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INTRODUCTION to
DYNAMIC MODELING of NEUROSENSORY SYSTEMS Robert B. Northrop
CRC Press Boca Raton London New York Washington, D.C.
© 2001 by CRC Press LLC
Library of Congress Cataloging-in-Publication Data Northrop, Robert B. Introduction to dynamic modeling of neuro-sensory systems / Robert B. Northrop. p. cm. (Biomedical engineering series) Includes bibliographical references and index. ISBN 0-8493-0814-3 (alk. paper) Biomedical engineering series (CRC Press) QP363.3.N67 2000 573.8′0285′632—dc21 00-063015 CIP
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© 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0814-3 Library of Congress Card Number 00-063015 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper
Note: All the Simnon programs used in the text and the problems are available online at the CRC Press Web site at http://www.crcpress.com/ under the title of this text. When downloaded, a given program can be cut and pasted and saved as a “Text Only,” *.t file to be used by Simnon 3.2 or Simnon 3.0/PCW.
Biomedical Engineering Series Edited by Michael R. Neuman
Medical Imaging: Techniques and Technology, Martin Fox Artificial Neural Networks in Cancer Diagnosis: Prognosis and Treatment, R.N.G. Naguib and G.V. Sherbet Biomedical Image Analysis, Rangaraj M. Rangayyan Endogenous and Exogenous Regulation and Control of Physiological Systems, Robert B. Northrop
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Preface This text is about neural modeling, i.e., about neurons and biological neural networks (BNNs) and how their dynamic behavior can be quantitatively described. It was written for graduate students in biomedical engineering, but will also be of interest to neurophysiologists, computational neurobiologists, and biophysicists who are concerned with how neural systems process information and how these processes can be modeled. What sort of academic background does the reader need to get the most out of this text? The author has assumed that readers are familiar with the formulation and solution of ordinary differential equations, also introductory probability theory, basic EE circuit theory, and that they have had an introductory course in neurobiology. This interdisciplinary background is not unusual for graduate students in biomedical engineering and biophysics. For the reader who wants to pursue any topic in greater depth, there are many references, some from the “classic period” in neurobiology (the 1960s and 1970s) and others from contemporary work. Neural modeling as a discipline (now known as computational neurobiology or computational neuroscience) has a long history, back at least to the 1952 groundbreaking kinetic model of Hodgkin and Huxley for the generation of the nerve action potential. The Hodgkin–Huxley model dealt with events at the molecular and ionic levels on a unit area of axon membrane. Other models have examined neural behavior on a more macroscopic level, preserving neural components such as synapses, dendrites, soma, axon, etc. In another approach, the bulk behavior of large sensory networks such as the vertebrate retina or the arthropod compound eye has been modeled using the linear mathematics of engineering systems analysis. Each approach is valid in the proper context. This text discusses tools and methods that can describe and predict the dynamic behavior of single neurons, small assemblies of neurons devoted to a single task (e.g., central pattern generators), larger sensory arrays and their underlying neuropile (e.g., arthropod compound eyes, vertebrate retinas, olfactory systems, etc.), and, finally, very large assemblies of neurons (e.g., central nervous system structures). Neural modeling is now performed by solving large sets of nonlinear, ordinary differential equations (ODEs) on a digital computer. There is considerable art and science in setting up a computational model that can be validated from existing neurophysiological data. There are special, free computer programs available at Web sites that allow the user to set up and test neural models. There are also “component libraries” available on the Internet that supply the modeler with the parameters of different types of neurons. One goal of neural modeling is to examine the validity of putative signal processing algorithms that may be performed by interneurons that enable sensory arrays to perform feature extraction. The results of modeling can point to new experiments
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with living systems that may validate hypotheses. Experimental neurobiology and computational neuroscience are in fact synergistic; results of one should reinforce the results of the other. Some of the neural models considered in this text are based on the author’s own neurophysiological research and that of his graduate students. Its focus has been on small and medium-sized neural networks. Attention has been directed to visual feature extraction and, in particular, the compound eye visual system and the vertebrate retina. Other interesting sensory arrays are also treated, and their behavior modeled or described mathematically, where applicable. These include the gravity receptor arrays of the cockroach Arenivaga sp., electroreceptors of fish, magnetoreceptors on many diverse animals, and the two angular rate sensors on dipteran flies, etc. Introduction to Dynamic Modeling of Neuro-Sensory Systems is organized into nine chapters plus several appendices containing large computer programs. There is also an extensive Bibliography/References section, which includes relevant Web site URLs. Chapter 1, Introduction to Neurons, begins by describing the anatomy of the major classes of neurons and their roles in the nervous system. The molecular bases for the resting potential of the neuron and its action potential are considered, along with ion pumps and the many, diverse, gated ion channels. Next, the bulk properties of the passive core conductor (the dendrite) are presented, and an equivalent, lumpedparameter RC transmission line circuit is developed to model how postsynaptic potentials propagate down dendrites to active regions of the neuron membrane. How chemical synapses (excitatory and inhibitory) work is described. Rectifying and nonrectifying electrical synapses are considered, as well. The generation of the nerve action potential by the active nerve membrane is described; the original Hodgkin–Huxley equations are presented and modeled with Simnon™. Factors affecting the propagation velocity of the action potential are discussed. Models of complete neural circuits are presented (e.g., a spinal reflex arc) and discussed in terms of neural “components.” The behavior of the Hodgkin–Huxley model under voltage clamp conditions is simulated. The second chapter is Selected Examples of Sensory Receptors and Small Receptor Arrays. The generalized receptor as a transducer is presented first. The power-law or log relation between stimulus and receptor response (not all receptors generate nerve spikes directly) is described. Factors limiting receptor response, including their membrane dynamics, noise, and dynamic range, are considered. Properties of selected receptors are considered in detail: • Certain mechanoreceptors (Pacinian corpuscles, muscle spindles, Golgi tendon organs, chordotonal organs in arthropods, sensory hairs, etc.) • Magnetoreceptors (putative magnetite-based mechanosensors and Lorentz force–based models) • Electroreceptors (in fish and the platypus) • Gravity vector sensors (tricholiths in Arenivaga sp., statocysts in crustaceans and mollusks) • Angular rate sensors used in dipteran flight (halteres)
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• Chemoreceptors (amazing threshold sensitivities) • Basic photoreceptors (the “eye” of Mytilus edulis) that convert photon energy to changes in nerve resting potential The third chapter, Electronic Models of Neurons: A Historical Perspective, introduces the neuromime, an electronic analog circuit first used to simulate the behavior of single neurons, and small groups of interacting neurons. Neuromime circuits use the phenomenological locus approach to emulate the behavior of single biological neurons and small numbers of biological neurons. Neural locus theory is described; spike generator loci, excitatory and inhibitory synaptic potentials, delay operators, summing points, and low-pass filters to model lossy propagation are introduced and used in examples. The locus architecture is extended to numerical simulation of neurons. Chapter 4, Simulation of the Behavior of Small Assemblies of Neurons, continues to examine the building blocks of locus theory in detail. The ODEs required to model ordinary excitatory postsynaptic potential (epsp) and inhibitory postsynaptic potential (ipsp) generation are derived, as well as those for nonlinear synapses, including facilitating and antifacilitating dynamics. Dendrites are considered over which epsps and ipsps are summed spatiotemporally. Dendrites are modeled as RC transmission lines; however, computational simplicity accrues when simple delay functions are used along with multipole low-pass filters to model their behavior. Rather than use the detailed Hodgkin–Huxley model for spike generation, the spike generator locus (SGL) can be modeled with either integral pulse frequency modulation (IPFM) or relaxation pulse frequency modulation (RPFM) (the leaky integrator spike generator). Considerable computational economy results when the RPFM SGL is used. ODEs and nonlinear equations are developed to model adaptation and neural fatigue. Basic signal-processing characteristics of small assemblies of neurons are treated. Locus models are shown to be effective in simulating simple two- and threeneuron circuits that exhibit pulse frequency selectivity (tuning) for a specific range of frequencies (the band detector), as well as high-pass and low-pass characteristics. The T-neuron is shown to be a pulse coincidence gate (an AND gate with an input memory). Finally, locus models are used to investigate the dynamics of reciprocal inhibition and neural circuits capable of behaving like central pattern generators (CPGs). Ring circuits with positive feedback are shown to make poor CPGs. Negative feedback rings with delays in their loop gains are shown to be capable of generating twophase burst patterns. Chapter 5, Large Arrays of Interacting Receptors: The Compound Eye, introduces the large neurosensory array. How arrays have certain properties that enhance their effectiveness over single receptors is shown. Such enhancements have to do with linear and nonlinear signal processing in the underlying neuropile. The arthropod compound eye was chosen as the first array example because the author has spent many years doing research on insect (grasshopper) compound eye vision. Compound eye optics and neuroanatomy are described. The spatial resolution of the compound eye is considered; repeated measurements have shown that certain
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neurons in the optic lobes respond to visual objects so small that theoretically they should not be “seen.” This theory is based on the analysis of the optics of a single dioptric unit (photoreceptor cluster) called an ommatidium. A theoretical model providing a basis for “anomalous resolution” was developed by the author based on multiplicative processing between adjacent ommatidia and the generation of an equivalent synthetic aperture receptor. Early studies of the simple compound eye of the common horseshoe crab, Limulus, revealed that an underlying neural network had the property of lateral inhibition (LI), in which adjacent photoreceptors inhibited each other when stimulated. LI is shown to produce spatial high-pass filtering for the compound eye system, enhancing the contrast of high spatial frequencies in an object presented to the eye. Perceptual evidence for the presence of some form of LI is observed in the human visual system. Finally, Chapter 5 considers feature extraction (FE) by the visual systems of grasshoppers, flies, and certain crustacean compound eye systems. FE is shown to be the result of neural preprocessing the spatiotemporal properties of an object (intensity, motion) to lighten the cognitive burden on the animal’s central nervous system. Neural units from the optic lobes of insects respond selectively to either contrasting edge or spot objects moved in a preferred direction. It is argued that these directionally sensitive (DS) units are used by the insect for flight stabilization and guidance, or for the detection of predators or other insects. DS units have been found in all arthropods with compound eyes in which the DS units have been sought. Large Arrays of Interacting Receptors: The Vertebrate Retina is the subject of Chapter 6. First, the anatomy and neurophysiology of a the vertebrate retina are described; cell types and functions are given, including the differences between rods and cones. FE was first shown to occur in the frog’s retina by Lettvin et al. in 1959. Vastly different in neural architecture than the arthropod compound eye, FE has evolved in the retina in several congruent patterns with compound eye systems. FE properties in the frog are contrasted to FE observed in the retinas of other vertebrates, such as the pigeon and the rabbit. Minimal FE takes place in the retinas of primates, where the burden of FE has been shifted to the well-developed visual cortex. Chapter 7, Theoretical Models of Information Processing and Feature Extraction in Visual Sensory Arrays, considers various models that have been proposed to describe FE in vertebrates’ retinas and the optic lobes of insects. One- and twodimensional, linear, Fourier spatial filters are introduced and it is shown how visual array circuits can be considered to perform spatial filtering, including the edgeenhancement function of the Limulus LI. More complex interconnections are shown to yield edge orientation filters and spot filters. The static spatial filtering models of Fukushima and the spatiotemporal filters of Zorkoczy are described. The spatial matched filter is introduced. Models for neural matched filters and how they might figure in visual pattern recognition are considered. Characterization of Neuro-Sensory Systems is treated in Chapter 8. First, the classic means for identifying linear systems (LSs), including cross-correlation techniques to extract an LS weighting function or transfer function, is reviewed. Next, methods of deriving canonical, anatomical models for the connection architecture
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small arrays of spiking neurons based on the joint peri-stimulus time (JPST) technique is considered. The mathematical basis for triggered correlation, developed to characterize the frequency selectivity of cat eighth nerve units by de Boer, is rigorously derived. The works of de Boer and of Wu are described. Finally, the white noise method of Marmarelis for the characterization of nonlinear physiological systems is reviewed, and its application to the goldfish retina by Naka is described. Chapter 9, Software for Simulation of Neural Systems, reviews the currently available simulation packages with which neurons can be modeled: NEURON, GENESIS, XNBC v8, EONS, SNNAP, SONN, and Nodus 3.2. All these programs are in English, and are free to university students. Most of them run on UNIX-type OS computers with Xwindows and have a graphical user interface. Some also run on personal computers with Windows 95, 98, or NT4. Nodus 3.2 runs on Apple Macintosh systems. The author also makes a strong case for the use of the general, powerful, nonlinear ODE solver, Simnon, for some basic tasks in neural modeling. Quantitative problems are included with Chapters 1, 2, 4, 5, and 7. In addition to their obvious pedagogical value for student readers using this text, solved problems can be used as teaching examples by instructors. A Solutions Manual is available from the publisher. Also available online at the CRC Press Web site at http://www.crcpress.com/ under the title of this text are all the Simnon programs used in the text and in the problems. When downloaded, a given program can be cut and pasted and saved as a “Text Only,” *.t file to be used by Simnon 3.2 or Simnon 3.0/PCW. This procedure saves the modeler the effort of copying files out of the text line by line and saving them in text only, *.t format. Robert B. Northrop
© 2001 by CRC Press LLC
Author Robert B. Northrop was born in White Plains, NY, in 1935. After graduating from Staples High School in Westport, CT, he majored in electrical engineering at MIT, graduating with a bachelor’s degree in 1956. At the University of Connecticut (UCONN), he received a master’s degree in control engineering in 1958. As the result of long-standing interest in physiology, he entered a Ph.D. program at UCONN in physiology, doing research on the neuromuscular physiology of catch muscles. He received his Ph.D. in 1964. In 1963, he rejoined the UCONN Electrical Engineering Department as a Lecturer, and was hired as an Assistant Professor of Electrical Engineering in 1964. In collaboration with his Ph.D. advisor, Dr. Edward G. Boettiger, he secured a 5-year training grant in 1965 from NIGMS (NIH), and started one of the first interdisciplinary biomedical engineering graduate training programs in New England. UCONN awards M.S. and Ph.D. degrees in this field of study. Throughout his career, Dr. Northrop’s areas of research have been broad and interdisciplinary and have been centered around biomedical engineering. He has done sponsored research on the neurophysiology of insect vision and theoretical models for visual neural signal processing. He also did sponsored research on electrofishing and developed, in collaboration with Northeast Utilities, effective working systems for fish guidance and control in hydroelectric plant waterways using underwater electric fields. Still another area of sponsored research has been in the design and simulation of nonlinear, adaptive digital controllers to regulate in vivo drug concentrations or physiological parameters, such as pain, blood pressure, or blood glucose in diabetics. An outgrowth of this research led to his development of mathematical models for the dynamics of the human immune system, which have been used to investigate theoretical therapies for autoimmune diseases, cancer, and HIV infection. Biomedical instrumentation has also been an active research area. An NIH grant supported studies on the use of the ocular pulse to detect obstructions in the carotid arteries. Minute pulsations of the cornea from arterial circulation in the eyeball were sensed using a no-touch ultrasound technique. Ocular pulse waveforms were shown to be related to cerebral blood flow in rabbits and humans. Most recently, he has addressed the problem of noninvasive blood glucose measurement for diabetics. Starting with a Phase I SBIR grant, Dr. Northrop has been developing a means of estimating blood glucose by reflecting a beam of polarized light off the front surface of the lens of the eye, and measuring the very small optical rotation resulting from glucose in the aqueous humor, which in turn is proportional to blood glucose. As an offshoot of techniques developed in micropolarimetry, he developed a sample chamber for glucose measurement in
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biotechnology applications. Another approach being developed will use percutaneous, long-wave infrared light in a nondispersive spectrometer to measure blood glucose noninvasively. He has written four textbooks: one on analog electronic circuits, and others on instrumentation and measurements, physiological control systems, and neural modeling. Dr. Northrop was a member of the Electrical and Systems Engineering faculty at UCONN until his retirement in June 1997. Throughout this time, he was program director of the Biomedical Engineering Graduate Program. As Emeritus Professor, he now teaches graduate courses in biomedical engineering, writes texts, sails, and travels. He lives in Chaplin, CT, with his wife and two cats.
