................................................... 38 2.2.1. Behavioral procedures............................................................................ 40 2.2.2. Spike sorting and unit classification....................................................... 40 2.2.3. Computing and Analyzing Spike Directivity ......................................... 41 2.2.4. Upcoming Choices, Decision and Spike Directivity .............................. 43 2.2.5. Learning: One Spike Seems Enough ...................................................... 48 CHAPTER 3. Information and Computation .......................................................... 55 3.1. What is Computation? ................................................................................... 55 3.1.1. Coding and decoding.............................................................................. 56 3.1.2. Are Turing Machines Universal Models of Computation? .................... 57 3.1.3. Von Neumann Architectures .................................................................. 59 3.1.4. Reversible Computation ......................................................................... 60 3.1.5. Models of Brain Computation ................................................................ 63 3.1.6. Building the Internal Model ................................................................... 66 3.2. Neurons as information engines.................................................................... 67 3.2.1. Maximum Entropy Principle .................................................................. 67 3.2.2. Mutual Information ................................................................................ 69 3.2.3. Information transfer................................................................................ 72 3.2.4. Efficiency of Information Transfer......................................................... 73 3.2.5. The Nature of Things – Functional Density ........................................... 76 3.2.6. Charge Distribution and Probability Density ......................................... 77 3.2.7. The Laws of Physics............................................................................... 79 3.2.8. Minimum Description Length ................................................................ 83 3.2.9. Is Hypercomputation a Myth? ................................................................ 85 CHAPTER 4. Models of Brain Computation ........................................................... 95 4.1. Dynamics Based Computation – The Power of Charges ............................ 95 4.1.1. Space and hidden dimensions................................................................. 96 4.1.2. Minimum Path Description – The Principle of Least Action ................. 97 4.1.3. Natural Computation – Abstract Physical Machines.............................. 99 4.1.4. Back to Building Brains ....................................................................... 104 4.1.5. Charge Movement Model – A Computational Approach..................... 109 4.1.6. Properties of Computation with Charges.............................................. 131 4.2. Quantum Model – Combining Many Worlds ............................................ 137 4.2.1. Many Worlds in a Single Spike............................................................ 138 4.2.2. Spike Models – A Quantum Formalism ............................................... 139 4.2.3. Quantum models................................................................................... 142 4.2.4. Brain the Physical Computer................................................................ 145 4.3. The Thermodynamic Model of Computation ............................................ 147 4.3.1. The Maxwell Daemon .......................................................................... 147 4.3.2. Computation in a Thermodynamic Engine........................................... 148 4.3.3. Entropy in Neuronal Spike ................................................................... 151 4.3.4. Thermodynamic Entropy and Information ........................................... 153 4.3.5. Synaptic spikes ..................................................................................... 155 4.3.6. Spikes as Thermodynamic Engines...................................................... 156

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CHAPTER 5. From Brain Language to Artificial Intelligence............................. 167 5.1. How are memories stored? .......................................................................... 167 5.1.1. Are proteins the key?............................................................................ 169 5.1.2. What sleep is for? ................................................................................. 183 5.2. Spike Timing – An Incomplete Description ............................................... 185 5.2.1. Models of Auditory Processing ............................................................ 187 5.2.2. Models of Schizophrenia...................................................................... 189 5.2.3. Models of Learning .............................................................................. 192 5.2.4. Models of Parkinson’s disease and Dyskinesia .................................... 195 5.2.5. Brain Computer Interfaces ................................................................... 196 5.2.6. Moving from Spike Timing to NeuroElectroDynamics ....................... 199 5.2.7. Computation, Cognition and Artificial intelligence ............................. 208 5.3. Instead of Discussion.................................................................................... 211

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Neuroelectrodynamics D. Aur and M.S. Jog IOS Press, 2010 © 2010 The authors and IOS Press. All rights reserved.

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CHAPTER 1. Understanding the Brain Language 1.1.1. The importance of asking the right questions It is believed that progress in science is determined by asking the right questions at the right moment in time. At the beginning of the twentieth century Hilbert asked if mathematics could be completely built based on an axiomatic formalism. The answer came three decades later. Gödel's incompleteness theorems (Gödel, 1931) showed that Hilbert's idea for a consistent set of axioms for mathematics was not possible. Maxwell addressed a similar and important question in physics regarding the second law of thermodynamics. Known today as Maxwell’s Daemon, the proposed thought experiment regarding controversy between thermodynamic and information entropy and was tackled later by many scientists. A remarkable advance made by Leo Szilard was followed by Brillouin and Landauer who showed that the Daemon can be defined in terms of information (Brillouin, 1956, Landauer, 1961). In the artificial intelligence field, the right question was asked by Searle (Searle, 1980). Searle questioned the current developments in AI where the main issue is with regard to acquiring semantics using an algorithmic approach. The thought experiment imagined by Searle assumes that a programmed computer is able to behave similarly to a human who understands Chinese and passes the Turing test. The computer uses Chinese characters as the input and syntactic rules to generate other Chinese characters. Searle’s thought experiment asked an important question; Is a computer which runs the above algorithm able to “understand” Chinese language? In other words, is the manipulation of symbols using syntactic rules performed by a computer capable of providing the required semantics in a manner similar to a person that speaks Chinese? With an exemplar philosophical clarity Searle showed that computers are only able to follow the syntactic rules and they do not understand the meaning of the Chinese language (Searle, 1980). Since the Chinese speaker has acquired in advance the semantics of words or symbols, he already knows what the symbol means. However, this process does not occur for computers, which manipulate symbols and follow syntactic rules. Many critics do not understand the gist of Searle’s philosophical argument. The problem of acquiring meaning within a system that manipulates symbols is termed symbol grounding. It is commonly believed that grounding can be achieved during learning or evolution. Despite strong opposition and criticisms (Sprevak, 2007) there isn’t a complete answer and yet no final algorithmic solution to grounded semantics. However, there is no question that Searle plays the same role in artificial intelligence as Maxwell played in physics or Hilbert in mathematics in terms of asking the right questions. 1.1.2. Does neuroscience ask the right questions? In neuroscience, we are at a critical cross-roads similar to the problem that the above three examples have demonstrated. The issue of recognizing the meaning within the neural code is closely represented by the problem in Searle’s thought experiment. Understanding the brain code using neuronal recordings faces a similary enormous problem of semantic representation within a symbolic framework. What is the language that the brain really uses to represent meaning? Is this language in terms of firing rate,

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Chapter 1: Understanding the Brain Language

spike timing, recorded oscillations or a hitherto undiscovered rhythm? Truly looking deeply at this question and attempting to understand it is akin to figuring out the semantics of a different language that we do not speak (Chinese as in Searle example). Any method that attempts to explain function of the nervous system must therefore intrinsicially be able to reveal how meaning or sematincs of behavior may be represented. The issue of representing neural function in spike timing faces the same crucial problem of having a symbolic representation without revealing how meaning is represented. Increasing or decreasing the firing rate may not have any meaning for us or at least not the same meaning that it has for neurons. In other words we are not able to understand neuronal language, since spiking seems to be a symbolic way of communication as the Chinese language is for a computer or for a non-Chinese speaker. Therefore, the main point that Searle made about digital computers can be extended to brain analyses based on neuronal recordings. Additionally, although the binary revolution has brought immense benefits, it has created a state of mind that makes us see just digital bits 0 and 1 almost everywhere. For many scientists the theory of computation is reduced to bit level operations. This has tremendously influenced the interpretation of neuronal activity. Action potentials are currently approximated almost everywhere as all or none events (stereotyped “one” for an action potential and “zero” values for the absence of an action potential). However, there is no experimental or theoretical evidence that such approximation correctly describes neuronal activity. Furthermore, if neuronal computation is reduced to symbolic manipulation then indeed it seems that there is no way to understand the intricacies of neuronal electrophysiology as a language representing meaning. A series of stereotyped action potentials and their timing can be simulated and programmed on almost any digital computer. However such an approach in the time domain does not seem to provide the richness required to build a human like information processing system and the required semantic level. 1.1.3. Cognition, Computation and Dynamical Systems Cognitive awareness and the resulting actions required to interact with the environment are a behavioral manifestation of an intrinsic semantic representation in brain. For several decades cognitive representation has been understood as a straightforward manipulation of internal symbols. Despite Searle’s demonstration, symbolic computational paradigms still dominate cognitive science. Regarding this issue, there are several claims that the brain computes differently than actual digital computers, though there is no clear understanding of how such computations are carried out. Additionally, dynamical systems theory plays an important role in emergent cognitive processes. This theory is a mathematical framework which describes how systems change over time. The early idea of the dynamical hypothesis in cognition came from Ashby's “design for a brain” (Ashby, 1952) where for the first time dynamical systems theory was used to assess cognitive behavior. The fact that intelligence is a real time embedded characteristic was also hypothesized by Ashby. More recently this idea was fully developed in a book by Port and Van Gelder (Port and Van Gelder, 1995). As presented in both books, cognition is indeed a dynamical process, which necessarily has a representation in neuronal activity. The critical point here is that the theoretical analysis of cognition requires a separation between computation and system dynamics. However, in general, such theoretical development

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pictures the theory of cognition in two distinct and opposed views one that reflects the theory of computation and another one with models of dynamic systems. Moreover, the dynamical system hypothesis presents a general theoretical approach that can be applied to several types of systems in different areas of science. This mathematical framework alone cannot explain how cognition arises or develops. A deeper understanding of the dynamics can come only from an intrinsic analysis of brain processes correlated with changes in developed semantics. Since mental processes are supposed to express the emerging behavior, the understanding of semantics should come from studying brain processes in relation to emerging changes in behavior. These interdependent, symbiotic issues that are intrinsically interconnected have not been presented anywhere yet to our knowledge. David Hume expressed this view of “mental mechanics” more than two hundred years ago (Hume, 1978). Inspired by Newtonian mechanics he developed an analogy between mental processes and the laws of motion. At that time Hume didn’t have any idea about electrophysiology, ion channels, movement of charges. This view of “mental mechanics” requires a deep understanding of brain processes related to intricacies of neuronal function. However, such data were not available two centuries ago. Extending Hume’s analogy we will show that mental processes are not only described by physical laws, they are intrinsically built using principles of motion. Hence, there should not be any contrast between a computational hypothesis and dynamic hypothesis regarding the emergence of behavioral semantics. The new development presented in this book not only ties the link between dynamical systems and brain computation, it shows that the dynamic framework is a distinct way of performing computation which shapes all cognitive processes in brain. A fundamental axiom is that the measurement of brain activity using behavioral experiments can be used to prove that animals acquire semantics during behavioral learning phases. Acquisition of semantics occurs naturally and can be expressed as changes in the emergent dynamics of electrical charges (Aur and Jog 2007a). We have demonstrated, for the first time to our knowledge, that computations performed and shaped by the dynamics of charges are radically different than computations in digital computers. Therefore, instead of a list of symbol manipulations without semantics, the representation in memory requires a physical determined structure fashioned by the intrinsic dynamical laws of charge movement which should be physically and directly associated to the represented behavior. This way of writing into memory facilitates the later retrieval of semantically connected concepts or events. Importantly, the critique that computational models are different than dynamic models is based upon only the current computational techniques algorithmized in von Neumann type machines. Since any coupled dynamic system could be used to perform computations (Ditto et al., 1993) the issue of symbolic manipulation in terms of Searle’s thought experiments should not affect the entire computational theory. Every physical realization of a computable system indeed involves a certain dynamics, whether or not the proposed framework develops a cognitive system. As an example even the implementation of von Neumann architecture of digital computers requires a dynamics for electron charges. Additionally, not all dynamical systems are able to embed semantics. This issue is now well perceived by the artificial intelligence community which is striving to build new principles for AI development. The important point to make here is that building a complex cognitive system requires particular dynamic structures and intrinsic interactions specific to neuronal activity.

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Therefore, this book proposes a broader view of the dynamical system hypothesis which directly connecting the theory of systems and physical laws of computation. The book relates this to neural mechanisms. Fundamentally, chapters 2 through 5 attempt to show that there is indeed an answer to Searle’s question posed above with the answer coming directly from an understanding of brain computation. Further, since cognition cannot be achieved as a simple manipulation of symbols, the understanding of the semantics carried by neuronal activity is highlighted as an important prerequisite. This analysis may provide a solution to build new types of intelligent systems using current digital computers that solve the grounding problem in terms of Searle’s thoughts. Each of the topics that are presented below is a subject that requires extensive discussion. This overview chapter does not make any attempt to provide a comprehensive look at the many years of work and thought that have gone into the development of these ideas. The purpose here is to state the problems and issues that neuroscience faces today. Many of these concepts will become the focal points of the chapters within the book. Neuroscience is presently at a stage where the development of classical physics found itself over a century ago. The rapidity of collection of intricate, factual data such as gene and protein identification in molecular biology and timing/rate of coding in electrophysiology and neural computation, has not answered the crucial issues alluded to above. The very focus of this book is to tackle this impasse and propose an important yet intuitive approach through a series of provocative chapters that will handle many of the issues which will be raised in the text below. 1.2. Myths, Controversies and Current Challenges The fact that neural code and information is transmitted by neuronal firing rate goes back to the very start of experimental recordings. Neurophysiologists believe that information is a function of spike time occurrence which when related to an event can be completely described by a Post-Stimulus Time Histogram (PSTH) (Winter and Palmer, 1990). Even today, almost all existing models of neural systems take for granted that information is transmitted by neurons in the form of interspike interval (ISI) or firing rate. Such myths have shaped the current theory of computation. Ignoring the fact that an AP is not a stereotyped phenomenon, several controversies and challenges are tied to this spike time analysis. Current advances in brain computations are then intrinsically related to the following four myths that are supposed to explain computation and memory storage. 1.2.1. The inadequacies of temporal coding There are several speculations regarding the nature of the neuronal code. Earlier analyses that started with electrical recordings proposed the firing rate as a coding method in neural networks (Adrian 1926, Adrian et al., 1932). The firing rate is determined as the average over time of the number of spikes generated by a nerve cell. After more than half a century of research in this field there are numerous views regarding the spike timing code. The coincidences of presynaptic events are considered to determine accurately the time of postsynaptic spikes and therefore the spike trains are supposed to represent the precise timing of presynaptic events (Abeles, 1991; Mainen and Sejnowski, 1995). Another view is that presynaptic input may alter the probability of the postsynaptic spike and therefore claims of “precise timing” are due to random occurrences (Shadlen and Newsome, 1994). Hence, Shadlen and Newsome

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rejected the idea that spike time patterns and intervals convey information. In their view only the firing rate is a relevant measure of neuronal code. The ‘synfire’ approach suggests the existence of cell ensembles that fire synchronously (Abeles 1982; Abeles, 1991). However, the synchrony mechanism cannot explain coding at low firing rates in the striatum or hippocampus and currently this mechanism is considered to be limited to certain brain areas. Additionally, other studies (Dinse, 2002) seriously called this view into question. They suggest that the “synfire” approach can explain information propagation in neural ensembles but it cannot explain information processing. Assessing these theories Thorpe, et al., (Thorpe, et al., 2001) proposed another scheme where information is encoded in the relative timing of spikes across ensembles of cells determined by the order in which the neurons fire. This mechanism of coding has also been rejected since to reach such neurons, information about the stimuli would need to travel several layers of neuronal connectivity. Then, the time of processing becomes critical for the proposed rank-order coding algorithm. However, recently, criticizing firing rate as a “true” measure of neural code several scientists claim that they had instead found neural coding in the interspike interval (ISI) (Rieke et al., 1997, Gerstner and Kistler 2002a, Maass, 1997). Their argument is that the nervous system does not have enough time to perform temporal averages since transmission times in several systems are too short, of the order of tens of milliseconds, to result in goal achievement (Rieke et al., 1997). Even though their arguments seem plausible, this issue regarding firing rate does not necessarily imply that the model based on ISI can accurately describe the incoming stimuli or that the timing of action potential (AP) is the “natural” code used by the neuron. Together, all of the above controversies and challenges demonstrate that ultimately the temporal aspect surrounding the occurrence of spikes may be an inadequate or at least an incomplete or partial description of neuronal coding. 1.2.2. The cable myth The axon and dendrites are frequently simulated as electrically conductive cables using the Hodgkin-Huxley parameterization. Based on available electrophysiological data several scientists were eager to estimate the passive cable parameters such as internal resistance, which is the membrane resistance at resting voltage using frequency-domain analysis (Koch and Poggio 1985; Meyer et al., 1997). However, reducing neuronal activity to electrical propagation in cables is an over simplification which cannot work as a general model of computation by neurons. Moving charges in cables (Figure 1) cannot account for complex processes that occur within the neuronal membrane, dendrite, soma or axons. It is unlikely that these structures act simply as conductors without involving additional mechanisms to allow information processing. While these initial studies enhanced the role of dendritic arborization, they spread confusion regarding the possible computational mechanisms.

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Aa

J

Ab

Δl

Figure 1 Schematic representation of a cable

Additionally, the cell membrane which separates the interior of the cell from the extracellular environment is most likely performing complex functions beyond the well claimed “capacitor approximation”. Treating the axons and dendrites as cables ignores all these possibilities. 1.2.3. The dilemma of synapses Synapses are considered to play a key role in information processing. Changes in the transmission properties of synapses are supposed to be directly connected with learning and memory. Another hypothesis that results from the spike timing theory is that synapses are capable of supporting computations based on a highly structured temporal code. This view of spike timing dependent plasticity (STDP) has been encouraged by existent temporal correlations on a millisecond level time scale between neuronal spikes. The synaptic weight is assumed to depend on the “relative timing between spikes” and is presented as a new Hebbian learning paradigm (Gerstner and Kistler, 2002b). The origins of STDP can be found in the early work of Grossberg (Grossberg, 1969), which was adapted to describe synaptic depression by Markram and colleagues (Markram et al., 1997). Current dogma hypothesizes that memories are stored through long-lasting modifications of synaptic strengths (structural modifications of synaptic connections). On the other hand several experiments show a dynamic change of synaptic strengths which, contradicts the synaptic long-term memory storage assumption (Khanbabaie et al, 2007; Banitt et al., 2005; Lewis and Maler, 2004). Therefore, this model using work done on the memory system should display only a short lifetime instead of the hypothesized lifetime memory preservation (Arshavsky, 2006). The link with molecular mechanisms starting from a timing description (see Clopath et al., 2009) needs a long series of additional hypotheses which are likely to make the timing model irrelevant in this case. Thus, alteration in synapse formation/elimination and changes in their ‘strength’ alone cannot fully explain the richness of information processing and especially that of memory storage. 1.2.4. The Hebbian approach of connectivity The connectionist view started with the historical remark of Hebb: “When an axon of cell A is near enough to excite cell B or repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased” (Hebb, 1949). Connectionist models assume that computation and representation is carried by cognitive units (neurons) organized in networks. Changes in their connectivity are

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supposed to provide an adaptive transformation between input and output of the system. Since this mechanism was translated into a mathematical formalism several attempts have been made to apply this formalism to mimic brain computations. This view was clearly opposed to symbolic models where computation is determined by symbol handling using inference rules. Both models have been developed separately. Unfortunately, this progress generates a huge gap between so called “high level semantics” acquired by symbolic development and lower-level connectionist representations in neural networks. 1.2.5. The myth of neurotransmitters The neuron doctrine defines the nerve cell as an embryological, metabolically distinct unit (Cajal, 1909, 1910) where the law of ‘dynamic polarization’ requires that information flows from dendrites to axon and its terminal branches. Since neurons are physically separated how any information is passed between two neurons? It is now known that synapses allow the transfer of information. In this context, the dynamics of neurotransmitters and their release in a stochastic manner was felt to describe this process of information transmission between neurons. Therefore the control of behavior was reduced to neurotransmitters’ role to alter the properties of neurons. Based solely on these observations regarding neurotransmitters many followers of spike timing theory rejected the importance of electric interactions claiming that “neurons do not interact through extracellular fields”. Even though these interactions are letely being reconsidered (see Milstein and Koch, 2008), computation cannot be linked/connected to intracellular processes. Most of the information processing takes place intracellularly and at the most fundamental level, a local sudden increase in charge density occurs within neurons which can be demonstrated within every spike delivered by a neuron (see Chapter 2). This description of computation and information processing in neurons would indeed be incomplete if it is reduced to AP occurrence or chemical synapse. Information transfer at the gap junction level is another important component that is directly carried by ions over a shorter distance, about 3.5 nm (Kandel et al. 2000, Bennett and Zukin, 2004). Additionally, the ionic flux is well recognized to have an important, active role in information processing at electrochemical synapse level (Abbott and Regehr, 2004) even though in this case the situation is somewhat more complex. If the presynaptic neuron develops an action potential, then a dynamic flow of charges occurs within the nerve cell which changes the internal and external environment without delay. 1.2.6. The myth of molecular biology The gene-centred view of evolution, seen as a sort of general philosophy capable of explaining almost every aspect of biological life is also a matter of concern. The full relationship between genes and the organism is missing and the genetic reductionism is not able to explain “how whole creatures work” (Ford, 1999). Known as the central dogma of molecular biology this model states that the flow of information is directed from genes by transcription translation processes to protein structures. Once proteins are synthesized information cannot be transferred back from protein to either protein or nucleic acid (Crick, 1958). This ‘dogma’ assigned a unique direction for information flow. Despite this earlier scheme, the reverse flow from RNA to DNA was later demonstrated by Baltimore (Baltimore, 1970) and Crick had to include this new aspect into another scheme (Crick, 1970). The intricacy of processes

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that operate at gene and protein level indicates a more complex regulatory mechanism of information flow than earlier proposed. 1.3. Spike Timing– Current Challenges 1.3.1. Is the neuron an integrator of incoming inputs - Temporal integration or coincidence detection? It had long been assumed that neurons behave as temporal integrators. Put simplistically, measures of neuronal activity firing rates have to be proportional to the time integral of stimulus inputs (Aksay et al., 2001; Loewenstein and Sompolinsky, 2003). Therefore, the probability of AP occurrence depends on the synchronization in neuronal ensembles (Roy and Alloway, 2001) and additionally neuronal synchronization should play a critical role especially in sensory information transmission. Konig challenged this view and considered coincidence detection as the main operational mode of cortical neurons (Konig et al., 1996). They used simulations to prove that neurons act more as a detector for dendritic activation with millisecond resolution than as temporal integrators (Softky and Koch, 1993). Also, they showed that simple models such as the leaky integrate-and-fire models are not able to explain the variability of cell firing in vivo. This issue poses a serious problem and weakens the hypothesis of the neuron as an integrator of incoming inputs. 1.3.2. Firing rate or ISI? Which technique is better to quantify neuronal activity? Typically, neurons receive thousands of synaptic inputs and the probability of the postsynaptic neuronal firing depends on the arrival of spikes from different inputs. The time between the resulting successive action potentials is known as the interspike interval (ISI). The relative spike timing and ISI duration is supposed to be the coding mechanism for sensory input neurons such as the visual neurons that “are well equipped to decode stimulus-related information” (Reich et al., 2000). On the other hand neurophysiologists generally assume that spike generation is a Poisson process and this approximation can provide a realistic description of information transmission. In the cortex the passive membrane time constant is about 15 ms, therefore the spiking rate should be a relevant measure (Shadlen and Newsome, 1994).This issue regarding firing rate or ISI as the more appropriate technique to quantify neuronal activity has generated a strong debate. 1.3.3. Do irregular spikes convey more or less information than the regular ones? If external events are represented by neuronal spiking activity then there should be a relationship between input stimuli and spikes. However, each neuron receives large numbers of synaptic inputs and displays a broad variability in temporal distribution of spikes. It is assumed that alterations in synaptic strength as a measure of connectivity should generate plasticity phenomena in neural circuits. Even if this connection between input stimuli, alteration in synaptic weights as proposed by Hebbian hypothesis and summation of all of these stimuli by the neuron (as an integrator) is truly robust and rich, a fundamental question still remains. How are these characteristics transformed and represented within the temporal sequences of spikes?

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The main issue here is that we don’t know how neurons integrate these synaptic inputs to produce the spike output. After presentation, neurons usually respond selectively to complex stimuli in only 100-150 ms (Thorpe and Imbert, 1989). If an adequate length of the stimulus sequence is specified, then spike trains are highly reproducible for adequate lengths of input stimuli and their standard deviation is of the order of 1-2 ms (Liu et al., 2001). However, similar stimuli may generate a variable number of AP (Britten et al 1993, Vogels et al., 1989). This source of irregularity has serious repercussions for neural information processing which seem to be unknown and have, to a large extent, been ignored. Several hypotheses ascribed this variability to NMDA conductance nonlinearity (Durstewitz and Gabriel 2007) while others hypothesized that spike-timing dependent plasticity alters the synaptic connections (Song, 2000) but no intrinsic explanation of the mechanism has been offered. Similarly, experiments in vivo and in vitro provide confirmation of varying responses of neurons to similar stimuli. If a neuron in vitro is stimulated, it fires a regular train of action potentials. However, by contrast in vivo recordings show that neurons from many areas will generate irregular trains of spikes to the same sensory stimulus. Given this property, Steveninck et al. showed that the patterns of action potentials have to be variable in order to provide information about dynamic sensory stimuli (Steveninck et al, 1997). Specifically as described by Mainen and Sejnowski, “constant stimuli led to imprecise spike trains, whereas stimuli with fluctuations produced spike trains with timing reproducible to less than 1 millisecond” (Mainen and Sejnowski, 1995). This view was opposed to the Shadlen and Newsome hypothesis that insists on the fact that an irregular ISI may reflect little information about the inputs (Shadlen and Newsome, 1994; van Vreeswijk and Sompolinsky, 1996). There are further contradictory results about information transfer and regularity or irregularity of spiking time. Davies et al. found that irregularity in time patterns reflects “the rich bandwidth” of information transfer (Davies et al., 2006). By contrast Hoebeek et al., demonstrate that irregularity is not needed and regular firing is important in Purkinje cells for information processing. As was shown by Hoebeek et al., the variability in firing rate of Purkinje cells is associated with severe motor deficits (Hoebeek et al., 2005). Also, spontaneous, irregular spikes can occur even in the absence of external inputs or extraneous sources of noise (van Vreeswijk and Sompolinsky, 1996; Brunel and Hakim, 1999). Trying to keep a balance between these two views Manwani showed that the variability of spike occurrence could either enhance or impede the decoding performance (Manwani, 2002). 1.3.4. Are synchrony and oscillations important? Moving from single neurons to network level theories, Edelman hypothesized that synchrony and oscillations are crucial for perceptual categorization, memory and consciousness (Edelman, 1989). Firing rates and their response profiles change under the influence of attention, memory activation or behavioral context. If synchrony or oscillations are so important then their temporal representation in the firing rate also is clearly an important phenomenon. Therefore, synchrony of firing is considered to be a way to encode signals with high temporal fidelity (Stevens and Zador, 1998) and is assumed to be dependent on the millisecond precision of spiking as shown by Mainen and Sejnowski (Mainen and Sejnowski, 1995). Taking the typical example of the visual system, “the binding model” suggests that neural synchrony is required for object recognition (Malsburg, 1981). This hypothesis,

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if confirmed, would imply that neurons should respond to the same sensory object and fire in temporal synchrony with precision in the millisecond range. Brunel and Hakim showed that such synchrony may be controlled by synaptic times and depend on the characteristics of the external input (Brunel and Hakim 1999). However, synchrony occurrence seems to be intrinsically generated in neuronal ensembles and can sometimes be independent of external stimuli. Also, high-frequency oscillations do not depend on synaptic transmission; they seem to be more related to direct electrotonic coupling between neurons (Draguhn et al., 1998). At this level several gap junctions assist chemical synaptic signaling in determining oscillatory synchronization with high frequency. Additionally, an increase in slow wave oscillations occurs usually in states associated with the loss of consciousness (Buzsáki and Draguhn, 2004) and these oscillatory patterns can be found also during sleep periods (Wilson and McNaughton, 2004). 1.3.5. Across-fiber pattern, grandmother cell, or labeled-line The nervous system is able to perceive the stimulus input in terms of intensity and quality. Over a half century of research in neuronal computation and brain coding has been largely focused on understanding how information about stimulus quality is processed by neurons by analyzing the spike-timing domain of single neurons or neuronal circuitry. There is not yet a unanimous answer to this coding problem and the issue has generated a long-standing controversy between several types of coding. The “labeled-line” model supports the idea that a specific characteristic such as taste, colour etc have a particular set of neurons in the brain (Hellekant et al., 1998; Dacey, 1996). Additionally, some experiments identified visual single units that respond selectively to a specific image of a familiar object. Jerzy Konorski called such cells “gnostic” units (Konorski, 1948). These neurons are better known as “grandmother” cells since the analyzed neuron is activated only by a specific meaningful stimulus (e.g. grand mother image). Many scientists have been critical regarding this grand mother hypothesis since one to one correspondence is not a plausible mechanism for neuronal coding, although, this correspondence between neurons and objects continued to be theoretically speculated (Edelman and Poggio, 1992). Despite the criticisms against the Konorski view, however, recent Quiroga et al., experimental evidence has supported the existence of these neurons which respond as exemplified in their paper only to “Jennifer Aniston” image as exemplified in their paper (Quiroga et al., 2005). Fearing the ridicule of the “grandmother” story the authors tried to explain such characteristics as a resultant of a “sparse distribution coding” in the neuronal ensemble. However, they were unable to prove and identify such circuitry or the hypothesized coding mechanism. The "across-fiber patterning" was first hypothesized by Pfaffmann who was trying to identify how neurons respond to specific tastes (Pfaffmann, 1941). Since he observed that the same fiber responds to different chemical substances he hypothesized that for a specific taste several neurons are activated and the coding takes place as “across-fiber patterning”. This model of distributed computation was later confirmed by several studies (Erickson, 1967, Erickson, 1968, Erickson, 1982, Caicedo et al., 2002, Scott, 2004). However, since the same neuron might be activated by other characteristics, there seems to be no clear ability of neuronal ensembles to detect a certain specific taste or color.