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Table of Contents Chapter 1 Introduction to Neurons Introduction 1.1 Types of Neurons 1.1.1 Motoneurons 1.1.2 Vertebrate Peripheral Sensory Neurons 1.1.3 Neuroendocrine Cells 1.1.4 Interneurons 1.1.5 Discussion 1.2 Electrical Properties of Nerve Membrane 1.2.1 The Source of UM Electrical Parameters 1.2.2 Decremental Conduction on Dendrites: The Space Constant 1.2.3 Active Membrane: The Nerve Spike 1.2.4 Saltatory Conduction on Myelinated Axons 1.2.5 Discussion 1.3 Synapses: epsps and ipsps 1.3.1 Chemical Synapses 1.3.2 Electrical Synapses 1.3.3 epsps and ipsps 1.3.4 Quantal Release of Neurotransmitter 1.3.5 Discussion 1.4 Models for the Nerve Action Potential 1.4.1 The 1952 Hodgkin–Huxley Model for Action Potential Generation 1.4.2 Properties of the Hodgkin–Huxley Model 1.4.3 Extended Hodgkin–Huxley Models 1.4.4 Discussion 1.5 Chapter Summary Problems Chapter 2 Selected Examples of Sensory Receptors and Small Receptor Arrays Introduction 2.1 The Generalized Receptor 2.1.1 Dynamic Response 2.1.2 Receptor Nonlinearity 2.1.3 Receptor Sensitivity 2.1.4 A Model for Optimum Firing Threshold 2.1.5 Simulation of a Model Receptor with a Continuously Variable Firing Threshold © 2001 by CRC Press LLC
2.1.6 Discussion Chemoreceptors 2.2.1 The Vertebrate Olfactory Chemoreceptor 2.2.2 Olfaction in Arthropods 2.2.3 Discussion 2.3 Mechanoreceptors 2.3.1 Insect Trichoid Hairs 2.3.2 Insect Campaniform Sensilla 2.3.3 Muscle Length Receptors 2.3.4 Muscle Force Receptors 2.3.5 Statocysts 2.3.6 Pacinian Corpuscles 2.3.7 Discussion 2.4 Magnetoreceptors 2.4.1 Behavioral Evidence for Magnetic Sensing 2.4.2 The Putative Magnetoreceptor Neurons of Tritonia 2.4.3 Models for Magnetoreceptors 2.4.4 Discussion 2.5 Electroreceptors 2.5.1 Ampullary Receptors 2.5.2 Weakly Electric Fish and Knollenorgans 2.5.3 Discussion 2.6 Gravity Sensors of the Cockroach, Arenivaga sp. 2.6.1 Hartman’s Methods 2.6.2 Hartman’s Results 2.6.3 CNS Unit Activity Induced in Arenivaga by Roll and Pitch 2.6.4 Willey’s Methods 2.6.5 Willey’s Results 2.6.6 A Tentative Model for PCP Unit Narrow Sensitivity 2.6.7 Discussion 2.7 The Dipteran Haltere 2.7.1 The Torsional Vibrating Mass Gyro 2.7.2 Discussion 2.8 The Simple “Eye” of Mytilus 2.8.1 Eye Morphology 2.8.2 Physiology of the Eye 2.8.3 Discussion 2.9 Chapter Summary Problems 2.2
Chapter 3 Electronic Models of Neurons: A Historical Perspective Introduction 3.1 Neccesary Attributes of Small- and Medium-Scale Neural Models 3.2 Electronic Neural Models (Neuromimes)
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3.3
Discussion
Chapter 4 Simulation of the Behavior of Small Assemblies of Neurons Introduction 4.1 Simulation of Synaptic Loci 4.1.1 A Linear Model for psp Generation 4.1.2 A Model for epsp Production Based on Chemical Kinetics 4.1.3 A Model for a Facilitating Synapse 4.1.4 A Model for an Antifacilitating Synapse 4.1.5 Inhibitory Synapses 4.1.6 Discussion 4.2 Dendrites and Local Response Loci 4.2.1 The Core-Conductor Transmission Line 4.2.2 Discussion 4.3 Integral and Relaxation Pulse Frequency Modulation Models for the Spike Generator Locus (SGL) 4.3.1 IPFM 4.3.2 RPFM 4.3.3 Modeling Adaptation 4.3.4 Modeling Neural Fatigue 4.3.5 Discussion 4.4 Theoretical Models for Neural Signal Conditioning 4.4.1 The T-Neuron 4.4.2 A Theoretical Band-Pass Structure: The Band Detector 4.4.3 Discussion 4.5 Recurrent Inhibition and Spike Train Pattern Generation 4.5.1 The Basic RI System 4.5.2 Szentagothai’s RI CIrcuit 4.5.3 A Simple Burst Generator 4.5.4 A Ring CPG Model with Negative Feedback 4.5.5 Discussion 4.6 Chapter Summary Problems Chaper 5 Large Arrays of Interacting Receptors: The Compound Eye Introduction 5.1 Anatomy of the Arthropod Compound Eye Visual System 5.1.1 Retinula Cells and Rhabdoms 5.1.2 The Optic Lobes 5.1.3 The Optics of the Compound Eye 5.1.4 Discussion 5.2 Spatial Resolution of the Compound Eye 5.2.1 The Compound Eye as a Two-Dimensional, Spatial Sampling Array
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5.2.2 5.2.3 5.2.4
Calculation of Intensity Contrast “Anomalous Resolution” A Model for Contrast Enhancement in the Insect Visual System 5.2.5 A Hypothetical Model for Spatial Resolution Improvement in the Compound Eye by Synthetic Aperture 5.2.6 Discussion 5.3 Lateral Inhibition in the Eye of Limulus 5.3.1 Evidence for Lateral Inhibition 5.3.2 Modeling Lateral Inhibition as a Spatial Filter for Objects 5.3.3 Discussion 5.4 Feature Extraction by the Compound Eye System 5.4.1 Feature Extraction by Optic Lobe Units of Romalea 5.4.2 Feature Extraction by Optic Lobe Units in Flies 5.4.3 Eye Movements and Visual Tracking in Flies 5.4.4 Feature Extraction by Optic Lobe Units of Crustaceans 5.4.5 Discussion 5.5 Chapter Summary Problems Chapter 6 Large Arrays of Interacting Receptors: The Vertebrate Retina Introduction 6.1 Review of the Anatomy and Physiology of the Vertebrate Retina 6.2 Feature Extraction by the Frog’s Retina 6.2.1 Early Work 6.2.2 Directionally Sensitive Neurons in the Frog’s Brain 6.3 Feature Extraction by Other Vertebrate Retinas 6.3.1 The Pigeon Retina 6.3.2 The Rabbit Retina 6.4 Chapter Summary Chapter 7 Theoretical Models of Information Processing and Feature Extraction in Visual Sensory Arrays Introduction 7.1 Models for Neural Spatial Filters and Feature Extraction in Retinas 7.1.1 The Logic-Based, Spatiotemporal Filter Approach of Zorkoczy 7.1.2 Analog Models for Motion Detection in Insects 7.1.3 Continuous, Layered Visual Feature Extraction Filters 7.1.4 Discussion 7.2 Models for Neural Matched Filters in Vision 7.2.1 The Continuous, One-Dimensional, Spatial Matched Filter 7.2.2 The Continuous, Two-Dimensional Spatial Matched Filter
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7.3
7.4
7.2.3 Discussion Models for Parallel Processing: Artificial Neural Networks 7.3.1 Rosenblatt’s Perceptron 7.3.2 Widrow’s ADALINE and MADALINE 7.3.3 Fukushima’s Neocognitron 7.3.4 Discussion Chapter Summary Problems
Chapter 8 Characterization of Neuro-Sensory Systems Review of Characterization and Identification Means for Linear Systems 8.1 Parsimonious Models for Neural Connectivity Based on Time Series Analysis of Spike Sequences 8.1.1 The JPST Diagram 8.1.2 Discussion 8.2 Triggered Correlation Applied to the Auditory System 8.2.1 Development for an Expression for the Conditional Expectation, x+(τ) 8.2.2 Optimum Conditions for the Application of the TC Algorithm 8.2.3 Electronic Model Studies of TC 8.2.4 Neurophysiological Studies of Auditory Systems Using TC 8.2.5 Summary 8.3 The White Noise Method of Characterizing Nonlinear Systems 8.3.1 The Lee–Schetzen Approach to White Noise Analysis 8.3.2 Practical Aspects of Implementing the Lee–Schetzen White Noise Analysis 8.3.3 Applications of the White Noise Method to Neurobiological Systems 8.3.4 Discussion 8.4 Chapter Summary Chapter 9 Software for Simulation of Neural Systems Introduction 9.1 XNBC v8 9.2 Neural Network Simulation Language, or NSL 9.3 Neuron 9.4 GENESIS 9.5 Other Neural Simulation Programs 9.5.1 EONS 9.5.2 SNNAP 9.5.3 SONN 9.5.4 Nodus 3.2
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9.6
9.7
Neural Modeling with General, Nonlinear System Simulation Software 9.6.1 Simnon 9.6.2 Simulink Conclusion
Bibliography and References Appendix 1 Appendix 2 Appendix 3
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1
Introduction to Neurons
INTRODUCTION This first chapter reviews the basic structures, molecular physiology, and electrophysiology of nerve cells found in the central nervous system (CNS) and in the peripheral nervous system, including motoneurons, sensory neurons, and interneurons. First considered is general neuroanatomy, and the ionic and electrical properties of passive and active nerve membrane. Decremental conduction of electrical transients on passive dendrites, spatial summation of dendritic potentials, spike generation and propagation on unmyelinated and myelinated axons are described in general terms. The anatomy and electrical properties of chemical and electrical synapses are reviewed, including the generation of excitatory postsynaptic potentials (epsps), inhibitory postsynaptic potentials (ipsps), and the quantal release of neurotransmitter at synapses and neuromuscular junctions. The 1952 Hodgkin and Huxley (HH) dynamic, mathematical model of action potential generation is described and simulated using the general, nonlinear ordinary differential equation (ODE) solver program Simnon™. Simnon is described in Chapter 9, and is used throughout this text to model the electrical behavior of neurons and small, biological neural networks (BNNs). Simnon has been used because it has little user overhead in learning to run models on it efficiently. Simnon is also ideally suited to run chemical kinetic and pharmacokinetic (compartmental) models. Spike generation by the HH model is shown to be a nonlinear, current-to-frequency conversion process where the steady-state frequency is described by an equation of the form: f = c1 + c2 冨Iin冨m, where Iin is the dc input current to the HH model, and m is an exponent < 1. The chapter illustrates how the basic HH model using K+ and Na+ voltage-dependent conductances can be extended to include many other types of transmembrane ion channels. The chapter further explains how the HH model can be simply modified to create a voltage clamp system, in which negative feedback causes the transmembrane voltage, Vm, to follow a command input, Vs. HH system parameters are examined under voltage clamp conditions for ± ramp and step inputs.
1.1
TYPES OF NEURONS
Neurons, or nerve cells, can be categorized both by their function and by their anatomical features. In terms of function there are sensory neurons, motor neurons, local interneurons (found in dense nervous tissues such as the brain and retina), projection interneurons (carrying information to and from the CNS), and neuroendocrine cells (which secrete hormones and signaling substances directly into the
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microcirculation of the CNS, or into the general circulation upon neural stimulation). Exocrine glands are stimulated to secrete by the synaptic action of acetylcholine (ACh) secreted by postganglionic autonomic neurons. The adrenal medullae which secrete epinephrine and norepinephrine into the blood are innervated by sympathetic preganglionic neurons. All neurons are found in association with various types of glial cells. In terms of anatomical features, all neurons have a cell body (also called a soma or perykaryon) with a nucleus. The output element of a neuron is its axon. The axon generally carries information in the form of propagated action potentials (spikes) from an originating site (the spike generator locus) to the output end (motor end plates in the case of a motoneuron, synaptic boutons for just about every interneuron with chemical synapses). To complicate things, there are neurons that do not generate spikes; they conduct signals electrotonically on their axons. Such nonspiking neurons can be found in the vertebrate retina or the optic lobes of arthropod compound eyes. Depending on the number of axonal processes that originate on the soma, neurons are called unipolar, pseudo-unipolar, bipolar, or multipolar (Kandel et al., 1991). See Figure 1.1-1 for a general description of neuron anatomical features.
1.1.1
MOTONEURONS
A vertebrate motoneuron is a good example of a unipolar neuron. Its cell body lies in the spinal cord. Dendrites (branching, treelike processes) arise from the soma and serve as points for synaptic inputs from presynaptic excitatory and inhibitory neurons. Analog epsp and ipsp transients are summed in space and time on the dendrites to determine a generator potential at the point where the axon joins the soma. This region is also called the spike generator locus (SGL) because it is where the all-or-none nerve action potential (spike) originates if the generator potential exceeds the firing threshold for the neuron. The spike propagates along the axon, away from the soma (orthodromically), toward the muscle motor unit the motorneuron activates. The end of the axon branches to end in a number of highly specialized structures called motor end plates, which are in intimate contact with the muscle membrane. The arrival of the spike causes the excitatory neurotransmitter ACh to be released from the motor end plates; ACh diffuses to receptors on the muscle membrane where it binds, causing the muscle membrane to depolarize, triggering the contraction process. Mammalian motoneurons generally have a myelin sheath. The myelin sheath is composed of specialized, flattened glial cells called Schwann cells, which wrap themselves around the axon the way that one would wrap a thin pancake around a pencil. That is, many turns of each Schwann cell cover the axon. Between each Schwann cell, along the axon, is a small gap called a node of Ranvier. Each Schwann cell forms a cylindrical, myelin “bead” of about 1 to 3 mm in length; the dense, multiturn wrapping around the axon is about 0.3 µm thick, and each nodal gap is about 2 to 3 µm wide. The myelin coating has several important functions: One is to increase the spike conduction velocity on the axon. Another is mechanical and biochemical support of the axon. Still another is to provide electrical isolation of the axon in a large bundle of nerve fibers, so that its action currents during spike propagation do not cause any “cross talk” with other fibers.
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A Unipolar cell
B Pseudo-unipolar cell
C Bipolar cell Dendrites
Dendrite
Central axon Single process
Axon
Cell body Peripheral axon to skin and muscle
Cell body
Invertebrate neuron
Dorsal root ganglion cell
Cell body Axon
Retinal bipolar cell
D Three types of multipolar cells
Dendrites Apical dendrite
Cell body
Cell body Basal dendrite Axon Dendrites
Cell body
Axon Axon
Spinal motor neuron
Hippocampal pyramidal cell
Purkinje cell of cerebellum
FIGURE 1.1-1 A sampler of neuron anatomy. Neurons are broadly classified as unipolar (A and B), bipolar (C), or multipolar (D). Unipolar neurons are found in invertebrate nervous systems. Pseudo-bipolar cells are typical of vertebrate sensory neurons. Their cell bodies are found in the dorsal root ganglia of the spinal cord. Many different types of sensory endings exist. Spinal motor neurons cell bodies lie in the spinal cord; their axons travel to the muscles they innervate. Very complex multipolar architectures are found in CNS interneurons. The dendritic field of the cerebellar Purkinje cells is amazingly complex. (From Kandel, E.R. et al., 1991. Principles of Neural Science, 3rd. ed., Appleton & Lange, Norwalk, CT. With permission from the McGraw-Hill Companies.)
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1.1.2
VERTEBRATE PERIPHERAL SENSORY NEURONS
Vertebrate peripheral sensory neurons are generally described by their sensory ending anatomy. They may or may not be myelinated. Parameters sensed are heat (T above ambient body temperature), cold, pain, touch, pressure, strain (muscle tension), and muscle length. Chemoreceptors (for taste, odor, pH, pO2. etc.), photoreceptors (rods and cones), and hearing receptors (hair cells) are generally considered to be part of the large receptor arrays having direct connection with the vertebrate brain. Peripheral receptors have a common plan. They all are of pseudo-unipolar design. The sensory ending is where the transduction of the physical quantity sensed causes membrane depolarization to take place. Depolarization leads to the generation of nerve spikes that propagate down the axon to synaptic connections in the spinal cord. Sensory neuron spikes generally originate in the axon near the sensory ending. Sensory endings differ widely in design, ranging from naked dendrites on pain sensors (nociceptors) to complex, single structures such as Pacinian corpuscles (pc, rate of change of pressure), Meissner’s corpuscles (flutter), Ruffini corpuscles (steady skin indentation), Krause’s corpuscles, Merkel’s disks (steady skin indentation). Muscle spindles (muscle length) and Golgi tendon organs (GTO, muscle force) have simple, encapsulated, endings in intimate contact with the tissues in which they sense strain. Figure 1.1-2 shows some of the receptor endings in situ. Hairy skin
Glabrous skin Papillary ridges Septa
Epidermal-dermal junction
Stratum corneum Epidermis
Merkel's receptor Meissner's corpuscle
Dermis
Bare nerve ending
Subpapillary plexus
Hair receptor Ruffini's corpuscle Pacinian corpuscle
FIGURE 1.1-2 Sensory receptors found in the hairy and hairless (glabrous) skin of primates. Not specifically labeled are the smooth nerve endings that respond to heat, cold, and pain (nociceptors). (From Kandel, E.R. et al., 1991. Principles of Neural Science, 3rd. ed., Appleton & Lange, Norwalk, CT. With permission from the McGraw-Hill Companies.)
All peripheral sensors exhibit some kind of proportional plus derivative response to their stimulus. That is, if a step of stimulus is applied, the receptor responds first by firing at a high initial rate, and then its spike frequency output slows down to a lower, steady-state value. For receptors such as the PC, the steady-state frequency
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FIGURE 1.1-3 Schematic cross section of a Pacinian corpuscle. The onionlike wrappings around the naked sensory ending cause this sensor to fire bursts for increasing or decreasing pressure in the tissues around it. The bulblike ending is about 1.5 mm in length.
is zero for an applied step of pressure. When the step of pressure is removed, the PC again fires a burst. As Figure 1.1-3 shows, the PC is a single, tapered nerve ending surrounded by layers of lamellae like the layers of an onion. In the steady state, internal pressure is uniform around the nerve ending, causing no stimulation. When a step of pressure is applied to the tissues surrounding the PC, it is generally asymmetrical. Forces are transferred through the lamellae into the viscous fluid directly surrounding the neuron tip, causing shear forces to distort one side of the tip, which in turn opens Na+ channels and depolarizes the tip, leading to a burst of spikes. Within hundreds of milliseconds, the pressure around the tip equilibrates, removing the shear force that opened the ion channels and the spikes cease. When the stimulus pressure is removed, the process repeats itself; there is a redistribution of external forces on the lamellae of the PC causing a transient distortion of the tip and another burst of spikes. A slower decrease in the overall response of a PC may be caused by neural accommodation. Accommodation may involve the mechanically sensitive ion channels on the tip losing their sensitivity to repeated stimuli, or a temporary exhaustion of Na+ ions in the volume surrounding the tip. A vertebrate, peripheral sensory neuron has a soma connected by a short neurite to the axon, the soma lies in the dorsal root ganglia outside the segments of the spinal cord. The axon enters through a dorsal root nerve trunk into the spinal cord, where it synapses with an appropriate, ascending, sensory projection interneuron. Some peripheral sensory neuron axons are myelinated (the PC, sharp pain, spindles, hair receptors, etc.); others are not (slow pain fibers, some warm and cold fibers, etc.). It is not the purpose here to describe the details of sensory neuron endings and their transduction processes; the interested reader who wishes to pursue these details should see chapters 24 through 27 in Kandel et al. (1991) and chapters 46 and 47 in Guyton (1991).
1.1.3
NEUROENDOCRINE CELLS
The peptidergic neuroendocrine cell bodies are located in the hypothalamus of the brain. They are neurons that release the endocrine peptide hormones oxytocin and vasopressin (also known as antidiuretic hormone) upon neural stimulation. These
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hormones are nine amino acid peptides; they enter the general circulation from the neurosecretory endings located in the posterior pituitary (neurohypophysis). Oxytocin initiates uterine contractions and the production of milk in nursing mothers. Vasopressin controls water resorption in the kidneys, and is a component of the feedback system that controls Na+ ion concentration in the blood (Northrop, 1999). Neurosecretory cells in the anterior pituitary secrete a number of glycoprotein endocrine hormones. These cells are part of the adenohypophyseal system. Unlike the neurohypophyseal neurosecretory cells, the adenohypophyseal cells are stimulated to secrete not by nerve inputs but by local hormonal signals derived from neurosecretory cells in the anterior hypothalamus. These hormones are listed in Table 1.1-1.
TABLE 1.1-1 Anterior Pituitary Hormones and Their Control Substances Anterior Pituitary Hormones Thyrotropin Prolactin Gonadotropin, leutinizing hormone, folliclestimulating hormone Adrenocorticotropin (ACTH), β-lipotropin Growth hormone (GH) Melanocyte-stimulating hormone (MSH), β-endorphin
Controlling Hormones Thyrotropin-releasing hormone (TRH) Prolactin-releasing factor (PRF), prolactin releaseinhibiting hormone (PIH), dopamine Gonadotropin-releasing factor (GnRF) Corticotropin-releasing hormone (CRH) Growth hormone-releasing hormone (GHRH), GH release-inhibiting hormone (GIH, somatostatin) MSH-releasing factor (MRF), MSH-release inhibiting factor (MIF)
The amino acid sequences are known for the peptide control hormones, GHRH, CRH, somatostatin, LHRH, and TRH (Kandel et al., 1991). The substances affecting the release of the anterior pituitary hormones are carried to the neurosecretory cells by blood in the local, hypophyseal-portal vessels running between capillary beds in the median eminence of the pituitary and the cells. To discuss the target organs of the anterior pituitary hormones and their functions is too far afield from the subject here. The interested reader should refer to Chapter 75 in Guyton (1991) or Chapter 6 in Northrop (1999).