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1.3.6. Optimal efficiency or a high metabolic cost of information transmission? Energy demands are recognized as an important factor in mediating neuronal computation. It is unanimously recognized that the rapid, sustained trains of action potentials and the high rates of synaptic activity require high levels of energy. The major source of cellular energy, ATP, is generated by mitochondria. An important role is played by neuron-astrocyte metabolic interactions where astrocytes regulate the flux of energy required by neurons (Escartin et al., 2006). Slight dysfunctions at the mitochondrial level may determine the occurrence of severe neurodegenerative disorders. Importantly, protein expression changes (determined with a proteomic analysis) may also underlie alterations in efficient energy production. These phenomena can involve several mechanisms such protein degradation, DNA damage and changes in intracellular signal transduction. In order to guarantee correct function calcium homeostasis dependent energy production is important for neurons. Since neurons are highly vulnerable to energy deficit several neuroprotective mechanisms are intrinsically present that ameliorate an eventual energy deprivation. The relationship between this high-energy demand of the brain and the neuronal function is not really understood. Laughlin et al., showed that the costs to transmit a bit at a chemical synapse is 104 ATP molecules, and 106 - 107 ATP for spike coding (Laughlin et al., 1998). However, neuroscience has not yet grasped how this energy and functional constraints translate into information processing. There is currently no explanation for the substantial differences in the metabolic costs required at the chemical synapse and spike level. Moreover, there is no explanation between the so called optimal efficiency of spike time coding in the fly’s visual system (Rieke et al., 1997) and the observed excessive energy demands. 1.4. Issues of Brain Computation 1.4.1. Brain Computations--Classical or Quantum? Several scientists have hypothesized that brain computation lies in a quantum framework as a result of effects inside cellular structures such as the microtubules (Penrose, 1989; Penrose, 1994; Hameroff and Schneiker, 1987) and cannot be fully explained in a classical framework. However, the existence of quantum effects was severely criticized by Tegmark who demonstrated that quantum behavior requires longer decoherence times than are supported by the laws of classical physics (Tegmark, 2000). The collisions with other ions, with water molecules and several Coulomb interactions may very quickly induce decoherence that would destroy any required superposition to perform quantum computation. Progress in quantum theory generated an open debate between classical and quantum concepts regarding cognitive processes including consciousness. Actual images of spikes show electrical activities with micron level precision. However, if one goes close to the nanometer-size dimensions the observed physical laws are different from the classical ones. Automatically, at this scale, laws and theories of quantum mechanics turn out to be relevant and important. There is an ongoing debate wether several phenomena in brain including computation can be classically explained or they require a quantum framework. The advocates for quantum theory, Penrose and Hameroff insist that brain and mind have to be explained in a quantum framework.

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1.4.2. Brain computation and AI One of the goals in understanding brain computations is to develop human-like intelligent machines. Artificial intelligence has been viewed as a computational model that traditionally included a simple network of neurons, symbolic systems based on inference and probabilistic rules. However, difficulties of finding solutions to several problems in a human-like fashion do not seem to be solved by designing algorithms on current computers. Several years ago many scientists claimed that the key algorithm that has humanoid characteristics had been developed and only required computational power to be able to run the program. Since there is currently an easy availability of such computing power, the common agreement is that all these attempts have been mere speculations. Our understanding is that the basic principles that govern human kind of intelligence have not yet been revealed. These principles still seem to be buried in this unknown description of brain computations. 1.4.3. Is computation in the brain closed to a reversible process? Neurons are biological entities that function in a milieu. During each spike the neuron processes information and communicates with other neurons and also with its milieu. Since physical computation needs to be explained by physical laws of electric field propagation, charge motion, ion fluxes seen as charges in motion describe spatial trajectories that can be perceived and recorded in each real spike (Aur and Jog 2006). This new view of computation highlights the possibility of applying reversibility principles developed by processes at microscopic scale (Day and Daggett, 2007). 1.4.4. Is ‘noise’ useful for brain computations? Noise is omnipresent and is expected to influence the computation in neurons. Seen as strong stochastic membrane potential fluctuations (Pare et al., 1998) to random input stimuli, the presence of noise may be responsible in modulating neuronal activity. Almost everywhere in our daily life, noise is considered to be an undesired phenomenon. However, recent results of Salinas, if confirmed, suggest that noise can be useful for neuronal computations since the addition of noise may allow neuronal ensembles to maintain an “optimal performance” bandwidth (Salinas, 2006). 1.4.5. What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? The presence of metabolic subunits within the neuron, defies the idea of a simple indivisible neuron. The current neuron doctrine assumes a passive role for dendrites in acquiring information. The role of passive dendrites has been contradicted by several patch-clamp experiments performed in dendrites. The dendrites and soma have voltage gated ion channels that generate electrical potentials at the origin of diverse recorded electrical activities. The existence of dendritic spikes based on opening of voltage-gated sodium and calcium channels show the dynamic role of dendrites in information processing. Their diversity in information processing is responsible for specialized functions of different neurons (Häusser et al.,2000) The existence of several forms of dendritic excitability and multi-compartmentalization in dendrites was explained as being necessary for multiple sites of integration of synaptic inputs that may enhance the computational power of single neurons (Poirazi and Mel, 2001). The importance of

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dendritic morphology for AP propagation has recently been demonstrated (Vetter et al., 2001). 1.5. How can Information be Represented in the Brain 1.5.1. Is postsynaptic activity required for dendritic development? A biological means by which information representation has been thought to occur is by changes in the synapses in “connected” neurons, as discussed above. This could also be seen as happening at the dendritic level. Dendritic growth is seen as an adaptive process to better respond to incoming signals (Konur and Ghosh, 2005; Wong and Wong, 2000). The role of excitatory neurotransmitters for such growth seems important since cells in cultures have less developed dendritic trees than in vivo, where, all kinds of inputs shape growth and arborization. However, in case of Purkinje cells dendritic development can be achieved by a predetermined growth program (Adcock et al., 2004). 1.5.2. How and where are memories stored? Declarative memory representation is supposed to have a distinct place in the hippocampus (Eichenbaum, 2004). Additional working memory is shown to be located in the prefrontal cortex (PFC) and somehow connected with the activity in the basal ganglia. However, there is a continuous debate whether memory access in PFC is related to a remembered stimulus or is preparatory for a decision (Boussaoud and Wise, 1993). Some studies show that neuronal responses correlate with the outcome of a perceptual decision (Shadlen, and Newsome, 1996). Other, electrophysiological recordings show an involvement of PFC by mediating working memory phenomena (Jonides et al.1993, Funahashi, et al., 1993). The short term memory capacity depends on the number of objects required to be memorized and on their complexity (Xu and Chun, 2006). However, it is expected that neural coding for stimuli is similar for the senses such as vision, hearing, and touch. 1.5.3. The key of the topographic map Brain-imaging measures suggest that a strong relationship exists between brain structure and function. It has been demonstrated that during a simple decision task functional connectivity occurs between several brain areas (Boorman et al., 2007). However, currently there is no plausible explanation for exactly how this topographic projection of information actually occurs within the brain. While topography can result from a wiring optimization process (Buzsaki, 2004) Hebb claimed that such topography is an artifact and in fact that connectivity (synaptic connectivity) can and should exclude the role of topography. The above discussion has only touched upon the substantial difficulties faced by neuroscience and a deeper understanding of neural computation remains out of our reach. These controversies and challenges show that the “brain language” is a complex and strange way of communication, which is hard to understand especially from the limited spike timing approach. The assumption of spike stereotypy and homogeneity between neurons remains an extremely important problem present everywhere in theoretical neuroscience. Since neuronal spikes are not stereotyped events (Aur et al.,

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2005, Aur and Jog 2006, Aur and Jog, 2007a, Aur and Jog, 2007b) they have richness and semantics, similar to words in a well-defined language. Electrophysiological recordings in behaving animals have revealed considerable information about the nature of coding. Recent new methods and techniques have for the first time provided a glimpse into the relationship between neuronal activity and emergent semantics (Aur and Jog, 2007a). These intrinsic neuronal activations seen as movement of charges could be the building blocks for higher order functions such as learning, memory and cognition. The movement of charges can play the role of letters in composing “words” in terms of brain language. With this new level of abstraction, which operates explicitly at neuron level and implicitly at network ensembles levels, we will be able to explore the full power of brain computation. Therefore, it is appropriate to settle these controversies by building a new model of neural computation using classical and quantum physical laws. However, understanding this complex mechanism of information processing requires knowledge from several disciplines. 1.6. Interdisciplinary Formalism The fundamentals of mathematical thought are often developed using several axioms accepted without proof. The challenge of building a theoretical framework for the brain computations in the same manner does not seem to be solved yet. Whether brain computations can be built on such fundamental notions which are similar to the approach taken by mathematical science, still remains an open question. However, since the approach of thought proposed in this book requires an interdisciplinary systemic approach, mathematics and the laws of physics are the essential tools for implementation. 1.6.1. Sets and functions Many important concepts in mathematics are based on the idea of a set. A set can be defined as a collection of distinct objects that may include numbers, letters, symbols, or other sets. Sets are listed as members inside parenthesis X = {4, 9, 3, 10} where the set X contains four elements 4, 9, 3 and 10. In order to show that 3 is an element of the set X one can write: 3∈ X (1) While for the element 5 that is not contained by X one can write: 5∉ X

(2)

If for two sets X and Y, all the members of X are also members of Y this can be represented as: X ⊂Y (3) or that X is a subset of Y. The union of two sets X ∪ Y is the set that includes all members of sets X and Y while the intersection X ∩ Y contains only the elements that are common for X and Y.

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The mathematical concept of a function shows an input-output relationship between two or several quantities. A function can be given by a graph, an algorithm or usually by a formula: y = f (x) (4) where f is the function with the output value y and the input x . The function f is called a transformation or mapping and can be written using the notation ƒ:X→Y where X is the domain and Y is the codomain of two given sets. Occasionally, these important concepts of set theory will be applied to characterize computation performed by charges. The required mathematical framework will be further developed in chapter 4 of this book. 1.6.2. Dynamical Systems The understanding of complex systems behaviour over time is the main focus of system theory. Ideas of system theory have grown in several scientific areas from engineering to sociology, biology and economics. The states of a general system are described as a state variable. All state variables and their combinations can be used to express the state space of the system. This formulation in state space allows us to represent almost every system in an n-dimensional space where n is the number of states S1 , S 2 ,...S n that describe the system. As an example, a single particle moving along the x axis can be considered to have two degrees of freedom namely position and velocity which can be represented in 2-dimensional space. Such state definitions may represent a particular combination of physical properties or can be more abstract. These states are thus the required set that describes the specific conditions for a particular system. The evolution of the system is seen then as a succession of stages from one state to another state. If the system does not satisfy the superposition principle then the representation has to be written in terms of nonlinear functions. In case the transition between states is perceived to be in continuous time then the system is considered to be a continuous time system and described as a continuous set of ordinary differential equations: dS1 = f1 ( S1 , S 2 ,...S n ) dt dS 2 = f 2 ( S1 , S 2 ,...S n ) (5) dt . dS n = f n ( S1 , S 2 ,...S n ) dt This description is well known and very common since almost any physical system can be modelled as a set of differential equations. In a similar fashion, every neuron can be regarded as a dynamical system with inputs and outputs and neuronal state (Figure 2)

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Inputs

Outputs Neuronal State

Figure 2 Schematic representation of a neuron as an input-output dynamical system

The dynamics of electric charges are determined by the laws that govern how the system evolves over time. Additionally, this evolution depends on selected state variables and their order. The inputs and the outputs of the system can be selected based upon any biochemical or electrical outcome that is relevant to neuronal function. This dynamical state dependent evolving system level approach is exceedingly important and has begun to be described in relation with information transfer in neurons (Aur et al., 2006). A detailed construct of this framework will be presented in subsequent chapters. A much simpler example can then be considered using the case of the particle (charge) dynamics. The state space description of the particle can be considered in the form: dX = f 1 ( X ,V ) dt (6) dV = f 2 ( X ,V ) dt where X is the position and V is the velocity of the particle. A plot of position as a function of time across the velocity variable for the particle is called a phase diagram (Figure 3). 0.8 0.6 0.4 V

0.2 0 -0.2 -0.4 -0.6 1

2

3 X

4

5

Figure 3: Phase diagram for a charged particle, which moves with velocity V over X-axis and describes damped amortized motion

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If the transition is seen in discrete steps then the description is in terms of a discrete time system as a set of difference equations: S1 (k + 1) = f 1 ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....) S 2 (k + 1) = f 2 ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....) .

(7)

S n (k + 1) = f n ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....)

Such reduction of a continuous description to this discrete form is performed in von Neumann computers and has several consequences. In case of smaller sampling rates under the Nyquist rate, information is always lost from the original continuous description (Cover and Thomas, 1991; Sobczyk, 2001; Aur et al., 2006). There is a similar terminology regarding the state of a machine in computer science. The transition of states can describe the behavior of a finite state machine, such as a Turing machine. Since usually the state term defines a particular set of instructions that are executed the sense of state and the considered transitions can be restrictive and sometimes significantly different than the terminology used in dynamical systems. 1.6.3. The laws of thermodynamics A deeper understanding of natural processes including life, brain function and cognition should come from well recognized general principles. The law of entropy is a general principle known as the second law of thermodynamics, which can be applied from physics to biology and evolutionary theory. Viewed as a 'law of disorder' in a system the second law of thermodynamics makes the link between physics and statistical properties of systems. Boltzmann defines this concept in the terms of entropy, H B : H B = k B ln Ω (8) where k B is the Boltzmann constant and Ω is the number of microstates. The second law states that in an isolated system which is not in equilibrium the entropy tends to increase over time, approaching a maximum value at equilibrium (Halliday et al., 2004). Importantly, the second law also establishes the direction of heat transfer. The first law of thermodynamics states that the total energy of a system does not change. Different forms of energy, mechanical, electrical or thermal energy can be converted from one to another, however, their sum must remain constant. This may also demonstrate the equivalence between heat and other forms of energy. The system can change its internal energy ΔWi only by work or heat exchanges: ΔWi = Q − L (9) where L represents the work and Q is the heat exchange. Importantly, the second law establishes the direction of heat transfer.

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QC

QH

W

Cold Reservoir

Hot Reservoir

Figure 4 Heat engine scheme

Additionally it can be successfully shown that it is not possible to convert all thermal energy Q to do work L . This result was extended in engineering and thermodynamics to the theoretical concept of heat engines (see Figure 4). Physical implementation of such an input-output system is based on a specific thermodynamic cycle and transforms the heat input to an output, which usually is mechanical work (e.g. steam engines, diesel engines). The efficiency of the thermodynamic engine can be written: η=

QH − QC QH

( 10 )

where QH is the amount of heat from hot reservoir and QC is exhausted at the cold reservoir. Depending on their interaction with the environment these heat engines can have open or closed cycles. The major revolution in science was the establishment of the link between the "law of maximum entropy” statistical mechanics and information theory (Shannon, 1948; Jaynes, 1957). It can be successfully shown that information encoding can be better performed within a distribution, which maximizes the information entropy. 1.6.4. Conclusion An integrated application of such universal laws will help us to understand how the brain computes. Therefore, in order to understand the brain language, neuroscience has to ask the right questions. The main issue yet unsolved is to forge a link between cognition with information processing and theoretical computation. The cable myth, the dilemma of synapses, the Hebbian approach of connectivity and spike stereotypy reflect the actual myth that pointed to spike timing as a single path in explaining neural computation. The temporal coding approach seems to be inadequate to solve important issues integral to brain processing begging for questions like. Do irregular spikes convey more or less information than the regular ones? Is the neuron an integrator of incoming inputs? Which technique is better to quantify neuronal activity? Firing rate or ISI? Across-fiber pattern, grandmother cell, or labeled-line?

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Other questions should aim to tackle the inherent aspect of computation: Is noise useful for brain computations? Is computation in the brain a quantum process? What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? Information representation in brain is also a critical issue. How is information represented in the brain? How and where are memories stored? Is there any meaning behind the topographic map? Solving all these issues requires an interdisciplinary formalism where the laws of physics should play a greater role than spike timing model. The view proposed here is that, ultimately, spike timing is a natural outcome of a much richer process that defines the brain language. However, this mechanism of computation may never be understood if the entire analysis is restricted to spike time patterns observed in single neurons or neuronal ensembles. Since such formalism based study of the brain may allow finding the link between the semantics of computation and emergent behavior an analysis of both phenomena is desirable. However, before developing these principles it is important to have a construct of the current advances in experimental data analysis. Therefore, in order to study the computational power we have to switch back and carefully re-analyze behavioral experiments and the strength of interpretations of earlier recordings performed in-vivo. This is developed thoroughly in the next chapter. Summary of Principal Concepts • • • • • • •

• • •

Brain computes differently than current digital computers The problem of acquiring semantics using an algorithmic approach has no solution as yet in terms of a symbolic framework The hypothesis of spike stereotypy is at the origins of several controversies in neuroscience and neural computation Understanding the brain code using neuronal recordings faces the problem of semantics. If neural computation is reduced to symbols manipulation (stereotype action potentials) then semantic issues may arise in Searle’s terms. Spikes (action potentials) as ‘binary pulses’ cannot be considered to be fundamental units for computation in the brain. This view regarding spikes determines several controversies in terms of information transmission, integration of input stimuli, synchrony, memory storage and retrieval. Neuronal spike, seen as a binary pulse has historically been the cause of much scientific debate: if the neuron is an integrator of incoming inputs or if it performs coincidence detection, if irregular spikes convey more or less information than the regular ones or if firing rate is better than ISI to quantify neuronal activity. Spike timing theory is a radical view that attempts to explain brain computations in terms of an existent temporal code. The description of brain computation has to be made in terms of universal laws of nature Brain analysis requires an interdisciplinary formalism where computation can be described in terms of dynamical systems theory and thermodynamic laws

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Stevens C. F and Zador A.M., (1998) Input synchrony and the irregular firing of cortical neurons, Nature Neuroscience 1, 210 - 217. Steveninck De R.V., R. R. Lewen, G. D. Strong, S. P. Koberle, R. Bialek, W. (1997) Reproducibility and Variability in Neural Spike Trains, Science, , 5307, pp. 1805. Thorpe S. J., and Imbert, M., (1989) Biological constraints on connectionist models. In R. van Vreeswijk C. and Sompolinsky H. (1996) Chaos in Neuronal Networks with Balanced Excitatory and Inhibitory Activity, Science 6 ,Vol. 274. no. 5293, pp. 1724 – 1726. Vogels R., Spileers W., Orban G.A., (1989) The response variability of striate cortical neurons in the behaving monkey. Exp Brain Res 77:432-436. Von der Malsburg, C. (1981) The Correlation Theory of Brain Function, Internal Report 81-2, Max Planck Institute for Biophysical Chemistry, Gottingen, Germany Vetter P., Roth A. and Hausser M., (2001) Propagation of action potentials in dendrites depends on dendritic morphology, J Neurophysiol 85 pp. 926–937. Xu Y. and Chun M. M., (2006) Dissociable neural mechanisms supporting visual short-term memory for objects Nature 440, 91-95 Wilson M. A., McNaughton B. L., (1994) Reactivation of hippocampal ensemble memories during sleep Science 265, 676. Winter IM, Palmer AR. (1990) Responses of single units in the anteroventral cochlear nucleus of the guinea pig. Hear Res. Mar;44(2-3):161-78. Wong W. T and Wong R. O.L., (2000) Rapid dendritic movements during synapse formation and rearrangement, Current Opinion in Neurobiology, Volume 10, Issue 1, Pages 118-124.

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CHAPTER 2. Imaging Spikes – Challenges for Brain Computations The Copernicus model generated the greatest revolution in astronomy and questioned the beliefs of Aristotle and Ptolemy that the Earth is at the center of the Universe. The initial failure to accept Copernicus’s theory was due in part to an inadequate ability to see far away in space. “Seeing is believing”, and a revolution in thinking was possible only after several improvements to the telescope led to observations of Jupiter’s moons. This is a classic example where the progress in technology confirmed earlier theoretical ideas. We face the same challenge today in neuroscience. Imaging techniques such as magnetic resonance imaging (MRI), functional MRI (fMRI) along with electro-encephalography (EEG), play an important role in monitoring brain activity. From sensations to body movements, from memory to language in general, imaging techniques have significantly changed our perception about brain functions. Recording of electrical activity of individual and networks of neurons can be thought of as imaging at a much finer level. Activity patterns revealed in such recordings can be studied at levels of slice recordings up to multiple cells in vivo where the activity is elegantly related to observed behavior. The correlation of large scale imaging such as fMRI to the electrophysiological substrate is however sorely lacking and a significant gap exists between imaging techniques at larger scales and electrophysiological recordings in single cells. Action potentials were assumed to be uniform pulses of electricity and we believe that the origins of this view are buried in the early single electrode recordings. Since neurophysiologists could see only similar shapes of action potentials, this occurrence of electrical signals in nerve cells was considered to be a stereotyped uniform pulse signal. This revelation came about from hearing such electrical impulses from the optic nerve as described by Adrian (Adrian 1928): "The room was nearly dark and I was puzzled to hear repeated noises in the loudspeaker attached to the amplifier, noises indicating that a great deal of impulse activity was going on. It was not until I compared the noises with my own movements around the room that I realized I was in the field of vision of the toad's eye and that it was signaling what I was doing." These early recordings claimed that action potentials are uniform pulses of electricity and their variability and importance consist only in their time of occurrence. Interestingly, this viewpoint has essentially remained unchallenged for a long time. Newly developed techniques of multi-tip electrodes however provide a tool for powerful analysis of electrical propagation during single action potentials and shows that the earlier claims of action potential stereotypy are not valid. 2.1. Methods, Models and Techniques Initiated by Ramón y Cajal in the late 19th century, the neuron doctrine was developed based on single electrode recordings and anatomical tracing studies of nerve cells in the brain. This doctrine states that neurons are distinct structural and functional units. Neurons were considered to carry information through electrochemical processes having an intrinsic ability to integrate both aspects as a part of their signaling capability. From an electrical perspective, two important assumptions that have shaped

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

the field of neurophysiology were made: that an AP propagates predominantly in the axon and the spike can be considered a stereotyped phenomenon. Given the electrochemical nature of processing, the ability to integrate a rapid (electrical) and a much slower (chemical) transmission in order for neurons to communicate became an important problem, especially since neurons are spatially separated. This led to an important controversy that started in the 1930s regarding neural communication. The issue of whether nerve cells communicate with each other chemically or electrically generated a strong debate between scientists in the field. After a long dispute the prior theory of “continuous nature of nervous transmission” (Fulton, 1926) was replaced. A new theory for information coding in the nervous system was developed and the neuron was regarded as the basic information-processing unit. Additional theoretical work of McCulloch and Pitts contributed to formalize brain processes where the neuron was considered the key element for processing and communication in the brain (McCulloch and Pitts, 1943). Initial supporters of the electrical theory suddenly changed their viewpoint in favor of a new approach. This change in interpretation was determined by images of synaptic vesicles that contain neurotransmitters and theoretical approaches regarding chemical mediators (Eccles, 1957). Neurotransmitters such adrenaline, acetylcholine, norepinephrine, glutamate or serotonin were associated with the inhibition or initiation of AP while the synapse was felt to be the substrate of neuronal communication. Over time, learning and memory phenomena also became closely related to synaptic plasticity (Kandel et al., 2000). Accumulation of experimental data determined the development of theoretical models with the main goal of studying electrophysiological properties of neurons. An important step in understanding the electrical nature of the AP was made by the Hodgkin-Huxley model (Hodgkin andHuxley, 1952). This model showed that almost every portion of the cell membrane is far more complex than the switching device model proposed early by McCulloch and Pitts. A new theoretical framework able to explain brain activity was required. Barlow (Barlow, 1972) made the first theoretical attempt to define rigorously the field of neuroscience and neuronal computation. He proposed a set of five dogmas that are considered even today to be the foundation of research in neuroscience. The “first dogma” states that we can understand the relationship between the brain and mind by simply analyzing cell interactions. This was the primary goal of Hubel and Wiesel and their studies expanded our knowledge regarding the visual system. They showed for the first time that visual experience in early life influences neural connections (Hubel and Wiesel, 1962; Hubel and Wiesel, 1968). Recently, neuronal activity in individual neurons or network of neurons has been analyzed using information theoretical measures (Mainen and Sejnowski 1995, Rieke et al., 1996). Several scientists have tried to describe the mechanism of coding in neuronal ensembles and new theories regarding interspike interval (ISI) are considered the basis of the “third generation” of neural networks (Gerstner and Kistler, 2002, Maass, 1997). However, all these analyses and proposed theories have their origins in an important assumption that was stated above, namely the spike stereotypy hypothesis. Neuronal activity and thus neuronal coding of information is analyzed predominantly in the temporal domain. This viewpoint has recently been challenged by the demonstration of information transfer in a much smaller time window, within milliseconds during every AP. Using simulations of Hodgkin-Huxley model it can be shown that the values of mutual information between input stimulus and sodium fluxes

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are more than double compared to mutual information between the membrane voltage and stimulus input (Aur et al., 2005). Such theoretical postulation that information is physically transferred during spike occurrence by the movement and interaction of electrical charges has implications and may define a new model of computation. All these issues regarding information transfer are presented in detail in the next chapter. However, before elaborating this new model we have to verify this theoretical outcome using experimental data. 2.1.1. Tetrodes Since cell bodies of neurons are packed at distances of tens of microns, single extracellular electrodes are able to record signals simultaneously from several neurons. Electrical activity in neurons can be revealed using multi-tip recording devices followed by data analysis using several processing techniques. The needle-like electrodes are brought in the proximity of the cell. As a result the amplitude of the recorded signal is slightly larger than the amplitude of neighbour neurons. This separation of amplitudes of recorded spikes and their correspondence to a specific individual cell is an important step in analysing the activity of neuronal ensembles. However, despite sophisticated hardware and software methodologies, it is difficult to assign recorded spikes effectively to corresponding neurons. More recently McNaughton et al., utilized a new implantable device with multiple electrodes packed together named stereotrodes and later tetrodes (McNaughton et al., 1983, Wilson and McNaughton, 1993). Since an electrode receives signals from several neurons it is difficult to assign a certain spike to the corresponding neuron. Finding this correspondence between cells and spikes is important if one analyses how neurons work together in the network. A tetrode uses inexpensive material and can be made virtually in any laboratory manually by twisting four insulated wires into a bundle (Figure 1). This manual assembly can produce a usual spacing between the geometric locations of wires on average between 20 and 30 micrometers and can be in an ensemble, which can be used to record from several areas simultaneously (Jog et al., 1999). Therefore, with this device the firing pattern of neurons in a very small region of tissue of about 100micrometer radius can be recorded.

Figure 1: Ensemble of several tetrodes

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

A band-pass filter is then used to pre-process the recorded signal followed by spike threshold detection. This detection procedure can be implemented using a hardware device or a software program. Then the amplitudes of the spike recorded by each channel of a tetrode are considered to generate a feature vector that is used to "cluster" the spikes. Clustering techniques group spikes with similar feature vectors and assign each group to the corresponding neuron. Improvements in the spike sorting methods of a recorded population of neurons using multi-tip electrodes were carried out in the early 1990s and is at the origin of fabrication of multi-site electrodes (McNaughton et al., 1983, O'Keefe and Reece, 1993). If this process of spike sorting is done manually, it is time consuming, and may be affected by subjective factors. It has been proposed that automatic methods may significantly speed up the clustering process, and reduce the effect of subjective factors (Fee et al. 1996; Lewicki, 1998; Sahani, 1999; Jog et al., 1999). However, despite these advances clustering is done manually in many laboratories. Even though the tetrode recordings were designed to improve spike-sorting techniques they can ideally be used to analyze the pattern of electrical activities during every AP. The methodology that allows this analysis of electrical patterns has been recently developed as a straightforward computational approach (Aur et al., 2005, Aur and Jog 2006). Two distinct processing techniques have to be applied in order to allow the visualization of electrical patterns during APs. These methods are discussed in detail elsewhere (Aur and Jog 2006). In short, they require assignment of charges or group or charges with charge points and then to perform independent component analysis (ICA) and triangulation techniques to spatially localize the position of charges. 2.1.2. The Necessity for a New Model Current models of brain activity are highly spike-time oriented and they omit some other important aspects of neural activity, including the intrinsic electrical communication with electrical charges, gap junction transmission, active dendrites, ion fluxes etc. Action potentials are indeed carriers of information; however they are not uniform (stereotyped) pulses of electricity. At the nerve cell scale, for dimensions between 100 to 300 micrometers, every action potential (AP) can be well described by mapping electrical patterns of activities in a three dimensional space. Using new techniques we provide evidence that APs are not stereotyped events and their analysis even for mathematical modeling purpose cannot be reduced to spike timing occurrences without loosing the essence of what is contained within spiking activity. The representation of independent signal sources in three-dimensional space shows that charges are preferentially spread during every AP across certain areas of the nerve cell where ion channels are likely to be more active and generate electrical patterns of activations (Aur and Jog 2006). The new model of charges in movement highlights all these aspects and provides a better interpretation of computation at a smaller scale within every action potential. 2.1.3. The Model of Charges in Movement Electrophysiologists have known for a long time that the amplitude of extracellularly recorded AP attenuates strongly and inversely with distance. Therefore, several experiments suggested that a “point source model” would be able to approximate well enough the extracellularly recorded AP (Drake et al., 1988). The point approximation does not seem to perform well in real recordings due to several

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issues presented below including noise level. Additionally, since every spike is a complex electrical phenomenon (Aur et al., 2005) neurons cannot be considered as point sources during the action potential occurrence. An important implication of this aspect is that neurons cannot be practically separated based on their spatial location (Recce and O'Keefe, 1989). Lately, this issue has become important since several researchers have tried to determine AP propagation. They anticipated that this point approximation with a single charge does not seem to be an adequate description of an AP. Calling this view into question Holt and Koch approximated the occurrence of AP with a line of charges named “the line source model” (Holt and Koch, 1999). Supporting earlier studies including the view that an axon can be approximated with a cable; recently, this model was recently extended to extracellular recordings by Gold et al (Gold et al., 2006). They assumed that the extracellularly recorded AP can also be treated as a line of point sources and then the resulting voltage can be computed as the sum of potentials generated in this “line” of point sources. By contrast, another proposed model (Aur et al., 2005) shows that even this “line” of point sources or the “point source model” alone is a poor descriptor of electrical phenomena during an AP. Simply, if one considers tetrode recordings, where the distance between tips is about 20 microns, then the spatial occurrence of charge movement during every AP is about five to ten times wider (or higher) than this distance between tips. Therefore, it is unlikely that electrical activities within an active neuron could be well approximated with a single point source or even with a simple line of electrical sources. This demonstration of spatial geometry shows that we may need a different model to describe the complexity of the electrical phenomenon within a spike. Since biological neurons are complex three-dimensional structures with dendrites and axonal arborizations every spike can be considered a superposition of several charges in movement. The electric field and charges that move in space are the sources for generated signals in electrode tips. This simple view has important implications for modeling extracellular measurements and consequently the electrical events such as spikes can only be well described as results of dynamics of many charges (Aur et al., 2005, Aur and Jog, 2006). Due to several factors already mentioned above, action potentials are considered to be stereotypical events. This view of spike stereotypy has remained unaltered for a long time. The factor that has contributed to a slow progress in this field is the beautiful mathematical theory that can be developed assuming spike stereotypy. Spike timing theory has attracted some of the best minds and any suggestion that may show a completely different view by contrast to the stereotypical, all or none AP event is stopped in advance (Scott, 1999). There are also several practical reasons why this spike timing theory has remained so popular. First, the spike phenomenon occurs within a short time about one millisecond. Second, the powerful propagation of electrical wave in the axon buries other electric phenomena, which take place in dendrites. Hence, these activities remained unobservable for a long period of time and indeed, as mentioned by Gold et al., dendritic morphology has very little impact on the extracellularly recorded waveform (Gold et al., 2006). Despite the currently prevalent view, we have been able to show that the biological and architectural details of dendritic arbors and axonal branches slightly yet significantly modulate the extracellular AP. However, these minor changes in AP waveforms can have a strong impact in determining electrical ‘connectivity’. We will demonstrate that (a) this weak activity in dendrites is extremely important for neuronal function, (b) this small variability can tremendously change patterns of activations in a

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neuron and (c) it is likely that such changes will alter the function of neuronal ensembles. Additionally, even though well accepted, axons cannot be approximated with simple (linear) cables since they always have a certain degree of arborization and contain numerous spatially distributed voltage-gated channels (Debanne, 2004). Further, the membrane itself is a complex electrical structure that is hard to approximate as a linear cable. Modulation of proteins and changes in their characteristics at ion channels or receptors level may have a strong impact in electrical signal transmission and implicitly on the interaction with synapses, other dendrites, or axonal coupling. The fundamental postulates of this book are based upon providing a mathematical formulation and an experimental base to understand brain computations. The issues raised in chapter one and the beginning of chapter two will be directly tackled by applying physical principles to allow an integrative view from macromolecular alterations seen in proteins to modulations of APs. These subtle changes that occur during neuronal activity will then be related with information transfer and emergent behavior. 2.1.4. The Triangulation technique As a technique, triangulation was intensively applied during and after the Second World War for detecting submarines with noisy diesel engines. The system called SOSUS (Sound Ocean Surveillance System) used the property that sound spreads in the ocean uniformly in all directions, with known attenuation. Since sound detectors are distributed in space, then the position of each submarine can be obtained by processing the signal received from detectors using as a simple geometrical procedure called triangulation (Figure 2).