1.1.4
INTERNEURONS
Interneurons are by far the most numerous type of neuron. This is because of their massive use in the CNS to process and store information. CNS interneurons tend to be small, and highly dendritic. They have diverse shapes depending on their functions and where they are found in the CNS. Some of the best known and most widely studied interneurons are found in the spinal cord controlling motoneurons. Each motoneuron receives many excitatory and inhibitory inputs via interneurons activated through the CNS, and through local reflexes involving spindles and GTOs. An interesting interneuron associated with each motoneuron is the Renshaw cell. A Renshaw cell
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provides a local negative feedback loop around each motoneuron. The motoneuron sends a recurrent fiber to activate the Renshaw cell, which in turn sends inhibitory signals to the motoneuron, and also to a type Ia inhibitory interneuron that inhibits a motoneuron serving the antagonist muscle (i.e., it inhibits an inhibitor). The direct negative feedback from the Renshaw cell reduces its firing sensitivity and tends to linearize its input/output firing characteristics from its various excitatory inputs. The Ia inhibitory interneuron is excited by a gamma afferent from a spindle in the agonist (e.g., flexor) muscle, thereby inhibiting the antagonist muscle. The agonist spindle gamma afferent also directly stimulates the agonist of the α-motoneuron muscle. Output from a GTO force sensor excites a type Ib inhibitory interneuron, which in turn inhibits the agonist of the α-motoneuron muscle. Figure 1.14 illustrates this system. Stretch reflex pathways involving the GTOs, spindles, inhibitory interneurons and motoneurons are collectively called a myotatic unit (see Kandel et al., 1991, Ch. 38). Because structure and function are intimately related in physiological systems, the CNS offers considerable challenge in understanding its function, given the diverse array of interneuron morphology found in the layered components of the brain (e.g., the cortex, the cerebellum). For example, in the cerebral cortex primates, there are about six layers containing about eight types of interneuron, including pyramidal cells (the output elements of the cortex), basket cells (inhibitory on pyramidal cells), chandelier cells, neurogliaform cells, arcade cells, bouquet cells, double bouquet cells, and long stringy cells (see Kandel et al., 1991, Ch. 50). The cerebellum is another specialized portion of the CNS with a highly organized structure. The role of the cerebellum is to coordinate motor actions by comparing what happens in terms of movement with what the CNS intended. The cerebellum has only 10% of the volume of the human brain; however, it contains over half the neurons found in the CNS. For example, there are 1011 granular cell neurons in the cerebellum, more than the total number in the cerebral cortex (Ghez, 1991). The cerebellum has three layers, and five types of interneuron, including Purkinje cells, Golgi cells (type II), basket cells, stellate cells, and granule cells. Figure 1.1-5 shows cerebellum features schematically. The outermost layer is called the molecular layer, the middle layer is the Purkinje layer, and the inner layer is the granular layer. In the molecular layer are parallel fibers from the granular cells, stellate cells, basket cells, the extensive dendrites from the Purkinje cells, and terminal arborizations from climbing fibers (inputs to the cerebellum). The Purkinje layer is relatively thin; it contains the closely packed, large (50 to 80 µm diameter) cell bodies of the Purkinje cells. The innermost granular layer contains granular cell cell bodies, terminal branches from the mossy (major input) fibers. The granular cells send axons to the outer surface of the molecular layer where they “T” into the parallel fibers. The Golgi cells have their cell bodies in the outer granular layer, and send extensive dendritic trees out into the molecular layer to receive excitatory synaptic inputs from the parallel fibers. In the granular layer, the output arborizations of Golgi cells make extensive inhibitory axodendritic synapses with granular cells, suppressing their excitation from mossy fibers. γ-Aminobutyric acid (GABA) is the inhibitory neurotransmitter used by Golgi cells. In the molecular layer, the stellate and basket cells act to inhibitor Purkinje cells. To quote Ghez (1991):
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Like the Purkinje cells, stellate and basket cells receive excitatory connections from the parallel fibers (granule cell axons). Stellate cells have short axons that contact nearby dendrites of Purkinje cells in the molecular layer, while basket cell axons run perpendicular to the parallel fibers and contact the cell bodies of more distant Purkinje cells. As a result, when a group of parallel fibers excites a row of Purkinje neurons and neighboring basket cells, the excited basket cells inhibit the Purkinje cells outside the beam of excitation [sic]. This results in a field of activity that resembles the centersurround antagonism that we have encountered in sensory neurons [e.g., in the retina].
1.1.5
DISCUSSION
This section has provided an overview of the many types of neurons used by an animal to survive in its environment. Sensory neurons respond to certain physical parameters its environment, both internal and external. In most cases, transduction of a physical quantity leads to the generation of nerve impulses, the frequency of which is a nonlinear, increasing function of the stimulus intensity. Receptors also code for rate of change of the stimulus; the output frequency thus can be thought of as having proportional-plus-derivative components. Motorneurons make muscles contract; the higher their frequency, the higher the tension developed or the quicker the contraction. To a crude approximation, the conversion of motoneuron spike frequency to muscle tension can be modeled by treating the muscle as an electrical low-pass filter, where output voltage is proportional to tension. Autonomic, effector neurons that innervate glands behave in much the same way as motoneurons, where the low-pass filter output voltage models the rate of release of a hormone. The most plentiful type of neuron is clearly the interneuron. Interneurons serve as control and computational elements for output (effector and motor) neurons in the spinal cord. In the CNS they have too many roles to list here. The cerebellum, which regulates the fine details of the motoneurons outputs, was seen to have more interneurons than the cerebral cortex. Interneurons are either excitatory or inhibitory, but their function lies in the details of their anatomies, synaptic inputs and outputs, and properties of their membranes.
1.2
ELECTRICAL PROPERTIES OF NERVE MEMBRANE
The basic structure of the unit membrane (UM) covering all nerve cells is the ubiquitous lipid bilayer, found on cells everywhere in the animal kingdom. The lipid bilayer is about 7.5 to 10 nm thick (how thick it appears under the transmission electron microscope often depends on the fixation techniques used to prepare the specimen). The lipid bilayer is composed of facing ordered arrays of phospholipid + cholesterol molecules. The fatty acid radicals on the molecules are hydrophobic and face each other to form the center of the membrane. The outer surfaces of the membrane are hydrophilic (attract water molecules) and contain the phosphate radicals of the phospholipid molecules. The lipid bilayer of the UM is a fluid, not a solid, because the UM can flow, much like the surface of a soap bubble, to adjust to internal and external forces on it. It can even reseal itself if slightly damaged.
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FIGURE 1.1-4 A schematic circuit diagram of a very simplified spinal myotatic reflex system. There are many parallel motoneurons, spindles and GTOs. Inhibition is represented by black ball synapses, excitation by arrows. αF is an α-motoneuron going to the flexor muscle. RF is a Renshaw inhibitory interneuron providing a local negative feedback loop around the motoneuron. Renshaw cells themselves have excitatory and inhibitory “gain control” inputs. Neurons labeled IbF and IaE are inhibitory interneurons. When the flexor muscle is stretched passively, the spindle output is active, causing a spinal reflex motor activation of the flexor, resisting the stretch, and a reflex inhibition of the α-motoneurons serving the extensor muscle. Increasing force on the flexor is sensed by the GTO, which inhibits the flexor α-motoneuron.
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A Stellate cell
Parallel fiber Molecular layer Purkinje cell
Basket cell
Purkinje cell layer
Granular layer Golgi cell Granule cell Purkinje cell
White matter
Climbing fiber
B
Purkinje cell axon
Glomerulus
Mossy fiber
Mossy fiber terminal Golgi cell axon
Granule cell dendrite
Mossy fiber
FIGURE 1.1-5 (A) Schematic sections through the cerebellar cortex of a primate brain. It has three anatomical layers and five types of interneuron. (B) Schematic structure of a cerebellar glomerulus found in the granular layer. In the glomerulus, intimate synaptic contact is made between a mossy fiber synapse, granule cell dendrites and a Golgi cell axon (not the GTO). (From Kandel, E.R. et al., 1991. Principles of Neural Science, 3rd. ed., Appleton & Lange, Norwalk, CT. With permission from the McGraw-Hill Companies.)
One factor that differentiates each different type of cell in the body are the many proteins and glycoproteins made by the internal biochemical machinery of the cell and embedded in the UM of the cell. Most of these large molecular weight proteins pass through the UM, projecting parts both on the inside and on the outside of the cell. The glycoproteins embedded in the UM of a cell have many functions. Some contain molecular receptors (binding affinity regions) on the outside of the cell where signaling molecules can dock and trigger configurational changes in the glycoproteins. In the case of neurons, some signaling substances are neurotransmitters released by other neurons. Neurotransmitters can either trigger direct configurational changes of the glycoproteins, allowing certain external or internal ions to pass easily through the membrane, or trigger a cascade of intracellular chemical reactions in which a second-messenger molecule reacts inside the cell with a target glycoprotein, causing it to pass certain ions through the membrane. As will be seen, the passage of ions such as Na+, K+, Cl–, Ca++, etc., across the cell membrane will change its
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resting potential, either hyperpolarizing the membrane (driving the inside more negative with respect to the outside) or depolarizing it (making the inside less negative), depending on the ionic currents. Not all membrane-bound glycoproteins are associated with gating ions. Some are adenosine triphosphate (ATP)-driven pumps that actively expel ions such as Na+ from inside the cell, or actively transport low-molecular-weight signaling substances to the inside of the cell. Still other membrane-bound glycoproteins are associated with electrical synapses, also known as gap junctions.
1.2.1
THE SOURCE
OF
UM ELECTRICAL PARAMETERS
The electrical behavior of a UM can be modeled electrically by a lumped-parameter, RC model. This model can be used to describe the behavior of dendrites with passive membrane (see Section 4.2) and also, with some modification, can be used to describe the generation of the nerve impulse (see Section 1.4). A very important property of nerve UM is its electrical capacitance. Because the hydrophobic lipid center of the UM acts like an insulator (neglecting ions leaking through the membrane-bound glycoproteins), and the internal and external liquids surrounding the membrane generally have relatively high conductances because of the dissolved ions, the membrane behaves like a parallel-plate capacitor. The plates are the inside surface of the UM in contact with the conductive axoplasm on the inside of the cell, and the outside surface of the UM in contact with the extracellular fluid. A patch of UM from a squid giant axon, for example, has a measured capacitance of about 1 µF/cm2. In MKS units, this is 10–2 F/m2. If one assumes that a UM patch is modeled by a parallel-plate capacitor with capacitance, then
C=
Aε o κ d
F
1.2-1
where C is the capacitance in farads, A is the plate area in m2, d is the plate separation in m, εo is the permittivity of free space in F/m, and κ is the dielectric constant. By substituting C = 10–6 F, A = 10–4 m2, d = 7.5 × 10–9 m, and εo = 8.85 × 10–12 F/m, the equivalent dielectric constant for the UM is κ = 8.47, which is a bit high for long-chain lipids, but not so for hydrated proteins (Plonsey, 1969, Ch. 3). Assuming the resting potential across the nerve cell membrane is 65 mV, the electric field across the dielectric is approximately E = 0.065/(7.5 × 10–9) = 8.67 × 106 V/m, or 8.67 × 104 V/cm, quite high. If one takes the dielectric thickness as 5 nm (50 Å), then κ = 5.65, and E = 1.30 × 105 V/cm. Because of this high electric field in the membrane, it would not be surprising to find that membrane capacitance is in fact a function of the transmembrane potential, Vm. Membrane capacitance is important because any potential change across the neuron UM cannot occur unless the capacitance is supplied with a current density ˙ (t) amp/m2 (C is the capacitance in F/m2). according to the relation: Jc(t) = Cm V m m Jc is generally supplied by ions flowing through gated-ion channel proteins, and axial ion currents outside and within the axon or dendrite.
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In formulating mathematical models of how passive dendrites behave electrically when subjected to transient changes of input voltage, it is more convenient to describe cylindrical dendrites in terms of per-unit-length parameters, including capacitance. It is easy to show that for a dendrite or axon, cm = Cm π D
F/m
1.2-2
where Cm is the membrane capacitance in F/m2, and D is the dendrite diameter in meters. When cm is multiplied by the length, L, of a cylindrical section of dendrite, one again obtains Cm F. There are three constant resistive parameters used along with cm needed to describe the electrical behavior of a dendrite with a passive membrane. (Passive nerve membrane has transmembrane conductances that are constant over the range of membrane voltage of interest.) First, an expression for ri, the internal longitudinal or axial resistance of the dendrite in ohm s/m will be derived. Assume that the axoplasm inside the dendrite has a net resistivity of ρi ohm cm. (Resistivities are commonly given in ohm cm.) The net internal resistance of a tube of axoplasm L cm long and D cm in diameter is Ri = ρ L/A = ρi L/(π D2/4)
ohm
1.2-3
Changing the length units to meters and dividing by L, ri = 0.04 ρi/(π D2)
ohm/m
1.2-4
(ρi is still in ohm cm.) Thus (ri ∆x) is a resistance in series with (internal) axial current flow in the lumped parameter model. The external longitudinal (spreading) resistance per unit length, ro, can also be found. ro is also in series with the external axial current near the dendrite. ro is generally small compared with ri, and its exact value depends on the tissue structures and extracellular fluid composition surrounding the dendrite. (In the case of the squid giant axon, the external medium is seawater which has a low resistivity, ρo). The passive membrane has an equivalent constant radial leakage conductance for the various ions found in the axoplasm and extracellular fluid. The total net leakage conductance is gm S/m in parallel with the membrane capacitance per unit length, cm F/m. In general, the leakage conductance for K+ is higher than that for Na+.
1.2.2
DECREMENTAL CONDUCTION THE SPACE CONSTANT
ON
DENDRITES:
Figure 1.2-1 illustrates the lumped-parameter, linear RC circuit that is used to model a cylindrical dendrite having a passive cell membrane. Ie(x) is a current sunk at a point x on the surface of the dendrite into an external electrode. Ie(x) has the units of amp/m. iL(x, t) is the net longitudinal current in the axoplasm or outside the dendrite’s surface. Most generally, iL is a function of displacement x and time; it has the units of amp. im(x, t) is the net transverse current through the membrane © 2001 by CRC Press LLC
FIGURE 1.2-1 Equivalent lumped-parameter model of a passive-membrane dendrite. The sections form a lossy RC transmission line.
capacitance and conductance; its units are amp/m. vm(x, t) is the transmembrane potential change (superimposed on the dc resting potential, Vm0). Use of Kirchoff’s voltage law yields: vm(x + ∆x) = vm(x) – iL(x)(ro + ri) ∆x
V
1.2-5
Vm
1.2-6
By rearranging terms and dividing by ∆x,
v m (x + ∆x) – v m (x) = – i L (x)( ro + ri ) ∆x In the limit, ∆x → 0, and
∂v m = – i L (x)( ro + ri ) ∂x
Vm
1.2-7
Next, Kirchoff’s current law is used to write: iL(x) = iL(x + ∆x) + im(x, t) ∆x – Ie ∆x
1.2-8
Now by Ohm’s law, im(x, t) is im(x, t) = vm(x + ∆x) gm + cm ¹[vm(x + ∆x, t)]/¹t
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1.2-9
Use of the first term in the Taylor’s series expansion for vm(x + ∆x, t) yields vm(x). Thus,
i m (x, t ) = v m g m + v˙ m c m
amp m
1.2-10
Equation 1.2-10 can be substituted into Equation 1.2-8 to obtain:
i L (x + ∆x) – i L (x) = I e – ( v m g m + c m ∂v m ∂t ) ∆x
1.2-11
which becomes, on letting ∆x → 0, ¹iL/¹x = Ie – (vm gm + cm ¹vm/¹t)
amp/m
1.2-12
Now taking the second derivative of Equation 1.2-7 with respect to x and substituting Equation 1.2-12 yields
[
]
∂ 2 v m ∂x 2 = – (∂i L ∂x)( ro + ri ) = – ( ro + ri ) I e – ( v m g m + c m ∂v m ∂t ) 1.2-13 ↓
∂ 2 v m ∂x 2 – v m g m ( ro + ri ) = – ( ro + ri )(I e – v˙ m c m )
v m2
1.2-14
Consider the dc voltage distribution on the dendrite in the temporal steady state, where v· m → 0. Laplace transforming Equation 1.2-14 with respect to the length variable, x, yields Vm(s)[s2 – gm(ro + ri)] = (ro + ri) Ie(s)
1.2-15
This linear second-order ODE has the general solution:
[
]
[
]
v m (x) = A exp – x g m ( ro + ri ) + B exp x g m ( ro + ri ) + K
1.2-16
[ g (r + r ) ] is defined as the space constant, λ (dimensions;
In Equation 1.2-16, 1
m
o
i
meters) of the dendrite. Thus Equation 1.2-16 can be rewritten more economically as vm(x) = A exp[–x/λ] + B exp[+x/λ] + K
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1.2-17
As a simple example, assume a dendrite of diameter D stretching from x = 0 to x = ×. At x = 0, a voltage source holds vm(0) at Vm0 V. vm(×) = 0, Ie ≡ 0. Thus it is clear that B = K = 0, and A = Vm0. The dc voltage along the dendrite decays exponentially as x → ×. The space constant, λ, of a dendrite is an important parameter because it is a measure of the steady-state electrotonic spread of epsps and ipsps. It can be shown that the bulk parameters of the UM and the axoplasm are related to λ by
λ≅
D 2
(G m ρ i )
cm
1.2-18
where D is the diameter of the dendrite in cm, GM is the leakage conductivity of the membrane in S/cm2, and ρi is the axoplasm resistivity in ohm cm. Note that λ increases with the square root of the diameter of the dendrite, other bulk parameters being held constant. The rate of spread of a transient voltage induced on a passive dendrite is inversely proportional to the product, (ro + ri)cm (Kandel et al., 1991). If one assumes that ri Ⰷ ro, and expresses (ri cm)–1 in terms of the bulk parameters of the membrane, then
(ri c m )
–1
=
D π D2 = 4 ρ i (C m π D) 4 ρ i C m
cm 2 s
1.2-19
where D is the dendrite diameter in cm, ρi is the axoplasm resistivity in ohm cm, and Cm is the bulk membrane capacitance in F/cm2. Thus, the larger the dendrite diameter, the faster the passive spread velocity.
1.2.3
ACTIVE MEMBRANE: THE NERVE SPIKE
The major difference between passive and active nerve membranes is that the transmembrane proteins of active membrane increase their conductances in response to signals. The signals can be electrical (sodium channels open and allow an inrush of sodium ions when the transmembrane potential, Vm, reaches the threshold voltage for depolarization, Vϕ, e.g., +10 mV above the resting potential). Signals can also be chemical, where neurotransmitters or second messenger molecules bind with specific ion-channel gates, opening them. The abrupt increase in specific ionic conductances causes ion currents to pass through the membrane in directions determined by the combined factors of the transmembrane potential and the concentration gradient of the ion across the membrane. The specific ionic conductance is defined in the case of sodium by gNa ≡ JNa/(Vm – VNa)
S/cm2
1.2-20
Vm (t) is the actual transmembrane potential, and VNa is the Nernst potential for sodium ions, defined by:
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[ [
⎛ Na + VNa ≡ ( RT F) ln ⎜ ⎜ Na + ⎝
]⎞ ] ⎟⎟⎠ o
V
1.2-21
i
where R is the gas constant, 8.314 J/mol K; F is the Faraday constant, 9.65 × 104 C/mol; T is the Kelvin temperature; [Na+]o and [Na+]i are the concentrations of sodium ions outside (460 mM) and inside (46 mM) the membrane, respectively, in mol/l for squid axon. ENa < 0 means the electrical force on Na+ is inward. For squid axon at 293 K (20°C), VNa can be calculated: VNa = (8.314 × 293/9.65 × 104) ln (460/46) = +57.4 mV
1.2-22
As Section 1.4.1 will show, the net potential energy forcing Na+ ions into the axoplasm is defined as ENa ≡ (Vm – VNa) electron volts (eV). ENa = –127.4 meV for squid axon where the resting potential, Vm0 = 70 mV. The sequence of events required for the initiation (and propagation) of a nerve impulse is as follows: Some combination of presynaptic events (or stimulus transduction in the case of sensory neurons) causes the transmembrane potential to depolarize, reaching the spike initiation threshold in a region of active membrane (e.g., the spike generator locus). There, voltage-triggered sodium channels abruptly increase their conductance for sodium ions, allowing enough Na+ ions to rush in to cause the local transmembrane potential to depolarize. This strong inward flow of Na+ ions causes Na+ and K+ ions on the outside of the membrane in front of the initiation region to move toward the region where JNa is highest through the membrane. gNa (xo, t) continues to rise, reaches a peak of about 30 mS/cm2 at about 0.5 ms, then spontaneously deactivates and returns to its resting level of about 5 µS/cm2 in about 1 to 2 ms. At the same time, depolarization-activated potassium ion channels open more slowly, reaching a peak in about 1 ms, allowing K+ ions to flow out of the core axoplasm, helping to restore the axon resting potential. gK has a resting value of about 0.5 mS, and its peak is about 13 mS. As a result of the outward JK, Vm(t) falls below the resting potential as much as 7 mV for about 1.5 to 5 ms, slowly returning to Vm0. From voltage clamp studies, it is known that gK remains high (potassium ion channels remain open) as long as Vm is depolarized; it does not spontaneously deactivate like gNa. So how does the spike propagate, and what factors control its velocity? Assume that some event has depolarized Vm at x = xo to the threshold voltage ϕ > Vm0 and the gNa begins to increase. Because of the initial inrush of Na+, the membrane depolarizes further, causing gNa to increase rapidly. Vm(t) now goes positive in response to the strong, inward JNa . gNa spontaneously, rapidly returns to its resting value. Depolarization-activated K+ channels allow potassium ion to leave the axon in front and in back of the sodium-active area. The inward JNa and the outward JK are accompanied by axial currents inside and outside the axon, as ions respond to electrostatic and diffusion forces caused by the radial ion currents. The membrane capacitance acts like a low-pass filter in limiting the rate of change of Vm. Cm must ˙ (t) in order that V change. charge with some net current density, Jc(t) = Cm V m m
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Because of the axial currents, iL(x, t), Vm reaches the firing threshold in front of the region of peak gNa activation, causing the Na channels here to open. The Na channels in back of the active region now close. Thus, the spike moves forward in a traveling wave. If an axon is artificially depolarized in its middle, the action potential will propagate in both directions. This is an unnatural situation, however. Artificial, antidromic stimulation can be used as an electrophysiological tool to relate axons to cell bodies in dense nervous tissue such as the spinal cord. The velocity of propagation of the action potential has been found experimentally to be proportional to the square root of the unmyelinated axon diameter. That is, v ≅ ko + k D . Thus, evolution has selected certain motoneuron axons to be large and to have high conduction velocities whenever survival requires a rapid escape reflex. The prime example of the need for a giant axon can be seen in the squid, whose giant motor axons can be as large as 1 mm diameter. (A 0.2-mm-diameter squid axon conducts at about 20 m/s.) The crayfish also has giant motor axons to effect its quick tail-flip escape maneuver. In vertebrates, one also can see a grading of nerve axon diameters that can be correlated with the survival importance of the information rapidly reaching the target organs. Figure 1.2-2 illustrates the recording setup for measuring the transmembrane potential, Vm(t), and the surface potential, Vs(t), during a propagated action potential on an unmyelinated axon. Figure 1.2-3 illustrates Vm(xo, t) as a spike is initiated at x = xo, and also the conductances, gNa(xo, t) and gK(xo, t). The approximate distribution of Vm(x, to), gNa (x, to) and gK(x, to) along the x axis at t = to for a propagating spike are illustrated in Figure 1.2-4. Note that the transmembrane voltage reaches the threshold for sodium ion channel opening in front of the spike peak, keeping the spike propagating at a constant velocity, v.