D

A

B

Figure 2: The distance D to the ship can be estimated if the spatial coordinates of sound detectors A and B are known

2.1.5. Charges and Electric potential The charge movement model (CMM) has been developed in order to address many of the concerns raised above (Aur et al., 2005). This model challenges current view which relates spiking activity to several myths and suggests a new development where the remaining formalism of spike timing can still be maintained and demonstrated as a special case with particular characteristics. This CMM formalism is critical for

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understanding how information is electrically communicated to downstream brain structures and ultimately how they are related to spatiotemporal spiking activities. The electric potential, V generated by every charge, q and recorded by tetrode tips depends on the distance r from the charge (Figure 3a): q V= (1) 4πε r and ε is the electrical permittivity of the medium. Additionally, for a line of charges, the electric potential can be computed by summing elementary potentials determined by single charges (Figure3, b): qi V =∑ ∑ (2) 4πε ri i In the proposed model the main hypothesis is that different charges or groups of charges under an electrical field have distinct movements. Therefore, in order to apply independent component analysis (ICA), the AP is assumed to be the output of several independent sources of signals generated by charges that move in space. Hence, for recorded AP signals, using ICA, a blind source separation can be performed. This technique generates a linear transformation which minimizes the statistical dependence between components (Comon, 1994). a

b -3

-3

x 10 Electric Potential [Vm-1 ]

Electric Potential [Vm-1 ]

x 10 10 8 6 4 2 5

5 0 Y [um]

0 -5

-5

X [um]

15 10

5

5 5 0 Y [um]

0 -5

-5

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Figure 3: The representation of electric potential a. b.

The electric potential representation for a single charge, The electric potential representation for a line of charges obtained by summation of elementary potentials

Consequently, the localization of charges or group of charges is obtained performing triangulation. Importantly, the distribution of charges can then be related to selective activation of ion channels during AP and spatial propagation of electrical signal (Aur and Jog, 2006). 2.1.6. The Projective Field: Spatial Directivity of Charges The activity of the sensory system has been a preferred area of investigation and several studies are concentrated in analyzing visual, auditory and tactile sensory properties. A classic example in this area is the early work of Hubel and Wiesel who studied the dependency of retinal receptive fields on the input stimuli characteristics (Hubel and Wiesel, 1962).

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Additionally, beyond the study of patterns in receptive fields it is important to understand the spatial path where these neurons project the electrical signal. The idea of a projective field is not new. In the 1930's Penfield, the famous Canadian neurosurgeon was able to show that specific muscles can be activated if particular areas from motor cortex are electrically stimulated. Therefore, we should be able to prove the existence of certain electrical paths where information is selectively transferred. Since the projective field characterizes in general the activity of neuronal ensembles, we anticipate that this type of information should somehow originate in neurons. However, to identify where in space specific nerve cell projects information is not a trivial analysis. Staining and labeling studies are unlikely to trace these electric pathways since these projections are rapidly altered within millisecond time range by the occurrences of spikes. For this purpose we have developed the new model that is able to analyze changes in electrical propagation at microscale level within recorded action potentials. The simplest model that we were able to simulate describes the movement of several charges in space during the action potential (AP) generation. If a frame of reference is considered in space as the origin of measurement, then the arrangement of these charges can be characterized globally by their “directivity” over the selected reference system (Figure 4 a, b, Figure 5 a, b).

a

b -3

x 10 15

-3

Electric Potential [V]

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x 10 15 10 5

5 0

5 0 Y [um]

-5

-5

X [um]

10 5

5 5

0 Y [um]

0 -5

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Figure 4: Representation of charge directivity as a measure of spatial distribution of charges over the reference .The frame of reference has the coordinate origin of X axis. Charges are represented in blue color and the directivity is represented in magenta. a Representation of a line of charges in blue and computed directivity in magenta b Since the position of charges over the reference is changed from a, then the computed directivity has an opposite direction

The method for computing the directivity of group of charges and their application to compute spike directivity was detailed in (Aur et al., 2005). Briefly, the method consists of singular value decomp osition (SVD) for the matrix that contains the coordinates of charges in space. The position of charges is determined from recorded voltages using the triangulation technique assuming the charge point approximation.

Chapter 2: Imaging Spikes – Challenges for Brain Computations

a

33

b x 10

-3

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Figure 5: Representative changes in computed “directivity” are determined by different arrangements of charges in space. Charges are represented in blue color and the directivity of spatially distributed charges is represented in magenta. a. An example of spatial distribution of charges in blue color and computed directivity b. A different spatial distribution of charges in blue color and computed directivity

2.1.7. What is “Spike Directivity”? Spike directivity is a new concept that describes electrical activity in a biological neuron. However, since the mathematical formalism is not always easily understood a physical interpretation of spike directivity measure is important. The computational approach underlying “charge movement model” offers such a physical representation in terms of a vectorial representation of every spike. Small differences in recorded voltage of APs from the same neuron (Figure 6, a,b) can be translated into specific spatiotemporal patterns of electrical activations (Figure 7 a,b). Additionally, spike directivity can be estimated for every spike.

-4

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Figure 6 Representation of two sequential spike waveforms recorded from the same nerve cell with the same tetrode. a, Four recorded waveforms and their corresponding electrical patterns of activation are represented in Figure 7a b, Four recorded waveforms and their corresponding electrical patterns of activation are represented in Figure 7b

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

Figure 7 Representative electric patterns of activation from the same neuron in red and computed directivity in blue. The computational processes occur along electrical spatial patterns of activation in dendrites, soma and axon. The axon appears to be strongly involved in electrical flux within each spike. a, Specific electric patterns of activation corresponding to waveforms from Figure 6a b, Specific electric patterns of activation corresponding to waveforms from Figure 6b

These spatio-temporal patterns during each AP are likely to be related with a preferential activation of ion channels in specific regions of nerve cell membrane, dendrites or axon. Indeed, Oesch et al., 2005, using patch-clamp recordings and two-photon calcium imaging demonstrated that dendrites are selective to AP directivity. Equally, the dendritic architecture has an influence on the cell response and these slight spatial changes in spike directivity in several spikes may be determined by a preferential activation of dendritic tree during each AP (Aur and Jog, 2006). In a simplified manner spike directivity can be seen as a reflection of the distribution of charges in space. Since the recordings are made with tetrodes, these spatial distribution of patterns describe electrical characteristics revealed over the frame of reference (tetrode position). The spike directivity term can be described by a preferred direction of propagation of electrical signal during each action potential (AP) and approximated with a vector (Aur et al., 2005). This characteristic can be obtained using tetrodes or any other type of multi-tip receiver (with number of tips n≥4) but cannot be measured based on single electrode recording techniques. Such analyses add an important level of subtlety to information processing by these neurons (Aur et al., 2006). Therefore, when simplified, every spike can be represented as a vector, which characterizes spatial distribution of charges in space. Following this rationale, the group of neurons that exhibits spiking activities can be described as a set of vectors spatially oriented at distinct moments in time. This model even in this simplified form offers a much richer description of neural activity than any current spike timing models. Therefore, beyond their time of occurrence, each spike can be characterized by a new feature called spike directivity. 2.1.8. Triangulation and Independent Component Analysis Together Now, imagine that instead of submarines one would like to determine the position of charges in space during AP generation. Switching back to the issue of AP recordings, the tips of a tetrode function like the sound detectors used by submarines. The movements of charges in space generate electrical currents, which determine the

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potential difference ΔV at the tip of a tetrode. As a result the corresponding current i (t ) produced by charge q can be computed: 1 q 1 q − = ΔV 4πε rB (t ) 4πε rA (t )

(3)

where ε is medium permittivity, and rA (t ) , rB (t ) are distances from the charges to conductor ends. The total drop of potential for several charges in movement can then be seen as a linear result determined by several currents generated by these charges in movement: ΔV (t ) = ∑ α k ik (t ) i =1, N

(4)

where α k are constant. Therefore, the voltage recorded during an AP by the tetrode tip (k) can be written in a discrete form:

V ( k ) = [V ( k ) (1) V ( k ) (2) . . . V ( k ) (n)]

(5)

where n represents the number of points recorded in time from a single spike. Since the recorded voltages are hypothesized to be some mixture of signal sources generated by an ionic flux, then each spike recording can be rewritten for each tip in the simplified form: V = Ms + N n

(6)

where V is data obtained from tetrode tips, M ∈ Rmxn mixing matrix, s is the signal matrix and N n models the noise. In order to provide an internal model of the data, an independent component analysis (ICA) technique is used where for simplicity we have omitted the noise term. Writing sˆ for the estimate of s , and Mˆ for the estimate of M, the estimated signal sources can be obtained after ICA is performed: sˆ = Mˆ −1V

(7)

If the number of observed linear mixtures is at least as large as the number of independent components then Eq. 6 has a solution. Even in the case where m
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Chapter 2: Imaging Spikes – Challenges for Brain Computations

t io ( x − x1 ) 2 +( y − y1 ) 2 + ( z − z1 ) 2 = t i1 x2 + y2 + z2

t io ( x − x 2 ) 2 +( y − y 2 ) 2 + ( z − z 2 ) 2 = ti 2 x2+ y2 + z2

(8)

t io ( x − x3 ) 2 +( y − y3 ) 2 + ( z − z 3 ) 2 = ti3 x2 + y2 + z2

where xi,yi,zi, i=0,1,2,3 are the coordinates of the four channel tips. Several details for this algorithm have already been published in (Aur et al., 2005). The position p k of electrical sources (i.e. where recorded signals are coming from) is being generated can be obtained with an iterative Newton-Raphson algorithm that solves the nonlinear system Eq (8): p k +1 = p k - J −1 F(p k ), k ∈ N

(9)

where matrix F is: ⎡ a − ti1 (( p11 − x1 ) 2 + ( p21 − y1 ) 2 + ( p31 − z1 ) 2 ) ⎤ ⎢ ⎥ F = ⎢a − ti 2 (( p11 − x2 ) 2 + ( p21 − y2 ) 2 + ( p31 − z 2 ) 2 )⎥ ⎢ a − ti 3 (( p11 − x3 ) 2 + ( p21 − y3 ) 2 + ( p31 − z3 ) 2 ) ⎥ ⎣ ⎦

( 10 )

and the value of a is computed: a = ti 0 ( p112 + p212 + p312 )

( 11 )

The value of the Jacobian, J has to be computed at each step: ⎡ ti 0 p11 − ti1 ( p11 − x1 ) ti 0 p21 − ti1 ( p21 − y1 ) ti 0 p31 − ti1 ( p31 − z1 ) ⎤ J = ⎢⎢ti 0 p11 − ti 2 ( p11 − x2 ) ti 0 p21 − ti 2 ( p21 − y2 ) ti 0 p31 − ti 2 ( p31 − z2 ) ⎥⎥ ⎢⎣ti 0 p11 − ti 3 ( p11 − x3 ) ti 0 p21 − ti 3 ( p21 − y3 ) ti 0 p31 − ti3 ( p31 − z31 )⎥⎦

( 12 )

The number of solutions provided by the Newton-Raphson algorithm is equal with the number of blue dots represented in Figure 9. Therefore, in order to improve image resolutions a higher sampling frequency could be used for recording the data. Currently employed low frequencies between 20 and 30 KHz generate low-resolution images for electrical activity. However, if the sampling frequency is increased by about 10 times the resolution would improve significantly since there will be about 320 data values recorded for each spike. Due to several nonlinearities the computed trajectories are bent. Therefore a “debending” technique needs to be performed by first constructing the symmetric plot and subsequently rotating the image to get a better image of electrical patterns as in Figure 8 (Aur et al., 2005). Since during every AP occurrence, voltage dependent channels are activated over specific areas along dendrites soma and the axon, the charge spread reveals parts of the neuronal architecture (Figure 9). Indeed, the spatial displacement of charges during an AP is a way of providing an electrical image of activity of a neuron.

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Z

β = -α α β

Y X Figure 8 The computed and schematic representation of trajectory for linear charge movement in the tetrode space. The “de-bending” is performed in two steps. First the symmetric plot schematically shown in the figure is constructed and then in the second step the image is rotated in order to perceive trajectory patterns from the axon.

Figure 9 A 2D-view of electrical potential during an AP estimated on spatial position of charges represented in blue color. These patterns of activation (micromaps) occur during an action potential due to spatial distributions of charges. The division on the X and Y-axes are in microns.

In fact, changes in AP amplitude and width are correlated with spatial distribution of densities of charges in specific areas. It is likely that this variability in charge distribution is a result of the dynamics of ion channels, which are selectively activated during the electrical event of an AP. The patterns may occur along ion channels within dendrites, soma and axon. During every spike, the axon always appears to be strongly activated which seems to be facilitated by the highly conductible nature of the axonal membrane. In summary, the steps required for imaging electrical activities during an action potential are (Figure 10) a) Perform the ICA algorithm for all four channels b) Construct the transformation T1, as the law of cosines ui=T1(s). d) Compute the position pk of electrical sources. e) De-bending by constructing a symmetric plot.

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

T1

ICA

v

sˆ

p k+1 = pk - J− 1 F(pk )

u

p

Figure 10: A schematic representations of required phases for imaging electrical activities during AP

This picture along with other images of analyzed spikes of neurons suggests that the AP cannot be considered a static stereotypical event. Using this technique, spatial patterns determined by the movement of charges during several successive AP from the same neuron are slightly different. Therefore, viewed as flux of charges, spikes are clearly not the same. This is important because the variability that is clearly pointed out in charge flux from spike to spike is a window into the subtle and complex computations that are physically occurring in neuron. It is the first step in realizing that: 1. Simply considering the spike as a stereotypic, all-or-none phenomenon is inadequate 2. There are many intrinsic and deeper layers of information processing that are occurring within neuron that cannot be ignored. 3. Charge flux analysis is a start of opening this new window and is the first step in understanding neural information processing that is discussed further in subsequent chapters. The obvious questions then are: Why do these electric patterns of activity (micromaps) exist within action potentials? What is their role? Could it be that these specific electric patterns of activation within neuronal spikes were correlated with computational processes performed by this intrinsic dynamic and interaction of charges? Does the underlying spike propagation and morphology reflect the intracellular information processing? It is already known that charges that move in space in an electric field intrinsically perform computations (Moore, 1990). Therefore, at this microscopic level it is probable that computation can be linked to changes in electric field and this physical movement of charges. Based on these observations, we suggest that spatiotemporal activation of ion channels and interaction of electric charges can be used to account for fundamental computational aspects that are observed. The current models, based solely in terms of a temporal representation can be easily incorporated in the new general theoretical framework. Additionally, it is likely that this phenomenon of electrical patterns within AP can be related to other processes including neurotransmitters, synaptic transmission. These aspects will be discussed in detail in Chapter 4. 2.2. From Spikes to Behavior The idea that neurons communicate electrically, thereby causing the generation of spikes and the relationship of spikes to behavior has been the central theme of research for many years (Cajal, 1909; Eccles, 1982). Typically, spikes were perceived to have stereotyped waveforms that can be reduced to an all or none event (Bialek et al., 1993; Strong et al., 1998; Maass et al., 1999; Purwins et al., 2000; Gerstner and Kistler 2002;

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Brody and Hopfield, 2003; Koch et al., 2004). Such reduction allowed linking of neuronal activity to behavior on a temporal basis (Wilson and McNaughton, 1993; Jog et al., 1999, Barnes et al. 2005). Thus, temporal analysis of spike trains became a standard method to understand the relationship of spiking to behavior. However, outside of relating spikes to behavior, the idea of semantics or meaning within the neural activity, started to gain ground even earlier (O'Keefe and Dostrovsky, 1971). Spatial location of the rat was associated with changes in firing rates in so called "place cells" from hippocampus. However, recent investigations show that place cells in the hippocampus appear to alter their preferred firing even in the absence of any changes in the environment (Lee et al., 2006). The predictability of the animal’s location based on firing in the “place cells” may therefore not be as robust as once thought and such temporal analyses may be too restrictive in their ability to define the complexity of neuronal activity. Finally, the current view of behavioral representation is based on population coding and not based on the activity in single neurons (Passaglia et al., 1997). We have already demonstrated above that the spike is a complex observable phenomenon that is potentally rich in information (see chapter 4). In the next sections of this chapter we will continue to use the above model of charge dynamics and extend it to the concepts of semantics. Specifically, we will show that the activity in selected single neurons from the striatum reflects behavior and even more particularly that these neurons demonstrate semantic differences regarding the emerging behavior. Due to its critical importance in motor learning and the substantial amount of existing data, we will use the striatum in order to demonstrate the relationship between spike directivity and semantics of complex behavior during procedural learning. We feel that such techniques could be applied to analyze any other area of the nervous system. The role of the striatum in learning stimulus-response associations, action selection or habit formation is unquestionable and has been discussed and demonstrated in several papers (Alexander et al., 1986; Jog et al. 1999, Packard and Knowlton, 2002; Devan and White, 1999; Yin and Knowlton, 2004; Barnes et al. 2005). The dorsal striatum, which receives a substantial component of its input from the sensorimotor cortex, plays a central role in behavioral action selection studies (Jog et al., 1999; Matell et al., 2003; Barnes et al., 2005). It is for this reason that the dorso-lateral striatum was chosen in the current study to examine the relationship between observed behavior and spike directivity properties from neurons. The presence of the so-called “expert” neurons in the striatum has been advanced for some time (Jog et al., 1999, Apicella P., 2002, Kimura et al., 2003). The term “expert” neuron was used before by Barnes et al., relating spike timing of neurons from the striatum to meaningful events of the task on the T-maze (Barnes et al., 2005).We have already shown that using a charge movement model (CMM) it is possible to compute spike directivity for every action potential (Aur et al.,2005). Additionally, performing information theoretic analyses, we have been able to reveal that during rewarded T-maze learning tasks, spike directivities of “expert” neurons from the dorsolateral striatum become organized with behavioral learning (Aur and Jog, 2007a).The important question is what the newly described feature of spike directivity adds to our understanding of the relationship to behavior. The current discussion describes the value of this organization of spike directivity in terms of what it represents within behavior. In these recordings, every tetrode implanted in the dorsolateral striatum is considered to form a frame of reference over which the electrical flow of charges in space can be analyzed during each individual

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spike and trains of spikes. In this view, the directivity property of a spike is embedded in a vector that reflects the flow of charges over the frame of reference (tetrode) for every spike. Having already shown the organization of directivity as rats learnt a procedural task (Aur and Jog, 2006) we embarked on finding the semantic relevance of this organizational process by analyzing the T-maze data during the time that is of the highest value for action selection in order to obtain the reward. This is the time between the cue (tone) delivery and the beginning of turn where spiking activity of striatal neurons is reduced with learning (Jog et al., 1999, Barnes et al., 2005). We have shown (Aur and Jog, 2007a;Aur and Jog, 2007b) that in animals which have already acquired the task, after tone delivery, spatial modulation of spike directivity (during this time of highest value) predicts the correct turn that the animal would make in order to get the reward. The details of these experiments, the protocol and the techniques are summarized below. 2.2.1. Behavioral procedures The training procedure is summarized here. The T-maze learning sessions began by opening of the start gate. On the maze during each trial, one of two tones (2 and 6 kHz pure tones, ~70 dB) was presented. By their low or high tone these stimuli indicate the position of food (chocolate sprinkles), on the right or left arm of the maze and determined the direction of turn. After few training sessions rats are able to discriminate the food position by auditory instruction cues and turn appropriately. Tone-turn arm assignments were randomized. During each session approximately 40 trials separated by 1-2 minute inter-trial intervals were given. Each rat was required to reach the correct goal in at least 70% of trials in a session to attain the acquisition criterion for significant correct performance. The number of training sessions ranged from 15 to 24. Daily recordings were made for an average of 39 ± 3 trials for every session. Tetrodes were not moved within each session during the time of data acquisition. Tetrodes position was adjusted by few microns (no more than 20 to 40 microns) between sessions to improve data quality if necessary. 2.2.2. Spike sorting and unit classification We analyzed electrical activity of isolated units from the dorsolateral portion of the striatum of 3 Sprague-Dawley rats. Tetrode recordings were obtained with established methods (Wilson and McNaughton, 1993, Jog et al., 2002, Buzsáki, 2004) and data was captured at an acquisition rate of about 25 KHz per channel using a Neuralynx@ data acquisition system. On average, 6 tetrodes were available for analysis in each animal. Subsequent processing included clustering into putative neurons and de-noising of the data. The details of spike activity estimation and the spike directivity computation based on the charge movement model are briefly explained and presented below. The activity of neurons in the sensorimotor striatum was recorded chronically during behavioural training on a conditional reward-based T-maze task. All animals were maintained on feeding restriction of not less than 80% of baseline weight required for behavioral training. The animals were anesthetized, a burr hole was drilled for the purposes of tetrode penetration (for striatum: AP 9.2mm, DV 5.9mm, L 3.5mm) and dura was removed. The headstage drive was lowered such that the cannula holding the tetrodes just touched the surface of the brain (Jog et al., 2002). Upon the rat awakening

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postoperatively, the tetrodes were lowered out of the cannula. Tetrodes were advanced partially on each day so as to allow the brain to settle. The brain targets were reached by day 3 or 4 post-operatively. Recordings commenced after this was achieved. All procedures were approved by the animal care facilities at the University of Western Ontario, London, Ontario. After classifying spikes into putative cells it is necessary to determine whether the spikes are correctly assigned to the determined units (Jog et al., 2002).Measures of unit isolation quality, Lratio and isolation distance were used to evaluate the performance of the sorting technique (Schmitzer-Torbert et al., 2005). Within recording sessions the location of the cells relative to the tetrode was considered stable only for spike recordings where the AP amplitudes and their profiles did not change significantly. Only data from well-separated units were used in computing spike directivity. Additionally, the quality of sorting was tested by analyzing autocorrelograms and corresponding refractory period of single neurons. 2.2.3. Computing and Analyzing Spike Directivity The specific aspects of spike directivity computation based on the charge movement model are summarized above and presented in detail in (Aur et al., 2005). Briefly, the following steps were utilized. Using the triangulation method and the point charge model, the trajectory of electric field propagation (charge flow) was calculated for each spike. Then the obtained trajectory was used to estimate spike directivity. This trajectory was analyzed using singular value decomposition (SVD) for every spike in order to find the best linear approximation of the spike direction. In this scenario, the tetrode acts as a frame of reference. This SVD technique generates three cosines ( v1 , v 2 , v3 ). Higher singular values of the decomposition indicate dimensions with higher energy within the data. By analyzing SVD during each spike, it is clear that about 70-80% of the AP energy has a preferred direction for propagation. Therefore, each AP can be represented by a vector with directivity determined by three cosine angles v1 , v 2 , v3 obtained from SVD decomposition. In order to show the directivity orientation, a PCA analysis is performed for direction cosines in recorded spikes from selected neurons. Analyzing the distribution of direction cosines for the best linear approximation using principal component analysis (PCA) we found two high-density clusters (Aur and Jog, 2007b). These clusters are represented in specific colors for each type of turn (yellow and blue for right, red and magenta for left). Then, during a session, yellow and blue vectors can be plotted to represent these directivities for the right turn related trials on the maze. In a similar way red and magenta arrows are drawn for left turn related trials on the maze. To understand the variability of distributions of the cosine angles the probability distribution is estimated for each cosine angle v1 , v2 , v3 , separately within each of the clusters. This analysis is carried out on computed directivity for each and every neuron both before learning (free exploration) and after the learning stage is achieved. The analysis of spike directivity before and after learning in the selected neuron reveals substantial changes in the probability density function of cosine angles. The recordings used for this analysis have been selected for the same neuron prior to and after

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neuron after the animal had achieved a stable 75% correct turn response. The neuron performs better than random in choosing spiking directivity. a, An example of pdf for the

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rewarded learning on the T-maze task. Only the v3 cosine angle is shown in the Figure 11 and Figure 12. The blue color represents the probability density function (pdf) for the first cluster and the red color the second cluster. A clear difference between pdf shapes for both clustered spikes is visible in Figure 11 and Figure 12. The two graphs in Figure 11 (free exploration) show a higher level of randomness than those in Figure 12 (after learning). This difference in randomness can be quantified using Shannon information entropy that has almost double the value before learning than after T-maze learning. In the presented pdf examples before T-maze learning the mean estimated

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information entropy for v3 cosine angle is H S = 4.2 ± 0.13 bits, while after learning the information entropy decreases substantially around the mean of H S = 2.7 ± 0.13 bits for about the same number of points in the data. This important decrease in the Shannon information entropy although visible in the shape of the pdf reflects the random nature of spiking directivity before learning and substantial organization in “charge flow” after learning. More details about organization in spike directivity can be found in (Aur and Jog, 2006). Having represented AP’s as vectors, the angle between the directions or the semantic distance of two spike vectors by definition can be determined:

Δθ = arccos(

< v, v ' > ) | v || v ' |

( 13 )

In the selected “expert” neuron for several spikes in a session during all T-maze trials the average value of cosines angle v r is computed: vr = avg (vi ) , i ∈ T

( 14 ) L

To obtain the variations of spike directivity during a left trial θ ( j ) we used the computed average v r as the reference value: Δθ L ( j ) = arccos(

< v( j ), vr > ) j∈N | v( j ) || vr |

( 15 )

and in a similar manner for turning right on the maze: Δθ R ( j ) = arccos(

< v( j ), vr > ) j∈N | v( j ) || vr |

( 16 )

2.2.4. Upcoming Choices, Decision and Spike Directivity The recorded signal from one neuron is used to obtain s ( k ), k ∈ N which represents the occurrence or absence of a spike in a neuron during a trial. The occurrence of the spike is assigned a value of 1 ( s(k ) = 1 ) while for non-spiking activity a zero value is assigned ( s( k ) = 0 ). To obtain an estimation of spiking activity sB (k ), k ∈ N , a low pass Butterworth filter of the 2nd order with the cutoff frequency of 10Hz is used to process the spiking signal s (k ) . The result is a smooth curve that shows the variations of spiking activity during each trial. This estimation of spiking activity is performed only for demonstration purposes to compare it with changes in spike directivity shapes. The computed value is not used for analysis at any point afterward. During T-maze sessions (Figure 13, a) all three animals showed the expected improvement revealed in average percentages of correct responses during behavioral learning (Figure 13, b). We recorded a daily average of 30 separate units per animal with six tetrodes each. Events associated with hexadecimal values are recorded simultaneously with neuronal spikes during each trial and color coded for display purposes (Table1).

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Figure 13: Experimental T-maze and behavioral response. a, Schematic representation of the T maze experiment. Events are associated with colors as per Table 1. b, An example of behavioral response that shows average percentages of correct responses (APCR). The cyan color shows low levels of performance while the magenta color shows at least 70% correct responses in the three rats. The vertical bar shows the color scheme used to show the change in performance level across the number of training sessions.

The neurons showed an increase in spiking activity visibly correlated with behavioral events (tone cue, turning on T-maze, etc., Figure 14 a and b). The percentage of units that responded to events in the task increased from approximately 50% to a maximum of 85% of recorded units, corresponding to a χ2 value with P<0.001. The number of units that responded to more than one event rose from 40% to 60%. At the beginning of training, 27% of task-related units responded during turns while by the end of training only 14% responded (P<0.001). Between the events of tone delivery and turn beginning, with learning, the number of responsive units decreased as did the actual number of spikes per unit occurring in this time period confirming previously published results (Jog et al., 1999; Barnes et al., 2005) This decrease in activity has been recorded by others but there is no explanation for this decrease during what may be the most important phase of the task. It is during this phase that the animal has to make and commit itself to the decision of the turn direction. Yet, spike-timing analysis is unable to highlight the importance of this phase of behavioral performance. Directivity analysis of the remaining responsive neurons showed increased modulation in spatial directivity with learning as shown below. Table 1: Relationship between events and recorded values during left and right trials

Left

Right

Start Tone Turn Begins Turn Ends Goal

All data analyzed and shown below are after at least 70% correct response stage is achieved. An example of temporal representation of events as recorded from one of the many neurons is shown (Figure 14, a). The lower half of the frame shows event flags as a color-coded bar (per Table 1). The graph above represents actual spikes from this

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sB in red color.

b, An example of temporal representation of events (bottom) where high tone was associated with right arm reward. Occurrence of spikes for the expert neuron is shown in blue color and the estimation of spiking activity s B in red color.

neuron in a single trial, plotted in each behavioral period. Figure 14a shows neuronal activity from the same neuron for a left turn while Figure 14b shows the same activity during the right turn. The filtered version of unit activation sB for the selected neuron is plotted in red color. Neuronal activity occurs upon signal tone delivery, during turning movements on the right or on the left of the T-maze and before or during attaining the goal. It is clear that the same expert neuron shows activity for both turns. At the end of the learning period, all neurons that are linked to the learning activity of cued turn showed this dual turn related responsivity. Other neurons showed a diversity of activities including response to start, tone or goal (analyzed but not shown). Spiketiming analysis seems not rich enough to effectively show this fact nor can it explain the phenomenon. Spatial directivity is now calculated for neurons spiking during different phases of the T-maze task. Every spike yields only one vector (Aur et al., 2005). It is important to note that spike directivity analysis is carried out on neurons after the animal has already acquired the task and organization in spike directivity has occurred. The fact that directivity is reorganized during learning has already been demonstrated above by computing Shannon information entropy. High values of Shannon information entropy reflect the random nature of spiking directivity before learning and reduced values show substantial organization of spike directivity after learning (Aur and Jog, 2006). The change of direction cosines during the whole trial is first computed in selected neurons for left and right turns. An example of these shapes for cosines angle variations is presented in Figure 15a, b for the left and for the right turn trials on T-maze. At least two distinctive parts can be easily seen to exist in these plots for both the right and left turn trials. In each trace of the change in cosines angle during the task, the

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a

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Figure 15: Changes in cosines angle in an expert neuron from striatum as related to events (bottom) over time during the whole trial. Event flags are represented as a bar, color coded as in Table 1. a, An example of temporal representation of events (bottom) with low tone for turning left and variation of directivity angle Δθ in blue color. b, Temporal representation of events (bottom) with high tone for turning right and variation of L

directivity angle

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first change occurs in Δθ L or Δθ R angle during or just after tone cue event. The second distinctive change in Δθ L or Δθ R angle appears just before or during left or right turns. The change of angle during movement shows similar variability during several trials. This characteristic profile is akin to a signature. In order to highlight the powerful and unique aspect of spike directivity not detectable by using spike timing alone, the directivity vector of every spike occurring between start and tone (Figure 16, a) and tone and beginning of turn (Figure 16, b) is plotted for about 20 trials and between beginning of turn and goal (Figure 16, c) for a single trial. Sequential spikes from the same expert neuron during a specified turn show variability in directivity as shown in Figure 15. Rotation of the three obtained 3D images showed a separation for spike directivity associated with the left versus the right turn trials on the T-maze only in Figure 16b. This separation between left and right turn related spikes cannot be seen in Figure 16a or Figure 16c in any of the analyzed neurons. It is important to note that for Figure 16b directivity is calculated for spikes between tone delivery and before the turn has actually begun and not calculated for those spikes that occur during the actual turn. This analysis demonstrates that the portion of the trial that has the lowest firing rate (after the learning is achieved), is the period of time that is most representative. It is in this period of time that the animal is deciding what turn it is required to make. It is crucial to note that this feature of predictability of an upcoming event is not demonstrable by using the spike counts (firing rate analysis) or timing of spike occurrence. Once learning progresses, neurons from the striatum significantly reduced their number of spikes between when the gate is open and turn is performed while the number of spikes between the turn and goal phase show an increase. This temporal reorganization phenomenon of neuronal activity was previously described in detail (Jog et al., 2007, Barnes et al., 2005) and can be directly observed in the number of spikes represented in Figure 16c and Figure 15a, b. While spike directivity for the data between when the gate opening and turn can be computed for every spike and represented as vectors (Figure 16, b), a reliable statistical analysis of these data is considerably limited due to the low numbers of spikes.

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Figure 16: Representation of spike directivity vectors for spikes in an expert neuron. Left turn trial is represented by yellow and blue arrows while the right turn trial is represented by red and magenta arrows. a, Spike directivity vectors of all spikes between gate opens and tone in about 20 trials b, Spike directivity vectors of all spikes between tone and turn starts in about 20 trials c, Representative spike directivity vectors of all spikes between beginning of turn and goal. Only one trial is shown here for better visualization due to larger number of spikes.

Figure 17 a, b show the cosines angle changes from the same neuron in three successive trials for both right and left turns while Figure 17c and d show corresponding spikes (for which the directivity analysis was performed) in blue color and spike activity estimates in red color. An important observation here is that the profiles for left versus right turns are visibly similar with slight changes from trial to trial, both in the temporal and the spatial directivity plots. Finally, these changes in the cosines angle can be compared to what happens in the traditional, spike timing view during a trial. This comparison allows us to reveal powerful similarities in dynamic variations in spike directivity and temporal activity. A correspondence between changes in the firing rate and modulation of spike directivity appears to exist as seen in Figure 17a-d. Thus when a significant change is seen in the timing of spike activity during a trial, there is a corresponding change in cosines angle. plotted for spikes (Figure 17: a versus c; b versus d). Three such trials are represented for the left turn and right turn trials respectively. The variation in cosines angle for every trial is similar to the corresponding changes in timing of spike occurrence.

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2.2.5. Learning: One Spike Seems Enough In the absence of any reward or punishment the decrease in behavioral responsiveness follows a repetitive presentation of a stimulus. After several repetitions of a stimulus there is a general tendency to become familiar with the stimulus. This phenomenon of learning was observed from Aplysia (Bailey and Chen, 1983) to cats (Groves et al., 1969) and was called habituation. Thompson and Spencer tried to characterize habituation using their experimental analyses (Thompson and Spencer, 1966). They showed that the repeated application of a stimulus leads to a decreased response. This is like the animal that learns how to ignore an input that does not show any new information. In case the stimulus is withdrawn, the response improves over time. Many neurons habituate during a repeated stimulation. If the stimulus presentation is more rapid then the habituation is faster. Potentiation or faster habituation occurs also if training and recovery phases are used. Usually habituation has not been associated with synaptic accommodation determined by an exhaustion of neurotransmitter at the synapse, which also may result

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in a severe decrease in behavioral responsiveness. McFadden and Koshland showed that alteration of the habituation pattern is caused by activators of the protein kinase pathway and changes in the activity of voltage-gated calcium channels (McFadden and Koshland, 1990). These observations link habituation patterns at temporal scales of spike trains with significant changes that occur at a molecular level. However, longterm potentiation (LTP), a relatively long-lived increase in synaptic strength, remains the most popular model for the cellular process that may underlie information storage within neural systems. All these experiments showed reduced activity of neuronal firing. Different biophysical explanations that range from protein folding to neurotransmitter and ion fluxes modification have been hypothesized to explain this phenomenon. Classical explanation of changes in wiring or synaptic plasticity does not seem to address the intrinsic physical aspect and the real meaning in terms of a computational model for such mechanism is still in debate. The answer is simple and comes from an analysis of electrical flow in specific neurons that fire before and during the learning process. The reduction in firing rate is directly related to organization of spike directivity at the choice point as presented above and in (Aur and Jog 2006). Using the tetrode system as a frame of reference, we have shown that an analysis of the newly described feature of spike directivity has important implications to understanding neuronal activity (Aur and Jog, 2006; Aur and Jog, 2007a). Based on electrical recordings and computational methods we are able to show that in certain neurons, spike directivity changes occur in a determined, behaviorally dependent fashion with events on the T-maze. Spike directivity is a hidden characteristic. Unlike spike timing, spike directivity cannot be directly perceived by experimental observation; it has to be computed in advance to allow further analysis. Additionally, as spike counts may decrease with learning, firing rate dependent analyses and ISI statistics cannot be performed. Since spike directivity is computed for every spike, only few spikes are required to exploit the significance of this feature. Coupled with spike timing analysis, our work uses spike directivity to reveal a hitherto unrecognized yet important aspect of how behavior is expressed by neurons. The first result is that “expert” spiking units responded to more than one meaningful event. This suggests that spiking behavior of each cell is likely to represent more than one class of events/features and does not imply the existence of individual neurons coding uniquely for certain discrete events. Since neurons receive inputs from a variety of sources the response pattern will vary significantly depending upon the nature of the input. This provides a much larger range of response sensitivity for the ‘expert’ neuron which can be seen in the temporal patterns of response (Figure 14a and b) and can be equally well shown in the directivity analysis of spikes from the same neurons (Figure 15a and b). Such results confirm the published observations that neurons do respond by changing spiking activity in specific periods of the T-maze behavior. The changes in the cosines angle in spike directivity have similar inflections as spike activity plots, corresponding to the same phases in the same neuron. Therefore, alterations in spatial directivity occur simultaneously to changes in the spike timing. This could imply that spike directivity characteristics are inclusive of what is shown by spike timing analysis techniques. It is likely that immediately after delivery of the auditory tone, action selection of which turn is to be taken occurs, which is then followed by turn execution. However, this had never actually been demonstrated before as has been with spike directivity. Neurons that respond to meaningful events of the task appear to have robust firing rate

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changes relating to periods corresponding to “before” (seen as the “decision” period) and “during” the actual turn (seen as an “execution” period). This implies that action selection and execution can be represented in the same expert neuron as shown in both the temporal and spike directivity analyses. Such neurons can then be seen to combine spatial directivity of spiking with time contingent properties. Yet a crucial difference exists between what we can gain from spike time versus directivity analysis. If merely the spiking activity is considered, one can see clearly the occurrence of turn selection and execution in a successive manner. However, based on estimated spike timing activity (Figure 14 a and b, Figure 17c and d), an observer would be unlikely to determine the direction of the turn that the animal may select since there is no significant difference between any specific temporal spike activity corresponding to either direction of turn. This crucial piece of information that recognizes which turn selection is valuable, i.e. has the potential to result in reward is however elegantly revealed in the spike directivity characteristic. The current result demonstrates that spike directivity during the period between tone and beginning of turn highlights the arm to be selected, information which is of great value to the animal in order to obtain reward. Most significantly, this turn oriented separation in spike directivity does not occur in the previous phase, between gate opening and tone delivery (Figure 16a) and is also not present after the selection of turn between the turn start and when goal is achieved (Figure 16c). The conclusion is then obvious that upon tone delivery, the spike directivity distinguishes the left versus the right turn represents the selection phase. Since correct selection increases the chances of reward, this intriguing result implies that the spike directivity predicts which turn is subsequently to be executed. Summarizing then, we show that the same “expert” neurons respond to multiple events of the T-maze task, that they show response modulations in spiking frequency (temporal domain) and in spike directivity (spatial domain). This temporal and spatial response sensitivity occurs to similar events in the task. In addition, we show that spike directivity can be correlated with meaning in terms of which turn selection would have the highest likelihood of obtaining reward. We anticipate that these changes in spike directivity are a result and a manifestation of a form of neuronal plasticity. Since electrical flux of charges carry information (Aur et al., 2006) “expert” neurons can be seen as projecting information related to behavior, temporally and spatially, to different targets at certain moments in time. The directionality of charge flow as observed by the tetrodes is a reflection of intracellular and dendritic events (Figure 7 a,b) that we have been able to obtain using ICA techniques and these electrical patterns may to some extent be determined by the received inputs (Aur and Jog, 2006). Finally and most importantly, our work has shown that predictability of the upcoming behavior is an inherent property of analyzed neurons as demonstrated by their spatio-temporal dynamics. Based on measured electrical activity from tetrode recordings, the demonstration of such a subtle yet necessary feature is in essence equivalent to decoding the neural code. Current investigations of neuronal coding are restricted to time domain analyses where computation of firing rate or interspike interval assumes the stereotypy of action potential (AP). However, viewed with multi-tip receivers (tetrode) the sequential APs from one neuron are not alike. The present study reveals the existence of an intrinsic spatial code within neuronal spikes that predicts behavior. As rats learnt a T-maze procedural task, simultaneous changes in temporal occurrence of spikes and spike directivity are evidenced in “expert” neurons. While the number of spikes between the

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tone delivery and the beginning of turn phase reduced with learning, the generated spikes between these two events acquired behavioral meaning that is of highest value for action selection. If the tetrode is considered a frame of reference, spike directivity characterizes the propagation of charges in every recorded action potential. This hidden feature can be computed and then graphically represented as a vector pointing in space to the center of mass of charges, which move during an AP. Since this electric propagation in every spike transfers information, spike directivity provides a vectorial representation of information propagation. Thus the analysis of spike directivity reveals the semantics of each spike and in the current experiment, predicts the correct turn that the animal would subsequently make to obtain reward. Semantic representation of behavior can then be revealed as spatial modulations of spike directivity. The predictability of observed behavior based on these subtle changes in spike directivity represents an important step towards decoding and understanding the underlying neural code. Even though temporal aspects including synchrony or oscillations seem to be important often we are not able to obtain information regarding the intrinsic mechanism of neural coding. In this book we propose to demonstrate that for the first time it is now possible to relate the neuronal activities with animal behavior and importantly with behavioral semantics. These results are of value in determining representations of distributed spike computations. During the action selection phase, the number of spikes decreases with learning. However, the spike directivity analysis reveals inherent additional decision-making value in indicating the turn arm being selected. It is intriguing to suggest that the occurring spikes are now more efficient at providing information about the task. After learning is achieved by the rat, analyzing electrical characteristics of only one spike recorded from a selected expert neuron would be enough to know in advance the arm where the rat will move to get reward. One spike seems to be enough to know in advance the decision that the rat may make. In this context the cosine between two spike directivity vectors from the same neuron or just their scalar product can be considered to measure the semantic distance between neuronal spikes. The computed semantic distance of spikes shows the similarity of meaning between two spikes from the same neuron implying that the more separated the directivity between spikes, the more distinct the spikes are semantically. • Spike directivity and its modulations challenge the traditionally held fundamental idea of ‘binary pulses’ or spike stereotypy of action potentials Indeed one may predict the animals’ turn direction on the T-maze simply based upon the directivity of spikes in the selection period. Once organized, at a point when the animal has acquired the task, the rat “understands” the meaning of the low or high tone and selects the turning arm to maximize the way of getting the reward. In our work, the use of tetrodes provides a local frame of reference that allows the visualization of these subtle changes in time and space on a small single spike-level scale. Communication in the brain appears to be represented by such spatiotemporal electrical flux on a small scale, which contains behaviorally meaningful data that is exchanged between anatomically interconnected structures. Such electrical waves can be effectively measured in the brain in recordings such as field potentials or electroencephalography and identified as rhythms of the brain (Buzsáki, 2006).

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Summary of Principal Concepts •

• • • • • • • • • • •

The origins of several gaps in neuroscience are buried in early single electrode recordings (Adrian 1929; Eccles, 1957) where action potentials have been considered to be similar uniform pulses of electricity and only their temporal variability of occurrence was perceived. New technological ideas and developed techniques based on tetrode recordings prove that action potentials recorded from the same neuron display changes in electrical patterns not just temporal variability The action potentials as electric events can be spatially modulated and therefore “point source” models or the “line source” models are inadequate to describe neuronal activity A new approach, the model of charges in movement (charge movement model) is introduced as a description of electrical activity during AP generation in neurons The directivity of electrical propagation modulated by dendritic activity (axonal branches) within every spike (spike directivity) can be computed if the charge movement model is combined with a triangulation technique Additionally, using independent component analysis of recorded signals and triangulation techniques with this new model of charges in movement the electric patterns of activation within neuronal spikes can be represented. Such electrical patterns that occur during AP propagation are likely to be a result and reflect intra-cellular information processing Spike directivity can be used as an effective characteristic that can explain changes in neural in activity during learning a T-maze procedural task Spike directivity organizes with learning. The generated spikes show responsiveness to the acquired behavioral meaning that is of the highest value for action selection and decision mechanism. Spikes are not alike, the reduction in firing rate during learning is correlated with an increase of significance (value) of few delivered action potentials With learning spatiotemporal electrical patterns occurring within analyzed spikes from expert neurons reflect behaviorally meaningful data Since two different spikes from the same neuron can carry different semantic we cannot add spikes, we cannot average them or measure ISI without loosing important information regarding the semantics of the stimulus or behavior.

References Alexander, G. E., DeLong, M. R. and Strick, P. L. (1986) in Annu Rev Neurosci (eds. Cowan, W. M., Shooter, E. M., Stevens, C. F. & Thompson, R. F.) 357-381, Annual Reviews, Palo Alto. Amari, S. Chen T.P, Cichocki A., (1997) Stability analysis of learning algorithms for blind source separation, Neural Networks, Vol.10, No8, 1345-1351, Apicella P., (2002), Tonically active neurons in the primate striatum and their role in the processing of information about motivationally relevant events. Eur. J. Neurosci. Dec; 16(11):2017-26. Aur D., Connolly C.I., and Jog M.S., (2005) Computing spike directivity with tetrodes. J. Neurosci. Vol. 149, Issue 1, pp. 57-63. Aur D., Connolly C.I. and Jog M.S., (2006) Computing Information in Neuronal Spikes, Neural Processing Letters, 23:183-199. Aur D., Jog MS (2006), Building Spike Representation in Tetrodes, Journal of Neuroscience Methods, vol. 157, Issue 2, 364-373.

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Aur D., Jog, M., (2007a) Reading the Neural Code: What do Spikes Mean for Behavior? Nature Precedings, Aur D, Jog MS (2007b), Neuronal spatial learning, Neural Processing Letters, vol. 25, No 1, pp 31-47. Bailey C.H., Chen M. (1983) Morphological basis of long-term habituation and sensitization in Aplysia. Science 220:91–93 Barlow, H. B. (1972) Single units and sensation: a neuron doctrine for perceptual psychology. Perception, 1:371-394 Barnes T.D., Kubota Y., Hu D., Jin D.Z.and Graybiel A.M., (2005) Activity of striatal neurons reflects dynamic encoding and recoding of procedural memories, vol 437, Nr. 7062 pp. 1158-1161. Bialek W., DeWeese M., Rieke F. and Warland D., (1993) Bits and brains: Information flow in the nervous system Physica A: Statistical and Theoretical Physics, Volume 200, 1-4, 581-593 Brody C, D. and Hopfield J. J., (2003) Simple Networks for Spike-Timing-Based Computation, with Application to Olfactory Processing ,Neuron, Volume 37, Issue 5, 843-852. Buzsáki G., (2006) Rhythms of the Brain, Oxford University Press. Buzsáki G., (2004) Large-scale recording of neuronal ensembles. Nat. Neurosci., 7:5. Cajal S. R., (1909) Histology of the Nervous System of Man and Vertebrates (trans. N. Swanson and L. W. Swanson 1995), New York: Oxford University Press. Cardoso J.F., 1991.Super-symmetric decomposition of the fourth-order cumulant tensor, blind identification of more sources than sensors. In Proc. ICASSP'91, pp. 3109-3112, Comon P, (1994) Independent Component Analysis: a new concept? Signal Processing, Elsevier, 36(3):287-314 Debanne D., (2004) Information processing in the axon, Nature Reviews Neuroscience 5 (4), pp. 304-316. Devan, B. D. and White, N. M., (1999) Parallel information processing in the dorsal striatum: relation to hippocampal function. J Neurosci 19, 2789-2798. Drake KL, Wise KD, Farraye J, Anderson DJ, BeMent SL. 1988 Performance of planar multisite microprobes in recording extracellular single-unit intracortical activity. IEEE Trans Biomed. Eng.; 35: 719. Eccles JC. (1982) The synapse: from electrical to chemical transmission. Annu. Rev. Neurosci. 5:32539. Eccles J. C. (1957) The Physiology of Nerve Cells. Baltimore: Johns Hopkins Press. Fee M.S., Mitra P.P. and Kleinfeld D., (1996) Automatic sorting of multiple unit neuronal signals in the presence of anisotropic and non-Gaussian variability. J. Neurosci. Methods 69 (1996), pp. 175–188. Fulton, J. F., (1938) Physiology of the nervous system. (Oxford Med. Publ,) New York, OxfordUniv. Press. Gold C, Henze DA, Koch C, Buzsáki G., (2006) On the origin of the extracellular action potential waveform: amodeling study. J Neurophysiol 95: 3113-3128. Groves, P. M., Lee, D., and Thompson, R. F. (1969). Effects of stimulus frequency and intensityon habituation and sensitization in acute spinal cat. Physiology and Behavior, 4, 383-388. Hodgkin A., and Huxley, A., (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544. Hubel D. and Wiesel, T. (1962) Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. Journal of Physiology of London, 160:106—154. Hubel D and Wiesel, T., (1968). Receptive fields and functional architecture of monkey striate cortex. J. Physiol 195, 215–243. Gerstner W and Kistler W. M., (2002) Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press. Holt G.R. and Koch, C. (1999) Electrical Interactions via the Extracellular Potential Near Cell Bodies. Journal of Computational Neuroscience, 6 (2). pp. 169-184. Jog M.S., Aur, D., Connolly C.I., (2007) Is there a Tipping Point in Neuronal Ensembles during Learning? Neuroscience Letters Volume 412, Issue1, pp. 39-44 Jog M.S., Kubota,Y., Connolly,C.I., Hillegaart,V. and Graybiel, A.M., (1999) Building neural representations of habits. Science 286, 1745-1749. Jog M.S., Connolly C.I., Kubota Y., Iyengar D.R., Garrido L., Harlan R., Graybiel A.M., (2002). Tetrode technology: advances in implantable hardware, neuroimaging, and data analysis techniques. Journal of Neuroscience Methods 117,141-152. Jourjine A., Rickard S., and Yilmaz O., (2000) Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures, in Proc. Int. Conf.Acoust. Speech Signal Process. (ICASSP’00), Istanbul, Turkey, vol. 5, pp. 2985–2988. Kandel ER, Schwartz JH, Jessell TM 2000. Principles of Neural Science, 4th ed. McGraw-Hill, New York. Koch K., McLean J., Berry M., Sterling P., Balasubramanian V. and Freed M. A., (2004) Efficiency of Information Transmission by Retinal Ganglion Cells Current Biology, Vol. 14, Issue 17, pp 1523-1530. Kimura M., Matsumoto, N., Okahashi, K., Ueda, Y., Satoh, T., (2003) Goal-directed, serial and synchronous activation of neurons in the primate striatum. Neuroreport. 14(6):799-802.

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Lee I., Griffin A. L., Zilli E. A., Eichenbaum H. and Hasselmo M. E., (2006) Gradual Translocation of Spatial Correlates of Neuronal Firing in the Hippocampus toward Prospective Reward Locations, Issue 5, pp. 639-650. Lee T.W., Girolami M., Bell A.J., Sejnowski T., (2000) A unifying information-theoretic framework for independent component analysis, Int. J. Comput. Math. Appl. 39 ,11, 1–21, De Lathauwer L., De Moor B., Vandewalle J., (1999) ESAT, ICA techniques for more sources than sensors, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics: 121-124. Lewicki M.S. (1998) A review of methods for spike sorting: the detection and classification of neural action potentials. Network 9 (1998), pp. R53–R78 Lewicki M.S., and Sejnowski T.J., (2000) Learning overcomplete representations. Neural Computation, Vol 12; Part 2, pages 337-366 Maass W., (1997) Networks of Spiking Neurons: The Third Generation of Neural Network Models, Neural Networks -Oxford- Vol 10; Number 9, pages 1659-1671 Maass W., Bishop C.M., Sejnowski T.J., (1999) Pulsed neural networks, Cambridge, Mass, MIT Press. Mainen ZF, Sejnowski T.J., (1995) Reliability of spike timing in neocortical neurons, Science, Vol 268, Issue 5216, 1503-1506. Matell M.S., Meck W.H., Nicolelis M.A., (2003) Interval timing and the encoding of signal duration by ensembles of cortical and striatal neurons, Behav. Neurosci, Vol. 117, No. 4, 760–773. McFadden PN, Koshland DE., Jr (1990), Parallel pathways for habituation in repetitively stimulated PC12 cells. Neuron. Apr; 4(4):615–621 McNaughton BL, Barnes CA, O’Keefe (1983) The contributions of position, direction, and velocity to single unit activity in the hippocampus Exp. Brain Res. 52:41-49. McCulloch W. and Pitts W., (1943) A Logical Calculus of Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics 5:115-133. Moore C., (1990) Unpredictability and undecidability in dynamical systems, Physical Review Letters, 64(20):2354–2357. Oesch, N., Euler, T., and Taylor, W. R., (2005) Direction-selective dendritic action potentials in rabbit retina, Neuron 47(5) 739–750. O'Keefe J. and Dostrovsky, J., (1971) The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat, Brain Res. 34 (1971), pp. 171–175. O'Keefe J., Reece, M. L. (1993) Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3: 317-330, Packard, M.G. and Knowlton, B. J., (2002) Learning and memory functions of the basal ganglia) Annual Review of Neuroscience, Vol. 25: 563-593 Passaglia C, Dodge F., Herzog E., Jackson S.and Barlow R., (1997) Deciphering a neural code for vision, Proc. Natl. Acad. Sci. USA, Vol. 94, pp. 12649-12654, November Neurobiology Purwins H., Blankertz B., Obermayer K., (2000) Computing auditory perception, Organized Sound 5(3): 159–171. Recce and O'Keefe, (1989) The tetrode a new technique for multi-unit extracellular recording. Soc. Neurosci. Abstr 19,1250. Sahani M., (1999) Latent Variable Models for Neural Data Analysis. Ph.D. Dissertation. Computation and Neural Systems. California Institute of Technology, 1999. Schmitzer-Torbert N., Jackson, N., J., Henze, D. Harris K. and Redish A.D., (2005) Quantitative measures of cluster quality for use in extracellular recordings, Neuroscience Volume 131, Issue 1, Pages 1-11. Strong S.P., Koberle, R., Steveninck, R. R., Bialek, W., (1998), Entropy and information in neural spike trains. Phys. Rev. Let., 80, 197–200. Thompson R.F. and Spencer W.A, (1966) Habituation: A model phenomenon for the study of neuronal substrates of behavior, Psychology Review 73 (1966), pp. 16–43 Wilson M.A. and McNaughton, B.L. (1993). Dynamics of the hippocampal ensemble code for space. Science 261, 1055-1058. Yin H. H. and Knowlton, B. J. (2004) Contributions of striatal subregions to place and response learning. Learn Mem 11, 459-63.

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CHAPTER 3. Information and Computation Ever since technological advances have become intergral to science, brain computations have gradually been imagined as switches or nonlinear functions and lately as parallel digital computations. To represent high-level abstractions or complex biological functions (e.g. in vision, language, and other AI-level tasks), one first needs to gain a deeper understanding of the “brain structures”. A key concept of brain function is its ability to encode information. However, many past attempts to describe brain processing have ignored the subtle relationship between information processing and computation. Such ideas of subtle relationship have been prevalent in physics to link the geometry of space, energy and time with information (Bekenstein, 1981, Bekenstein and Schiffer, 1990). It is clear that this progress in an understanding of information is closely related to the existent laws of physics that govern all natural phenomena. We are still far from understanding the link between information and computation in the brain. It is still unclear whether there is a difference between living (neurons) and nonliving (silicon) matter in terms of information processing capability or if the brain intelligence involves more than mere computation (Pfeiffer, 1999). One limitation appears to be the current idea of computation. Chapter three will begin by exploring this vital concept compared to what we understand by it today. Currently our understanding of computational processes seems to be limited to theoretical principles developed by Turing which are supposed to have been built on as an “universal” model (Turing, 1939). 3.1. What is Computation? The study of computation started with the early work of Alan Turing and Kurt Gödel before the Second World War. Initially, this work was seen as a step to advance the field of cryptography. This initial development has evolved into a new area of research known as complexity analysis. However, it can be demonstrated that there is a close link between system computability and the study of complexity. Recent theoretical breakthroughs in both these areas are stepping stones that may help us to understand subtle processes that define brain computations. Usually, when one asks about the limits of computers or computation, such a question is with respect to computations performed by digital computers. This popular view shows that we are unable to picture different types of computation, other than those built within binary, digital machines. However, there are varieties of natural processes that perform complex computations. In fact one can view all processes in the universe as carrying out “computations” (Lloyd, 2006). Even though we are far from the Lloyd extension, the brain is a naturally shaped system where all processes have to be entirely governed by biophysical laws. Seen as a distinct “universe” determined by several steps of natural evolution, the brain is able to perform complex computations. The main issue then is whether we can envision the complexity of brain computations in terms of an extension of the accepted machine models of computation. As mentioned above, Turing machines are universally accepted as a general model that formalizes several types of computations. Can brain computations be assimilated with Turing Machines? Can such

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computational processes be simulated using the von Neumann architectures? These are important questions that need to be answered. The current chapter will begin to address these issues. 3.1.1. Coding and decoding The easiest way to see computation in terms of information processing is simply as the classic example of the “Caesar cipher”. This ciphering technique was used by Caesar to protect several transmitted messages during the war campaigns. A "cipher" line is obtained by shifting the letters of the alphabet. Such operations can be formalized as mathematical transformations. The techniques of encryption (coding) by replacing letters from the original message with letters from a "cipher" line can be formalized and described mathematically. However, even though the letters from the original message can still be read one cannot associate meaning to the new “words” due to the change of letters. This example should remind us of Searle’s experiment where messages written in Chinese are interpreted by the computer. One cannot interpret the semantics hidden in such an encrypted message. However, in this particular case of Caesar cipher the process of decoding (decryption) can be simply performed by obtaining the frequency distribution of the letters in the message. Then, by evaluating the differences between the usual frequency of appearance of letters in a regular text and in the coded text the shift in letters can be assessed. Therefore, after this technique is performed the text can be decoded and interpreted. Importantly, as in the Searle example, the intrinsic semantics can be understood only by the human interpreter after reading the message. Understanding of the coding and decoding processes performed in the brain is essential to model brain computation. In case of the Caesar cipher the decoding procedure can be performed very fast using current computers. However, there is a particular aspect that current computers cannot perform. Reading and interpreting the decoded message or, in other words extracting the semantics behind the text seems to be a unique capability of our brain. Searle showed that this characteristic couldn’t be achieved by any algorithmic processing in current computers. This simple example of ciphering shows that our brain is capable of performing additional tasks beyond simple coding or decoding of messages. While almost all theoretical frameworks in building models of the brain analyze the coding and decoding strategies, the topic of semantics is not extensively covered by a theoretical implementable approach. Is this saying that the interpreter has to be human? No, not necessarily; however it shows that the computational system may require that information processing/storage function needs to be performed differently than that being done in current computers in order to fulfill the criteria of the required “intelligence”. A simple approach to coding and decoding shows the importance of probability distributions (frequency of letters) as a way of addressing a certain code. An extension of this description in terms of frequency of occurrence of events or items representing X

T

Y

Figure 1 Schematic representation of coding as a set of transformations T required to achieve the representation Y from external input X.