1.2.4
SALTATORY CONDUCTION
ON
MYELINATED AXONS
A morphological adaptation of vertebrates that increases the spike propagation velocity on their axons is myelinization. An electron micrograph of the cross section of a myelinated axon is shown in Figure 1.2-5 (from a rat sciatic nerve, × 52,000). Where the axon is wrapped in the myelin “tape,” two important changes occur in the cable parameters. Because of the insulation of multiple, close-packed layers of UM, gm decreases, perhaps by a factor of 1/64. Because of the effective thickening of the axon wall, Cm and cm decrease by a factor of about 1/20. Thus, the effective space constant of the covered axon increases by about eight-fold, and the passive velocity of propagation, which is proportional to (ri cm)–1, increases by about 20-fold. Once a spike has been initiated at the SGL, it propagates down the axon to the first myelin “bead.” There it propagates electrotonically because the myelin blocks the high JNa and JK required for conventional regenerative propagation. Because of the increased space constant, there is little attenuation along the myelin bead, and the velocity of the electrotonic spread is higher than the conduction velocity on the unmyelinated axon. The nodes of Ranvier between the myelin beads on the axon allow the propagating spike to regenerate to its full height. At the nodes, conventional voltage-gated ion channels in the membrane can carry the normal ion currents required for the action potential. The process of fast electrotonic spread followed
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FIGURE 1.2-2 Figure showing how the transmembrane potential of a large axon is measured with a glass micropipette electrode and a capacitance-neutralized electrometer amplifier. A circuit for the unipolar measurement of the external potential at a point xo on the axon is also shown.
FIGURE 1.2-3 Plots of the axon transmembrane potential, Vm(t), the sodium ion conductance, gNa(t), and the transmembrane potassium conductance, gK(t), all measured at point xo on the axon. Note the rapid drop in gNa after Vm(t) reaches its peak. A plot of this type is easily obtained from a simulation of the HH (1952) model equations for a nerve action potential.
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FIGURE 1.2-4 Distribution of Vm(x), gNa(x), and gK(x) along the axon at a fixed time, to. Note high gK persists over a longer distance than does gNa.
by regeneration at each node is called saltatory conduction. The term saltatory is used because early neurophysiologists visualized the velocity of propagation speeding up at the nodes of Ranvier, then slowing down along each myelin-covered stretch of axon. This view is misleading, because the nodes are only about 0.002 mm in length, and serve to regenerate the spike height to make up for the passive attenuation that occurs in the electrotonic phase of propagation under the 1 to 2 mm myelin beads. It appears that the spike velocity is essentially constant, and above the velocity of the same axon without myelin because of the greatly reduced cm. In 1949, Huxlely and Stampfli (reported in Plonsey, 1969) measured saltatory conduction times on a myelinated axon: They found the internodal propagation time to be 0.02 ms, and the propagation time along a myelin bead to be 0.1 ms. If one assumes the node is 0.002 mm wide, then the nodal velocity is 0.1 mm/ms and the electrotonic velocity is 1 mm/0.1 s = 10 mm/ms. Thus, their myelinated axon conducted at slightly over 10 m/s, and the ratio of electrotonic velocity to regenerative velocity is about 100:1. There is a huge survival advantage for myelinization. A myelinated axon also has a metabolic advantage over a bare axon of the same diameter. Far less total Na+ enters and K+ leaves the axon at the nodes than would over the same length of bare axon conducting a spike. Thus, less metabolic energy in the form of ATP is required to drive the ion pumps that maintain the steady-state ion concentrations in the axon axoplasm. The analytical differential equation model of Fitzhugh (1962) for saltatory conduction on a myelinated axon has been described by Plonsey (1969), sec. 4.10. Fitzhugh used the continuous partial differential equation derived from the lumped parameter, per unit length RC transmission line to model electrotonic conduction under the myelin beads, and he used the Hodgkin–Huxley, regenerative model for spike generation to simulate what happens to vm at the nodes of Ranvier. The electrotonic differential equation (DE) used was
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FIGURE 1.2-5 Electron micrograph of the cross section of a myelinated nerve axon. A living glial (Schwann) cell wraps itself around a peripheral nerve axon much like one would wrap electrical tape around a bare wire. The myelin wrapping has two major effects: It speeds the conduction of the nerve action potential and it mechanically protects and insulates the axon. (From University of Delaware, Mammalian Histology B408 Web site www.udel.edu/Biology/Wags/histopage/histopage.htm.)
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∂vm ∂ 2 vm = ∂t ∂x 2
[(r + r )c ]
−1
o
i
m
– vm gm cm
1.2-23
(See Section 1.4.1 for the Hodgkin–Huxley equations.) Some seven or eight nodes were modeled. The parameters used were cm (myelinated axon) = 1.6 pF/mm, ri = 15 × 106 ohm s/mm, gm = 3.45 × 10–9 S/mm, the area of the node was 0.003 mm2, the capacitance of the node was 1.5 pF. Figure 1.2-6 illustrates a three-dimensional plot of the calculated transmembrane potential distribution at various times along a myelinated axon having an area of 0.003 mm2, and nodes every 2 mm (at the vertical lines). A 10 µs, 30 mA pulse was given at t = 0, x = 0. Note that the peak of the traveling wave has reached x = 10 mm in 1.32 ms, giving an approximate conduction velocity of v = 7.58 m/s to the right.
1.2.5
DISCUSSION
This section has shown that the basis for nearly all modern computational models for neurons and biological neural networks dates from the discovery in the early 1950s that information propagated along neural membranes is in the form of transient changes in the transmembrane voltage. These membrane voltage changes could be active, as in the case of nerve action potentials on axons, or passive, as in the case of dendrites. All nerve axons were shown to have a nearly constant capacitance per square centimeter, and to contain two types of voltage-activated, ion-conducting proteins that under certain conditions would pass either sodium or potassium ions in directions that were determined by concentration gradients and the transmembrane voltage. The behavior of these specific ion-channel proteins was shown to be modeled by nonlinear, voltage-dependent conductances. Dendrites can be modeled by fixed-parameter, lumped-parameter RC transmission lines, such as illustrated in Figure 1.2-1. Such transmission lines are linear circuits; they are most easily modeled by dividing the dendrite tube into sections of finite length, ∆x, and representing each section by a parallel RC circuit emulating the transmembrane leakage conductance and capacitance. The parallel RC sections are connected by resistors representing the longitudinal (axial) resistance inside the dendrite and also outside it. The presence of myelin beads was shown to increase conduction velocity on axons because high-speed electrotonic conduction occurs on the axon under each bead, and the action potential is regenerated at the nodes of Ranvier between the myelin beads. Both myelinated and unmyelinated axons can be modeled by modified transmission line models. A lumped-parameter model for a myelinated axon is shown in Figure 1.2-7. An active Hodgkin–Huxley circuit with its voltage-dependent sodium and potassium conductances marks the nodes at the left end of a myelin bead, and a passive RC ladder emulates the passive propagation of the action potential under the myelin bead. A circuit consisting of a string of seven or eight of the subunits of Figure 1.2-7 was used to obtain the plot of Figure 1.2-6.
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FIGURE 1.2-6 A three-dimensional (transmembrane potential vs. time and distance) plot of saltatory conduction on a myelinated axon model. The distance between nodes of Ranvier is 2 mm. A 30-mA, 10-µs pulse was given at t = x = 0. Each heavy line represents transmembrane potential calculated at and between nodes at a particular time. Note that full depolarization only occurs at the nodes; decremental (passive) propagation occurs under the myelin beads with little attenuation. (Derived from a two-dimensional graph in Plonsey, 1969.)
Modern computational models for spike generation and propagation still use the basic HH architecture, or modifications of it. The details of ODEs generating the specific ionic conductances are often changed, as are their voltage-dependent coefficients. When modeling synapses, ion channels other than for Na+ and K+ are used to reflect current details known about epsp and ipsp generation.
1.3
SYNAPSES: EPSPS AND IPSPS
Synapses are specialized structures attached to a neuron that enable the neuron to communicate with one or more other neurons rapidly, and also with muscles (here they are called motor end plates). When a nerve spike propagates down an axon, its
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FIGURE 1.2-7 A finite RCG transmission line model of a myelinated axon. One node of Ranvier is shown to the left with (active) voltage-dependent gNa and gK conductances; the other four linear RCG elements model the electrical properties of the axon under the myelin bead. Six or seven of the units shown can be connected in series to simulate the saltatory conduction illustrated in Figure 1.2-6. (Eight ODEs are needed to simulate one unit, four for the HH node and four for the myelinated axon.)
effect is felt on the next neuron in the chain of communication, the postsynaptic neuron. Depending on the type of synapse, the postsynaptic neuron can be brought closer to firing its own spike(s), or inhibited from firing. An interneuron or motoneuron generally has many input synapses, providing redundancy, and, as will be shown, noise reduction in the information transfer process. Generally, synapses terminate the branches of the terminal arborizations of the axon of the presynaptic neuron. Synapses may make contact with dendrites (axodendritic synapses), the soma (axosomatic synapses), and even with the boutons of the postsynaptic neuron (axo-axonic synapses). There are two types of synapses: chemical and electrical. Most synapses are chemical; however, there are examples of electrical coupling between neurons where speed and reliability of transmission are required. In Table 1.3-1, properties of chemical and electrical synapses are summarized:
TABLE 1.3.1 Summary Properties of Synapses Property
Electrical Synapses
Chemical Synapses
Distance between pre- and postsynaptic cell membranes Cytoplasmic continuity between pre- and postsynaptic neurons? Ultrastructural components
~ 3.5 nm.
16–20 nm
Yes
No
Gap junction channels, i.e., connexins
Mechanism of transmission
Ionic currents
Synaptic delay Direction of transmission
Negligible Generally bidirectional
Presynaptic vesicles, mitochondria, postsynaptic receptors Release and diffusion of chemical neurotransmitter 0.3–5 ms, depending Generally unidirectional
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1.3.1
CHEMICAL SYNAPSES
At the ends of the terminal arborizations of the presynaptic nerve axon are the bulbs or boutons of chemical synapses. Each synaptic bulb is from 1 to 2 µm in diameter; it is filled with many spherical, synaptic vesicles containing neurotransmitter, some of which are in intimate contact with the membrane of the bulb facing the synaptic cleft or gap (Eccles, 1964). The vesicles are from 20 to 60 µm in diameter and are constantly being made by the metabolic machinery in the bulb and axon terminal. The size and shape of vesicles depends on the neurotransmitter being used. The 20 to 60 nm spherical vesicles are thought to be associated with excitatory synaptic coupling and the neurotransmitter ACh. Certain inhibitory boutons in the CNS have flattened vesicles, while spherical vesicles in central adrenergic boutons are larger (60 to 80 nm) and have electron-dense cores. The details of presynaptic chemical synapse action are as follows: 1. In a resting bouton, a certain fraction of the vesicles filled with neurotransmitter molecules are docked to the inside of the active zone of the bouton (the membrane facing the synaptic cleft) at membrane fusion proteins. A small fraction of vesicles is free to move inside the bouton, but the majority of the vesicles are immobile, bound to one another and the inside of the bouton membrane by cytoskeletal filaments made from proteins called synapsins (Kandel et al., 1991, Ch. 13). 2. When a presynaptic action potential (spike) propagates down the terminal arborization to a bouton, the depolarization causes voltage-gated calcium channel proteins to “open,” allowing Ca2+ to flow inward through the walls of the bouton. During an action potential, the [Ca++] inside the bouton at the active zone can rise a 1000-fold from about 100 nM to 100 µM. 3. The local increase in [Ca++] activates the protein, calmodulin, which in turn activates the protein kinase enzymes. 4. Three transient events follow the creation of active protein kinases. The cytoskeletal filaments dissolve, allowing bound vesicles to move toward the active zone. Vesicle motion toward the fusion proteins may in fact be guided by low-molecular-weight G-proteins. Vesicles bind to membrane fusion proteins. The membrane fusion proteins dilate, dumping the neurotransmitter in the bound vesicles into the synaptic cleft (Kandel et al., 1991). This process is illustrated in Figure 1.3-1. The postsynaptic events of synaptic transmission begin with the nearly simultaneous release of the neurotransmitter (NT) from the bouton following the arrival of the presynaptic action potential. About 150 vesicles dump NT per presynaptic spike at a motor end plate, and from one to ten vesicles may be involved for interneuronal communication in the CNS. The “bolus” of NT diffuses rapidly across the 200-nm cleft. On the subsynaptic membrane (SSM) (e.g., on a dendrite) there are located many receptor proteins for that specific NT. Also present are molecules of an NT-esterase protein that rapidly destroys the free NT in the cleft. In the case of the NT ACh, cholinesterase breaks it down into acetate and choline. Choline is recycled by the metabolic machinery in the bouton; proteins in the bouton membrane actively transport
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FIGURE 1.3-1 (A) Schematic of a synaptic bouton before the arrival of an action potential. Note that some small amount of NT leaks randomly from the vesicles docked at the presynaptic membrane, causing noise in the postsynaptic transmembrane potential. (B) Events occurring at the arrival of an action potential. Note that voltage-gated Ca++ channels open, NT is released, Na+ passes through the SSM NT-gated Na+ channels, and an epsp is generated.
choline into the bouton where the enzyme cholineacetyltransferase synthesizes ACh from acetyl coenzyme A and the free choline. This process is more complex that it sounds, because the ACh must be put into the vesicles in just the right amount. When the nicotinic ACh receptor proteins on the SSM each bind with two ACh molecules, their conductance to [Na+] and [K+] increases transiently, allowing sodium ions to flow in and some potassium ions to flow out. The net result of about 200,000 receptor proteins being activated by ACh is the generation of a small transient depolarization of the SSM (an excitatory post synaptic potential or epsp). The peak amplitude of a normal epsp is about 5 mV. epsps are summed in time and space over a dendritic © 2001 by CRC Press LLC
tree, and if the instantaneous sum is large enough, the post synaptic neuron generates an action potential on its axon (Figure 1.3-2).
FIGURE 1.3-2 Schematic cross section of a nicotinic ACh NT synapse. Two molecules of ACh must bind to the proteins of the ACh-gated channels so that their conductance to Na+ and K+ increases. The resulting depolarization of the SSM causes voltage-gated Na+ channels to open, further depolarizing the SSM, forming an epsp.
ACh can also affect inhibitory postsynaptic potentials (ipsps) at SSMs having a muscarinic receptor system. In this case, 1 molecule of ACh in the cleft combines with a single site on a muscarinic protein that projects through the subsynaptic membrane. As a result of the ACh binding, the muscarinic protein causes the α-subunit of a threeunit, G-protein complex to dissociate and bind with a potassium ion channel protein. The α-G*K+-channel protein association causes the K+ channel to open, allowing [K+] to flow outward. The outward potassium current causes the postsynaptic membrane potential at the synapse to hyperpolarize (i.e., go more negative). After a few milliseconds, the α-G protein dissociates from the K+-channel protein, allowing it to close (Figure 1.33). This type of ipsp generation is known to occur when the autonomic fibers in the vagus nerve that synapse with pacemaker cells in the heart are active; the heart slows as a result of this action. Far from Nature being consistent in her designs, in smooth muscle in the stomach, when ACh binds to subsynaptic muscarinic receptors, other elements of the G-protein complex are released and bind to K+ channel proteins, closing them. The reduction of the slow, outward, K+ leakage current causes the postsynaptic membrane voltage to depolarize in this case, leading to muscle contraction (Fox, 1996).
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Another baroque scenario is seen in the action of the neurotransmitter norepinephrine (NEP), in both the central and peripheral nervous systems. In this case, one molecule of NEP in the synaptic cleft combines with its site on a receptor protein spanning the SSM. The binding of NEP with its site causes the α-subunit of the G-protein to dissociate. α-G then combines with the membrane-bound enzyme, adenylate cyclase, activating it. Adenylate cyclase causes the production of cyclic adenosine monophosphate (c-AMP) from ATP. c-AMP in turn activates protein kinase, which can open ion channels and produce other intracellular effects (Fox. 1996). Contrast the complexity of this six-step process with the simple, direct opening of nicotinic Na+ channels by ACh.
FIGURE 1.3-3 Schematic cross section of an SSM of an inhibitory synapse. One molecule of ACh binds to a muscarinic ACh receptor protein. This binding triggers the dissociation of an attached G-protein. The α portion of the G-protein diffuses to a receptor site on a potassium channel, where it causes an increase in gK. The outward flow of K+ ions causes the transmembrane potential to hyperpolarize toward the Nernst potential for potassium ions, causing an ipsp to be seen. This process is transient, and the resting conditions are slowly restored by enzymatic machinery that reassembles the G-protein and breaks down the ACh.
There are many neurotransmitters; some act excitatorily in one class of postsynaptic neuron and inhibitorily on others. Some NTs are well-identified, others are suspected, and some are found in invertebrates. They include but are not limited to ACh, epinephrine, NEP, dopamine, serotonin (5HT), histamine, glycine, glutamate, GABA, nitric oxide (NO, a gas), endorphins, and enkephalins. In addition, there are many neuroactive peptides found in the mammalian CNS, such as neuropeptide-Y (see Table 14.2 in Kandel et al., 1991). The vertebrate retina is a neurophysiologist’s garden of synapses and neurotransmitters. Known retinal inhibitory neurotransmitters include glycine and GABA. Excitatory NTs include ACh and glutamate; other substances, classed as neuromodulators, include dopamine and serotonin. Still other substances found in the retina whose roles have yet to be clarified include adenosine, substance P (a large-molecular-weight protein), NO, and somatostatin (also known as growth hormone
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inhibitory hormone, a short peptide usually associated with the hypothalamus) (Kolb et al., 1999). The neurotransmitters glycine and GABA are associated with the generation of ipsps and the inhibition of firing in the postsynaptic neuron. Both these NTs cause chloride channels to open. Open chloride channels tend to drive the transmembrane potential toward the Nernst potential for chloride (approximately –70 mV), assuming the internal concentration of [Cl–] stays constant. The increased gCl resulting from the open chloride channels tends to clamp Vm → –70 mV, counteracting any depolarization caused by sodium epsps. Glutamate is an excitatory NT in the CNS. There are four types of glutamate receptors, and of those there may be subtypes or variants. The first three are directly gated receptors. The NMDA receptor requires both a glutamate and a glycine molecule to open for Ca++ and Na+ (in), and K+ (out) currents. The Kainate receptor requires one glutamate molecule to allow Na+ (in) and K+ (out). The KainateQuisqualate-A receptor binds one glutamate molecule and has a site for Zn++ binding, too. It passes Na+ (in) and K+ (out). The quisqualate B receptor also binds one glutamate molecule, and uses a G-protein second-messenger system to open ion channels. Details on the biochemistry of these systems can be found in Kandel et al., (1991).
1.3.2
ELECTRICAL SYNAPSES
Chemical synaptic action is always accompanied by small delays associated with NT release, diffusion and binding, and the time it takes ion gate proteins to open and ions to move in or out of the postsynaptic membrane (PSM). The capacitance of the PSM must also be charged or discharged to realize an epsp voltage transient. Electrical synapses allow the depolarization or hyperpolarization of the presynaptic neuron to be directly coupled to the postsynaptic neuron without delays. Electron microscopy and X-ray diffraction studies have shown that an electrical synapse, also called a gap junction, consists of facing areas of the pre- and postsynaptic membranes separated by about 3.5 nm. Penetrating each membrane are curious protein structures called connexons. There are hundreds if not thousands of connexons in each gap junction. Each connexon is made up from six subunits called connexins, spaced around a common center (like the sections of a orange). Each connexin is 7.5 nm in length, and is directly opposite a corresponding connexin in the opposite membrane. The center axes of each pair of pre-and post synaptic connexons are aligned; the opposite pre- and postsynaptic connexons touch in the gap. When the presynaptic membrane is depolarized, a tube of ~1.5 nm diameter is formed in the centers of each pair of opposite connexons. This tube allows ions, amino acids, and molecules with molecular weights up to 1000 to pass freely, depending on electrostatic and diffusion potential energies. For example, Na+, K+, Cl, Ca++, c-AMP, and ATP could pass. All gap junctions can be classified as either bidirectional (most electrical synapses are) or rectifying (the classical example is the giant motor synapse of the crayfish). If two neurons, A and B, are connected by a nonrectifying gap junction, depolarization of A will produce an immediate, similar, smaller depolarization in
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B, and vice versa. In the case of a rectifying electrical synapse, depolarization of A will produce an immediate, similar, smaller depolarization in B, but depolarization of B has little effect on the membrane potential of A. Thus, the connexons of A cause the connexons of B to open, but not vice versa. Note that nonrectifying gap junctions permit two-way, neural communication. (It is worthy to note that bidirectional chemical synapses have been observed in the brains of crabs. They are quite uncommon, otherwise.) Gap junctions are found in the vertebrate retina. The horizontal cells (HCs) in the outer plexiform layer form a syncytium, connected to each other by gap junctions. Thus, any electrical activity induced in a central HC will spread rapidly and decrementally out from the center on the connected HCs. Neighboring rods and cones are also connected to one another by gap junctions (Kolb et al., 1999). Such electrical synapses allow rapid decremental interneuronal signaling. In heart muscle, which forms a syncytium, there are extensive arrays of gap functions between muscle cells to facilitate the propagation of the electrical wave synchronizing the coordinated muscle contraction required for pumping. Gap junctions serve a similar role in spreading depolarization-triggered contraction in visceral smooth muscle, such as found in the gut, bile ducts, uterus, and blood vessels. Gap junctions are also found in liver cells, presumably to expedite the passage of lowmolecular-weight chemicals between cells. Are there waves of electrical activity in liver cells? Why do electrical synapses exist? One property of gap junctions is their speed and reliability. There is no synaptic delay, and there is no need for complex metabolic machinery to manufacture a neurotransmitter and package it in vesicles ready for release by depolarization of the bouton. There is also no need for postsynaptic receptors and an enzyme to break down the neurotransmitter, and more metabolic machinery to recycle the transmitter. Chemical synapses are complex; gap junctions are simple. So why are there not more gap junctions in the nervous system? Gap junctions apparently require a large amount of presynaptic ion current to transmit large excitatory (or inhibitory) signals to a postsynaptic neuron. Thus, in the crayfish giant motor fiber, a very large presynaptic neuron supplies enough presynaptic depolarizing current through a rectifying electrical synapse to cause the giant motor fiber to fire in a 1:1 manner. A CNS motorneuron system that uses both electrical and chemical synapses is the Mauthner neuron system in fish. The electrical coupling is inhibitory (Eccles, 1964). Mauthner neurons are used to synchronize tail motoneurons, providing a strong swimming response. It is noteworthy that gap junctions are also seen in adjacent glial cells in the CNS. Perhaps they serve here to transmit low-molecular-weight regulatory molecules from cell to cell, or equalize extracellular cation concentrations around neurons.
1.3.3
EPSPS AND IPSPS
As has been seen, when an excitatory chemical synapse is stimulated by a presynaptic spike, it releases the NT contents of a number of vesicles into the cleft. The NT molecules diffuse rapidly to receptor sites on large protein molecules embedded in the subsynaptic membrane, where they combine and initiate either
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directly or by a second messenger cascade reaction the opening (or closing) of specific ionic channels in the subsynaptic membrane. The channels remain open for a short time then close again as enzymes in the cleft rapidly destroy the free and bound NT molecules. In the simplest case, the NT is ACh. Two ACh molecules bind with a nicotinic receptor embedded in the SSM, causing a center channel in the receptor protein complex (containing five protein subunits arranged around a common center) to open. The open channel permits Na+ to flow inward and K+ to flow outward at the same time. The net result is that the transmembrane potential of the SSM depolarizes (i.e., goes positive) as shown in Figure 1.3-4, creating an excitatory postsynaptic potential (epsp). Epsps are characterized by the delay between arrival of the presynaptic spike and the initiation of the epsp, the time to the peak, the peak height in millivolts above the resting potential, and the decay time constant. Since many excitatory synapses are made on the thin branching dendrites of the postsynaptic neuron, it is generally not possible to record a single epsp at the synapse because the dendrite is too thin to accommodate a glass micropipette electrode. Recording of the epsp is generally done from the soma of the PSN, or thick, basal portions of dendrites. See the diagram of epsp recording procedure in Figure 1.3-5. Note that as the presynaptic stimulus intensity increases, it recruits more and more axons to fire, increasing the height of the soma epsp as more and more synapses become active. At some high stimulus voltage, all presynaptic axons fire, a maximum epsp height is reached. There is a slight extra delay and attenuation of the epsp as it propagates from the site of the synapse to the recording site. Table 1.3-2, adapted from Eccles (1964), lists comparative figures for epsps. The peak voltage for a mammalian motoneuron epsp is about 5 to 7 mV above the normal resting potential for the postsynaptic neuron.
TABLE 1.3-2 Summary Properties of epsps from Chemical Synapses
Synaptic Type
Synaptic Delay (ms)
Mammalian motoneuron Frog motoneuron Toad motoneuron Frog sympathetic ganglion Squid giant synapse Lobster cardiac ganglion Aplysia, large ganglion cell
0.3 1.0–1.2 1.5 — 1–2 — —
Source: Adapted from Eccles, 1964.
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Time to Peak 1.2 (1.0–1.5) 2.5–3.0 2.1 2.5–3 1–1.5 10 20
Decay Time Constant
PSM Time Constant (ms)
4.9 (3.5–6.1) 6.5–10 8.0 12.1 1–1.5 20 120
3.2 (2.3–3.6) — 4.5 9.7 1–1.5 12 45
Spike Initiation Threshold (mV) 10 (5–18) 10 10 25 10–15 — —
FIGURE 1.3-4 Representative intracellular recordings of epsps from the soma of a motoneuron in the lumbar region of cat spinal cord. The medial gastrocnemius nerve was stimulated with progressively larger pulses, recruiting more and more fibers that excite the motoneuron being recorded from. (Based on data from Eccles, 1964.)
epsps, wherever on the postsynaptic neuron they are generated, sum in space (over dendrites and soma) and in time (by superposition) to generate a net generator potential (Vg) at the SGL. If the net Vg(t) at the SGL exceeds the spike initiation threshold, the postsynaptic neuron generates an action potential. The epsp “ballistic potential” is often modeled by a simple, two real-pole lowpass filter in which the input is a unit impulse coincident with the arrival of the presynaptic action potential, and the output is the epsp, vm(t), which is added to the dc resting potential of the postsynaptic neuron. In terms of Laplace transforms, this potential is vm(s) = K/[(s + a)(s + b)]. When a = b, the transfer function is called an α-function by some computational neurobiologists. Inhibition of spinal motoneurons is generally accomplished by second-messenger-type synapses on the soma that admit chloride ions. As the result of the transient
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FIGURE 1.3-5 Schematic of stimulation and epsp recording procedure from a spinal motoneuron. Multiple input fibers converge on the dendritic tree with excitatory synapses.
increase in influx of Cl–, an inhibitory postsynaptic potential (ipsp) is generated. In cat spinal motoneurons, the peak ipsp is about –2.3 to –3.0 mV below the normal resting potential of –65 mV. That chloride ions are involved in this ipsp can be shown by voltage-clamping the soma of the postsynaptic neuron to potentials below –70 mV, the Nernst potential for chloride ions for the motoneuron. When Vm is held at –100 mV (strong hyperpolarization), stimulation of the inhibitory synapse produces a large, positive-going ipsp, where the chloride ions flow out of the postsynaptic membrane. Eccles shows the normal motoneron ipsp to have a delay of about 1.2 ms, a peak at ~2.5 ms, and a decay time constant of about 1.8 ms. It is true that the negative ipsp transients subtract from the summed epsps, but the inhibition is more complex than simple subtractive superposition. The open chloride channels during inhibitory synaptic activation raise membrane conductance in their neighborhood, shunting the excitatory currents, attenuating epsps, as well as subtracting from them. By locating inhibitory synapses on the soma near the SGL, inhibitory inputs gain more “leverage” than if they were out on the dendrites of the motoneuron. Although it is tempting to treat neurons as “all-or-none,” discrete devices, it is clear from examining the synaptic control of motoneuron firing that the motoneuron acts as a complex, nonlinear analog threshold device. The decision to fire is simply determined by the generator potential, but the generator potential is the result of complex spatiotemporal summation of excitatory and inhibitory inputs over the dendrites and cell body.
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1.3.4
QUANTAL RELEASE
OF
NEUROTRANSMITTER
In 1952, Fatt and Katz, using extra- and intracellular recording from the subsynaptic membrane of resting muscle, observed what they called spontaneous miniature endplate potentials (SMEPPs) (Katz, 1966; Eccles, 1964). These SMEPPs were transient depolarizations of the membrane under the motor end plate having the same time course as a muscle twitch potential caused by the arrival of a single motoneuron spike at the motor end plate. The SMEPPs occurred randomly at a mean rate of about one per second. The most frequently observed amplitude was 0.4 mV peak. Long-term recording of SMEPPs from resting muscle subsynaptic membrane showed that the SMEPPs were quantized, i.e., most were about 0.4 mV, and with decreasing frequency of occurrence, some were 0.8. 1.2, 1.6, 2.0, and 2.4 mV peak. In each SMEPP height class, the amplitudes were distributed around the mean with an approximate normal distribution, with the standard deviations being proportional to the means. As neurophysiologists began to inquire about the source of SMEPPs, it became evident from the electron micrographs of motor end plates and chemical synapses, that chemical neurotransmission was due to the action potential–stimulated release of neurotransmitter from vesicles found in close proximity to the inside of the presynaptic membrane. Since all of the vesicles are of a fairly uniform diameter, and they contain neurotransmitter molecules, an arriving spike causes the release of the contents of about 150 vesicles, producing a normal (full-sized) motor end plate potential of about +70 mV (Kandel et al., 1991). It was estimated that the spontaneous release of the contents of one vesicle (one quantum) into the subsynaptic cleft would cause the basic amplitude (0.4 mV) SMEPP. Based on a vesicle diameter of 50 nm and an internal ACh concentration of 0.15 M (the isotonic concentration), about 6000 NT molecules are in a typical motor end plate vesicle. Of much lower probability is the (nearly) simultaneous, spontaneous release of the contents of two vesicles (two quanta), producing a SMEPP of 0.8 mV peak. The mean rate of SMEPPs was shown to be proportional to a low level of dc depolarization imposed on the presynaptic terminal by either electrical or external ion substitution methods. For example, when ammonium ions are substituted for extracellular Na+, the resultant presynaptic membrane depolarization causes a marked increase in the rate of SMEPPs (Eccles, 1964). Needless to say, spontaneous postsynaptic potentials are present in neuron–neuron chemical synapses as well as in muscle motor end plates (Kandel et al., 1991). Quantal noise on motoneron epsps was observed by Eccles (1964). Such noise is potentially more disturbing, because spike excitation of neural–neural synapses normally causes the release of far fewer quanta of neurotransmitter (1 to 10) than at a motor end plate (~150). So it would appear that convergent resting chemical synapses have the potential to inject large amounts of noise into the membrane potential of a postsynaptic neuron. However, the convergence of many synapses from the same presynaptic neuron on the dendrites of the target postsynaptic neuron offers a clue to how this noise is mitigated. What occurs is spatiotemporal averaging; conduction down the passive core-conductor of a dendrite low-pass filters and smoothes the spontaneous essps (SEPSPs) from a given
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synapse. Presumably, the SEPSPs from the other N – 1 synapses are all generated with the same statistics, but are uncorrelated. They, too are low-pass-filtered by the dendrites, and sum together on the soma to give a slightly raised dc resting potential without appreciable noise by what can be viewed as a spatial ensemble averaging process. In neuro-sensory systems, the spatiotemporal averaging process inherent in multisynaptic transmission can produce an improved signal-to-noise ratio, and permits detection of lower threshold stimuli than would be possible with fewer synapses (Northrop, 1975).
1.3.5
DISCUSSION
The synaptic connections between neurons and between neurons and muscles permit the high-speed transmission of information in an animal’s body. Truly fast interneuronal communication is accomplished by electrical synapses. Here the action currents associated with action potentials pass conductively from the pre- to postsynaptic neuron membranes, depolarizing the postsynaptic neuron Vm toward its firing threshold. There are few factors that can alter the electrical synapse coupling. Chemical synapses, on the other hand, offer a variety of interesting plastic behaviors. All chemical synapses release neurotransmitter molecules when stimulated by the arrival of a presynaptic action potentials. As has been seen, the neurotransmitter quickly diffuses across the synaptic cleft to receptor molecules on proteins protruding through the postsynaptic membrane. There they combine with receptor sites and initiate, depending on the transmitter/receptor variety, transient conductance increases for specific ions. Once in the cleft, the neurotransmitter is quickly destroyed (in milliseconds) by enzymatic hydrolysis or some other process so that each arriving spike will produce a new result. If the transmitter is not destroyed, the postsynaptic conductance increases will not die out, and the nervous system ceases to function normally. A class of poisons known as cholinesterase inhibitors have this effect on the common neurotransmitter, ACh. Depending on the ions, the voltage across the ssm can depolarize, producing an epsp, or hyperpolarize, giving an ipsp. In some cases, an inhibitory synapse can cause a general conductance increase in the SSM receptors that effectively clamps the subsynaptic Vm to a level below the firing threshold, effectively inhibiting the postsynaptic neuron from firing. If the clamping potential equals the resting potential, one does not see a postsynaptic potential. A question often asked is why does a presynaptic neuron need so many synapses to drive the dendrites of the postsynaptic neuron. There are several answers to this question. One is redundancy; if synapses are damaged by disease or injury, there is ample backup. Another is to reduce the effect of synaptic noise by spatiotemporal averaging over the dendritic field. (Synaptic noise arises from the random release of neurotransmitter from vesicles.) Still another is to provide a large area of dendritic membrane where interaction with other excitatory and inhibitory synapses can take place. Since synapses release their transmitter in an all-or-none manner at the arrival of a presynaptic spike, some factor that deletes synapses could serve to weaken the functional coupling between two neurons. A factor that stimulates the regrowth of
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synapses thus could strengthen interneuronal coupling. Such modulation of the strength of coupling could be associated with reflex conditioning, or “learning.”
1.4
MODELS FOR THE NERVE ACTION POTENTIAL
Probably no mathematical model in physiology has had more impact on the field of computational neurobiology than the Hodgkin–Huxley (HH) (1952) model for nerve impulse generation. The HH model does not describe the propagation of a nerve impulse down a myelinated or unmyelinated axon, but it can easily be modified to do so. The HH model simulates the what-was-then-known behavior of the two major ionic species, Na+ and K+, and describes how they pass through a “unit” patch of active membrane in terms of a set of nonlinear ODEs controlling their specific ionic conductances. The main HH ODE is basically a node equation, which by Kirchoff’s current law sums the radial currents passing through the membrane. There is an external (input) current, a capacitive current, and three ionic currents. One is a voltage-dependent Na+ current, the second is a voltage-dependent K+ current, and the third is a linear “leakage” current. The sodium current depolarizes the membrane patch; the potassium current hyperpolarizes it. Both ion currents are nonlinear functions of the transmembrane voltage; their dynamics are determined by three first-order ODEs with nonlinear, voltage-dependent coefficients. To adapt the HH model to describe spike propagation on an unmyelinated axon, one need only link HH “modules” with linear resistances representing the per-unit length resistance of the axoplasm and the external axial spreading resistance (Plonsey, 1969). Such a model is shown in Figure 1.4-1. To emulate saltatory spike propagation, one can modify the HH model to include the intervening electrotonic sections of myelin-wrapped axon. These are shown in Figure 1.4-2.
FIGURE 1.4-1 Lumped-parameter model of a nerve axon. HH “patches” are linked by longitudinal (axial) resistances.
The following sections describe the mathematical details of the HH model.
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FIGURE 1.4-2 Lumped-parameter model of a myelinated axon. The HH patches are at the nodes of Ranvier; the length of axon under the myelin bead is represented as a series of linked, passive electrotonic modules with constant parameters.