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or carrying information was originally developed by Shannon and later expanded into the theory of information (Shannon, 1948). Coding/decoding can be seen as a set of transformations to achieve a representation Y when an external input X is applied (Figure 1).Since coding/decoding is seen essentially as mathematical transformations, functions and operations between functions, can formalize this approach. Therefore coding/ decoding phases can be related to functions that operate on other functions. This basic model of computation shapes a formal description of cognition where the “representation law” is described in terms of coding and decoding functions (Newell, 1990) (1) decode[encode(T)(encode(X))] = T(X)

where X is the external situation or oncoming stimuli and T defines the set of transformations required to achieve the representation. Using the system theory formalism this law can be extended further to a set of internal brain states that match an external input X or which may be considered to convert the arbitrary input X to an observed/measured output Y= T (X). In this example, computation can be theoretically described as information processing or simply as a formal process of data manipulation. When this is applied to the brain, any type of computation that requires and involves physical processes implies that well known physical laws have to be involved in explaining brain computations. Therefore, as for any other processes in the brain, computations start within an arbitrary physical state and then end usually in a different “particular” state. Since the brain computations are physically driven, then it is logical that any theoretical framework applicable to machines can equally become a valid method required to clarify some important issues regarding brain computations. Initially, they can be formulated as a series of specific queries: How is information encoded and decoded in brain? What are the specific processes required in coding and decoding mechanisms? Does the brain function change in a way so as to determine the semantics of input stimuli or behavioral situations and if so, how is this represented in computational terms? Is there a computational formulation that allows us to extend brain coding into the realm of semantics and meaning? We will begin by first providing a brief grounding in computation theory which may help summarize the current knowledge in this field before we highlight specific details of brain and neural function. 3.1.2. Are Turing Machines Universal Models of Computation? As discussed in the introduction, current theoretical computational examples are generally restricted to examples of Turing machines considered to be the universal model of computation. Every Turing machine (TM) can be seen as a simple theoretical computer which transforms input strings into output strings using a sequence of state transitions. Physical aspects of implementation as well as energy requirements are not relevant for theoretical Turing machines (Figure 2). The TM has a finite set of internal states s (k ) ∈ Q , k ∈ N and a fixed design. It reads one binary symbol at a time

i (k ) ∈ Γ , supplied on a tape. Each machine’s action O(k + 1) and D(k + 1) depends only on the input symbol i(k) and internal state s(k) and moves the tape one place in direction D(k + 1) (left L or right R). This sequence is resumed by the next equations: s (k + 1) = G ( s (k ), i (k ))

(2)

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O( k + 1) = F ( s( k ), i ( k ))

D(k + 1) = D( s (k ), i (k ) )

1

0

1

0

(3) (4)

1

Read/Write Head

Figure 2 The Turing machine reads a symbol from the tape and moves the head one cell to the left L or to the right R depending on the internal states s (k ) and received input i(k). While the computations are carried out the Turing machine does not accept any inputs from the external environment. Each output O (k + 1) is determined only by inputs and internal machine states

The Turing machine can read and write on a tape by moving the read-write head to the left and to the right and the transition function μ which describes this phenomenon is: μ : QxΓ → QxΓx{L, R} (5) where Q is the set of states, Γ is the tape alphabet and {L, R} defines the transition to left or to right. The Turing machine follows the rules imposed by the transition function, may accept a certain state or reject a certain state and then the machine halts. For a program which follows the Turing model, halting can be seen as a decision problem and can be described as a sequence of mathematical objects where the algorithm may halt if a certain input i is presented: H = {( M , i ) | the machine M will halt if the input is i

(6)

Importantly, the halting issue is considered to be the first computational problem which has been proven to be undecidable since there is no universal method (or algorithm) that works for all programs and inputs to determine if an algorithm halts. However, there are cases when instead of halting, the machine repeats succesively a set of states and follows a specific loop (oscillations). Halting does not guarantee that the correct answer for a problem is found since sometimes the algorithm may not return any answer. The importance of Turing machines for computation theory derives from the fact that it can simulate any program on any type of von Neumann computers. This simulation provides the solution within a certain amount of time; usually, such computations are much slower. Some problems are easy to solve by TM, some of them are hard while others are unsolvable. Regarding this aspect in terms of complexity theory, several decision problems can be solved using a deterministic Turing machine in a polynomial computation time known as complexity class P. However, a more inclusive class is known to be the non-deterministic polynomial time decision class NP. Such problems can be solved in polynomial time by a

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nondeterministic Turing machine. Different types of algorithms can be simulated by Turing machines and in general, there is an equivalence in terms between recursive algorithms and Turing Machines . One important question is whether Turing machine models are capable of completely describing all forms of computation. The fact that several problems do not have an algorithmic solution was clear even for Turing. However, for several decades after Turing presented this formalism it was believed that TM can describe almost every type of computation. Additionally, these beliefs were strengthened by the Church-Turing thesis, which assumes that what is computable by an algorithm is also computable using a Turing machine. However, lately, it has been realized that this is not the case and several scientists claim that Turing machines are able to model only a small subset of computational possibilities. It is important to remember that TMs do not accept any external inputs while they compute.This considerably limits the capability of machine interaction with the environment and its history dependent behavior (Goldin, 2000). Additionally, in order to perform an operation a computing machine has to transfer information between its internal components. Such transfer has to be physically performed and requires discrete speed limits applicable for information transfer. However, computation time or energy required to perform certain computations are irrelevant for Turing machines. Therefore, it was projected that TM framework does not capture all forms of computations (Wegner, 1997). Since Turing machines compute via a series of discrete steps they provide only a discrete model of computation which cannot fully capture all physical aspects of computation. Recently, new models of TM have been proposed trying to break the limits of the conventional Turing Machine. These versions of Turing machines include fuzzy Turing machines (FTMs), superrecursive or inductive Turing Machines (Burgin and Klinger, 2004) and even quantum Turing machines. It is claimed that such machines can be very efficient and they might have more computational power than classical Turing machines (Wiedermann, 2002; Wiedermann, 2004; Wegner and Eberbach, 2004). Similar to physics at the beginning of the nineteenth century, the theory of computation appears to be evolving and currently there is a healthy debate regarding several forms of computation. However, some new proposed forms of computation are still strongly rejected by the mainstream of science (Cockshott and Michaelson, 2007). By analogy, it seems that a major issue is to limit brain processes to discrete Turing models of computation since such models constrain the analyses to be intrinsically discrete. Experimental evidence from neuroscience(Aur and Jog 2007, Aur et al., 2006) shows that computation can be performed continuously by a physical movement of charges in the brain which interact and intrinsically can carry and store information. This new understanding of computation in the brain challenges the classical assumption a conventional Turing model of computability by the brain. Understanding how the brain computes may also change our perception regarding TM and their supposed “universality”. 3.1.3. Von Neumann Architectures Current electronic computers that sequentially perform tasks are von Neumann architectures (VAN) which store data and instructions as binary values. In essence any such computer architecture has four important components: the memory, the arithmetic logic unit (ALU), the control unit (CPU) and the input-output devices (Figure 6). Von

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Neumann computations are defined by algorithms or programs that use a series of instructions. Every algorithm that runs on a von Neumann machine can compute the same class of functions as a universal Turing machine. The data are read or written from or to the memory in a sequential order. The memory stores data, intermediate results of computation and also the instructions. Therefore computation involves a frequent transfer of information between memory and control processing unit (CPU).

Input/Output

Memory

Control Unit Figure 3: Schematic representation of von Neumann architectures

ALU

This process of data transfer needs time to be performed and generates severe limitations regarding the speed of computation. However, since current computers do not use a tape as memory support unlike the classic Turing implementation, the machine can read and write at any location in memory. With higher data lengths, the time required to perform the task significantly increases since the memory has to be sequentially accessed. Such centralized control and sequential execution of instructions limits the speed of processing and is the main issue with von Neumann style architectures. Additionally, power dissipation and information storage capabilities are other critical issues. The finding of new alternatives for computation that use different principles may provide in the near future solutions for these issues. Inspired by biological life, in the late 1940s von Neumann's work was concentrated on developing novel computational principles that can adaptively change and self-replicate. This approach may challenge current theoretical view and generate in the future some useful, real-world models that may defy current computer architecture. 3.1.4. Reversible Computation Understanding the thermodynamic nature of computation took a long time and generated a strong debate. Landauer showed that the erasure of information in any computational device requires an increase in the thermodynamic entropy (Landauer, 1961). Another important aspect stressed by physicists is that certain computations can be theoretically performed without a required thermodynamic cost (Bennett, 2003). They are called logical reversible computations. Any logical transformation can be logically reversible if and only if it defines a one-to-one mapping. After a reversible

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logical transformation takes place the input state can be uniquely reconstructed based on information provided by the output state. There is a difference between a logical transformation which is a mathematical operation and the physical process required to perform this operation. The associated physical system has to go from an arbitrary physical state to another particular physical state. This physical process can be assimilated under certain conditions to become thermodynamically reversible without loss or dissipation energy. Landauer presented the theoretical solution regarding computation reversibility. If the computation is carried out slowly, then computers can dissipate arbitrarily small amounts of energy per step and therefore reversible computations may be carried out at each step without discarding information (Landauer, 1961; Landauer, 1988). An important aspect is that the laws of physics at microscopic scale are invariant under the reversal of time; therefore one should imagine that such processes could in principle be reversible. However, the transport of information in almost all devices where

0

a

0

0

b

a b

a+b

Output

Input Figure 4: A logically irreversible operations, an adder can be implemented in a mechanically ‘reversible’ way (adapted from Denker, 1994)

dissipative processes occur is an irreversible process. It is expected that if such computation can be carried out slowly enough the process can be physically considered to be close to an adiabatic process (Landauer, 1961; Landauer, 1988). If the speed of electrical propagation is higher then dissipation and energy requirement increases. However, the relationship between logical reversibility and thermodynamic reversibility is still unclear. Mathematically, logical reversibility has to map a given input pattern to a unique output pattern. This behavior cannot be easily determined just based on a thermodynamic analysis and this difference is explained in Figure 4. An adder even though it is not logically reversible can be made thermodynamically reversible (mechanically reversible) in the usual sense. This means that in this operation logical reversibility is not required and does not necessarily imply thermodynamic reversibility. One can see that “a” and “b” are not deleted during this operation and they are preserved (Denker, 1994). This relationship of reversibility between thermodynamic and logical operations is of great importance in the biological systems. The development of processes that are based upon the principles of reversibility of logical and thermodynamic operations have gradually evolved over time and have been shaped to improve process efficiency. These processes have resulted in higher order, yet efficient biological systems that presumably started off as individual

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systems and have now become much more complex. The nervous system can also be expected to have followed a similar process. As was discussed in the previous chapter and will be elaborated in detail in this chapter, electric charges can be envisioned to be the carriers upon which such logical and thermodynamic operations can be performed. From an information processing perspective, a trade off between these two properties results with evolution to maximize efficiency and minimize energy expenditure so as to produce the most informationally rich state. Therefore, one can envision the flow of electrical charges within neurons as an optimal trade off, achieved biologically, between required power for computation and limited sources of energy to provide such power. Mathematically, the processing of information can be seen as a transformation of the incoming signals. These transformations of the input signal may preserve Ndimensionality of the input in a possible reversible framework. Depending on design or in case of a biological system, connectivity, this transformation can map from the Ndimensional space to N-m a lower-dimensional space. Such an operation of dimensionality reduction results in an irreversible transformation and information is lost. This problem would be substantially compounded if the numbers of carriers which perform these operations are constrained. With a small number of charges computation would be unattainable and information transfer would be limited. However, within a given neuron and its environment, an extremely large number of charges exist that make possible to preserve the dimensionality of the process. Therefore, within the neuronal system, a high number of electrical charges are involved in computation. An interesting property is seen in terms of information entropy when a computation occurs. Computation can be seen as being closely related with changes in physical entropy. Although a computation may not intrinsically add ‘new’ information about the input, physical entropy must increase when information is communicated along with the movement of charges. Therefore, computations that require a dynamic transfer of information are commonly dissipative, these processes requiring deletion of information in one part while creating it in another, different part (Parker, and Walker 2007). To understand this computation performed by charges requires an integration of the knowledge gained from information theory with the physics of charge movement and electric interaction. A simple theoretical experiment can reveal the link between spatial movement of electrical charges and statistical properties of generated signals recorded by an electrode (Aur and Jog 2006). If several charges q1 , q2 ,,, qn are →

considered to be distributed in an electric field E then the force acting on each charge is dependent on the electrical field : r r Fi = qi E (7)

r

and can also be written as a product between the mass of charge ai is the acceleration: r r Fi = mi ai (8) →

→

→

→

→

→

Given the initial spatial position r01 , r02 ,…, rn3 and initial velocities v01 , v02 ,.., vn3 the equations of motion can be integrated numerically: r r i r i qi E (9) v = v0 + dt m

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r r r r i = r0i + v dt

63

( 10 )

and the trajectories can be plotted. Electrical signals generated by moving several charges in an electric field can be used to relate changes in information with changes in charge trajectories. These signals generated in the conductor can be statistically analyzed using mutual information (MI) measures (Aur, et al., 2006). This simple theoretical experiment above shows that if charges have similar trajectories in space they provide similar information while if their dynamics include trajectories that are far apart then the transmitted information may be different. In other words charges increase their statistical independence if they move far apart. Therefore, during every AP each charge or more accurately several group of charges that move independently in space could be treated as satisfying a certain degree of statistical independence. Additionally, the transfer of information between two charges is maximized if their trajectory is close enough. It is likely that charges that move in the same region in neuron carry similar information. Since writing down any information is a physical process a neuron or a group of neurons that is topologically located in certain location in space would have to theoretically share similar specific information. o Charges are able to read, carry and transfer information during their movement in space. In neurons the transferred information by a charge or a group of charges is directly related to their spatial trajectory. o Importantly, this discussion illustrates the dual role of a charge - as a sensor that can “perceive” the presence of other charges and as a carrier of information during their movement in space and electric/magnetic field generation. Additionally, two or more charges would provide more information about the electric field or the presence of other charges than one single charge. Theoretically, this aspect can be understood by introducing a new measure for information, namely multiinformation (Studeny, 1989; Studeny and Vejnarova, 1998). 3.1.5. Models of Brain Computation Currently a number of theories have been proposed to describe how computation is performed in the brain. The recent history of neural computation includes an abundance of theoretical models that started with a simple description of a neuron developed by McCulloch and Pitts (McCulloch and Pitts, 1943). This theoretical model was followed by the book of Donald Hebb (Hebb, 1949) that described the principle of plastic changes and connectivity between neurons. The first simulations of neuronal networks were developed by Frank Rosenblatt (Rosenblatt , 1958) building the well known perceptron model. Later, Bernard Widrow and Marcian Hoff introduced several adaptation algorithms in their paper “Adaptive Switching Circuits” (Widrow and Hoff, 1960) while Marvin Minsky and Seymour Papert have made important theoretical contributions to the development of neural learning and object recognition (Minsky and Papert, 1969). Other significant theories regarding network architecture include Adaptive Resonance Theory (ART) developed by Stephen Grossberg (Grossberg, 1976; Grossberg, 1987). The neocognitron model built by Kunihiko Fukushima (Fukushima, 1980) achieved recognition for simple visual patterns based on learning. Shun-Ichi Amari introduced the concept of information geometry and related it with the theory of the natural gradient (Amari, 1977; Amari, 1985). Inspired by the Ising model, John Hopfield developed, with binary

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threshold units, models of content addressable memory (Hopfield, 1984; Hopfield and Tank 1985, Hopfield and Tank 1986). Teuvo Kohonen has demonstrated that unsupervised learning is able to preserve the topological properties of the input signal and this novel theoretical approach was at the origin of self-organization algorithms (Kohonen, 1974; Kohonen, 1982; Kohonen, 1984). Mclelland and Rumelhart (McClelland and Rumelhart, 1988) challenged the dominant symbolic paradigm, which claimed “mental processes can be modeled as programs running on a digital computer". Cognitive abilities become in this context result of lower level processes, rather than the application of “higher level rules”. This is the most important hypothesis that shaped the theory of neural networks. Information is represented by patterns of activation over neuron-like "units" that perform parallel distributed processing where synapses provide the required connectivity. Since every neuron can process information, computation is distributed in the network and does not require the existence of a central processing unit as in current digital computers. Behavior needs to be described by adaptive neural networks where the connection weights are changed using the generalized delta rule as a result of interaction with stimuli from the environment. This is an important premise that shaped several theoretical developments and their view remained valid for a considerable length of time. Therefore one can surmise that: o “Many of the constructs of macrolevel descriptions such as schemata, prototypes, rules, productions, can be viewed as emerging out of interactions of the microstructure of distributed models" As mentioned above, the Adaptive Resonance Theory (ART) was developed by Stephen Grossberg (Grossberg, 1976; Grossberg, 1987). There are two general issues that Grossberg and collaborators themselves raised regarding connectionist implementations. First, the stability-plasticity dilemma enquires how new knowledge can be incorporated within an already trained network, essentially asking how can learning continue without resulting in catastrophic forgetting. This is an important problem encountered in building artificial neural networks where information is preserved in the set of weights. The second issue called the noise saturation dilemma questions how features are processed in noisy cells with finite input sites without being contaminated by either noise or saturation. Despite their effort and some partial solutions both problems seem to remain critical issues in the development of connectionist theory and artificial neural networks. Other approaches follow the Mclelland –Rumelhart development and are heavily rooted in ‘parallel distributed processing (PDP) framework in building neural networks (Anderson 1995; Gardner, 1993; O'Reilly and Munakata 2000). As an example O'Reilly identifies three categories of PDP principles (O'Reilly, 1998): o “The first principle, biological realism, is in a category by itself, providing a general overriding constraint on the framework. The next three principles, distributed representations, inhibitory competition, and bidirectional activation propagation (interactivity), are concerned with the architecture of the network and the general behavior of the neuron-like processing units within it. The final two principles, error driven task learning and Hebbian model learning, govern the way that learning occurs in the network.” Churchland and Sejnowski (Churchland and Sejnowski, 1992) following the same principles insisted on computational models of the dynamics of neural networks

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extending PDP models to a new level. Their work stressed the role of inhibitory/ excitatory connections and made distinction between stored (weights of connection) and non-stored representations (neural activations). These models represented a new attempt to understand distributed representations and similarities to mind-brain models. In a more recent attempt using the cable theory, Koch described the basics of ionic currents (Koch, 1999) including dendritic attenuation, interaction of synaptic inputs . However, this description missed some important phenomena such as learning. Koch’s vision regarding information processing by neurons can be summarized by the following paragraph: o “Individual neurons convert the incoming streams of binary pulses into analog, spatially distributed variables, the postsynaptic voltage and calcium distribution throughout the dendritic tree, soma, and axon. These appear to be the two primary variables regulating and controlling information flow, each with a distinct spatio-temporal signature and dynamic bandwidth. The transformation between the input and the postsynaptic Vm and [Ca2+], involves highly dynamic synapses that adapt to different time scales, from milliseconds to seconds and longer. Information is processed in the analog domain, using a menu of linear and nonlinear operations. Second-order and higher multiplication, amplification, sharpening and thresholding, saturation, temporal and spatial filtering, and coincidence detection are the major operations readily available in the dendritic cable structure augmented by voltage-dependent membrane and synaptic conductances” The importance of spikes beyond the standard firing rate model is also the main topic of the book “Spikes: Exploring the Neural Code”. The power of computation with spikes reveals the fact that “the individual spike, so often averaged in within neighbors, deserves more respect” (Rieke et al., 1997). Using information theoretic techniques the authors envisioned that information processing in the brain approaches the limits imposed by “physical limits”. This new path approach where the timing of occurrence of individual spikes plays a key role, is developed further by Gerstner and Kistler (Gerstner and Kistler, 2002). The fundamental assumption of the current development of spike timing theory is that an action potential is stereotyped (Maass, 1997; Maass and Bishop 1998; Mainen and Sejnowski, 1995). In this treatment, the spike is simply an event, either present or absent, and the interspike interval is considered important, performing better than the average firing rate in coding input signals. o “The exact timing of spikes can, in principle, convey a lot of information which is lost if only the temporally averaged mean firing rate is considered.” Instead of adopting an ”input-output unit with a nonlinear transfer function” as previously described, the neuron is therefore mathematically built to approximate the temporal activity of spiking activity simplifying previous conductance models (Hodgkin and Huxley, 1952). Importantly, the spike response model can incorporate temporal properties including the refractoriness (Jolivet et al., 2008). This view was earlier developed by (Eliasmith and Anderson, 2003) and summarized by Eliasmith based on the following three principles: o Neural representations are defined by the combination of nonlinear encoding (exemplified by neuron tuning curves, and neural spiking) and weighted linear decoding (over populations of neurons and over time).

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o

Transformations of neural representations are functions of the variables represented by neural populations. Transformations are determined using an alternately weighted linear decoding. o Neural dynamics are characterized by considering neural representations as control theoretic state variables. Thus, the dynamics of neurobiological systems can be analyzed using control theory. There are two main contributions that Eliasmith and Anderson stress in their approach. Firstly, they formalize computation as an input/output model where encoding and decoding phases are successively performed. The second aspect is that “neural representations are modeled as control theoretic state variables”. On the neurophysiological front, advances in electrophysiological recordings during behavior in rodents and primates allowed correlative analyses between the electrical properties and functions in the brain. The adaptation mechanism called Hebbian plasticity became applicable to these systems and was recently applied to spike-timing-dependent plasticity, which has become an active area of research for many neuroscientists. As already discussed previously, a stereotyped action potential formed the basis of current development of spike timing theory (Maass, 1997; Maass and Bishop 1998; Mainen and Sejnowski 1995). The interspike interval is postulated to perform better than the average firing rate in coding input signals. The dynamics of spiking models have lately been analyzed by Eugene Izhikevich (Izhikevich, 2007), bringing new theoretical analysis in the field. Other recent contributions may include the confabulation theory of Hecht-Nielsen as an attempt of reconciliation between connectionist view and symbolic paradigms (Hecht-Nielsen, 2007). All these models are examples of original research. Many of these computational models have theorized and algorithmized the adaptation process. Seen as plastic changes of synaptic strength or modulations in firing rate as an effect of neurotransmitters, adaptation has embraced a theoretical mathematical formalism. The common issue for all these contributions is that they mainly reflect and extend the scientific vision based on early electrical recordings performed by Adrian (Adrian, 1932) where the timing of spikes, either as to their firing rate or the associated interspike interval, was felt to fully describe neural processes. All of these theoretical constructs assume yet fail to describe an important, unforgettable fact: for any computation and transfer of information, underlying, physical information carriers are needed to perform the classical computation. As an example, electrons play the role of information carriers in order to physically perform simple computations in von Neumann machines. Temporal description of computation in the brain ignores or limits this vital requirement of a physical carrier for information. The predominant issue is that the temporal unit so valued to date, is not a rich physical quantity compared to complex dynamics and interactions between electric charges. Therefore, even though many things in nature occur in time, there is always a physical mechanism that is acting within this time domain. Spike timing theory limits this vital fact to the time where an action potential event occurs. 3.1.6. Building the Internal Model Developing conceptual models of complex real world entities and simulation of dynamic agent-based systems can help us to understand some basic phenomena and the development of cognition can be easily exemplified using the theory of cognitive agents (Lee and Loia, 2007). If one assumes that the environment E can be described

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using the state variables (e1,e2,…,en) then the agent function is to build an accurate model of the environment. Therefore, the agent A which is represented as a dynamic system with internal states (a1,a2,…an) can build during time an equivalent model E’ with state variables (e1’,e2’,…en’). Since the agent is sensorially coupled (S’) then it will provide motor commands M’ to the internal model exactly in the same manner as it receives sensorial inputs S and gives motor commands M when it interacts with the environment (Figure 5). In this case the internal model E’ acts as an emulator of the environment and its design depends on dynamical interaction with the environment. Therefore, the executive part, the controller C is directly dependent on the accuracy of the internal model M’. The study of cognitive agents shows that building a

M

M' E' S'

S

C

Environment Figure 5: The representation of information and cognitive system development determined by the dynamic interaction of agent –environment .See related text for symbol description.

representative model of the environment is not an easy task. An increase in complexity of the environment or in its interaction with the agent requires an increase in complexity of the internal model. This way of building cognitive agents can be extrapolated to our work of understanding how knowledge is embedded in our brain. Therefore, it is expected that in order to describe the dynamics of the external world, the brain should be able to provide a similar complex dynamic mechanism that acts as an emulator or an internal model of the external world. Understanding how these dynamic processes occur and how they are physically built is the essence of building every system that underlies “intelligent” computations. 3.2. Neurons as information engines It is generally established that neurons play an important role and are involved in brain information processing. The accepted view assumes that information is transferred from the axon of the presynaptic neuron to the dendrites of the postsynaptic neuron. Such transfer of information is generally biochemically modulated. However, the basic physical principles that govern this information transfer still remain unrevealed. 3.2.1. Maximum Entropy Principle Individual cells fire many action potentials and this firing activity can be measured using both extracellular and intracellular recording electrodes. As discussed above,

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neural activity cannot be completely described by or reduced to AP occurrence alone. Additionally, neural architecture is extremely complex with large numbers of spatially distributed synaptic inputs on every neuron and such bioelectrical activity determined by synaptic inputs is almost continuous and spatially spread out. The complexity of the relationship between information transfer and electrical properties of neurons can be seen to occur at various scales. At the highest level a moderate analysis has been developed for “wave coding” as a large-scale feature of a group of neurons that encode behavior in human fMRI studies (Cerf-Ducastel and Murphy, 2003; Poellinger et al., 2001). The second level at the cell scale involves spike coding usually viewed in the time domain, as discussed above. Known examples are well studied involving rate coding or spike coding (Maass, 1997; Gerstner and Kistler, 2002) that have been the premise of the strong development of artificial neural networks and artificial intelligence. Such processing can occur at the ensemble level where groups of neurons can be seen to encode for complex behavior (Wilson and McNaughton, 1993; Barnes et al., 2005). However, it is likely that information processing and transfer does also occur at an even more subtle scale, one that is based upon the dynamics of different ionic fluxes within neurons. This topic, along with experimental evidence has been discussed in detail in chapter 2. Such level of information processing may be the most fundamental and possibly the richest level of information processing for the entire nervous system. Our work has tackled the problem of information coding by neurons at this level. We have shown that neuronal spiking and the resulting charge flux is characterized by spatial directivity. These changes in spike directivity could result in important effects during every AP and may alter the neuron’s ability to communicate information (Aur et al., 2005; Aur et al., 2006; Aur and Jog 2006; Aur and Jog, 2007). When a neuron spikes, it considers all received inputs. Many of these inputs are synaptic while several other inputs are non-synaptic. At this time scale of a spike, these inputs come predominantly in the form of ionic flow and the spike results in its own ionic flux that can be measured electrophysiologically. During synaptic communication there is a substantial involvement of ion channels. As the AP invades the presynaptic terminal it generates a depolarization of the presynaptic neuronal membrane. Depolarization operates on several voltage-gated ion channels, such as Na +, Ca2+, K+ , Cl- channels. The probability of opening of the channels usually increases during depolarization (Meir et al., 1999). In the neostriatal cholinergic interneurons, spikes are produced spontaneously in the absence of synaptic inputs (Bennett et al., 2000). Another typical example can be found in the cerebellar Purkinje neurons where spontaneous firing appears in the absence of synaptic input (Bell and Grimm, 1969; Latham, 1971). Spontaneous firing can be maintained also in cerebellar slice preparations when synaptic activity is blocked (Häusser and Clark, 1997). Such spontaneous firing implies that in the absence of synaptic input, the extracellular milieu may serve as the medium through which Purkinje cells can be depolarized. Hence, in terms of information communication, the activity of the ionic channels on neuronal membranes may be modulated by synaptic and non-synaptic electrical events and result in generating the action potential. The work of Boltzmann, Gibbs, Shannon has led to the Jaynes Maximum Entropy Principle (MEP) (Jaynes, 1957). This principle applies to a large variety of nonequilibrium systems including large-scale climate changes to small scales of crystal growth or bacterial metabolism. Within every neuron there is a large number of components such as atoms, electric charges, molecules, which can be categorized into

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a certain number of classes. Such an ensemble of several components allows us to see each neuron as a self-contained and complete system. Boltzmann’s statistical physics approach assumes that all microstates of a system are equally probable. His definition of entropy: H B = k ln Ω ( 11 ) correlates elementary disorder in a given system with entropy. Low entropy is specific to an ordered system which should have only a few microscopic states while a less ordered system should increase the number of macrostates and also the entropy. The postulate of maximum entropy states that the entropy of a closed system never decreases in any process ( ΔH B ≥ 0 ). Additionally, Boltzmann’s formula can be obtained as a result of minimizing the “average index of probability of phase” ∑ pi log pi with respect to the microstate probabilities (Gibbs, 1902). In a similar i

fashion but with respect to information entropy, Shannon introduced the equation for information entropy, (Shannon, 1948) ∑ pi log pi as a measure of uncertainty i

associated with the probability distribution pi . Therefore, Shannon’s information entropy measures the level of our ignorance about the state of the system. Jaynes has further formalized this concept in terms of probability distribution related to information. “Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum entropy estimate. It is the least biased estimate possible on the given information” (Jaynes, 1957). This development of partial knowledge of a process can be related to Planck’s work in statistical thermodynamics where he hypothesized the equality of a priori probabilities for the states of a system (Planck, 1900). Based on this theoretical approach one has to select from among all possible distributions the initial states that show maximum entropy values (Tolman, 1979). It is quite likely that from such a broad spectrum of every possible outcome, Nature selects the most probable one which is the state that obeys the Maximum Entropy Principle. 3.2.2. Mutual Information It is therefore possible that with information entropy fluctuations, a neuron can transfer information at multiple levels during AP propagation and diffusion processes (Goychuk and Hanggi, 2000; Aur and Jog 2007).

X

H(X)

∫

Y

H(X|Y)

Figure 6 Schematic representation of dynamic model of information processing in the brain where H(X) is the entropy of the input variable X and H(X|Y) is the conditional entropy. Input and output variables can physically describe the flow of charges in the system.

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Voltage sensitive channels under the influence of electric fields and concentration gradients that allow charge transfer are responsible for this phenomenon. The voltagegated ion channels are distributed along dendrites, soma and axons and the ionic flow is greatly enriched almost everywhere during each spike. Even the process of diffusion in liquid–liquid systems “can be compared to information transfer in information systems of multiple components” (Ceruti and Rubin, 2007). Artificially we can generate signals that share large volumes of information, however in the real world in order to obtain large values of mutual information, information has to be transferred. Therefore the process of diffusion of ions during AP occurrence can be shown to process and transfer information within every spike. Information can thus be accumulated or removed from the neuron by electrical ionic flow during each spike. Therefore, the analysis of information transferred within AP’s by the electric field and flow of charges is critical. Information entropy of any continuous variable contains more information entropy than its discrete version. However, for high sampling rates, the information entropy of a discrete variable comes close to the continuous value (Sobczyk, 2001). This characteristic allows us to see the mutual information MI between two continuous variables as the limit of computed MI values for the discrete versions using the Hodgkin Huxley (HH) model. Since the mutual information measures arbitrary dependencies between random variables such measure is far superior to the autocorrelation function (Fraser and Swinney 1986). Additionally this measure can be used to analyze dynamical systems (Figure 6). Mutual information measures information shared by two selected variables X1 and X2 (Cover and Thomas, Gray, 2000): I(X1 ; X2 ) = H(X1 ) + H(X2 ) - H(X1 , X2 ) ( 12 ) where H(X1,X2) is the joint information entropy. Substituting I (Na+, K+) in Eq. 12, we can measure the reduction in uncertainty about the Na+ flux due to the knowledge of K+ flux. Considering the two main ionic fluxes Na+ and K+ the information transferred between them is: I spike = I ( Na + ; K + ) = H(Na + ) + H(K + ) - H(Na + , K + ) ( 13 ) spike represents mutual information between these two fluxes. By using values of where I ionic INa+ and IK+ fluxes from HH model it is possible to estimate the joint probability distribution and mutual information between Na+ and K+. o The total amount of correlations is given by computing mutual information, large correlations imply large mutual information in classical or quantum systems (see Horodecki et al., 2005) The concepts and the methods for computing distributions, entropy and mutual information have been presented elsewhere in detail (Aur et al., 2006). Since the analysis of ionic fluxes in vivo during AP activity is not an easy task we have used for this study extensive simulation of the Hodgkin-Huxley (HH) model. This model is close enough to biological cell activity to allow us to model the dynamics of ionic currents during AP (Appendix 1). Based on simulated input-output relationships the quantity of information transferred between input stimuli and ion fluxes can be estimated using general principles from signal processing and information theory. The interdependencies between input signal and the output are analyzed in the context of mutual information. Theoretically, mutual information measure does not provide the direction of information transfer (Kaiser and Schreiber, 2002). However, for physical processes the directionality in most cases is predictable.