1.4.1
THE 1952 HODGKIN–HUXLEY MODEL POTENTIAL GENERATION
FOR
ACTION
As seen in the previous section, one of the great leaps forward in neurophysiology was the basic understanding of the molecular mechanisms underlying the generation of nerve action potentials, or “spikes.” Most of the experiments leading to the understanding of nerve spike generation and propagation were done on the giant, unmyelinated nerve axons of the squid, Loligo sp. Squid giant axons were attractive because their large size (~0.5 mm diameter) made it easier to replace their (internal) axoplasm with an artificial ionic media of any desired composition. Of course, the composition of their external bathing solution could be made up in any desired manner, as well. The cell membrane of the squid axon was found to have a relatively high distributed capacitance of ~1 µF/cm2. The membrane is studded with protein molecules that penetrate it between the inside and outside of the axon; these proteins offer selective passage for major ionic species to enter or exit the axon interior, depending on the potential energy gradient across the membrane for a given ionic species. For example, the inside of a squid axon has a resting potential from 60 to 70 mV, inside negative (Bullock and Horridge, 1965, Ch. 3). The resting or equilibrium concentration of sodium ions is higher outside than inside the axon. Thus, both an electrical field and a concentration gradient provide thermodynamic potential energy, which tries to force Na+ inward through the membrane. The net inward potential energy in electron volts (eV) for Na+ ions is the sum of the electrical potential, Vm, and the Nernst potential for sodium ions. This potential energy is given by the well-known relation (Katz, 1966):
[ [
⎛ Na + E Na = Vm – ( RT F) ln ⎜ ⎜ Na + ⎝
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]⎞ ] ⎟⎟⎠ o i
eV
1.4-1
where Vm is the membrane resting potential in volts, –0.070 V for squid at 20°C (note Vm depends on temperature); R is the gas constant, 8.314 J/mol K; F is the Faraday constant, 9.65 × 104 C/mol; T is the Kelvin temperature; [Na+]o and [Na+]i are the concentrations of sodium ions outside (460 mM) and inside (46 mM) the membrane, respectively, in mol/l for squid axon. ENa < 0 means the net force on Na+ is inward. That is, a sodium ion outside the cell membrane has a total potential energy of ENa electronvolts acting to force it inward. For squid axon at 293 K (20°C), ENa can be calculated: ENa = – 0.070 – (8.314 × 293/9.65 × 104) ln (460/46) = –0.128 eV
1.4-2
Similarly, one can calculate the potential energy, EK, acting to force a K+ ion outward through the resting membrane: EK = – 0.070 – (8.314 × 293/9.65 × 104) ln(10/410) = + 0.024 eV
1.4-3
Na+ ions generally enter the axon through special voltage-gated Na+ channels. With the membrane unexcited and in its steady-state condition, there is a very low, random leakage of Na+ ions inward through the membrane. Potassium ions also leak out because the net potassium potential energy is dominated by the concentration gradient (the inside having a much higher potassium concentration than the outside, i.e., [K+]i Ⰷ [K+]o). Other small ions leak as well, being driven through the membrane by the potential energy difference between the membrane resting potential and the ion Nernst potential. At steady-state equilibrium, a condition known as electroneutrality exists in unit volumes inside and immediately outside the axon. That is, in each volume there are equal numbers of positive and negatively charged ions and molecules. Inside the axon, a significant percentage of negative charges are bound to large protein molecules that cannot pass through the membrane because of their sizes. Not specifically pertinent to the short-term solution to the HH model is the fact that nerve membrane (on soma, axon, and dendrites), and those of nearly all other types of cells, contains molecular “pumps” driven from the energy in ATP molecules which can eject Na+ from the interior of the cell against the potential energy barrier, ENa, often as an exchange operation with K+ being pumped in. Ionic pumps are ubiquitous in nature, and are responsible for the maintenance of the steady-state intracellular resting potentials of nerves, muscle cells, and all other cells. To try to describe and summarize the electrical events associated with nerve membrane, a simple, parallel, electrical circuit was developed (Hodgkin and Huxley, 1952; Katz, 1966) that includes specific ionic conductances for Na+, K+, and “other ions” (leakage) for a 1-cm2 area of axon membrane. Figure 1.4-3 illustrates the circuit. VNa and VK are the Nernst potentials for sodium and potassium ions, respectively, and VL is the equivalent Nernst potential for all leakage ions, including chloride. A specific ionic conductance is defined by the specific ionic current density divided by the difference between the actual transmembrane potential and the Nernst potential. Thus,
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FIGURE 1.4-3 Details of the HH membrane patch. Batteries represent the Nernst potentials for the Na+, K+, and “leakage” ions. Leakage includes Cl– and all other non-voltage-sensitive ions. Currents are given as current densities (units, A/cm2). Also shown are the metabolically driven ion pumps as current sources. The pumps slowly restore steady-state internal ion concentrations that maintain the Nernst potentials.
gK ≡ JK/(Vm – VK) S/cm2
1.4-4A
gNa ≡ JNa/(Vm – VNa) S/cm2
1.4-4B
gL ≡ JL/(Vm – VL) S/cm2
1.4-4C
For example, in resting squid axon, gNa = JNa/(–0.070 – 0.058). JNa is negative inward by definition, so gNa is positive as it should be. The mathematical model devised by Hodgkin and Huxley (1952) to describe the generation of the nerve action potential begins by writing a node equation for the simplified equivalent circuit of Figure 1.4-3:
J in = C m dv dt + J K + J Na + J L
1.4-5A
↓
C m v˙ = J in – J K – J Na – J L
1.4-5B
↓
v˙ = ( J in – J K – J Na – J L ) C m
1.4-5C
By definition, v ≡ (Vmr – Vm ). v is in millivolts. If v < 0, Vm is depolarizing (going positive from the resting Vmr = –70 mV). The leakage current density is assumed to obey Ohm’s law:
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JL = gLo (v – VL) A/cm2
1.4-6
gLo is taken as 0.3 mS/cm2, VL ≡ –10.613 mV. From previous studies with the squid axon membrane under electronic voltage clamp conditions (where Vm is forced to assume set values), where the potassium current was measured with the sodium channels blocked, it was observed that gK = JK /Vm = gK (Vm, t). From chemical kinetics considerations, Hodgkin and Huxley assumed that four “particles” must simultaneously occupy specific sites on the potassium gate protein in order to open it. Thus, they wrote: JK = gKo n4 (v – VK) µA/cm2
1.4-7
where n is the K+ activation parameter ; it is taken as the probability of a K gate opening particle being at the active site. gKo is taken to be 36 mS/cm2, VK = 12 mV. n is given by the ODE:
n˙ = – n(α n + β n ) + α n
1.4-8
αn and βn are exponential functions of membrane potential: αn = 0.01(v + 10)/[exp(0.1v + 1) – 1] βn = 0.125 exp(v/80)
1.4-9 1.4-10
The sodium current is activated once Vm reaches a depolarization threshold voltage. The sodium current or conductance deactivates spontaneously, once its channels have been open. Hodgkin and Huxley observed that sodium channel activation had kinetics that suggested three “particles” (not necessarily the particles for potassium channel activation) were required. Deactivation of the sodium channels suggested monomolecular kinetics, where a single event of probability (1 – h) caused inactivation. Thus, the probability of Na+ channels being open was m3h, and the sodium current density was JNa = gNao m3h(v – VNa) µA/cm2
1.4-11
gNao = 120 mS/cm2, VNa = – 115 mV. The ODEs for sodium activation and deactivation are
˙ = – m (α m + β m ) + α m m
1.4-12
h˙ = – h(α h + β h ) + α h
1.4-13
The four voltage-dependent parameters are αm = 0.1(v + 25)/[exp(0.1v + 2.5) – 1]
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1.4-14
βm = 4 exp(v/18)
1.4-15
αh = 0.07 exp(v/20)
1.4-16
βh = 1/[exp(0.1v + 3) + 1]
1.4-17
A Simnon™ program, Hodhux, to compute Vm = – (v + 70) mV, and the various current densities follows. Note that the current densities are in µA/cm2, voltages are in mV, conductances in mS/cm2, and the capacitance is in mF/cm2. The strange treatment of units and voltage signs follows from the original HH (1952) paper. continuous system HODHUX “ 6/06/99 “ Run w/ Euler integration w/ dt = 0.00001. STATE v m n h “ v in mV. v is depolarization if < 0. DER dv dm dn dh “ v = Vmo – Vm. Vm is actual transmembrane V. “ Vmo = resting potential = – 70 mV. TIME t “ t in ms. “ “ HH membrane patch ODE. (1 cm^2) dv = Jin/Cm – Jk/Cm – Jna/Cm – Jl/Cm “ “ Ionic current densities. Jk = gko*(n^4)*(v – Vk) Jna = gnao*(m^3)*h*(v – Vna) Jl = glo*(v – Vl) Jc = Cm*dv Jnet = Jc + Jk + Jna + Jl “ Microamps/cm^2 “ “ Ionic conductances: gk = gko*(n^4) “ Potassium conductance gna = gnao*(m^3)*h “ Sodium conductance gl = glo “ Leakage conductance gnet = gk + gna + gl “ Net membrane conductance negv = –v “ dn = – n*(an + bn) + an “ K+ activation parameter. dm = – m*(am + bm) + am “ Na+ activation parameter. dh = – h*(ah + bh) + ah “ Na+ inactivation parameter “ “ VOLTAGE-DEPENDENT PARAMETER FUNCTIONS: an = 0.01*(v + 10)/(exp(0.1*v + 1) – 1) bn = 0.125*exp(v/80) am = 0.1*(v + 25)/(exp(0.1*v + 2.5) – 1) bm = 4*exp(v/18) ah = 0.07*exp(v/20) bh = 1/(exp(0.1*v + 3) + 1) “ Vm = –(v + 70) Jksc = Jk/10 “ Scaled current densities Jnasc = Jna/10 Jcsc = 10*Jc “ Scaled capacitor current density. Vmsc = Vm/70 “ Scaled membrane voltage. “
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“ CONSTANTS: zero:0 Vk:12 “ mV Vna:-115 “ mV VL:-10.613 “ mV “ glo:0.3 “ milliS/cm^2. gko:36 “ “ gnao:120 “ “ Cm:.001 “ milliF/cm^2 = 1 microF/cm^2. f:0.10 “ Hz pi:3.14159 “ “ INPUTS: “Jin1 = IF t > to THEN K*(t - to) ELSE 0 “ Delayed current ramp input. “Jin2 = IF t > t2 THEN -K*(t - t2) ELSE 0 “ Jin in microamps/cm^2. t2:10 “ End of ramp to:1 “ Jin < 0 depolarizes membrane. K:-0.3 “ Jin inward is + charges inward Jin1 = IF t > to THEN Jino ELSE 0“ Delayed current pulse input. Jin2 = IF t > (to + delt) THEN –Jino ELSE 0 delt:5 “ Duration of current pulse, ms. Jino:–2 “ Microamps. Jinac:1 “ Jin1 = IF t > to THEN Jino ELSE 0“ Delayed current step input. “ Jin2 = Jinac*sin(2*pi*f*t) “ Sinusoidal current input. Jin = Jin1 + Jin2 “ “ INITIAL CONDS. (Also final values for v = 0.) m:0.052932 n:0.31768 h:0.59612 v:0 “ END
The following section examines how the nonlinear ODE system of Hodgkin and Huxley responds to depolarizing and hyperpolarizing input current densities. Of particular interest is the sensitivity of the system to the rate of change of Jin. Also, the system acts as a nonlinear current-to-frequency converter. Such steady-state oscillatory behavior is seen in nature in certain pacemaker neurons.
1.4.2
PROPERTIES
OF THE
HODGKIN–HUXLEY MODEL
This section reviews some of the results obtained using the Simnon program HODHUX.t given in the previous section. Using Hodgkin and Huxley’s original sign convention, currents (positive ion flow) entering the axon from outside, such as JNa, are positive. JK leaving the axon lumen is negative. Vm is shown as it would be measured; the resting Vmr ≡ –70 mV. Jin < 0 depolarizes the membrane, i.e., drives Vm positive. (Jin is a – ion current density.)
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Figure 1.4-4 illustrates an action potential produced when Jin = –2 µA for 5 ms. Also plotted are the scaled membrane capacitance current density, potassium ion current density and the sodium ion current density. Scaling details are in the figure caption. Figure 1.4-5 plots the auxiliary parameters n, m, and h when the same action potential (AP) is generated. The AP is scaled by 1/70 in order that the same scale can be used as for n, m, and h. Note that the sodium activation parameter, m, falls rapidly after reaching its peak near unity. Finally, in Figure 1.4-6, the specific ionic conductances, gK, gNa, and gnet are plotted with (Vm(t) + 70) mV. Note that gK dominates the recovery phase of Vm following the spike. 100
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FIGURE 1.4-4 Results of Simnon simulation of the HH model run with Euler integration; δt = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (2) Vm mV; (3) Jin = –2 µA/cm2 from t = 1 to 6 ms, else 0; (4) JCsc = 10 JC (scaled capacitive current density, µA/cm2); (5) JKsc = JK/10 (scaled potassium ion current density, µA/cm2); (6) JNasc = JNa /10 (scaled sodium ion current density, µA/cm2).
A little-appreciated property of the HH model is its propensity to fire on rebound from a prolonged hyperpolarization of Vm. Hodgkin and Huxley (1952) called this phenomenon anode break excitation. Figure 1.4-7 illustrates what happens when Jin is positive, forcing Vm more negative, hyperpolarizing the membrane. Rebound firing occurs when Jin → 0; Vm overshoots Vmr enough to cause a spike to occur for Jin = 2, 3, and 5 µA. There is not enough rebound in Vm for Jin = 1 or 1.5 µA to cause firing, however. Another property of the HH model is its nonlinear behavior as a current-tofrequency converter. In this case, a prolonged, negative Jin is applied. For the model parameters given, and Cm = 0.003 mF, it is found that for Jin ð –7 µA, the model
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FIGURE 1.4-5 Results of Simnon simulation of the HH model run with Euler integration; δt = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (1) Vm(t)/70 mV; (2) n(t); (3) m(t); (4) h(t); (5) zero.
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FIGURE 1.4-6 Results of Simnon simulation of the HH model run with Euler integration; δt = 0.00001, Cm = 0.01. Horizontal axis, time in milliseconds. Traces: (1) Jin µA/cm2; (2) [Vm(t) + 70] mV; (3) gK (t, Vm) mS/cm2; (4)gNa(t, Vm) mS/cm2; (5) gnet = gK + gNa + gL mS/cm2.
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FIGURE 1.4-7 Simulation of anode break excitation. HH model run with Euler integration; δt = 0.00001, Cm = 0.001. Horizontal axis, time in milliseconds. Top trace: Jin (hyperpolarizing current densities of 1, 1.5, 2, 3, 5 µA/cm2; no spike for Jin = 1 and 1.5).
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FIGURE 1.4-8 Simulation of current-to-frequency conversion in the HH model. HH model run with Euler integration; δt = 0.00001, Cm = 0.003. Horizontal axis, time in milliseconds. Jin = –7 µA/cm2.
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fires repetitively at a constant frequency; the frequency increases as Jin becomes more negative. Figures 1.4-8 through 1.4-10 illustrate this behavior. Note particularly that the peak-to-peak height of the periodic spikes decreases as the frequency increases. At Jin = –140 µA, the membrane oscillation was centered around –50 mV, and was nearly sinusoidal, having a peak-to-peak amplitude of about 12.5 mV. Hardly nerve spikes. The steady-state frequency of the HH model “oscillator” vs. Jin is plotted in Figure 1.4-11 on log–log coordinates. An approximate model that fits the plot is f = 57 + 6 冨 Jin冨0.64 pps. Thus, the HH model is not a linear current-to-voltage converter, but behaves in a power law manner over the range, –7 Š Jin Š –140 µA.
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FIGURE 1.4-9 Simulation of current-to-frequency conversion in the HH model. HH model run with Euler integration; δt = 0.00001, Cm = 0.003. Horizontal axis, time in milliseconds. Jin = –50 µA/cm2.
1.4.3
EXTENDED HODGKIN–HUXLEY MODELS
This section examines two embellishments on the basic HH model: (1) A computer model of a voltage clamp experiment where Vm is electronically forced to follow a desired input, Vs. (2) Models for other, recently described, gated ion channels in nerve membrane. Evidence from early voltage clamp experiments on squid axon on how the bulk current densities, JNa and JK, behave with Vm and dVm/dt provided the basis for Hodgkin and Huxley’s (1952) dynamic model for the electrical and ionic behavior of a patch of active nerve membrane. Using Simnon, it is relatively easy to simulate the control electronics of the voltage clamp apparatus along with the HH model for nerve membrane. A circuit for a simple voltage clamp is shown in Figure 1.4-12.
© 2001 by CRC Press LLC
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FIGURE 1.4-10 Simulation of current-to-frequency conversion in the HH model. HH model run with Euler integration; δt = 0.00001, Cm = 0.003. Horizontal axis, time in milliseconds. Jin = –105 µA/cm2.
FIGURE 1.4-11 Log–log plot of frequency vs. Jin for the HH model with Cm = 0.003 mF/cm2. See text for discussion.
A transconductance amplifier or voltage-controlled current source (VCCS) is used to force a known current, Iin, into the axon’s center. The membrane potential, Vm, is monitored by a glass micropipette microelectrode that penetrates the axon membrane and is connected to a unity-gain, buffer amplifier. The command signal, Vs, and Vm are compared at the summing junction of a fast, high-gain op amp with gain, –Kv. The op amp output, V2, drives an (ideal) VCCS with transconductance, Gc.
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FIGURE 1.4-12 Schematic of a voltage-clamp apparatus applied to a large (e.g., squid) axon tied at both ends. An Ag冨AgCl electrode is shown in the interior of the axon. Vm(t) is measured with a glass micropipette electrode and a capacitance-neutralized electrometer preamp.
FIGURE 1.4-13 Systems block diagram of the voltage-clamp system of Figure 1.4-12. Gm represents the parallel behavior of gK, gNa, and gL.
Referring to the systems block diagram in Figure 1.4-13 based on the circuit of Figure 1.4-12, it is easy to write the overall transfer function for Vm:
[
– (1 2)K v G c G m + (1 2)K v G c Vm (s) = Vs sC m G m + (1 2)K v G c + 1
[
]
]
1.4-18
The (1/2) gain factor comes from letting R1 = R2. If (1/2)Kv Gc Ⰷ Gm(max), then the dc gain of the closed-loop system → –1, and its time constant τc → Cm/[(1/2)KvGc] sec. Gm is the sum of [gL + gNa(Vm) + gK(Vm)] and varies with Vm and time. The closedloop system τcl is much smaller than [Cm/Gm]min. The voltage clamp controller computes: Jin = KvGc[VsG1/(G1 + G2) + VmG2/(G1 + G2)] © 2001 by CRC Press LLC
1.4-19
It is easy to include the voltage clamp in the HH patch model of Section 1.4.1. Note that Vm = –(v + 70) mV. Assume that the transconductance amplifier (VCVS) is ideal. That is, it has a zero Norton output conductance, and infinite bandwith (at least compared with the neuron membrane patch). The op amp is ideal except for finite gain, Kv, and the microelectrode is perfectly neutralized so Vm is measured exactly. The short Simnon program HHVCLAMP.t is listed below. continuous system HHVCLAMP “ 6/04/99 “ Run w/ Euler integration w/ dt = 0.0001. STATE v n m h “ v in mV. v is depolarization if < 0. DER dv dn dm dh “ v = Vmo - Vm. Vm is actual transmembrane V. “ Vmo = resting potential = - 70 mV. TIME t “ t in ms. “ “ HH membrane patch ODE. dv = Jin/Cm - Jl/Cm - Jk/Cm - Jna/Cm “ “ Ionic current densities. Jk = gko*(n^4)*(v - Vk) Jna = gnao*(m^3)*h*(v - Vna) Jl = glo*(v - Vl) Jc = Cm*dv Jnet = Jc + Jl + Jk “+ Jna “ Microamps/cm^2 “ dn = - n*(an + bn) + an “ K+ activation parameter ODE. dm = - m*(am + bm) + am “ Na+ activation parameter ODE. dh = - h*(ah + bh) + ah “ Na+ inactivation para-meter ODE. “ “ VOLTAGE-DEPENDENT PARAMETER FUNCTIONS: an = 0.01*(v + 10)/(exp(0.1*v + 1) - 1) bn = 0.125*exp(v/80) am = 0.1*(v + 25)/(exp(0.1*v + 2.5) - 1) bm = 4*exp(v/18) ah = 0.07*exp(v/20) bh = 1/(exp(0.1*v + 3) + 1) “ Vm = -(v + 70) Jksc = Jk/10 “ Scaled current densities Jnasc = Jna/10 Jcsc = 10*Jc “ Scaled capacitor current density. Vmsc = Vm/70 “ Scaled membrane voltage. “ “ VOLTAGE CLAMP CONTROL ALGORITHM: Jin = Kc*(Vs - v - 70) “ “ CONSTANTS: zero:0 Kc:1.E4 “ Controller gain, Kc = (1/2)Kv*Gc Vk:12 “ mV Vna:-115 “ mV VL:-10.613 “ mV glo:0.3 “ mS/cm^2.
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gko:36 “ “ gnao:120 “ “ Cm:1 “ mF/cm^2. “ “ INPUTS: Jin < 0 depolarizes membrane. Jin inward is + charges inward “Vs1 = IF t > to THEN Vso ELSE 0 “ Delayed voltage pulse input. “Vs2 = IF t > (to + delt) THEN -Vso ELSE 0 delt:14 “ Duration of voltage pulse, ms. to:1 “ ms. Vso:-20 “ mV. “Vs = Vs1 + Vs2 + 70 “ -Vso sets Vm Vs = 70 + ramp ramp = IF t > to THEN Kr*(t - to) ELSE 0 Kr:-3 “ “ INITIAL CONDS. (Also final values for v = 0.) m:0.052932 n:0.31768 h:0.59612 v:0 “ END _
Figure. 1.4-14 illustrates the behavior of the scaled JK and JNa as Vm is ramped up (depolarized) linearly from the –70 mV resting potential. Note that the initiation ˙ as of rise in JNa does not have a crisp threshold and that the threshold depends on V m well as Vm. Also note that the specific ion channels responsible for JNa deactivate ˙ . The response of J and J to step deporegardless of the depolarized Vm and V K Na m larizations in Vm is shown in Figure 1.4-15. The larger the depolarization, the larger JK, and the faster JNa responds, its peak coming earlier and decaying faster as Vm goes more positive. For Vm > 0, the peak in JNa actually decreases, and the steadystate JNa gets smaller. (Most full-featured, specialized neural modeling simulation programs, such as GENESIS, allow simulation of voltage clamping.) Basic research on the molecular basis for neurophysiological phenomena has made enormous strides since the formulation of the dynamic model for nerve spike generation by Hodgkin and Huxley in 1952. Using the patch-clamp technique, which uses glass micropipette electrodes to isolate single ion gate proteins, it has been possible to characterize the conductance dynamics of five different kinds of potassium channel in bullfrog sympathetic ganglion cells, two types of voltage-gated sodium channel, three types of calcium channel, and a voltage-gated chloride channel (Yamada, Koch et al., 1989). Instead of subjecting the whole outside of a nerve axon membrane to a common potential, the patch-clamp technique isolates a single ionic channel on the outside of the membrane with the tip of the microelectrode, which is then made part of an electronic feedback system that forces a known potential across the isolated channel. The very small, specific ionic current of that channel is measured under conditions of known trans-channel potential and external ionic
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100
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FIGURE 1.4-14 Simulation of the voltage-clamped HH model with Simnon. Vm(t) in mV caused to ramp up (depolarize). HH model run with Euler integration; δt = 0.00001, Cm = 1.0. Horizontal axis, time in milliseconds. Numbered traces are JNasc = JNa/10 µA/cm2; 1, 2, 3, 4 correspond to input voltage ramp slope Kr = –3, –6, –9, –12, respectively. Downwardcurving traces are the scaled potassium ion current densities, JKsc = JK/10 µA/cm2, for the corresponding Vm ramps. Note that the JNas deactivate, even though Vm keeps depolarizing.
100 43 2
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FIGURE 1.4-15 In this simulation, the voltage clamp input forces Vm to have a stepwise change to levels of –50, –30, –10, +10, +30 mV. These levels correspond to the scaled sodium current density (JNasc) traces of 1, 2, 3, 4, and 5, respectively. The unnumbered, downwardcurving traces are the scaled potassium current densities (outward flow) for the corresponding 5 Vm steps. (The steepest is for Vm = +30 mV.)
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composition (in the microelectrode). The patch-clamping technique provides a powerful tool for the electrical and pharmacological study of ion channel behavior. By using the same notation as Section 1.4.1 for the HH equations, the five bullfrog potassium channel models can be written (Yamada et al., 1989): 1. The transient outward IK:
l Kt = g Kto m Kt h Kt ( v – VK )
nA
1.4-20
˙ Kt = (m Kt∞ – m Kt ) τ mKt m
1.4-21
h˙ Kt = ( h Kt∞ – h Kt ) τ hKt
1.4-22
where gKto = 0.120 µS, VK = 25 ln([K+]o/[K+]i) mV, τmKt = 1.38 ms, v ≡ Vmr – Vm, Vmr ≡ –70 mV, mKt× = 1/{1 + exp[(v + 42)/13]}, hKt× = 1/{1 + exp[+(v + 110)/18]}, and τhKt = 50 ms if v < –80 mV, else 150 ms. 2. The noninactivating, muscarinic IK:
l Kn = g Kno m Kn ( v – VK )
nA
1.4-23
˙ Kn = (m Kn∞ – m Kn ) τ mKn m
1.4-24
where: gKno = 0.084 µS, τmKn = (1000/3.3)/{exp[(v + 35)/40] + exp[(v + 35)/20]}, and mKn× = 1/{1 + exp[(v + 35)/10]}. 3. Delayed, rectifying IK:
l Kdr = g Kdro m 2Kdr h Kdr ( v – VK )
nA
1.4-25
˙ Kdr = (m Kdr∞ – m Kdr ) τ mKdr m
1.4-26
h˙ Kdr = ( h Kdr∞ – h Kdr ) τ Kdr
1.4-27
where gKdro = 1.17 µS, τmKdr = 1/[αdr(v) + βdr(v)], hKdr× = 1/{1 + exp[+(v + 25)/4]}, mKdr× = αdr(v – 20)/[αdr(v – 20) + βdr(v – 20)], αdr(e) = –0.0047(e + 12)/{exp[–(e + 12)/12] – 1}, βdr(e) = exp[–(e + 147)/30], e = v, or (v – 20), and τKdr = 6.E3 ms if v < –25 mV, else 50 ms.
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4. Non-inactivating, Ca-dependent IK:
l KCad = g KCado m Cad ( v – VK )
nA
˙ Cad = (m Cad∞ – m Cad ) τ Cad m
1.4-28 1.4-29
where gKCado = 1.2 µS, τCad = 1/[f(v, Ca) + b(v)], mCad× = f(v, Ca)/[f(v, Ca) + b(V)], b(v) = exp(– v/24), and f(v, Ca) = 250[Ca++]i exp(v/24), [Ca++]i in mM. 5. Voltage-independent, Ca-dependent IK:
l Kvic = g Kvico m 2Kvic ( v – VK )
nA
˙ Kvic = (m Kvic∞ – m Kvic ) τ Kvic m
1.4-30 1.4-31
where gKvico = 0.054 µS, τKvic = 103/[f(Ca) + b] ms, mKvic× = f(Ca)/[f(Ca) + b], b = 2.5, and f(Ca) = 1.25 × 108 [Ca++]n2 (the subscipt n refers to [Ca++] just inside the axon membrane). There is one type of fast sodium channel in the bullfrog axon. Its dynamics are given by
l Na = g Nao m 2Na h Na ( v – VNa )
nA
1.4-32
˙ Na = (m Na∞ – m Na ) τ mNa (activation) m
1.4-33
h˙ Na = ( h Na∞ – h Na ) τ hNa (deactivation)
1.4-34
where gNao = 2 µS, τmNa = 2/(αm + βm) ms, mNa× = αm/(αm + βm), τhNa = 2/(αh + βh) ms, hNa× = αh/(αh + βh) ms, αm = 0.36(v + 33)/{1 – exp[–(v + 33)/3]}, βm = –0.4(v + 42)/{1 – exp[(v + 42)/20]}, αh = –0.1(v + 55)/{1 – exp[(v + 55)/6]}, and βh = 4.5/[1 + exp(–v/10)]. The bullfrog axon also has one type of fast, voltage-gated, calcium channel with dynamics given by:
l Ca = g Cao m Ca h Ca ( v – VCa )
nA
˙ Ca = (m Ca∞ – m Ca ) τ mCa (activation) m
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1.4-35 1.4-36
where gCao = 0.116 µS, [Ca++]o = 4 mM (fixed), hCa = 0.01/(0.01 + [Ca++]i) deactivation, mCa× = 1/{1 + exp[–(v – 3)/8]}, and τmCa = 7.8/{exp[(v + 6)/16] + exp[–(v + 6)/16]} ms. The leakage current is given by IL = gLo(v – VL)
nA
1.4-37
where gLo = 0.02 µS, VL = –10 mV, T = 295 K, Cm = 0.15 nF. The very small capacitance and the small conductances listed above are because the bullfrog ion currents were determined using a test membrane area considerably less than the standard 1-cm2 area used in the HH equations as originally formulated. The author estimated that the axon area used was between 50 and 225 × 10–6 cm2. Note that as Ca ions enter the membrane through their channels, [Ca++]i will rise. The local concentration of [Ca++]i just inside the membrane must be calculated using the diffusion equation and ICa. Finite-difference calcium diffusion equations can be found in Yamada et al. (1989), and are not given here. So why is such a detailed model for the bullfrog nerve axon voltage- and ionconcentration-dependent behavior needed? The extra detail allows one to examine and understand the effects of long-term changes in Vm by voltage clamp, and the effects of changing ion concentrations and replacing ions (e.g., Na+, K+, Ca++, Cl–) with nonpermeable equivalents. Channel blockers can also be emulated (e.g., tetrodotoxin to block Na+ channels) to isolate components of total membrane current under voltage clamp conditions. Apparently, nature is far more complex in its design of certain nerve axon membranes than originally described by Hodgkin and Huxley in 1952 for the squid. Still, their model has withstood the test of time and has provided the basis for subsequent dynamic models for spike generation in a variety of neurons.
1.4.4
DISCUSSION
This section has been devoted to a detailed description of the HH model for action potential generation in active nerve membrane. Most modern neural modeling software packages offer the option of a HH-type model for spike generation. Often, the user can modify the HH auxiliary equation structure to reflect simplifications, or to include other voltage-dependent ionic conductances (other than for K+ and N+). The HH model was stimulated using Simnon, the nonlinear ODE solver. The section has shown how the original HH model acts as a nonlinear, currentto-voltage converter, and how the HH equations can be simply modified to produce a voltage clamp in which the properties of the ionic conductances can be examined.
1.5
CHAPTER SUMMARY
This first chapter has reviewed the anatomy and functions of various different types of neurons. Next, the electrical properties of and models for dendrites having passive membrane were examined. The generation of the nerve action potential was next
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described, and the electrical properties of unmyelinated and myelinated nerve axons were shown. The different kinds of synaptic connections between neurons were described, as well as the postsynaptic potential changes resulting from chemical synapse action. Inhibitory synaptic action can be the result of summing hyperpolarizing ipsps or the clamping of membrane potential to a Nernst potential below the postsynaptic firing threshold voltage. Finally, the HH model for action potential generation was treated in detail, including the properties of the HH model as a current-to-frequency converter. The basis for the detailed electrical modeling all types of neurons was shown to be the distributed-parameter RC transmission line with various types of specific ionic, voltage-dependent shunting conductances in parallel with the transmenbrane capacitance. A complete model can be made by connecting many of the lumped-parameter circuits of various types together to emulate synapses, dendrites, the soma, and axons.
PROBLEMS 1.1. Assume that a section of squid axon membrane has a capacitance of 1 µF/cm2, a resting potential of Vm = –70 mV, and a dielectric constant of 8. Neglect the presence of transmembrane proteins. a. Calculate the pressure of electrostatic origin acting on the membrane. Assume a parallel-plate capacitor structure. Give your answer both in dyne/cm2 and in psi. b. Calculate the electric field within the membrane in V/cm (assume it is uniform). 1.2. A cylindrical dendrite is modeled by a cylinder of passive membrane with radius r = 0.5 µm, and length L = 200 µm. An excitatory synapse at the distal end can be considered to be an ideal voltage source, Vs(t). The proximal end joins a neuron cell body (soma) with a large radius, which can be considered a short-circuit across the membrane of the dendrite. The membrane has a net passive conductance of GM = 2 × 10–4 S/cm2. CM = 2 × 10–7 F/cm2, the axoplasm resistivity is ρi = 60 Ωcm. Let ro = 0 for simplicity. a. Find the cable parameters for the dendrite: cm F/cm, gm S/cm, ri Ω/cm. b. Find Zo(jω), the dendrite τ, its break frequency, fb = 1/2, and its dc space constant, λ, in µm. c. Find the steady-state short-circuit current at the soma, given Vs = 10 mV depolarization at the distal end of the dendrite model. 1.3. Every time a squid giant axon conducts an action potential, 3.5 pmol/cm2 of Na+ enter the axon, and 3.5 pM/cm2 exit the axon. a. Calculate the minimum metabolic energy in calories needed by the ion pump to restore the ionic balance over 100 cm of 1-mm-diameter axon, per action potential. Assume the resting transmembrane potential is Vm = 0.07 V, and the steady-state concentrations are [Na+]in = 0.05 M, [Na+]out = 0.460 M, [K+]in = 0.40 M, [K+]out = 0.01 M.
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b. What is the steady-state energy, in calories, stored in the electric field over the 100 cm length of 1-mm-diameter axon? Assume CM = 1 µF/cm2. 1.4. Consider the lumped-parameter circuit model for 1 cm2 of nerve axon membrane shown in Figure P1.4.
FIGURE P1.4
a. When the action potential reaches its peak (positive) value, it is known that gL = 0.3 mS/cm2, gK = 13 mS/cm2, and gNa = 32 mS/cm2. Calculate the membrane time constant for this condition. b. Assume CM = 0. Calculate Vm at the peak of the action potential. 1.5. Use the HH spike generation model with parameters given in program Hodhux.t in Section 1.4.1 to examine its steady-state current-to-frequency generation properties. Figures 1.4-8 to 1.4-11 in the text were made with Cm = 0.003 mF. See how different Cm values affect the values for A, B, and γ in the equation model, f = A + B 冨Jin冨γ, for steady-state spike frequency. Also, determine how different Cm values affect the minimum –Jin required to cause periodic spikes. Try Cm = 0.0005, 0.001, 0.005, and 0.01 mF. Use the other parameters given in Hodhux.t 1.6. Use the voltage clamp program HHVCLAMP.T to investigate the effect of a sinusoidal modulation of Vm on spike generation. Use Vs = 70 + Vspk sin(103 2πf t). Vspk is in mV (try different values, e.g., 1, 5, 10, 20 mV). Note that t is in ms, so one needs the 103 factor so f will be in hertz. Vary f over 1 to 100 Hz or so. Plot Vs, Vm, Jin, JNa, JK, JL, and JC. 1.7. This problem investigates a chemical kinetic model for the generation of epsps. Refer to Figure P1.7. At the arrival of a presynaptic nerve spike, the synaptic bouton releases a fraction, ko, of its stored NT, which diffuses across the synaptic cleft to the external surface of the postsynaptic membrane, where it encounters an excess of receptor molecules. One molecule of neurotransmitter combines with one receptor, forming a complex, bNT. Once the NT is bound, the receptor enables an ion channel to open, depolarizing the SSM. The depolarization voltage (epsp) is proportional to the density of bNT. The bound NT is hydrolized by an (excess) enzyme
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FIGURE P1.7
into an inactive molecule, Nbar. Nbar diffuses back to the presynaptic membrane of the bouton where it binds to an uptake molecule with density Ns. The bound Nbar is taken up at rate Ku by the bouton and resynthesized to stored NT. (Note that this model neglects the quantal nature of NT storage and release from the bouton.) The process model outlined above and in the figure can be modeled by a Simnon program. (Note that the kinetic process is linear in this problem.) The Simnon program uses an IPFM system as a source of periodic, presynaptic impulses. The frequency of the impulses is proportional to Vin. When Vin = 0.1 V, the frequency is 100 pps. The program is as follows: CONTINUOUS SYSTEM epsp“ 02/05/2000 Use Euler integration with delT = tau. STATE v psNT bNT Nbar Ns NT DER dv dpsNT dbNT dNbar dNs dNT TIME t “ t in milliseconds. “ “ IPFM SOURCE OF PERIODIC INPUT PULSES: dv = Vin - z w = IF v > phi THEN 1 ELSE 0 s = DELAY(w, tau) x = w - s y = IF x > 0 THEN x ELSE 0 yo = y/10 - .15 “ yo for plotting z = y*phi/tau Pin = y*Do/tau“ Input pulses. “ Ko is fraction of NT released at every pulse. dpsNT = -Kb*psNT + DELAY(Ko*NT*Pin, dT) “ Conc. NT @ SSM. Kb is mono“ molecular rate const for NT binding to receptors. dbNT = Kb*psNT - Kh*bNT “ Relative density of NT bound to (XS) “ receptors. Kh is hydrolysis rate constant. dNbar = Kh*bNT - Kd*Nbar “ Conc. inactive NT in cleft. “ Kd is diffusion rate constant. dNs = Kd*Nbar - Ku*Ns “ Relative density of Nbar bound to bouton “ membrane. Ku is rate of active Nbar uptake by bouton. dNT = c + Ku*Ns - Ko*NT*Pin “ Conc. stored neurotransmitter in bouton.
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“ Note that NT is recycled with rate Ku. Vm = Kv*bNT “ epsp is proportional to bNT. “ Vin = Vo + Kr*t “ Causes the freq. of y to ramp up. “ “ CONSTANTS: phi:1 tau:.001 Vo:.10 Do:8 Ko:.05 Kb:1. Kh:2. Kd:0.5 Kv:10 Ku:5 Kr:.00333 c:0 dT:0.2 “ ms. zero:0 “ “ IC: NT:.5 “ END
a. Run the simulation with the constants given. Use Euler integration with interval = 0.001. Plot yo, psNT, Vm, Nbar, Ns, and NT for 0 ð t ð 80 ms. Use a vertical scale of –0.15 to 0 .6. Note what happens to bNT(t) as the presynaptic input frequency is raised by increasing Vin. b. Now try varying system rate constants. Note that in the real world, concentrations and densities are non-negative quantities. What does it mean regarding the simulation if non- negativity is not preserved? 1.8. A linear transfer function of the form: H(s) = an/(s + a)n = (Yn/X)(s) can be used to model the dynamics of certain neurophysiological systems. When n = 2, the dynamics of the so-called alpha function that is often used to model excitatory postsynaptic potentials is obtained. Let the input to the transfer function, x, be a unit impulse. (In the simulation, make the pulse rectangular with width 0.001 ms, and its height 1000.) The system can be simulated by writing 10, simple, linear concatenated ODEs of the form:
y˙ k = a y k + a y k −1 where k = 1, 2, … 10, and y0 = x, the system input. a. Simulate and plot the 10 yk(t). Note what happens to the response as k → 10. b. What mathematical form does yk(t) take as k → ×? 1.9. A passive, cylindrical dendrite is to be modeled by an RCG transmission line model, as shown in Figure P1.9. The dendrite is 250 µm long, and
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has a 1-µm radius. A synapse at the far end introduces an epsp that can be modeled as an ideal voltage source of the form: vs(0, t) = 1 exp[–at] mV. where a = 200 r/s. This pulse propagates passively down the dendrite to the cell body, where it is measured by a glass micropipette electrode as vm(200, t). The dendrite membrane has a capacitance of CM = 1 µF/cm2, and a net passive conductance of GM = 2 × 10–4 S/cm2. The axoplasm resistivity is ρi = 630 ohm cm. The dendrite is to be modeled by a lumpedparameter RCG transmission line with five sections (one for every 50 µm of dendrite length). The soma is known to terminate the dendrite with a parallel CG of Cs = 78.54 pF, and Gs = 1.571 × 10–8 S. vm(200, t) is measured across CsGs. a. Calculate the appropriate ri, cm , and gm for each RCG section. b. Use a circuit simulation software package such as PSPICE™ or MicroCap™ to observe the vm(200, t) transient, given the vs above. (Note that the simulation of this linear system can also be done with Simnon or Matlab® after writing node equations for the circuit of Figure P1.9.) c. Plot the sinusoidal frequency response 20 log[Vm/Vs(jω)] of the dendrite model. d. Now examine the result when only two (instead of five) RCG transmission line sections are used. It is necessary to multiply ri, gm , and cm each by (5/2) to scale the two, 125-µm sections correctly. Use the same Rs and Cs to terminate the line model.