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Briefly, to perform this analysis the following steps are developed. First, a RungeKutta technique is used to compute ion currents in the HH model that occur when the input stimuli with variable frequency are applied. The accuracy for integration was tested for the 4th order Runge-Kutta method and we have found it to be accurate enough to perform the integration of the HH model. Since the HH model is described by known equations, the sampling rate of the discrete model is adjusted to keep certain dimensions for analyzed data. For computing MI we kept these dimensions between 330x330 and 2000x2000. The estimated average error is under 0.2 bits for all range of data between 330 and 2000 points and the standard deviation is below 0.15 bits. For these data dimensions the maximum error is less than 0.23 bits. The errors and standard deviation between estimated values with the histogram method and analytical estimated values are computed (see Appendix 2, and also Aur et al., 2006). Simulation of the HH model and mutual information estimation were implemented using a PC computer (Pentium 4, 2.8 GHz, 512 MB RAM) and Matlab - MathWorks, Inc. All the routines were custom developed or are freely available on the world-wideweb. Mutual information I IstimNa was estimated between the stimuli current I stim and sodium flux I Na : I IstimNa ( I stim ; I Na ) = H( I stim ) + H(I Na ) - H( I stim , I Na ) ( 14 ) and Ispike between I stim and membrane potential V: I spike ( I stim ; V) = H( I stim ) + H(V) - H( I stim , V)

( 15 ) and potassium

Additionally, mutual information I IstimK was estimated between I stim flux I K : I IstimK ( I stim ; I K ) = H( I stim ) + H(I K ) - H( I stim , I K ) ( 16 ) and mutual information I NaK between I Na and I K : I NaK ( I Na ; I K ) = H( I Na ) + H(I K ) - H( I Na , I K ) ( 17 ) During each spike, ions keep in contact through space and time and these correlations can be quantified based on MI estimations. Therefore, to find the information “encoded in spikes one has to analyze the dependencies in information entropies of several ionic fluxes. Different ionic fluxes convey information mutually because they are correlated. As discussed above, I(Na+, K+) measures the reduction in uncertainty about the Na+ flux due to the knowledge K+ flux: I(Na+; K+)=H(Na+)- H (Na+/ K+) ( 18 ) where H (Na+/ K+) is the conditional entropy. Considering only these two main ionic fluxes Na+ and K+ the “encoded information” is: Ispike = I (Na+ ; K + ) = H(Na+ ) + H(K+ ) - H(Na+ , K+ ) 2

( 19 ) By using values of ionic fluxes from HH model it is possible to find their joint probability distribution as presented in (Aur et al. 2006). Since mutual information is a positive quantity, it is expected that many fluxes together tell us more than two fluxes. Indeed neuronal processes include the activity of many ionic fluxes. Stochastic dependence between several variables can be characterized by the multi-information measure. Multi-information always increases

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by adding a new variable (Appendix1). Therefore, introducing Cl − flux, Eq.( 19 ) becomes: I3spike= I(Na+ ; K+ ;Cl− ) = I(Na+ ; K+ ) + I((Na+ , K+ );Cl− ) ( 20 ) The increase in multi-information is based on mutual information between Cl − and previously considered Na+ and K+ ionic fluxes. From Eq. 22 and 23 the inequality is: I 3spike ≥ I 2spike ( 21 ) Inequality 24 can be generalized by adding the transport of different ionic fluxes such as Ca2+, Zn2+, Mn2+,Co2+, Cd2+, Cu2+, Ni2+ (Colvin, 1998). Adding enumerated ionic fluxes inequalities can be written as: spike I nspike ≥,... ≥ I 3spike ≥ I 2spike ( 22 ) +1 ≥ I n showing mainly that considering several ionic fluxes in the spike model, multiinformation measure increases. Therefore, from informational point of view, it is clear that assemblies of ionic fluxes display more information within the system than each pair of ionic fluxes. These correlations between several ions can be quantified based on multiinformation estimations. The concepts and the methods for computing distributions, entropy and mutual information have been presented in several papers (Silverman 1986, Gray, 1990, N. Tishby et al, 1999, Moddemeijer, R. 2000, Dinh-Tuan Pham, 2004). The computing and assessment of the quality of MI estimations have been explained in detail (Aur et al., 2006). The following section describes the application of theoretical concepts discussed above. 3.2.3. Information transfer In the real world large correlations between a set of variables in any system are unlikely to occur unless information is physically transferred. A variety of results from HH model simulations and calculations of mutual information have already been presented in Aur and Jog, 2006. Here we would like to illustrate the importance of input phase modulation in information transfer. We are able to show this phenomenon since the values of MI transferred in sodium fluxes can be even higher than previously estimated at low frequencies. The values of mutual information depend on the input phase (Figure 7 a-d). This dependence on phase increases at low frequency. The lower the frequency of stimulus, the stronger is the dependence on phase. Therefore, to show this dependence we lowered the input frequency to 50 Hz and plotted the obtained values. Previously the estimated MI per spike was Ispike= 3.56 ± 0.2 bits and the maximum value for INa fluxes was 7.05 ± 0.2 bits at 100Hz. (Aur et al., 2006). If the frequency of Istim is lowered to 50 Hz the maximum value for IstimNa =8.7840 ±0.2 bits while for Ispike is 3.3264 ±0.2 bits. Since phase differences can be translated in time, this may explain why, by analyzing spike timing, the values of information are comparable with those obtained from Ispike estimations. This is an extremely important point as it is the beginning of our ability to show that the analysis that we have presented above can incorporate subtle concepts related to spike timing theory. Sodium and potassium currents are highly correlated. This is an indication that there is a strong information transfer between incoming Na+ and outgoing K+. Therefore, in terms of mutual information represented by Na+ and K+ components, one can estimate relevant information of Na+ ionic flux with respect to K+ flux and vice versa during each AP. Using Na+ influx with respect to K+ efflux during AP, the values

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Figure 7: The dependence between estimated values of mutual information IIstimNa and Ispike plotted across the phase of sinusoidal input at 50 Hz Istim(µA/cm2) during 22 ms. a, The 3D plot of IstimNa over amplitude of input stimuli and phase of sinusoidal input. b, The estimated mean of IstimNa over the phase of sinusoidal input. c, The 3D plot of Ispike over amplitude of input stimuli and the phase of sinusoidal input d, The estimated mean of Ispike over the phase of sinusoidal input

of mutual information are computed and represented in (Figure 7a) for different frequencies and amplitudes of the input stimuli. Mutual information encoded in Na+ and K+ fluxes have comparable values to information transferred from sinusoidal input signal to Na+ fluxes. o Information is transferred synaptically and non-synapticaly within ion fluxes and this electric interaction is able to transfer on average about 3.5 bits/spike and maximum values of 8.7840 ±0.2 bits o Temporal encoding (spike timing) can be seen as a natural outcome generated by changes in information transfer modulated by the phase of the stimulus input The dependence between estimated values of mutual information I NaK and frequencies for a sinusoidal stimulus Istim computed over 30 ms give values between 3 and 3.5 bits. 3.2.4. Efficiency of Information Transfer In order to compute the values of information transfer that occur from ionic fluxes and their dependence on input stimuli, we have used simulations of signal input and output in the Hodgkin-Huxley (HH) giant squid axon model (Aur et al., 2006). The

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applied inward current I stim could come from another neuron or can be artificially generated. Indeed, I stim from HH simulations can be viewed as a dendritic current, a synaptic current, or a current injected through a microelectrode. Before using the method to compute mutual information (MI) between ionic currents during a spike we have extensively analyzed the histogram method to compute MI for Gaussian noise signals (mean μ=0 and standard deviation σ=1) (see Appendix 2). In comparison, the estimated values of MI between ionic currents are at least ten times higher, > 2 bits. To compare information by spikes or by ion fluxes we use the same method therefore we expect robust results. One way to see efficiency within each spike is to consider the mutual information I stimNa between I stim and sodium flux as the input MI and mutual information

I spike between I stim and membrane potential as an output MI. Then efficiency can be computed: η1 =

I spike I stimNa

( 23 )

One can view this in terms of ionic fluxes alone. If I Na is seen as an intra-

neuronal carrier of information then the K+ outflow can be seen as an export of information. One can then compute the efficiency as the ratio between MI transfer from I stim to K+ and I stim to Na+: η2 =

I istimK I istimNa

( 24 )

Interestingly, in both cases the values of efficiency of information transfer in spike seem to have the same shape over the frequency of the input stimuli: η 2 ( f ) ≅ η1 ( f ) = η ( f )

( 25 )

where f is the frequency of the sinusoidal input. The upper bound in efficiency can attain values of η ≅ 0.7 in both cases for frequencies lower than 200 Hz. The maximum and mean values of efficiency decrease for frequencies higher than 200 Hz (Figure 8 ad). The dependence between efficiency values η1 , η 2 and frequencies of Istim is presented in Figure 9. The maximum and mean values of efficiency have been computed. One can see clearly that the efficiency of information transfer within a spike depends on input frequency. For lower frequencies less than 200Hz, neurons efficiently transfer information from input stimuli during each spike while this transfer is reduced significantly for higher input stimuli frequencies. Additionally, at lower frequencies less than 200Hz the MI value and computed efficiency values for information transfer present a nonlinear complex dependency over Istim phase (Figure 9 a-d) (Aur et al., 2006 ). Since the phase between Istim and the occurrence of AP can be translated into a temporal domain, this variability of MI values over phase shift is confirmed by the experimental observations of current spike timing theory. While the current spike time theory was built from observation of MI between stimuli and spikes occurrence our model reveals this dependence from a different theoretical approach by analyzing ion fluxes in HH model. Indeed, our approach provides a firm framework upon which the

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observations of spike timing theory can be justified. The broader platform is also capable of providing much more expandability as we show in this book. It is also clear that the notions of spike timing are limited in what they can offer to understanding neuronal function. Thus, unlike the spike-timing model, the new model we have developed is far richer and complex and yields a picture that is closer to the nature of neuronal computation than the spike-timing model can ever expect to achieve. An extremely powerful consequence of experimental results presented in Chapter 2 and the above development is that if sequential APs from neurons can now be treated as being distinct implying that (see Aur and Jog 2006) every spike may transfer slightly different amounts of information during their occurrence.

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3.2.5. The Nature of Things - Functional Density The major goal in cognitive neuroscience is to determine how information is processed in the brain. Mathematical formalism has been applied to mimic neuronal computation and provide basic tools for developing various theories of computation and neural coding. Although Putnam claimed that the brain could be adequately modeled by a classical Turing representation (Putnam, 1960) many other scientists have rejected this idea (see Eliasmith, 2002). However, since the current theories of neural computation fail to mimic the power of the brain, new concepts regarding computation need to be developed. Understanding the basic principles of brain computation can be a result of direct observation and analysis of several aspects that shape all natural processes. At a large scale, a view from an airplane may only show several cities and some roads connecting them. At a smaller scale in every city we are able to see details such as conglomerations of houses, traffic lights, cars and finally the people. The city has high density of buildings, streets where each component certainly plays a functional role. This increase in density is a natural way of increasing interaction that seems to work everywhere as a universal principle. Urban conglomeration plays a role since energy resources can be

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effectively used. This may include energy spent for transportation, human resources, labor force. The development of certain companies in the region may affect long term, production from another that has effects upon nearby economic activity. These observations regarding increased efficiency of resource utilization based upon increased density can be extended to several other natural phenomena and can be used to understand brain processing. Our universe contains several things that repeat in space or in time with or without obviously visible patterns. One can observe leaves, trees, or even neurons in the brain that are spread in space. This analysis can go to a microscopic scale where water molecules, atoms or electrons are evenly or randomly distributed in space. The same phenomenon can occur and can also be repeated in time as in the heart or in brain rhythms (Buzsáki, 2006). This property is called recursivity. The case of trees, leaves, migration and arborization of neurons are forms of spatial recursivity while brain and heart rhythms are typical cases of temporal recursivity. o Once derived as an appropriate structure or a suitable mechanism, the structure/mechanism is repeated in space or in time with small variability to complete certain functions. Nature itself, and not mathematics, found this way of building complex systems. Mathematical models described by equations, functions or systems of equations can in some measure represent these real-life phenomena. In theory, recursion as an important property occurs when a function calls itself (Odifreddi, 1989). Typical examples in mathematics are computing factorials, Fibonacci series or graphing Sierpinski triangles (Rich, 2007). Beyond nice-looking images spatial recursion is a principle that shows us how to put together constructs improving understanding of many complex phenomena. In quantum physics changes in density at atomic scale are described by so called density functional theory that replaced the many-body electron-based wave function with the electronic density. This replacement provides a way to develop a new theoretical framework that can be applied more easily than a many body wave function approach to study microscopic phenomena. 3.2.6. Charge Distribution and Probability Density It is well known that density estimation is a powerful technique for modeling distributions in case of sampling different types of data. The basic formulation specifies that the probability of a given variable in the interval between any two points a and b can be computed: b

P( x ∈ [a, b]) = ∫ p( x)dx

( 26 )

a

where p(x) is the probability density of a normalized function. Since information transfer is dependent on the movement of charges it is important to find a way to characterize their spatial distribution (Figure 10). The amount of uncertainty about a distribution of charges P =( p1 , p2 ,... pn ) can be computed using Shannon information entropy H: n

H ( p1 , p2 ,... pn ) = −∑ pk log 2 pk k

( 27 )

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aa

b

Figure 10: Schematic representation of two different distributions of charges across X axis. a, The charges are uniformly distributed b,The charges are non-uniformly distributed

If one considers charges evenly distributed in space (Figure 10,a) with the given probability distribution P1 ={1/3, 1/3, 1/3}then information entropy value is: n 1 1 H 1 = −∑ log 2 = 1.585 bits ( 28 ) 3 k 3 If the distribution of charges follows the schematic representation (Figure 10,b) with given P2 ={0.1,0.8,0.1} then the computed entropy is: H 2 = −2(0.1 log 2 0.1) − 0.8 log 2 0.8 = 0.921 bits

( 29 )

This simple example shows the relationship between spatial distribution of charges and information entropy values. Once the charges are non-uniformly distributed Shannon information entropy computed for this system composed by charges is reduced. This implies that (Figure 10, b) there is also a reduction in uncertainty since charge particles can be better located across the X axis. In other words for an external observer this distribution provides more “information” regarding the spatial location of charges across the X axis. This problem of linking information with spatial distribution of charges becomes complex if these charges move in space. These charges can be seen as points in a threedimensional space (Figure 11). Then, the observer has to further consider the time as an additional dimension when the measurement or the observation of charges in motion has to be performed. However, an accurate measurement of charge density in a certain region of space for such a dynamical system composed of several charges is not a trivial issue. In fact solving this issue provides the answer of how the brain decrypts information carried by electric field waves and charges. The information carried and delivered by these charges is likely the information which is read by biochemically modulated synapses. Therefore, neural activity can be artificially separated into two main phases. The first phase consists of generating the movement of charges or an uneven spatial distribution of charges and the second phase is the above problem of reading information embedded in such a distribution. Both phases should be physically realizable and must obey the laws of physics.

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Looking at the brain from small to large scales, the occurrence of changes in charge densities plays an important role.

S1

S2 S3

Figure 11: Representation of spatially distributed charges in blue and a set of solid spheres S1, S2….Sn with radius R. These spheres can cover with a certain approximation the regions with high density of charges

A certain arrangement of spheres or in a general case of ellipsoids is able to pick up the regions with high densities of charges (Figure 11). Choosing an optimal number of spheres/ellipsoids, their radius and/or location in space to minimize the number of charges left out is a delicate problem. The theoretical model can be reduced to the Khachiyan problem (Khachiyan,1996). Since the charge points should be inside the ellipsoids/spheres then one can write: (x i − c)T M ( xi − c) < 1, i = 1, N

( 30 )

where x i are the coordinates of the N charges, M and c are variables that need to be determined. Finding the optimal ellipsoid/sphere with the minimum volume that covers the position of charge points can be obtained by solving the Khachiyan problem (Khachiyan, 1996). If instead of ellipsoids we use solid spheres with similar radius R the Khachiyan problem can be simplified (Figure 11). This problem becomes more complex when these charges move and interact in space which in fact may be the role played by the active synapses if computations are performed by electric charges at synaptic cleft level. Additionally, considering charges as basic elements, “neural computation” becomes in fact computation by all elements that make up the brain, since several other mechanisms including glia and other buffering systems dynamically regulate the distribution/redistribution of charges in space (Kofuji and Newman, 2004). 3.2.7. The Laws of Physics The laws of motion are determined by the existence of fields and forces and several other factors involved in the physical process. The key to solving this subtle problem of dynamics provides the answer of how computation is built from nanoscale level as a process of diffusion. This also shows why neurons have several types of synapses

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spatially distributed all over. Furthermore when and where the observation (measurement) is performed provides insight about reading certain specific “information” processed further in the nervous system. Starting from this simple presentation several models of computation in brain can be imagined. The simplest model can start with a charge density model expressed via the Dirac function:

ρ ( r ) = ∑ q iδ ( r − ri ) i

( 31 )

where ri gives the position of the ith charge. This simple description of a group of charges allows us to relate the movement and interaction of these charges with signal processing techniques and the theory of dynamical systems. Importantly the main difference between charges and their counterpart as simple particles is that charges are r always at the origin of the generated electric field E that mediates electrical interactions and information transfer. As a classic example the electric potential of a group of charges distributed in space can be computed using the Poisson law: ∂2 ∂2 ∂2 ρ ( x, y , z ) ( 2 + 2 + 2 )Φ ( x , y , z ) = − ( 32 ) ∂x ∂y ∂z ε The equations of classical mechanics can be used to describe the motion of charges in a multidimensional potential energy surface. The force that acts on charge qi

r

depends on the intensity of the electric field E : r r Fi = qi E

( 33 )

where for each charge within a certain volume V the intensity of electric field is:

r 1 E (r ) = 4πε

∫

V

ρ (r ) r r dV r2

( 34 )

r Additionally the force Fi can be written as a product between the mass of charge r and the acceleration. ai : r r Fi = mi ai ( 35 )

Naturally these charges will redistribute themselves by moving away one from another or attracting each other until they achieve the lowest state of electrostatic potential energy. Classical description provides the evolution and location of the charges as points in three dimensional space (Figure 12). The attraction or repulsion force depends inversely on the square of distance between charges: r r N r q (r − r ) Fi = ∑ j i j 3 ( 36 ) 4πε (r − r ) j =1 j ≠i

i

j

In case of N charges their kinetic energy can then be written: 1 N K = ∑ mi vi2 ( 37 ) 2 i=1 and their potential energy P in the electrostatic Φ i potential can be described: 1 N P = ∑ qi Φ i ( 38 ) 2 i =1

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Xb, Vb

Xa, Va

Figure 12 : The movement of charge from point A to point B deletes information in point A and transfers information to point B. Energy is required to perform such movement to transfer information in space.

where the electrostatic potential Φ i for charges q j ,j=1,2,…N is: N

Φi = ∑ j =1 j ≠i

qj 4πε | ri − rj |

( 39 )

The effective potential includes the effects of the Coulomb interactions between charges that can be represented by different types of ions, electrons or even polarized molecules. The easiest way to find the distribution of charges in space is to minimize their electrostatic potential energy P . The process of obtaining the position of charges in space can then be viewed as a mathematical multidimensional optimization procedure. However, if the number of charges increases the modeling and the simulation of charge movement on current computers becomes problematic since it is not easy to describe all interactions that occur between such charges in movement. At microscopic scales the attractions between individual atoms, molecules etc can be expressed as van der Waals forces. Since the electrons are not evenly distributed in atoms and molecules they may determine polarizations. The resulting forces display an attractive or a repulsive force determined by an electrostatic interaction between charges, polarization or dispersion. Sometimes van der Waals forces are assumed to be only attractive. However, the repulsive forces are equally important components since they prevent molecules from collapsing. Once the distance between constituents becomes smaller the interaction sharply increases (see Figure 13) and the trajectory of charges can be described in a probabilistic fashion using quantum theory (Schrödinger, 1926). The charged particle can be seen as a wave ψ that moves in a potential well, known as Schrödinger Equation (Schrödinger, 1926): ∂ψ h 2 ∂ 2ψ ih =− + Vψ ( 40 ) ∂t 2m ∂x 2 where h is the reduced Planck constant: h h= ( 41 ) 2π -34 2 and h =6.626068 × 10 m kg / s.

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For a plane wave, ψ ( x, t ) : ψ ( x, t ) = ψ 0 e i ( kx−ω t )

( 42 )

one can write: −

h ∂ψ = hωψ i ∂t

( 43 )

Since:

K = hω ( 44 ) then the Schrödinger wave equation can be rewritten as: h2 2 Kψ = − Δ ψ + Pψ ( 45 ) 2m where ψ is the wave function, P is the wave potential and K is the kinetic energy. At this microscopic level it can be shown that information communication can be seen also to be intrinsic to the treatment of charges behaving either as ‘particles’ or with a wave like behavior. Since: 2 ( 46 ) ∫ |ψ ( x) | dx = 1 R3

then information that is transferred from one location to another spatial location can be seen in terms of a normalized wave function: dH S = ∫ | ψ (r ) |2 log 2 | ψ (r ) |2 dr ( 47 ) R3

(Parker and Walker, 2007) At atomic scales the strong nuclear force that occurs cannot be modeled in terms of a Coulomb force law and such interactions depend on potential energy and internuclear distance. There is no straightforward formula to describe such dependence (Lippincott and Schroeder, 1955). As represented in (see Figure 13) interaction depends on the distance between particles and varies in a nonlinear manner from weak interaction to strong interaction (Landau and Lifshitz, 1960). Since computation is related to the movement of charges it is expected that such changes in interaction would alter the movement of charges and information transfer. o Information is communicated by electric interaction; any charge or group of charges is able to alter the movement/dynamics of closed charges due to their electric field. In this case the electric field becomes the physical entity involved in information transfer. The applicable laws of physics also change with quantum mechanics and probabilistic laws of movement replacing at the microscopic level classical dependence 1 where interaction depends inversely on the square of distance 2 . Since movement and r interaction are different depending on distance it is likely that computational laws are applicable differently depending on the scale where they are described. Many of these models have to include subtle phenomena at atomic scales described by quantum computation (Nielsen and Chuang 2000). Such description requires a quantum mechanics formalism where the probability for a charge to follow a certain path depends on the difference between kinetic energy and potential energy. Based on the Schrödinger equation this probability can be mathematically expressed in terms of the wave state function | ψ (t ) > (Feynman and Hibbs, 1965):

| ψ ( t ) >= all

∑

paths

e

−

i h/

∫[

m x&

2

2

(τ )

− P ( x ( τ ))] d τ

( 48 )

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where m is the mass of the charge, x is the position and P is the electric potential. Such description relates charge path trajectory to charge energy. In the same time the electric potential can be translated into the required work to obtain a certain spatial position for a certain charge. This wavelike behavior at microscopic level is at the origin of the uncertainty principle (Heisenberg, 1930) and represents a direct consequence of unmodeled strong interactions at microscopic level.

Interaction energy

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Figure 13: Representation of weak interaction and strong interaction energy depending on van der Waals interaction, electrostatic attraction forces and London dispersion forces.

For an external observer this principle provides theoretical limits in predicting particle trajectory in contrast with classical approach of Newtonian principles where the real trajectory is determined with enough confidence. Science faces an ill posed problem trying to find which model is better suited to model neural computations since both the quantum and classical models coexist together cannot be separated (see Figure 13). Their coexistence reflects our limits to find a unique model to describe natural phenomena at different scales rather than the superiority of one model over another. 3.2.8. Minimum Description Length Statistical models are generally used to find regularities in observed data. The main inconvenience is that the same set of data can generate several different descriptions and therefore several models. Comparing these models is not an easy task and one may use complexity analyses to discriminate between many competing models (Li and Vitanyi, 1997). Kolmogorov has defined complexity of a sequence as the length of the shortest program that prints the sequence and then halts. Finding the right model which provides the shortest data description or “minimum description length” (MDL) is often required in science. Ray Solomonoff suggested the use of computer code as a complexity measure (Solomonoff, 1964). Such description of complexity has been extensively studied and described by Rissanen:

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“The number of digits it takes to write down an observed sequence x1, …, xN of a time series depends on the model with its parameters that one assumes to have generated the observed data. Accordingly, by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters” (Rissanen, 1978). Seen as regularity in a given set of data MDL, allows the largest compression of the data (length in bits). Minimum description length is also an information theoretic measure. In other words the MDL principle provides the solution for selecting the model in the model class which achieves the shortest code length for the data and the model. Since computations in brain are performed by a continuous dynamic of charges which builds a required model, then the main question is if we can find a theoretical principle similar to MDL that can depict the process. It has been shown that charges that move are also able to perform universal computations (Moore, 1990). Therefore, it is likely correct to assume that there should be a ‘minimum description’ for such computations. Achieving a certain goal or reaching a certain point in space can be described at a microscopic scale as “a many path description” which can be a model built in a quantum mechanics framework. This approach is consistent with the Occam’s razor principle since among all hypotheses/paths consistent with the observations we have to select the simplest one which can be a straight line or a curve depending on external fields. Therefore, it is expected that the selection of the optimal path for charge movement is consistent with a minimum path description (MPD) principle. The charge displacement should result in the shortest dynamics which may generate the more accurate representation or model of the input: t r r Δr = arg min ∫ ( F (r , E ( Stimulus, ε M ) + n(t ))dt ( 49 ) f

εM

t0

r where Δr is the infinitesimal displacement of the charged particle, and the force F depends on the electric field. The noise source n(t ) and the error between the input and

the model ε M is :

ε M = StimulusData − Model

( 50 ) Additionally the MPD principle asserts that the best model of given input data is the one that minimizes the combined cost of describing the model in terms of number of charges by minimizing the error between the model and input stimuli data. Therefore MPD should also minimize the number N of charges involved in such description by considering the misfit between input stimulus data and the internal model built by the distribution of electric charges : N * = arg min ( StimulusData − Model ( N )) ( 51 ) N

This approach when applied to biological systems reveals a very crucial and important feature. In the brain computation this natural phenomenon can be seen as a self-organization process or learning. In a dynamic model the shortest ‘program’ has to be equivalent to producing a charge trajectory that reflects the “minimum path description”. It is important to show that once these dynamic patterns occur in the movement of charges within action potential, synaptic spikes they may be perceived at

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a macroscopic level and recorded as resulting temporal patterns of AP or synaptic activities. Therefore, temporal patterns or dynamics of AP in neurons or networks of neurons as described in the literature may be a direct outcome of electric field propagation and spatial charge displacement. It is therefore likely that the analyses of such patterns provide less information than one may obtain by analyzing the movement and interaction of charges. Given a current distribution of electric charges and a performance index that measures the error between the ‘model’ and stimuli inputs it is likely that the MPD principle can be reduced to find the minimum displacement of charges able to provide an accurate model of the input. 3.2.9. Is Hypercomputation a Myth? The theory of computation founded by the work of Church, and Alain Turing is still in early stages of development. Theoretical models and algorithmic techniques have been developed mathematically in great detail. At their heart the ideas that information and computation are related have their origins from analyses in physics and computer science. These developments are mainly related to the work of Feynman on quantum physics and new progress in quantum information processing (Nielsen and Chuang 2000; Marinescu and Marinescu, 2005; Angelakis, 2006). The early development of computation related Turing Machines with brain computation capabilities (Putnam, 1960). Computation was generally perceived as a function-based transformation of an input to an output. However, recently it has been suggested that such models cannot describe all forms of computation (Goldin and Wegner, 2008). They show that the definition of Turing computability is highly restrictive since it constrains computation to discrete sequences that generate values in a finite number of steps. Indeed, the Turing model was not constructed to be a general model for computation (Cleland, 2004) and “there is nothing in mathematics or physics to prevent the implementation of a hypercomputational systems”(Stannett, 2006). Even Turing himself suggested that human intelligence “requires more than the standard version of computation described by his machines”(Stannett, 2006). There are several examples where computation cannot be expressed in terms of Turing machines. The behavior of a quantum system cannot be reduced to a Turing model (Komar, 1994; Copeland, 2002; Bringsjord and Zenzen, 2003). Additionally, it has been suggested that “membrane computing” an inspired bio-computational model developed by Paun, (Paun, 2002) can be a new solution for hypercomputation. Using classical physics examples, several nonrecursive computational phenomena can be demonstrated (Scapellini, 2003). Recursive functions are a class of computable functions. Such functions can be expressed in term of themselves and this process is known as "recurrence" or "recursion". As presented above, many natural processes are recursive either in time or space. However the existence of several physical systems whose time evolution is not describable by recursive functions (e.g. wave equation of classical physics) are known to exist (Siegelmann and Sontag, 1992). Scapellini suggested “that the brain may rely on analogue processes for certain types of computation and decision-making”. Possible candidates which may give rise to such processes are the axons of nerve cells.” Even though Scapellini does not describe issues that define new computational paradigms, the proposed hypercomputational model in brain encourages the idea that our brains’ capabilities are beyond those of simple Turing models.

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The new model of computation that includes charges, their movement and interaction show why we cannot reduce brain computation to Turing models. 1. First, the charges move and interact continuously rather than in “isolated jumps” as computation is pictured in the Turing model. Such continuous behavior is clearly a non-Turing behavior. 2. Second the number of charges involved in computation is high enough to provide high degrees of freedom and high dimensionality of processing that can be hardly emulated by single TMs. 3. Third, such movement of charges determines changes in electric field, intrinsic interactions and high levels of parallelism which cannot be easily modeled by TM. Accumulation of information in different forms and the resultant history dependency cannot be also be naturally included in Turing models. Therefore, physical computation in each spike (action potential/synaptic spike) seen as a propagation of electric wave and diffusion process involves the interaction and dynamics of various types of charges. The laws of physics including electric field propagation, charge motion and interaction would then define the general rules of computation. Since charge fluxes are information carriers in neurons, information transfer occurs during an action potential (AP) via the voltage gated ion channels in membranes and the same physical mechanism can be extended to various types of synapses. The action potential can be regarded as providing the required inputoutput dynamics to perform computation by a neuron (Aur et al., 2006). Sodium ions that enter during the depolarization phase can be considered as an input variable while potassium current can be considered an output variable. By computing mutual information between these to variables one can measure the interaction between these two components and the efficiency in information transfer as shown above in section 3.2 of this chapter. This reductionist scheme of charges (ions) as carriers of information represents the required physical machinery for a dynamic process of information transfer (Figure 14) that is substantially limited in the current spike-timing description. Yn

∫ Y2

X

∫ ∫

Y1

Figure 14: Schematic representation of coding as a feedback mechanism that uses the dynamics of charges to compute and build a model during an action potential. Multidimensional input X is dynamically transformed in a spatial distribution Y1, Y2, … Yn of outputs via ion channels

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As a powerful example the visual system has been presented as a highly efficient system that processes information from the environment. Photoreceptors transduce information from received photons into electrical signals that propagate in the network of neurons in the retina, then to the primary visual cortex. The representation of information can be evidenced in distributed electrical activity of neurons and generates models of perceived objects. The intrinsic processing phase is performed by a continuous movement of charges, which corresponds to a permanent information flow. As presented above such processes are governed by the laws of motion and electrodynamics. Charges become the common denominator into which every sensorial input (sound, vision etc) is transformed and processed. Therefore, an adequate representation of information in the brain is shaped by the density of charges and their continuous interaction. At the microscopic level, the laws of quantum mechanics can describe these interactions between charges. However, a detailed analysis of interactions and effective potential energies of the system cannot always be achieved due to high degrees of freedom of charges.

Figure 15 Schematic representation of decoding activity. Electrical synapses in blue and chemical synapses in red as solid spheres allow the transfer of information between neurons.

Information is distributed spatially during every action potential within ion channel activities in axons, membranes and dendrites which represents the coding phase. The mechanism is repeated at larger scales where activation spreads from sensory input areas to “output” areas due to a continuous flow of charges. Between cells this information is carried by electrical synapses (gap junctions) and chemical synapses (Figure 15) and the neuron is able to “read” the input which represents the decoding phase. Therefore, the role of spheres with radius R in (Figure 15) can be seen as a tool to measure the concentration/density of charges. Such activity is repeatedly performed in the brain by synaptic connections. In chemical synapses, considered to be either excitatory or inhibitory, ion channel kinetics are controlled by several neurotransmitters and are considered to be either excitatory or inhibitory. The neurotransmitter release and the activation of ionic conductances in synapses depend on ion concentration

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(Zenisek and Matthews, 2000). As a general mechanism excitation and inhibition is determined at small scales (synapses) and large scales (group of neurons) by the dynamic changes in the densities of charges and their interaction mediated by several neurotransmitters which regulate the computation performed by charges. We can reinforce the fact that once derived this appropriate structure /mechanism that uses the movement and interaction of charges to compute (code/ decode). The structure/mechanism is repeated in space with small variability to design neural networks and in time to complete certain brain functions perceived or measured as brain rhythms. Summary of Principal Concepts •

• • • •

• • • • • • • •

The movement of charges from one point in space to another point in space deletes information in the first point and transfers information to the second point in space. Every charge can get information regarding other charges through the electric field. The erasure of information in any computational device requires an increase in the thermodynamic entropy (Landauer, 1961). Moving charges across the nerve cell membrane determines changes of thermodynamic entropy. These variations of thermodynamic entropy can be directly related with changes in Shannon information entropy (Brioullin,1956; Zurek, 1989) Therefore, using the HH model one can demonstrate that high levels of information can be transferred between the input signals and electrical charges that move (ionic fluxes) within a millisecond-level time domain (Aur et al., 2006, Aur and Jog, 2006) Since these charges are carriers of information then the movement and interaction of electrical charges can be seen as natural forms of physical computation. Therefore, computation in the brain is performed by a dynamical movement and interaction of electrical charges. The optimal path for charge movement is consistent with a Minimum Path Description (MPD) principle. The charge displacement should result in the shortest dynamics which may generate the most accurate representation or model of the input. Learning (self-organization) processes can be seen as an optimization phase which minimizes the number of charges that describe or build a certain model for a set of given inputs. The role of interaction between electrical charges during computation is critical since information is shared between several electric fluxes. Information is distributed spatially during every action potential with ion channel activations in axons, membrane and dendrites which can be described as a process of coding. The distribution of several types of synapses in space is used to “read” and decode the spatial distribution of charges. This approach provides a unifying description of neuronal activity in computational terms at a more fundamental level unachieved by current spiketiming theory.

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• • • •

89

At the microscopic scale quantum probabilistic laws govern the movement of charges their interaction and information processing and communication Turing computability constrains computation to discrete sequences and seems to be inadequate to describe highly interactive systems Brain computations display a non-Turing behavior, the continuous interaction and dynamics of charges provide a high degree of freedom and high dimensionality of information processing Classical and quantum laws and models based on either laws do not compete together, they both coexist and provide a description of computation at different scales

Appendix 1: The Hodgkin-Huxley model describes the action potential in a giant squid axon:

Cm

dV = I leak + I Na + I K + I stim dt

( 52 )

where Cm is the specific membrane capacitance, INa is the voltage-dependent fast-inactivating sodium, IK is a delayed rectifier potassium current, and Ileak is the leakage current. The inward current coming to cell body Istim could be viewed as a dendritic current, a synaptic current, or a current injected through a microelectrode. Then, I leak ,

I Na and I K can be written:

I leak = g leak (V − Eleak )

( 53 )

I Na = g Na m 3 h(V − ENa )

( 54 )

I K = g K n 4 (V − EK )

( 55 )

g K , g Na , g leak are the maximum conductances for currents of the K+, Na+ and respectively of leak currents, while E K , E Na , Eleak are the reversal potentials determined by the Nernst equation. where

A probabilistic model of changes in ionic conductances was presented by Hodgkin and Huxley (Hodgkin and Huxley, 1952) is outlined by the following equations:

dm = α m (1 − m) − β m dt

( 56 )

dn = α n (1 − n) − β n n dt

( 57 )

dh = α h (1 − h) − β h h dt

( 58 )

The conductance variables m, n, and h take values between 0 and 1 and they approach steady state values m∞ (V ) ,

n∞ (V )

and

h∞ (V )

with time constants τ m (V ) ,

τ n (V )

and τ h (V ) :

dm = m∞ (V ) − m dt

( 59 )

τ n (V )

dn = n∞ (V ) − n dt

( 60 )

τ h (V )

dh = h∞ (V ) − h dt

( 61 )

τ m (V )

90

Chapter 3: Information and Computation Appendix 2: Consider a linear input -output model:

y = as + n ( 62 ) where n the noise has a Gaussian distribution (mean μ=0 and standard deviation σ=1). To compute MI for a Gaussian channel we assume that the signal s has also a Gaussian distribution. From Rieke et al., A12, (Rieke et al., 1997 [27]) mutual information can be computed: I = 0.5 log 2 (1 +

a2 < s2 > ) < n2 >

( 63 )

This model is used to asses the errors of the histogram method in computing MI for a Gaussian input. Appendix 3 For three variables X1, X2, X3 by definition multi-information is the difference between marginal distribution and joint distribution: I(X1 ; X 2 ; X 3 ) = H(X 1 ) + H(X 2 ) + H(X 3 ) - H(X 1 , X 2 , X 3 )

( 64 )

Then from Eq.16 and 36 multi-information is: I(X 1 ; X 2 ; X 3 ) = I(X 1 ; X 2 ) + H(X 1 , X 2 ) + H(X 3 ) - H(X 1 , X 2 , X 3 )

( 65 )

Eq 37 can be written then:

I(X 1 ; X 2 ; X 3 ) = I(X1 ; X 2 ) + I((X 1 , X 2 ); X 3 ) ( 66 ) Since mutual information is always positive, multi-information always increases by introducing a new variable. In addition Eq. 37 can be generalized by induction for several variables. References Adrian E., (1932) The Mechanism of Nervous Action: Electrical Studies of the Neuron. Philadelphia: University of Pennsylvania Press. Amari S. (1977) Neural Theory of Association and Concept Formation, Biological Cybernetics, Vol. 26, pp. 175–185. Amari S., (1985) Differential-geometrical methods in statistics, Lecture notes in statistics, SpringerVerlag, Berlin. Anderson J.A., (1995) An Introduction to Neural Networks, The MIT Press. Angelakis D.G (2006) Quantum information processing: from theory to experiment. Proceedings of the NATO Advanced Study Institute on Quantum Computation and Quantum Information, Chania, Crete, Greece, 2-13 May 2005 IOS Press. Aur D., Connolly C.I., and Jog M.S., (2005) Computing spike directivity with tetrodes. J. Neurosci. Vol. 149, Issue 1, pp. 57-63. Aur D., Connolly C.I. and Jog M.S., (2006) Computing Information in Neuronal Spikes, Neural Processing Letters, 23:183-199. Aur D., Jog MS (2006) Building Spike Representation in Tetrodes, Journal of Neuroscience Methods, vol. 157, Issue 2, 364-373. Aur D, Jog MS (2007) Neuronal spatial learning, Neural Processing Letters, vol. 25, No 1, pp 31-47. Barnes T.D., Kubota Y., Hu D., Jin D.Z. and Graybiel A.M., (2005) Activity of striatal neurons reflects dynamic encoding and recoding of procedural memories, Nature, vol 437, Nr. 7062 pp. 1158-1161. Bell C.C., Grimm R.J., (1969) Discharge properties of Purkinje cells recorded on single and double microelectrodes. J Neurophysiol, 32:1044-1055. Bennett C.H., (1973), Logical Reversibility of Computation,IBM J. Research and Develop. 17 525-532 Bennett C.H., (2003) Notes on Landauer's principle, reversible computation, and Maxwell's demon, Studies in History and Philosophy of Modern Physics, Volume: 34, (2003), pp. 501—510. Bennett B. D., Callaway J.C., and Wilson C. J., (2000) Intrinsic Membrane Properties Underlying Spontaneous Tonic Firing in Neostriatal Cholinergic Interneurons, The Journal of Neuroscience, November 15, 20(22):8493-8503. Bekenstein, Jacob D. (1981) Universal upper bound on the entropy-to-energy ratio for bounded systems. Physical Review DD 23 (215).

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Bekenstein J. D. and Schiffer M., (1990) Quantum Limitations on the Storage and Transmission of Information, Int. J. of Modern Physics 1:355-422. Bringsjord S., Zenzen M., (2003) Superminds: People Harness Hypercomputation, and More, Kluwer Academic Publishers, Dordrecht. Burgin, M., Klinger, A., (2004) Three aspects of super-recursive algorithms and hypercomputation or finding black swans, Theoretical Computer Science 317 (1-3), pp. 1-11. Buzsáki G., (2006) Rhythms of the Brain, Oxford Press. Cerf-Ducastel B., and Murphy C., (2003) FMRI brain activation in response to odors is reduced in primary olfactory areas of elderly subjects, Brain Res. 986, pp. 39–53 Colvin R. A., (1998) Characterization of a plasma membrane zinc transporter in rat brain, Neuroscience Letters, 247 147–150. Cover T.M., Thomas J.A., (1991) Elements of Information Theory, John Wiley & Sons, Inc. Copeland, B.J., (2002) Hypercomputation , Minds and Machines 12 (4), pp. 461-502. Jaynes E.T., (1957) Information Theory and Statistical Mechanics, in Physical Review, Vol.106, pp. 620-630. Cleland C., (2004) The concept of computability, Theor. Comput. Sci., Volume: 317, pp. 209—225. Ceruti M.G., Rubin S.H., (2007) Infodynamics: Analogical analysis of states of matter and information Information Sciences, 177 (4), pp. 969-987. Cockshott P. and Michaelson G., (2007) Are there new models of computation? Reply to Wegner and Eberbach , The computer journal, vol:50 iss:2 pg:232 -247. Churchland P.S. and Sejnowski T. J., (1992) The Computational Brain, MIT Press. Denker, J.S., (1994) A review of adiabatic computing, Low Power Electronics, Digest of Technical Papers., IEEE Symposium 10-12, Page(s):94 – 97. Eliasmith C., (2002) The myth of the Turing machine: the failings of functionalism and related theses,Journal of Experimental & Theoretical Artificial Intelligence, Volume 14, Issue 1 , pages 1 – 8. Eliasmith C. and Anderson C. H., (2003) Neural Engineering: Computation, MIT Press. Feynman R. P., and Hibbs, A. R., (1965) Quantum Physics and Path Integrals, New York: McGrawHill. Fukushima K., (1980) Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position, Biological Cybernetics, 36[4], pp. 193-202. Fraser AM., Swinney H.L., (1986) Independent coordinates for strange attractors from mutual information Phys. Rev. A 33, 1134 – 1140. Gardner D., (1993)The Neurobiology of Neural Networks., MIT Press. Gerstner W., and Kistler W.M., (2002) Spiking Neuron Models Single Neurons, Populations, Plasticity Cambridge University Press. Gibbs J. W., (1902): Elementary Principles in Statistical Mechanics, Yale University Press. Goldin, D. Q., (2000) Persistent Turing Machines as a Model of Interactive Computation, Lecture Notes In Computer Science, ISSU 1762, pages 116-135. Goldin D., Wegner, P., (2008) The interactive nature of computing: Refuting the strong Church-Turing Thesis , Minds and Machines 18 (1), pp. 17-38. Gray R.M., (2000) Entropy and Information Theory, Springer-Verlag New York. Grossberg, S., (1976) Adaptive Pattern Classification and Universal Pattern Recoding: I. Parallel Development and Coding of Neural Feature Detectors, Biological Cybernetics, Vol. 23, pp. 121–134. Grossberg S., (1987), Competitive learning: From interactive activation to adaptive resonance, Cognitive Science (Publication), 11, 23-63. Goychuk I Hanggi P., (2000) Stochastic resonance in ion channels characterized by information theoryPhys. Rev. E 61, 4272 - 4280. Häusser M., Clark B.A. (1997)Tonic synaptic inhibition modulates neuronal output pattern and spatiotemporal synaptic integration. Neuron, 19:665-678. Hebb D.O., (1949) The organization of behavior, New York: Wiley Hecht-Nielsen R., (2007) Confabulation Theory the Mechanism of Thought, .Springer Verlag. Heisenberg W., (1930) Die Physikalischen Prinzipien der Quantenmechanik (Leipzig: Hirzel). English translation The Physical Principles of Quantum Theory University of Chicago Press. Horodecki M., Horodecki P., Horodecki R., Oppenheim J., Sen A., Sen U., Synak-Radtke B., (2005) Local versus nonlocal information in quantum-information theory: Formalism and phenomena Physical Review A - Atomic, Molecular, and Optical Physics 71 (6), art. no. 062307, pp. 1-25 Hodgkin A. L. and Huxley, A.: (1952) Quantitative description of ion currents and its applications to conduction and excitation in nerve membranes, J. Physiol. (Lond.), 117.500–544. Hopfield J., and D. Tank (1985) Neural Computations of Decisions in Optimization Problems, Biological Cybernetics, Vol. 52, pp. 141–152.

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Hopfield J., (1984) Neurons with Graded Response Have Collective Computational Properties Like those of Two-State Neurons, Proceedings of the Proc Natl Acad Sci U S A. 1984 May, 81(10): 3088–3092. Hopfield, J., and Tank D., (1986) Computing with Neural Circuits, Science, Vol. 233, pp. 625–633. Izhikevich E. M., (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Jolivet R, Schürmann F., T. Berger, R. Naud, W. Gerstner, A. Roth (2008) The quantitative singleneuron modeling competition, Biological Cybernetics, Vol. 99, No. 4, pp. 417-426. Kaiser A., Schreiber, T., (2002) Information transfer in continuous processes, Physica D: Nonlinear Phenomena 166 (1-2), pp. 43-62. Koch C., (1999) Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press: New York, New York. Kohonen T., (1972) Correlation Matrix Memories, IEEE Transactions on Computers, Vol. C-21, pp. 353–359. Kohonen T., (1982) Self-Organized Formation of Topologically Correct Feature Maps, Biological Cybernetics, Vol. 43, pp. 59–69. Kohonen T., (1984) Self-Organization and Associative Memory, Springer Verlag, Berlin. Komar A., (1964) Undecidability of Macroscopically Distinguishable States in Quantum Field, Theory, Physical Review, second series, 133B, pp. 542–544. Kofuji P., and Newman E.A., (2004) Potassium buffering in the central nervous system, Neuroscience Volume 129, Issue 4, Pages 1043-1054. Khachiyan L. G., (1996) Rounding of polytopes in the real number model of computation,Mathematics of Operations Research 21 307, 320. Landau L. D. and Lifshitz E. M., (1960) Electrodynamics of Continuous Media, Pergamon, Oxford, pp. 368–376. Landauer R., (1961) Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, Volume: 5, (1961), pp. 183--191 (Reprinted in Leff & Rex (1990). Lee R.S. T., Loia V., (2007) Computational Intelligence for Agent-Based Systems, Springer. Landauer, R., (1961) Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 5, 183-191. Landauer R., (1988) Dissipation and noise immunity in computation and communication, Nature 335 (6193), pp. 779-784. Latham A., Paul D.H., (1971) Spontaneous activity of cerebellar Purkinje cells and their responses to impulses in climbing fibres. J Physiol ,(Lond) 213:135-156. Li M. and Vitanyi P. M B., (2007) An introduction to Kolmogorov complexity and its applications. Springer, New York, 2nd edition 1997, and 3rd edition. Lippincott E R. and Schroeder R., (1955) General Relation between Potential Energy and Internuclear Distance for Diatomic and Polyatomic Molecules. I J. Chem. Phys. 23, 1131. Lloyd S, (2006) Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos. Marinescu D. C and Marinescu G. M. (2005) Approaching Quantum Computing, Prentice Hall. Maass W., (1997) Networks of spiking neurons: the third generation of neural network models. Neural Networks, 10:1659-1671. Maass W. and Bishop C.M., (1998) Pulsed Neural Networks, MIT Press. Mainen ZF, Sejnowski TJ., (1995) Reliability of spike timing in neocortical neurons, Science, Vol 268, Issue 5216, 1503-1506. McClelland J.L. and Rumelhart D.E., (1988) Exploration in Parallel Distributing Processing, Brandford Books, MIT Press, Cambridge, MA. McCulloch W. and Pitts W., (1943) A Logical Calculus of Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics 5:115-133. Meir A., Ginsburg S., Butkevich A., Kachalsky S. G., Kaiserman I., Ahdut R., Mesulam, M., (1998) From sensation to cognition, Brain 121 (6), pp. 1013-1052. Minsky, M. and Papert S., (1969) Perceptrons: An introduction to Computational Geometry, MIT Press. Moore C., (1996) Recursion theory on the reals and continuous-time computation, Theor. Comput. Sci., Volume: 162, pp. 23-44. Moore C., (1990) Predictability and Undecidability in Dynamical Systems, Physical Review Letters, Vol. 64, pp. 2354-2357. Nielsen M. A., I Chuang L.I., (2000) Quantum computation and quantum information, Cambridge University Press. Newell A., (1990) Unified Theories of Cognition. Cambridge, Massachusetts: Harvard University Press. O'Reilly C. R. and Munakata Y., (2002) Computational Explorations in Cognitive Neuroscience Paun G., (2002) Membrane Computing. An Introduction. Springer-Verlag, Berlin.

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CHAPTER 4. Models of Brain Computation Transposing biological information into algorithms that mimic biological computations in neurons has proved to be a difficult task and current work in neuronal computation is still influenced by the McCulloch-Pitts model considered to be the first model in neural computation. The firing rate and sigmoid neural activation function are at the origin of implementation of the second generation of neural networks. More recently the spike timing approach is considered the third generation of models (Maass, 1997, Rieke et al., 1997; Gerstner and Kistler, 2002; Mainen and Sejnowski 1995). However, it is already accepted that, as a whole, the brain can be regarded as a physical computer. This is not necessarily a new theoretical concept since many developments in physics and mathematics showed that complex computations can be achieved by moving billiard balls when they are modeled as particles within a dynamical system (Bunimovich and Sinai, 1980). As presented by Moore (Moore, 1990), a classical example shows that a single particle which moves in a threedimensional potential is able to perform computation and is theoretically equivalent to a Turing machine. Computation is, therefore, performed in this case following the Newtonian laws of motion where the state-space system that describes the dynamics of single particles has definite inputs and can generate outputs. It is believed that coding and decoding of information in the brain is governed by classical physics; however, at microscopic level many interacting particles within molecular machines, namely the proteins, may require a different model. Quantum properties seem to occur almost anywhere at microscopic scales (e.g. Brooks, 2005) and sometimes such characteristics can be macroscopically perceived (Julsgaard et al, 2001). Does the brain exploit these quantum features? Undoubtedly the macroscopic outcome recorded as a train of spikes cannot be generated without this infrastructure that obeys the quantum laws of physics. However, when we talk about models, the problem of classic or quantum model seems to be ill posed since no matter how complex, models that are mathematical descriptions cannot fully characterize the real world phenomena. 4.1. Dynamics Based Computation – The Power of Charges The details of electrical patterns within a spike and the relationship with emerging behavior have been observed during T-maze procedural experiments (Aur and Jog, 2007b). Each spike can be reduced to a vector representation (see spike directivity) that is spatiotemporally modulated during learning. This view provides a natural interpretation for vector representation or vector coding (see Georgopoulos et al. 1993); however this model although very useful is an over simplification for computation with charges. Experimental data, neurophysiological recordings, the model of information transfer based on Hodgkin-Huxley description (Aur et al., 2006), prediction about semantics (Aur and Jog 2007a) are consistent with and support the idea that computation and information transfer is carried out by electrical charges.

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4.1.1. Space and hidden dimensions Our common view of space is shaped by a 3D view of the world. Frequently, in physics, time can be added as the fourth dimension. In quantum mechanics or string theory extra dimensions are used to describe new concepts since there is no demonstration which limits any space analysis to three dimensions. The general theory in mathematics is based on the concept of an n-dimensional Euclidian space E n. This ndimensional presentation where n>3 can be extended to every incoming stimulus s(t) or its discretized version s(k), (k ∈ N) which can be seen as a point in an n-dimensional

Figure 1 Schematic representation of a three-dimensional sphere and a three-dimensional cube. If these three- dimensional objects are projected in a single dimension both become lines

space. The point Sk={s(k), s(k+1), ..., s(k+n)} corresponds to n-discrete samples of the signal. Since, usually the incoming stimulus is not constant the variation in signal amplitude can be seen as a movement or a trajectory of the point in space (Figure 2). This system where particles may operate under Newtonian laws can well describe complex computational processes (Beggs and Tucker, 2007). In this case the dimension of processing space becomes critical. A very simple example shows the importance of space dimension selection. If the space dimension is lowered from 3D to 2D and subsequently to 1D, some relevant characteristics required for object perception are lost. It is likely that with dimension reduction, the information required for object recognition is lost. A cube or a sphere projected onto 1D space cannot be distinguished since both objects become lines in the single dimension. If information is lost, the cube or the sphere cannot be reconstructed. This simple example shows that projecting onto higher dimensional space may be a better alternative than reducing dimensionality for solving object recognition problems. Theoretically, the Support Vector Machine is a good abstract example of increasing dimensionality for learning classifiers (Vapnik and Chervonenkis, 1974; Vapnik et al., 1997). o The dimension of processing space is critical in terms of information resources required to obtain/find solutions for certain problems. In the brain it involves electric charges as a way of building a higher dimensional space. This simple thought experiment reveals the fact that the brain seen as a collection of electrical charges that move and interact may hide information since a human observer used to a three dimensional view of the world cannot perceive the depth and richness of higher dimensional processing with charges. These characteristics seem to

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be essential for several processes that occur simultaneously in the brain. In this case the principles of brain computation can be related to an application of fundamental physical principles such as the principle of least action or second law of thermodynamics. Therefore, the physical mechanism of coding should map the incoming signals/features into the dynamics of charges or group of charges in a multidimensional space. The X

S(t) Voltage

Xi

Time

Figure 2: The incoming signal properties can be translated to a distribution of charges in space. An n-dimensional input signal has an equivalent representation in brain as a distribution of electrical charges in space. A multi-dimensional input signal s(t) can be represented in the brain using a spatial distribution of electrical charges

process can be experimentally evidenced by measuring the spatial distribution of electrical activities within certain regions of the brain starting with smaller microscopic scales during spike occurrences at the level of ion channels. These characteristics of many charges that have a certain dynamics show the importance played by unperceived hidden dimensions in the brain for processing since there is no natural constraint to limit any space analysis to three dimensions (Randall, 2007). Additionally, the assumption of a Euclidian space is restrictive and in contrast to the realm of space curvature determined by the presence of charges and electric fields that are likely to build a non-Euclidean universe in the brain. 4.1.2. Minimum Path Description - The Principle of Least Action The minimum description length concept has been discussed in Chapter 3. This principle is used to find an optimal way to describe data or models (Rissanen, 1984; Barron et al., 1998). As shown in the previous chapter this model that describes the dynamics of charges may require a similar mathematical formalism since the physics of charge movement and interaction can be described in terms of universal physical laws. The application of physical principles should provide answers to several questions. What is the trajectory of charged particles? Can we predict electric propagation of charges knowing some constraints? To find a way to express the movement of a charge in physical space we have to exploit the laws that govern energy constraints. For at least two hundred years this law is known as the principle of least action (Maupertuis, 1774; Hanc et al., 2005). The analysis of this dynamic system can be made therefore in terms of a Lagrangian L : L=K −P

(1)

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where P is the potential energy and K is the kinetic energy of the system. If one considers that each charged particle i has the dimensional coordinates ri , then, to express the movement of charges we can introduce the generalized coordinates q1 , q2 ,..., qm : ri = ri (q1 , q2 ,..., qm , t ) (2) Therefore, the trajectory of charges can be obtained by writing the Lagrange equations: d ∂L ∂L , i=1,2…m ( )− dt ∂q&i ∂qi and the evolution in time of the system can be summarized: 0=

(3)

tf

S = ∫ L(q1 , q2 ,...., qm , q&1 , q& 2 ,...., q& m , t )dt

(4)

ti

where S is the action. Another way to express the evolution in time is to write the Hamiltonian as the Legendre transform of the Lagrangian: H (qi , p i , t ) = ∑ q& j p j − L( qi , pi , t ) j

(5)

where qi , pi are the canonical coordinates in phase space (Marsden and Ratiu,1999) The theory known as the principle of least action states that the charged particle r should follow the “worldline” r for which the average kinetic energy minus the average potential energy is minimum and can be written in the form where the optimal *

action S : tf

r r S * = arg min ∫ ( K (r ) − P(r ))dt r

(6)

ti

The “path description” formalism represents, in this case, the physical law that r requires a certain path for the charged particle that has to follow the “worldline” r representing the “least action”. Therefore, using this principle of least action (de Maupertuis 1744, Hanc et al., 2005) we are able to predict the trajectory of the particle/charge. Since the energy is not a vectorial quantity, the easiest way to figure out the trajectory is to consider a simple example where the kinetic energy is zero. In this case the charged particle will move along the force direction: r F = − grad (P) (7) The sharper the gradient slope, the faster the charge movement in the steepest direction since nature selects the lowest possible value of the kinetic energy which is required for the movement between the two selected points. The displacement of the particle or group of particles can indeed be represented as a vector in 3D space (see Chapter2) and the minimum path description highlights the principle where every charge moves only along the selected “worldline”. o The Minimum Path Description states that the electric propagation (charge propagation) follows the principle of least action in selecting the trajectory where the average kinetic energy minus the average potential energy is minimum as long as these charges follow a given topology

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This basic principle can be directly related to changes that occur in dendritic topology, development of synapses, ion channel distribution and neuronal topology along with selective genetic changes at molecular level. Such modifications are results of emergent adaptations required to follow the principle of least action and to represent with a minimum cost certain inputs and stimulus features. Since topological properties of the electric fields created by point charges require a non-Euclidean description, in a Riemannian manifold the curves that correspond to extremal paths (geodesics) are locally of minimal length and then the shortest possible path is connected along the geodesics. o In a Riemannian space the Minimum Path Description follows the curve of minimal length which minimizes the energy function (see do Carmo, 1992). This can be interpreted as a natural extension of the Minimum Path Description to the framework of general Riemannian manifolds and it is likely to be related with morphological changes at neuronal level that allow locally the shortest possible paths for electric charges. Other physical constraints may derive from the concept of free energy (Gibbs, 1873) which reflects the thermodynamic perspective. 4.1.3. Natural Computation –Abstract Physical Machines The idea of computation by a physical system is not new. Recent research accredited the idea that physical systems can compute more than a machine which performs computation based on algorithms (Beggs et al., 2008). Examples of such systems include the abstract physical machines that may compute noncomputable/ computable Turing functions using experimental computational models. Basically any physical or biological entity as a part of nature can be seen as a system which can be transformed, reduced and simulated using algorithmic procedures. These forms of natural computation may include variants of the analogue-digital scatter machine, DNA computing (Adleman, 1994; Kari and Rozenberg, 2008), membrane computation which may display some hypercomputational powers (Calude and Păun, 2004; Beggs et al., 2008). •

Membrane computing

Membranes represent physical barriers for charges and molecules to move and interact. Every charge can escape from this trap due to certain potential configurations. The result is that charges/molecules cross the membrane and determine the nature of neuronal activity in that region. Theoretical membrane computation combines some abstract computing ideas which formalize the biological substrate of (excitable) polarized membranes (Păun 2002; Păun 2000). The power of computation of the membranes consists of using rewriting rules analogous to reactions taking place in the cell that encode local transitions in cells with specific mechanisms (e.g. division mechanism). Called P-systems (from Păun) they can be used to solve problems in an algorithmic manner where computations are sequences of synchronous transitions. These models are equal in power with Turing machines and can be generated by a deterministic Turing machine in polynomial time. The membrane can be seen as an abstract machine: M = ( A, μ , ω1 ,..., ω m , E , R1 ,..., Rm ) (8) that includes an alphabet of objects A , a membrane structure μ , ω1 ,..., ω m the strings that represent the set of objects E present in the environment and a finite set of rules

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R1 ,..., Rm that define the membrane transport properties. Importantly, these rules apply to transport phenomena when an “object” enters or leaves the membrane and can be labeled as ‘uniport’ rules (a, in), (a, out), ‘symport’ rules where both objects ab enter together or leave the membrane (ab,in),(ab, out) or antiport rules when the object a enters and object b leaves the membrane (a, in; b out). Such mathematical rules can formalize changes that occur during polarization and depolarization phases in neurons where ions Na+ and K+, Ca2+ cross the neuronal membrane. Some new membrane models include dynamic membrane structures with rules that define membrane “creation”. Additionally, some new advances in this field incorporate distributed computing models, neural like systems based on membrane development (Ionescu et al., 2006; Chen et al., 2008)

•

Particle Computation

The model proposed by Eberhart and Kennedy in 1995 was inspired from models of social behaviors in birds (Eberhart and Kennedy, 1995). In optimization problems this model seems to perform better than evolutionary algorithms (genetic algorithms). Given a multivariable optimization problem: Minimize f ( x1 , x2 ,...., xn ) (9) x1 , x2 ,...., xn

the solution is found by the moving particles in the search space to find the best solution, i.e. the moving particles perform the searching. The theoretical model assumes an n-dimensional search space S, and the j-th, particle position is an ndimensional vector: x j = ( x j1 , x j 2 ,....x jn )T ( 10 ) where the velocity of this particle is also an n-dimensional vector v j = (v j1 , v j 2 ,....v jn )T

( 11 ) and j=1,2,….N. The matrix of velocities of particles can be updated at each discrete step k: Vk +1 = WVk + c1 A( Pb − X k ) + c2 B( Pg − X k ) ( 12 ) where Pb is the memory of the best solution and Pg is the best position group of neighborhood particles, W is the inertia weight matrix, c1 , c2 two positive coefficients, and A and B two random matrices. The position of particles is obtained: X k +1 = Vk + X k ( 13 ) based on their previous position X k . Besides the standard version proposed by Eberhart and Kennedy in 1995 this technique can be used to find solutions for several optimization problems and since it has received constant interest from scientists, numerous versions and improvements have been recently developed (Parsopoulos and Vrahatis, 2004; Li and Liu, 2002; Coello et al., 2004; Jie et al., 2006) •

Computation with Charges –An Example

As seen above, computation can be seen as an intrinsic physical property of a natural system to solve an optimization problem by searching in the space for the best solutions using the movement of several particles/charges. A simple example considers

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a quadratic function V which represents an electric potential that takes the values of a given function: V ( x, y ) = ( x − 0.5) 2 + ( y − 0.7) 2 − 9 ( 14 ) The optimization problem can be seen as a mathematical problem to find the values x * , y * for which the potential V reaches the minimum value. There are several possible paths that one charge may follow to reach the goal. Here we are interested to reveal a possible physical way to implement such a procedure given the fact that V is an electric potential. In order to find the minimum value for the function V one can apply several theoretical techniques. However, the best answer will be provided by the charge or group of charges with the path that provides the correct solution (the optimal path) following the principle of least action. The electric field across each axis can then be computed: ∂V = 2( x − 0.5) Ex = − ( 15 ) ∂x ∂V Ey = − = 2( y − 1.5) ( 16 ) ∂y ∂V =0 Ez = − ( 17 ) ∂z As an example, when we consider the conservation of energy for every charge qi with mass mi over the x axis we can then write: Wi =

1 2 mi vi + qiV ( x) 2

( 18 )

where the velocity of the charge is: 2 (Wi − qiV ( x) ) mi If there is no dissipation, in terms of conservation energy then dW=0 and: .. dV mi x = −qi = − qi E x dx vi =

( 19 )

( 20 )

In this particular case the electric field generates an electric force that can be assimilated with an elastic force determining periodic movements of the charges. The total energy W can be rewritten: 1 1 2 mv x + k E x 2 2 2 where the oscillatory solution has the form: W=

x=±

2W sin ω x t k

( 21 )

( 22 )

and ω x = k E / m . The above technique can be applied similarly for the y and z axes. In the presence of a dissipation force, the charge will always move to the minimum potential energy and will remain in that position which in fact provides the expected solution of the proposed problem to find the coordinates x* , y * , z * where the potential

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reaches the minimum value V * . Since almost all processes are dissipative then this natural way of searching and finding solutions should work. The time required to find/search the optimal solution and the robustness of this solution are two other important issues. Under dissipative constraints a rapid, stable response can be obtained if several charges are involved in the process. However, if dissipation is weak, once these charges have reached their spatial position for the minimum value, they come close enough to generate an electrostatic force. Since all charges have the same sign, they will move far apart and the process of finding the minimum potential is repeated indefinitely. This simple example shows a natural way of computing with charges which follows the “least action” principle. One can find a similitude between the movement of charges during action potential and this description where several charges move to reach the solution (minimum value for electric potential). There are several important ideas which can be drawn from this theoretical experiment: First, under certain conditions the electric force can be compared (in a parabolic field) with an elastic force. Second, if the number of charges is increased then the space exploration is dramatically improved and the solution can be found faster (parallel search=the power of many) (see Figure 8, a). Additionally, many charges can provide a richer representation and the error in representing the feature/stimulus tends to decrease exponentially with the number of charges involved in the process (Figure 8 b). Third, the random motion of charges usually seen as a noisy process in some cases may help to achieve a better solution (Figure 7) since the charge ci can be anywhere in the threedimensional space. For every individual charge ci or a group of charges ci we can define a function V ( xci , y ci , z ci ) which characterizes the quality of solution reached by the charge or the group of charges ci in providing the minimum electric potential V . If the charge cj is better placed than ci then V ( xci , yci , z ci ) > V ( xcj , ycj , zcj ) One can see that this issue in case of charge computation gives another perspective for noise processes and challenges the current view regarding “noise” in brain processes (see Eliasmith and Anderson, 2003) Importantly, this natural process can be modeled on a computer performing the following algorithmic steps: Step 1: Charges are distributed randomly in space, they are characterized by a mass center position xc, yc, zc, and their velocity vx , vy, vz Step 2: Their velocity and position are computed using a Eq ( 11 ) and Eq.( 13 ) respectively Step 3: Groups of charges are found from the population of charges which are better placed than other charges in the group ( V * ( xcg , y cg , z cg )
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received. Indeed, the solution can be found faster if in time these charges are constrained to follow certain paths. One way to constrain a charge or a group of charges to pursue a certain trajectory is to develop electrical or mechanical barriers with certain configurations. Or the electric field should be configured in such a way so as to constrain the movement of charges. Importantly, the existence of neurons with membranes, receptors, dendrites, axons and ion channels plays this role since channel gating

Figure 3: Schematic representation of electric field lines and the required direction in red for the group of charges in blue color to reach the point in three- dimensional space where the potential has the minimum value V

*

( x, y, z ) .

processes can be considered to be part of this mechanism of controlling the movement of electric charges in space. Other cellular structures including glia also regulate this extracellular flux (MacVicar, 1984; Volterra and Steinhäuser, 2004). As an example the astrocytes provide potassium homeostasis by removing the excess of ions (Nielsen et al., 1997). Since glial cells outnumber neurons (Steinhaeuser and Gallo 1996) their role in charge motion (or flux), homeostasis and computation cannot be ignored. The existence of such refined regulatory mechanisms at the brain scale proves that the dynamics of electrical charges is an essential component in this process and is not solely the attribute of neurons. o Rhythmic processes (e.g. action potentials, synaptic spikes) can be seen as a way of periodically restarting (reiterating) the computation performed by moving electrical charges. o Computation with charges may use the “noise” as a better way to improve the finding/searching strategy in the space. o Neurons with spatially distributed synapses/ ion channels restrict/ constrain the diffusion of charges to certain spatial topologies. o Since the homeostatic processes including ion regulation in the brain is provided by glial cells, a deeper understanding of their role in assisting and maintaining normal ionic environment is essential and cannot be ignored. Therefore, the term ‘neural computation’ doesn’t seem to fully describe the complexity and dynamics of electric charges in brain. This is the main reason why we

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prefer to avoid the term “neural computation” and we think that ‘brain computations’ or ‘brain language’ as a metaphoric expression are more appropriate to describe computational processes in the brain. 4.1.4. Back to Building Brains Seen as movement and interaction of charges, brain computations can be regarded as a set of transformations that map input signals acquired from different sensory systems (audio, visual, tactile) into spatio-temporal distribution of charges at neuron level. Due to their multidimensional intrinsic behavior and parallelism charges are likely to achieve a strong computational power. Such computations could provide the required capability to process complex input signals. These signals/input stimuli tend to be fully reconstructed (decoded) if the inverse set of transformations T-1 is performed. This inverse set of transformations can be carried out even in the absence of the external input stimuli. Theoretically, as a result of bijectivity, this set of inverse transformations carried out by charges could be seen as the basic phenomenon that helps us to remember/recreate previous events, places, names or imagine data, objects. Additionally, it is likely that this distribution of charges and their intrinsic interaction can provide a rich representation of the solution space. While these transformations are physically performed, we would like to further analyze how such transformations can be modeled at the micro scale level. It is already known that every action potential is a result of the high number of charges that move in space with trajectories controlled by the cell machinery, spatial architecture of neurons, T CODE

INPUT SPACE

CHARGE SPACE

DECODE

T- 1

Figure 4: Computation in the brain can be reduced to a set of transformations T between input signals and spatio-temporal distribution of charges. Such transformations may define a morphism between the signal space Sn and the charge space Cm. Brain mapping can be described as set of transformations between the input signal space and spatial distribution of charges which can be homeomorphic if the map is a bijection. The input signals/stimuli tend to be fully recovered after the inverse set of transformations T-1 is performed.

distribution of ion channels and several external constraints including electric fields. Therefore, instead of defining only two distinct states for a neuron (presence or absence of action potential, “Up and Down states”), computation can be described by the

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dynamics and interaction of electrical charges. A systemic approach of the phenomenon should identify the following data: 1. The set of inputs with initial conditions r ∈ R (stimuli, intensity of electric field) 2. The set of outputs obtained during measurements at a certain time (e.g. coordinates of selected charged particles, x ∈ X ); 3. The states of the system that define the properties of the smallest set of variables which characterize the dynamics of the system; in this case the set of positions xi and velocities vi of electric charges; 4. The state transition is determined by physical laws of movement (Newtonian, quantum electrodynamics, Poisson-Nernst-Planck equations) The motion of charges in an electric potential can be simply modeled using the Langevin equation: r r r r dvi mi = −miγ i vi + Fi R (t ) + qi Ei ( 23 ) dt where mi is the mass of the qi charge. The average frictional force that acts on the charge qi is dependent on the velocity v of the charge: r r Fi D = miγ i vi ( 24 ) where γ a coefficient is a function of the viscosity of the fluid. The random collision

term Fi R (t) is added and can be assimilated within as white noise. At a given position r r where the charge qi is located, the force exerted by the electric field is Fi e = qi Ei . Working with large numbers of charges may generate severe computational difficulties, therefore, to simulate the movement of charges and their interaction we have to select r r only a few charges. For every two selected charges at distances r1 and r2 respectively in space the Coulomb interaction depends on the value of charges q1 and q2 and the distance between them: r r r (r − r ) F = kq1q2 r1 r 2 3 ( 25 ) | r1 − r2 | where k= 4πε0 εr where ε0 =8.854x10-12 F/m, εr=80 . If the two charges have the same sign, they cannot come very close since the repulsive force will drive them apart. This feedback model tries to resume the effect of the electric field where charges are mainly driven by the dynamics of the input signals generated by stimuli and/or spatial distribution of other charges within macromolecular formations such as protein structures/polarization. The electric force that acts on each charge (Figure 5) depends on its position xi in the field. Therefore, under certain conditions (electric field generated by other charges) the electric interaction between charges can be seen as described in the above example as an elastic force that drives the charge to a certain r position ri in space. The previous equation can be rewritten: mi

dvi = −miγ i vi + Fi R (t ) + k E (ri − xi ) dt

( 26 )

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FR r

+

kE

+

dv/dt

∫

-

-

V

X

∫

mγ

Figure 5: Schematic representation of

the feedback system which drives the charge qi to

position xi in space. This plot formalizes the effect of a three dimensional potential where computation can be seen to be performed by the movement of charges.

Xb, Vb

Xa, Va

Fe

Figure 6 Schematic representation of an elastic force which occurs as a result of electrical interactions. This force can be repulsive or attractive depending on the sign of charges. For two positive ions the electrostatic force is repulsive

where the elastic force depends on the error in displacement ri − xi and the spring constant k E (Figure 6). If the velocity of the charge is known, then the position xi of the charge qi can be obtained: t

xi = ∫ vi dt

( 27 )

0

In order to simulate this model the simplest way is to use the Euler integration method (Ascher and Petzold, 1998) and the dynamics of charges can be seen as a map: X k +1 = f ( X k , Rk )

( 28 ) where the function f specifies the dynamics of the system, X k and Rk are the set of positions, respectively the set of references at k step ( k ∈ N ). In this context the physical description required to reach the final position in a minimum number of steps

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is related to minimum action principle and can be expressed and easily translated into an objective function: J = arg min ∑ ( Rk − X k ) 2 ( 29 ) k k The number of steps required in simulation to achieve the goal depends on several factors that should include initial position of charges, their initial velocities, and the existence of collisions. Importantly, the analysis of the number of steps shows that collision dependent noise can play an important role and sometimes these random events may allow the goal to be reached faster (see Figure 7) b

a

2.5

2

Z

Z

2 1.5

1.5 1 2

1 2

3

3 1.5

1.5

2 1

1 0

Y

2 1 0

Y

X

1 X

Figure 7 : The representation of a single charge in movement where the probability to reach the goal in a certain number of steps depends on initial position, velocity and the existence of collision generated noise. a. Without noise the charge reaches the goal in 33 steps ( J <0.01) b. With added white noise the charge reaches the goal just in 28 steps

Another factor that may determine how to reach the goal faster is the number of charges involved in this dynamic movement. If the number of charges is increased, the time required to find a certain solution may be significantly reduced since many charges can cover and explore a larger area/volume better. These simulations indicate an exponential decrease of required time to find a certain solution with an increase in the number of charges. However, these charges will require far more energy to move and interact in space. -3

2.2

a

x 10

0.3

b

2

0.29

1.8

0.28

Time[ms]

Error

1.6 1.4 1.2 1

0.26 0.25 0.24

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700

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Figure 8: Searching /finding solutions with particles/charges depends on the number of charges involved in computation. a, The error in feature representation decreases exponentially with the number of particles/ charges b, If the error ε is kept constant the time required to find the solution decreases with the number of particles/ charges that perform the task.

1000

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o

The solution can be found faster if the number of charges involved in computation increases o The increase in the number of charges that perform computation requires a growth in energy resources. o The increase in complexity of problems will always require an increase in computational complexity (Ding-Zhu and Ker-I, 2000) and an activation of larger volume areas in the brain (neuronal circuits). Therefore, for new problems a direct correspondence is expected to exist between computational complexity and activation of volume/area in brain. The trajectory of a charge can be also plotted in the phase space if all possible states of the system are represented. If the system is a charged particle in one dimension, then its position in phase space can be given by the pair ( xk , v k ) where xk is the position of the particle and vk is its velocity at discrete moment k ∈ N . Given enough time, some set of states can be repeated and may define a subset of the phase space, called an attracting set A. The neighborhood of A for the dynamical system is called the basin of attraction B(A). If the pair ( xk , vk ) is an element of the phase space, then ( xk , vk ) totally specifies the state of the system at some instant, k ∈ N . 1.5

a

b

1

1

0.5

0

V

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0.5

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-0.5

-0.5 -1 -1.5 -2

0

2

4

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-1 0

1

X

2

3 X

4

5

6

Figure 9: Phase space representation of a charge movement a, The solution is found in the absence of random input variables b, A slightly better solution can be found if a random, Brownian motion is considered

The subset of phase space, the attracting set A, can be a point, a group of points, a curve, a manifold or a complex shape. The “point attractor” is the simplest type of attractor which may direct the charges to a single stable point (Figure 10). Since the patterns of electrical activity occur in every action potential, they can be mathematically modeled as temporal sequences of point attractors (see Figure 10, chapter 2). It is likely that electrical fluctuations generated by this movement of charges can be reflected in fluctuations of firing rate or spike timing. As presented above (Figure 10) such fluctuations and oscillations can be perceived as a direct way of exploring/searching in the solution space within the volume/area where the solutions for some class of problems can be found while the motion of particles/charges, the attractor model can be seen to be the physical root of electrical patterns.

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4

Z

3 2 1 4 6 2 Y

4 0 0

2 X

Figure 10 Representation of a “point attractor” in three-dimensional space in red. This amortized oscillatory trajectory in space of a selected charge may play an important role in finding/searching solutions

4.1.5. Charge Movement Model - A Computational Approach Brain activity shows complex patterns either in simple spikes or in neuronal ensembles and we have had difficulty understanding what these activities represent. Our ability to interpret or make decisions is generally diminished by the increase in complexity (Zadeh, 1968). Therefore, before moving to a complex framework it is important to analyze fundamental characteristics that define brain computations which may help us refine our understanding of information transmission and computation in brain. Since we cannot assume that brain processing can be simply performed by “uniform pulses of electricity” (Bains, 2003) this subchapter is an attempt to develop new theoretical ideas and to investigate some new models of computation that can better mimic some processes in brain. Within each collection of neurons, cells or their subdivisions one can define a processing space where the movement/trajectories of charges characterize/represent different computational levels carried on in order to find solutions. It is clear for us that modeling with current generation of computers cannot account for the required richness provided by the natural system. However, these models may allow us to better understand the outcomes observed in several experiments regarding dynamics and plasticity of the neural processes including the occurrence of the highly studied spike timing patterns. Our previous approach highlighted the important role played by ion fluxes as main carriers of information. However, it is expected that the number of positive and negative charges within the brain (e.g. electrons) is equal since the brain has to be electrically neutral. Their local distribution in space can be significantly changed for short periods of time and this process of changing spatial charge distribution defines the essence of computation, since as shown above the movement of charges in an electric field is able to provide natural solutions. Therefore, grouping or increasing the density of charges for short periods of time in a specific spatial region defines the coding phase and this step can be considered to be the first phase of computation where information is projected in space by an intrinsic movement of charges as a result of electric field propagation. Too often several assumptions of spike timing theory reject the importance of charges or electric interactions claiming that “neurons do not interact through extracellular fields”. However, the main issue regarding computation is not

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intercellular interaction since most of this processing occurs within every cell. At a fundamental level, local increase in charge density occurs within neurons and can be demonstrated within every spike delivered by a neuron (see Chapter 2). Additionally reaction-diffusion phenomena lead to significant changes in charge concentration. Even in case of chemical synapses information transfer and the interaction between neurons is greatly determined by ion fluxes. Calcium (Ca2+) regulates the process of neurotransmitter release that furthermore allows ions to move in and out determining the occurrence of the postsynaptic potential. Several other phenomena highlight the importance of electrical charges in brain computations: • Important quantities of information are transferred by ionic fluxes during every spike within a millisecond time scale (maximum average values over 89 bits/spike). Using the HH model, this phenomenon has been proved by computing mutual information between input stimulus and sodium fluxes (Aur et al., 2006). These values are more than double compared to mutual information between the membrane voltage and stimulus input. A similar computational model of mutual information was previously used to claim that spike timing and ISI embed information regarding stimuli/behavior. This mainly demonstrates that information regarding stimuli/behavior is distributed in several observable/measurable quantities; however, the significant values of mutual information provide a higher status in this hierarchy for sodium/potassium fluxes. • The study of atypical alterations that occur in terms of changes in local electrical properties as well as information transfer within brain areas can be strongly connected with the occurrence of symptoms of schizophrenia / Parkinson’s disease in animal models (see Chapter 5- Parkinson’s and schizophrenia). • Electrical charges are traditional carriers of information in many biophysical systems (see electron transport chain). The processing of extracellular recordings (e.g. spike directivity) demonstrates that the striatum uses this dynamics of charges to represent semantics during procedural learning in Tmaze tests (Aur and Jog 2007a). • Charge conservation principle states that electric charges cannot be created or destroyed; therefore these charges under a set of conditions represent a valuable resource for computation. Even though in the brain the number of positive charges should equal the number of negative charges their topology/spatial distribution is continuously changing. In its resting state each neuron contains a small excess of negative charges, therefore, an inherent change of density of positive ions (e.g. Na+) close to the membrane may determine voltage gated sodium channels to open and neuron to generate an action potential. We have shown that electrical patterns of activation occur within every spike as a result of diffusion and charge dynamics at micro-scale level within neurons (Aur and Jog, 2006). • Since within the electric field inherent electrostatic interactions play a crucial role at a micro-scale level by directly influencing protein folding and aggregation, the changes in spatial distribution of charges can be directly related with alterations in the internal structure of the cell. Therefore computation with charges is able to resolve the current gap between molecular

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description, neuronal function and spike timing patterns analysed in several experiments. • The electric/electrostatic interactions between charges are extremely important in providing a triple role for charges in brain computations. They are able to read information, transfer information by their movement and store information within a given spatial topology of charges within macromolecular formations (e.g. proteins). • As a consequence of learning, the organization of spatial charge distribution at micro-scale level determines several observable effects at neuronal scale: o The antagonistic surround in the receptive fields observed in the activity of cells of the lateral geniculate nucleus becomes stronger with learning shaping the Mexican hat form in terms of firing rate measure (Ohshiro and Weliky, 2006) o The millisecond occurrence of spikes within neurons (Mainen and Sejnowski, 1995) o The organization of spike timing within neuronal ensembles (Jog et al., 2007) o Overrepresentation, with learning larger areas/volumes are assigned to familiar stimuli inputs (Penfield and Rasmussen, 1950). All these observables can be demonstrated to occur in simulation as outcomes of reorganization in charge distribution: • Seen as particles that have a certain dynamics, these charges during their movement from an initial state to a later state can carry out complex mathematical transformation and serve as direct means for computation. • Several properties attributed to brain/neural computations such as temporal patterns, multidimensionality, parallelism, fuzziness, topological fractality can be naturally seen as direct effects of computation with charges. • Importantly the laws of physics are well known for electrical charges, and there is no need to discover other physical principles in order to build a basic model of computation. All these aspects summarized above show why we need to develop a new, different model beyond the current limited view provided by spike timing framework in order to reflect the inherent natural richness of capacity for complex computations within the brain. The essence and richness of brain computations in terms of charge dynamics and their interaction can be structured within six fundamental principles that form the basis of brain computation. 1. The Principle of Information: To understand brain computations one needs to look at the brain’s fine grained structure that includes electrical charges. Information processing and transfer occur naturally by moving these charges in space (e.g. Na +, K+, Ca2+) intra and extracellularly as a result of electrical wave propagation, selective diffusion process during action potentials, synaptic spikes. Intracellularly, where information processing predominantly occurs, the generated electric field is the fundamental component that allows a fast transfer of information and mediates the interaction between electrically charged particles at a molecular level. Importantly, the information carried by charges does not relate solely to stimuli or behavior. At a fundamental molecular level information carried by electric field and charges can be selectively transferred, and this information operates at gene level to determine intricate regulatory mechanisms required to build appropriate proteins intracellulary. Several

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other intrinsic physiological regulatory mechanisms require information transfer (e.g. homeostasis). 2. Brain Computation Principles: Since information travels with the propagation of electric field and movement of charges, neurons and other cells (e.g. astrocytes) can be considered essential units in this complex process that spatially modulates the flow and interaction between charges. During every AP/synaptic spikes the spatial trajectories of electric field and charges are determined by a selective activation of ion channels and interaction with molecular structures. Computations can be reduced to a set of transformations that define a morphism between the input signal/stimuli space and electric patterns represented by the ‘charge space’. The inverse set of transformations carried out in the presence/absence of the input stimuli may help us to identify or remember/recreate previous events, place names. 2.1 Semantic Acquisition Principle: Computation with electric charges is mediated by many neurotransmitters and neuromodulators which can alter the ion channel function and directly relate neuronal propagation of charges with behavioural or stimulus input semantics (e.g. see changes in spike directivity). This principle provides several answers regarding non-deterministic computations and intrinsic properties of electric field propagation and charge movement. An increased spatial charge density within molecular formations, their emergent spatial topology and interaction with charges in movement is at the origin of distance-based semantics. 2.2 Rhythm Dependency Principle: Specific phases of computation require different brain rhythms. As an example, physical phenomenon of ‘writing’ in memory during sleep time is frequency dependent since changing polarization at molecular level and also conformational changes at protein scale are voltage and frequency dependent events that may determine a preferential expression level of certain genes. 2.3 Fundamental Thermodynamic Principle of Computation: The understanding of thermodynamic nature of computations relies on the analysis of thermodynamic entropy. The transmembrane movement of charges is correlated with changes in thermodynamic entropy and information transfer during spike activity (action potentials, synaptic spikes). Reading and writing or deleting information at molecular level has to follow the fundamental thermodynamic principle (e.g. second law of thermodynamics, minimum entropy production principle, Landauer principle). Computation has a thermodynamic cost that has to be paid when information is erased. 2.3 The Principle of Coding: A charge or a group of charges that move during an AP/synaptic spike are able to read (decode) and write (code) information at or in the same time. The role of the “Turing tape” is played by molecular structures including ion channels where the redistributions of charges within macromolecular formations determine changes in polarizations, conformational changes in proteins and selective gene expression. These physical processes briefly described above are carried out by several repetitive electrical phenomena (synaptic spikes, action potentials). These reiterating electrical phenomena are essentially modulated by changes that occur at the molecular level. 2.4 The Principle of Decoding: The patterns of external events, stimuli are directly reflected in changes of electric patterns that occur in neurons determined by the dynamics of electric charges. This “reading” of information (about stimulus, or behaviour) is altered by a continuous interaction with already existent data stored in memory (in terms of conformational states, polarizations). This view can simply describe the basic mechanism of perception where the input information from stimuli is modulated by previously accumulated and stored information in memory.

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2.5 The Principle of Parallel Processing and Multidimensionality: The richness of computation in the brain relies on parallel processing performed by the movement and interaction of many types of charges. Automatically, parallelism and multidimensionality occur directly in terms of physical laws that govern the motion of charges. Synchronous neuronal firing can be explained as a requirement to increase interaction and the number of charges involved in information processing in a particular region which directly increases the power of computation and decreases the required time to find the solution (see above section 4.1.4). Thus, local environmental charge enrichment occurs in order to increase computational efficiency. 2.6 Global Computational Principle: The richness of computation with charges is lost if the inputs/stimuli/other processing are seen as local disconnected events, analyzed in vitro in brain slices. Computationally, the dynamics and interaction of charges involve the whole brain as an entire machine tuned to unravel the emerging behavioral aspect or the semantics of incoming stimuli. 2.7 Biomolecular computation: At the molecular scale biological information follows a direct path from DNA to RNA through to protein synthesis. It is likely that a reverse path in information flow is possible through various types of interactions at the molecular and electric level required by an appropriate gene selection for protein synthesis. This reverse path involves electric interactions and provides a feedback mechanism critical for maintaining, consolidating and generating new memories. 3. The Principle of Representation: “Constructs at ‘macrolevel scale’ (image representations, movement control) are results of emergent dynamics and interactions between electric charges that alter and build charge configuration at molecular scale. From motor output to visual processing, mental representations are essentially built from smaller to larger scales by a spatial distribution of charges which temporarily increases their density in specific regions and can be seen then as natural, physical solutions (electrical patterns within action potentials, synaptic spikes). At a molecular level spatial distribution of charges is modulated by changes that occur in gene selection for new protein synthesis which contribute to an intrinsic memory formation. 3.1 The Principle of Optimality: During learning the computational process in the brain is organised to achieve the required representation of a certain feature with the minimum number of electrical patterns (charges) in minimum time. The “minimum path description” follows the physical principle of least action and has in this case to be adapted to the cell’s and ion channel’s morphology. This process can be evidenced in terms of spatio-temporal dynamics of charges within neurons and neuronal ensembles. Spikes (action potentials, synaptic spikes) can be considered as ‘discrete’ steps that carry out in time this optimization process based on an active fundamental loop of molecular computations (changes in polarization, folding mechanism, changes in gene expression) able to provide the required dynamic adaptation. 3.2. The Learning Principle: Learning represents a natural solution for reducing energy expenditure required to solve repeatable, complex problems. The rhythmic occurrence of an increase in spatial distributions of charges determines in time alterations/organization of charges within macromolecular structures (e.g. proteins) which in the long term tend to preserve information regarding the metric properties of represented scenes, objects or events. This process is deeply rooted within specific bimolecular mechanisms of ‘computation’ that include gene selection, regulation and expression related to DNA-RNA transcription, protein regeneration and new protein synthesis in response to a wide variety of extracellular signals and electrical interactions. As a result learning changes occur at the microscopic scale and determine

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effective changes in molecular-genetic mechanisms of protein biosynthesis, proteinprotein associations mediated by electrical interactions and then determine organization/re-organization of spike patterns observed/measured by common electrophysiological recording techniques and revealed as forms of learning (habituation, sensitization, classical conditioning). With learning the electrical maps related to frequently used inputs become overrepresented (see the hand and the mouth in the Penfield 'homunculus'). 4. Scalability Principle: The model of the dynamics of charges has the property to function and effectively accommodate at different scale/sizes from macromolecules/ proteins to neurons and networks of neurons where these models can be built and analyzed. (See the computational cube, Chapter 4) 5. Quantum Computing Principle: Computation with charges occurs naturally at the microscopic scale within macromolecular formations. When one goes deeper at the nano-scale level, various quantum phenomena such as time-dependent tunneling, Coulomb blockade, stochastic resonance can be considered to physically drive computations and information exchange. This quantum scale is possibly the most fundamental level of computation. 6. The Unity Principle: The electrodynamic model of computation unifies current observables at macro scale level (spike timing patterns, firing rate) with macromolecular changes in terms of spatial distribution of charges that occur at ion channel and protein level (within neuronal membranes, inside synapses) and their dynamics. Based on these six principles we have to build a new model of computation which should better reflect what happens in brain. In order to be simulated on current computers we are forced to use a reductionist approach and these models are simplified and discretized. Therefore, only some of the relevant aspects of computation with charges can be exposed in the development below. o The coding/writing phase is a process where information regarding an input feature/stimulus/behavior is embedded in a spatial distribution of charges that can be seen at different scale levels from AP to macromolecular structures, proteins. An example of this new model of computation at the neuronal level provides the solution in terms of charge distribution that is “read” by the dynamics of other charges in certain spatial regions defined as synaptic boundaries, gap junctions and ion channels. Due to synaptic and non-synaptic connectivity and high impedance in these structures, neurons are able to “read” this spatial distribution of charges. The phase of receiving information regarding spatial distribution of charges can be seen as the “reading” or decoding phase, a phase where information is received by neuronal structures (e.g. synapses) that spatially lie or present certain type of connectivity in a definite specific region. o The decoding/reading phase is a process of acquiring information regarding the spatial distribution of charges The increased number of charges involved in this process and their spatial trajectories reflect the multidimensional aspect of computation. In fact McClelland and Rumelhart in their seminal work (McClelland and Rumelhart 1988) stepped up to the next level of analyzing firing rate and missed this fundamental “lower level” where parallel distributed processes occur as results of dynamics and interaction between electric charges.

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Despite common thought regarding noise, the random movement of charges during the process of diffusion (Brownian motion) can drive charged particles to appropriate and sometimes faster solutions which may increase the performance of the computing system. Incorporating all these observations into an algorithm should allow us to describe mathematically and simulate the main characteristics of computations in brain on current computers. This separation in two main phases (coding and decoding) is presented here only for modeling purposes. In reality the process of computation is unlikely to be strictly separable in phases since reading and writing information are continous and simultaneous processes driven by the movement and interaction of charges. There are some important differences between the current algorithm and how computation emerges in biological neurons. The movement of charges during AP/synaptic spikes and their interaction has two main purposes; to read and to write information and both activities are simultaneously achieved. Therefore, there is no real separation between the coding/writing information and reading/decoding information as presented above when one considers a biological system. Another issue is to artificially introduce the input features by another group of charges. Such separation between charges that “model” the input and charges that “compute” does not exist in the brain and is introduced just for demonstrative purposes. This model of computation with charges has to incorporate the most essential characteristics of the system; it should include the spatio-temporal distribution of charges, spatial distribution of synapses/neurons and importantly the dynamical aspects of computation. The computational model presented below is in two forms, a static mapping model and a dynamic mapping model representing a general framework where the main goal is to build a theoretical approach that can explain brain computations starting from spike occurrence including synaptic role and learning. This model can be modified in several ways to fit data from brain recordings and other experiments; however it remains just a model which cannot replace the real brain functions. In other words quoting Stevan Harnard, “the virtual brain is no more able to think than a virtual plane is able to fly” (Harnard, 2001) •

The Static Mapping Model

The recent progress in machine vision faces several challenges. The ability of current computers to recognize classes of objects based on their shape is as yet unsolved. Standard optical devices provide just a partial view of objects; therefore solving the problem using partial information is difficult due to large variations in shapes and uncertainty. The more intuitive example is to design a model to process and recognize visual stimuli; therefore, we have chosen the object recognition problem as an example. Since every el