FIGURE P1.9
1.10. This problem simulates saltatory conduction on a section of myelinated nerve axon. The approach is to model the active nerve membrane at the nodes of Ranvier by a slightly modified set of HH equations (see Plonsey, 1969, Section 4.10). As was seen in Section 1.4, the original HH equations described the behavior of a 1-cm2 patch of (active) axon membrane. They were written and scaled so that time was in milliseconds, voltages were in millivolts, and current densities were in µamps/cm2. To obtain this curious scaling, the membrane capacitance is in mF/cm2, and all conduc-
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tances are in mS/cm2. In addition, the depolarization voltage, v, used in the HH ODE goes negative for membrane depolarization and spiking (which it is accustomary to view as positive). From this viewpoint, the complete transmembrane voltage is given by Vm = (–v + Vmo). Vmo is the resting transmembrane potential, often taken as –70 mV. Between the nodes of Ranvier are 2-mm-long, myelin “beads,” which respond passively to the voltages at the active nodes of Ranvier. This problem models the linear, distributed-parameter RCG transmission line formed by the myelin-covered axon by a five-internal-node, lumpedparameter RCG transmission line (TL) circuit, as shown in Figure P1.10A and B. Because the HH system operates with a millisecond timescale, the myelin transmission line sections must be given a similar timescale; thus, millifarads must be used for Cm. The Rs and Gs of the TL are not scaled to preserve millivolts everywhere in the linear circuit portions of the model. Model parameters are taken from Plonsey (1969) and are based on measured data from Tasaki (1955, Am. J. Physiol. 181: 63). Beginning with the input HH (v0) node, one can use Kirchoffs current law to write the main HH ODE as
FIGURE P1.10 “ HH membrane patch ODE for 1st Node of Ranvier (the SGL). dv1 = A*(Jin0 – Jk1 – Jna1 – JL1)/Ch + (–v1 + V11)*3.333E-4/Ch “ mV/ms “ “ “ “
3.333E-4 is milliSiemens so current will be microA. v1 is in mV. HH convention has v1 < 0 if it is depolarization. A is area of N of R in cm^2. Ionic current densities are in microamps/cm^2.
Jin0 is a negative input current pulse needed to initiate the nerve spike at the SGL (v1) node. A is an area scaling factor: A = 3 × 10–5 cm2. All J
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turns are current densities in µA/cm2. Thus, multiplication by A converts dv1/dt to mV/ms. The right-hand term gives the µA current leaving the v0 node. Take Ch = 1 × 10–4 (mF). The series resitance between the v1 HH node and the first TL (V11) node is 3 Mohms. If one were dealing in volts, amps, ohms, farads, and seconds, the conductance used in the v1 node equation would be 3.3333 × 10–7 S. However, all voltages are in mV, and the current must be in µA to be compatible with the HH equation. Thus the conductance must be multiplied by 106 and divided by 103, giving a scaled conductance of 3.333 × 10–4 in the HH node equation. The rest of the HH format is standard, given below: Jk1 = gko*(n1^4)*(v1 – VK) “ microamps/cm^2. JNa1 = gnao*(m1^3)*h1*(v1 – VNa) JL1 = glo*(v1 – VL) “ dn1 = – n1*(an1 + bn1) + an1 “ K+ activation para meter. dm1 = – m1*(am1 + bm1) + am1 “ Na+ activation para meter. dh1 = – h1*(ah1 + bh1) + ah1 “ Na+ inactivation parameter “ “ HH VOLTAGE-DEPENDENT PARAMETER FUNCTIONS for v0 node: an1 = .010*(v1 + 10)/(exp(0.1*v1 + 1) – 1) bn1 = .125*exp(v1/80) am1 = .100*(v1 + 25)/(exp(0.1*v1 + 2.5) – 1) bm1 = 4*exp(v1/18) ah1 = .07*exp(v1/20) bh1 = 1/(exp(0.1*v1 + 3) + 1) “ Vm1 = –(v1 + 70) negv1 = –v1
The (first) V11 node equation of the myelin TL can be written: Cm*dV11/dt + V11*(gm + 1/6M + 1/3M) + v1/3M – V12/6M = 0 “ Amps
or dV11 = –V11*(gm + 3/6M)/Cm – (v1/3M – V12/6M)/Cm
From Plonsey, gm = 1.37931 × 10–9 S, Cm (time-scaled so ODE has ms time) = 6.40 × 10–10 F. The total series resistance between nodes of Ranvier is 30 Mohms. Thus, each of the five RCG TL sections is separated by 6 Mohms, with 3 M on the ends. Hence the five node equations for myelin bead 1 are dV11 dV12 dV13 dV14
= = = =
–V11*7.83405E2 –V12*5.22989E2 –V13*5.22989E2 –V14*5.22989E2
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+ + + +
v1*5.20833E2 V11*2.6042E2 V12*2.6042E2 V13*2.6042E2
+ + + +
V12*2.6042E2 V13*2.6042E2 V14*2.6042E2 V15*2.6042E2
dV15 = –V15*7.83405E2 + V14*2.6042E2 + v2*5.20833E2
v2 is the node voltage at the second node of Ranvier. The node equations for the second HH node are “ HH membrane patch ODE. (Node #2 = v2 node). dv2 = –A*(Jk2 + JNa2 + JL2)/Cm + (–2*v2 + V21 + V15)*3.333E-4/Cm “ “ Ionic current densities in microamps/cm^2. Jk2 = gko*(n2^4)*(v2 - VK) JNa2 = gnao*(m2^3)*h2*(v2 – VNa) JL2 = glo*(v2 – VL) “ dn2 = – n2*(an2 + bn2) + an2 “ K+ activation parameter. dm2 = – m2*(am2 + bm2) + am2 “ Na+ activation parameter. dh2 = – h2*(ah2 + bh2) + ah2 “ Na+ inactivation parameter “ “ VOLTAGE-DEPENDENT PARAMETER FUNCTIONS: an2 = .010*(v2 + 10)/(exp(0.1*v2 + 1) – 1) bn2 = .125*exp(v2/80) am2 = .100*(v2 + 25)/(exp(0.1*v2 + 2.5) – 1) bm2 = 4*exp(v2/18) ah2 = .07*exp(v2/20) bh2 = 1/(exp(0.1*v2 + 3) + 1)
Note that both adjacent myelin beads figure in the v2 node equation. The five node equations for the second bead are dV21 dV22 dV23 dV24 dV25
= = = = =
–V21*7.83405E2 –V22*5.22989E2 –V23*5.22989E2 –V24*5.22989E2 –V25*7.83405E2
+ + + + +
v2*5.20833E2 V21*2.6042E2 V22*2.6042E2 V23*2.6042E2 V24*2.6042E2
+ + + + +
V22*2.6042E2 V23*2.6042E2 V24*2.6042E2 V25*2.6042E2 v3*5.20833E2
Note that v3 is the voltage at the third node of Ranvier, etc. One can now write the 45 ODEs describing the model. Note that each of the HH systems has the same initial conditions, vital for correct simulation. In general, for k = 1 to 5: mk:0.052932 nk:0.31768 hk:0.59612
a. Write the complete Simnon program to simulate the five-node, fivemyelin-bead model. b. Using Simnon with Euler integration with delT = 0.0001 ms, with Cm = 1.E-4, and Jino = –300 (µA), run the simulation and observe the five depolarization voltages at the nodes of Ranvier. Note that they are
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negative. Use a vertical axis from –130 to +20 mV, and a timescale from 0 to 10 ms. (To observe the true transmembrane potentials, compute and plot Vmk = Vmo – vk, Vmo = –70 mV.) See how the time between vk peaks changes with different Cm values. Plot and observe the TL node voltages, Vjk , for the model (j = 0, 1, … 4, k = 1, 2, … 5). c. Make a three-dimensional plot of the five vks vs. time in ms (0 to 10 ms) and distance along the axon (0 to 10 mm).
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2
Selected Examples of Sensory Receptors and Small Receptor Arrays
INTRODUCTION A neuron that responds to a non-neural, physical stimulus is a sensory receptor. Exteroreceptors sense stimuli that are external to the animal; interoreceptors respond to stimuli from within the body. Stimuli include, but are not limited to electromagnetic radiation — visible, infrared (IR), ultraviolet (UV); internal mechanical inputs such as muscle stretch and tension, joint rotation, and their first and second derivatives; chemical inputs including pH, pCO2, osmotic pressure, concentration of K+, amino acids, various odorants, pheremones, etc; sound and vibration; the Earth’s magnetic field vector; electric field intensity; external mechanical inputs such as angular velocity and acceleration and linear acceleration including the Earth’s gravity field. There are many amazing sensory modalities that neurons and sensory systems respond to. The following sections consider some of the more unusual ones, as well as problems associated with threshold sensitivity. That is, what are the factors that determine the least resolvable stimulus (LRS)? Most sensory receptors respond to an increasing stimulus by an increasing rate of firing, or if the receptor itself does not spike, a depolarizing (positive-going) generator potential. But, as will be seen, there are some receptors that fire at the cessation of a stimulus (OFF response), or, if nonspiking, by an hyperpolarizing membrane potential at ON of the stimulus. Of interest is the dynamic responses of sensory receptors because they can be modeled mathematically, and they provide information about the information-processing properties of the nervous system. The following sections first examine external chemoreceptors in vertebrates and arthropods. The next section describes the properties of certain mechanoreceptors — insect trichoid hairs and campaniform sensillae, vertebrate muscle length receptors (spindles), muscle force sensors (Golgi tendon organs), invertebrate gravity and acceleration sensory organs (statocysts), and vertebrate internal pressure sensors (pacinian corpuscles). There is a large body of behavioral evidence that certain vertebrates and invertebrates can sense their body orientation in the Earth’s magnetic field. Section 2.4 reviews some of this evidence and examines putative magnetoreceptors and some theoretical models for animal magnetoreceptors.
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Electroreceptors (sensory neurons that respond to an external electric field) are found both in certain saltwater and freshwater fish. These electroreceptors are found in arrays, an ubiquitous organizational modality that allows increased sensitivity over that for single receptors. Some electroreceptors are specialized to sense lowfrequency changes in the electric field around the fish (~0.1 to 10 Hz), and are used to passively locate prey, and even for dc magnetic field sensing. Other electroreceptors respond to audiofrequency, amplitude, and frequency modulations in the electric field around mormyrid and gymnotid fish, which produce weak ac electric fields for navigation and communication. The unique, gravity-sensing organs (tricholiths) found on the cerci of certain burrowing desert cockroaches are described in Section 2.6. (Insects as a rule do not have specific gravity-sensing organs or neurons; the cockroach Arenevaga sp. violates this principle.) The Arenevaga gravity-sensing system is also unique because it is an example of a simple sensory array that sends only four afferent axons to the brain, where central processing sharpens the detection of the animal’s roll and pitch angles. Dipteran flies generally have short, stubby, nonaerodynamic bodies, and two wings. Flight stabilization appears to be the result of the interaction of visual information, wind pressure on sensory hairs and the antennae, and the sensory outputs of mechanoreceptors in the bases of a pair of vibrating gyroscopes, the halteres. A mechanical model of the haltere vibrating gyroscope is analyzed, and it is shown that torques are generated at their bases proportional to roll, pitch, and yaw angular rate and accelerations. Finally, the curious electrophysiological behavior of the simple, multireceptor eye of the plecypod mollusk Mytilus edulis is examined.
2.1
THE GENERALIZED RECEPTOR
All sensory receptor cells are transducers; they respond to the physical quantity under measurement (QUM) by a change in their transmembrane potential, Vm. In most cases, Vm goes positive (depolarization) as a nonlinear, monotonic function of the QUM. Depolarization of Vm generally leads to the production of nerve spikes at the spike generator locus (SGL) of the cell. However, some receptor cells do not produce spikes; their depolarization is coupled by either electrical or chemical synapses to a sensory interneuron that does spike. Certain receptors, such as vertebrate rods and cones, hyperpolarize with increasing stimulus intensity (absorbed light power) and affect a complex of signal processing cells in the retina, the underlying neuropile in the vertebrate eye. The outputs of the retina are spikes on the axons of the ganglion cells that form the optic nerve, which carries spike signals to the CNS. Still other receptors, such as the photoreceptors in the eye of the plecypod mollusk, Mytilus edulis, respond to light only at OFF. That is, they fire a burst when the illumination dims or goes off, and do not fire for ON or brightening at all (see Section 2.9).
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2.1.1
DYNAMIC RESPONSE
Most receptors that respond positively to increasing QUM do so in a manner that suggests that the QUM is acted on by physical processes in which the receptor’s spike frequency is driven by an approximate proportional plus derivative operation on the QUM as a function of time. That is, at QUM ON, the instantaneous spike frequency of the receptor jumps to a peak value, then slowly declines to a lower steady-state value, or even to zero. This proportional plus derivative-like response is called adaptation of response by neurophysiologists. Often, the spike instantaneous frequency (IF) of certain adapting receptors in response to a step ON of the QUM is like that of a linear system where r(t) = Ae–αt + B. There may be more than one exponential term, so, for example, r(t) = Ae–αt + Be–βt + C. In this latter case, the peak IF is (A + B + C), and the steady-state IF is C pps. The IF at ON for other receptors may be fit better by a plot of linear frequency vs. log(time) coordinates. That is, the step response IF is fit by a mathematical model of the form: r(t) = Kt–β. After Laplace transformation r(t) and multiplication of this transform by s, the impulse response of the receptor is H(s) = K Γ(1 – β)sβ, where Γ(x) is the gamma function. In one example in Milsum (1966), a cockroach mechanoreceptor was described with K = 23 and β = 0.76. Very often the response of a receptor exhibits unidirectional rate sensitivity. That is, the strong derivative component in the step (ON) response is lacking in the OFF response. This may be true even if the receptor has a steady-state firing rate, ro, for a zero-QUM. The firing rate drops quickly to ro with no undershoot to zero. That ON and OFF response dynamics differ is probably due to the fact that the physical/chemical processes mediating the ON response are different from those involved with the OFF dynamics. A very rapidly adapting receptor (e.g., the pacinian corpuscle) fires a short burst of spikes at ON of the QUM (pressure), and another short burst at OFF. In other words, the changes in pressure are important to these sensors. A conceptual model for this sensor is shown in Figure 2.1-1. Note that the absolute value of the derivative term is used to describe the ON and OFF response of this receptor.
FIGURE 2.1-1 Block diagram describing the instantaneous frequency vs. input (pressure) for a PC. In most cases, Kp → 0.
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2.1.2
RECEPTOR NONLINEARITY
Many receptors can operate at the theoretical limit of signal detection, and have an enormous dynamic range. In an engineering context, a large dynamic range can be obtained by using a nonlinear, gain compression-type nonlinearity, such as using the logarithm of the QUM at the front end of the receptor. For certain neuro-sensory systems, it is speculated that the CNS sends an efferent signal that adjusts the SGL firing threshold of the receptor so that, in the absence of the QUM, the threshold is made very low, such that the receptor is maximally sensitive, occasionally firing on membrane noise. In the presence of a large QUM, the sensor spike output causes the CNS to send an efferent signal to the receptor to raise its threshold so that the output spike rate does not saturate. (Few receptors can fire over 500 pps because of basic nerve membrane dynamics.) Many receptors exhibit a log/linear transfer characteristic over a sizable potion of their dynamic range, both for their initial firing rate at ON, and for the rate sampled later after adaptation has occurred. That is: r ≅ Kr log(I – Io) + ro
2.1-1
where r is the IF of the receptor, Kr and ro are positive constants, and Io is the intensity threshold. This relation is basically the mathematical result of the Weber–Fechner law for perception which says that the just noticeable difference in a QUM is proportional to (I – Io) (Milsum, 1966). Another mathematical model for the perceived output of receptors comes from Stevens (1964). Stevens noted that the perceived intensity of a QUM, Ψ, can be modeled by Ψ ≅ K(I – Io)ν
2.1-2
The exponent, ν, ranges between 0.33 and 3.5, the larger values generally associated with noxious QUMs such as electroshock, heat, or pain. Of course, r and Ψ are nonnegative quantities. If the full dynamic range of a typical receptor, e.g., a touch receptor, is examined, the output frequency actually has four ranges with respect to the intensity of the QUM. These are shown in Figure 2.1-2. At QUM intensities ranging from zero to Iϕ, r = 0. This is in effect a dead zone. From Iϕ to Io, r increases with a slope less than Kr log(I – Io), and in the range Io ð I ð Iu, r is approximated by Kr log(I – Io). For I > Iu, r flattens out and saturates at rmax. The dead zone and the low slope range (0 < I < Io) give the receptor a certain robustness against noise. As mentioned above, certain receptors fire slowly and randomly for I = 0. That is, a certain noisiness in their spike output is tolerated to obtain enhanced sensitivity to very low I > 0.
2.1.3
RECEPTOR SENSITIVITY
As is the case with all physical measurements, the ultimate limit to detectability of the QUM is noise arising in the receptor as well as noise accompanying the QUM
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FIGURE 2.1-2 Graph showing the instantaneous output frequency of a typical receptor’s vs. stimulus intensity. (Note that some receptors do not generate output spikes, but rather a generator potential.) If this curve represents the instantaneous frequency (IF) at ON of a step stimulus, the steady-state IF vs. stimulus intensity will have a similar shape, but a much lower slope. See text for discussion.
(environmental noise). Assume for now that the environmental noise is zero, that is, the signal-to-noise ratio (SNR) at the input is infinite. Now the smallest detectable signal depends on the inherent noisiness of Vm with QUM = 0. Noise in Vm can come from several sources: (1) The random leakage of ions (e.g., Na+ inward, K+ outward, etc.) through the cell membrane specific ion channel proteins. (2) Thermal (Johnson) noise current arising from the bulk, resting, membrane conductance. This is given by: 2inm = 4kTGm B mean squared amp/cm2. B is the hertz bandwidth over which the noise is viewed, and Gm is the bulk membrane conductivity in S/cm2. If the thermal noise plus leakage noise [vn (t)] plus the resting potential (Vmr) exceeds the sensory neuron SGL firing threshold (Vϕ), it will fire, giving a false-positive output. A false negative can occur as well when QUM > 0 so that Vm would normally exceed the firing threshold, but [Vmr + vn (t)] < Vϕ. Now when vn (t) goes positive, the receptor will fire but at a higher rate than if vn ≡ 0. One of the challenges in sensory neurophysiology is to try to understand how an animal detects the change in the random firing pattern of a sensory neuron axon from the zero-QUM condition to the firing statistics present when there is a threshold level of depolarization caused by a nonzero, threshold QUM intensity. (The probabilistic approach to threshold sensory perception is treated rigorously by Reike et al. (1997) in their book on the neuro-sensory code, Spikes.) The use of efferent feedback to optimize or maximize detection probability while minimizing false-positive spikes was mentioned above. It is worth noting that many neuro-sensory systems capable of great sensitivity are known to have efferent fibers in their nerves. These include both vertebrate “camera” eyes and arthropod compound eyes, the vertebrate cochlea, the statocysts of octopus, etc.
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2.1.4
A MODEL
FOR
OPTIMUM FIRING THRESHOLD
This section, using basic probability theory, derives conditions for an optimum SGL firing threshold that will minimize the operating “cost” of a sensory neuron, defined below. The firing threshold is a fixed, positive voltage defined by ψo = Vϕ – Vmr
2.1-3
Assume that a bandwidth-limited Gaussian noise, vn, is added to Vmr. vn is defined
{ }
to have zero mean, E[vn] ≡ 0, and a variance, σ 2n = E v 2n . The derivative of vn also
{ }
has zero mean and variance, σ 2d = E v˙ 2n . Another required property of the membrane voltage noise autocorrelation function, Rnn(τ) is
dR nn ( τ) = 0, dτ
for τ = 0
2.1-4
Assume that the QUM occurs randomly as short, infrequent pulses of amplitude A. For an input pulse to cause the sensory neuron to fire, d vn(t)/dt > 0, and [vn(t) + AP(t)] Š ψo. P(t) is defined as a pulse of short width, ε s, and peak height 1. When the neuron fires in response to an input pulse, the event is called a true positive; its firing rate due to AP(t) is RTP. From probability theory, RTP = λ Pr{ψo ð (vn + A) < (ψo + A)}. λ is the mean rate of occurrence of input pulses. Because it is assumed that vn is described by a Gaussian probability density function, it is easy to show that
{ [
)]
(
[
(
RTP = (λ 2) erf ( ψ o ) σ n 2 – erf ( ψ o – A) σ n 2
)]}
2.1-5
where erf(z) is the error function, defined as
(
erf (z) ≡ 2
π
)∫ exp(– t ) dt z
2
2.1-6
0
Note that erf(0) = 0, and erf(×) = 1. Two types of error can occur for this type of threshold event detector. A false positive, where an output spike occurs because vn has crossed the firing threshold with positive slope, giving a rate of false-positive (RFP) outputs. The second type of error is a false negative, where the instantaneous sum (vn + A) < ψo, or vn > ψo. The RFP is given by (Papoulis, 1965):
RFP =
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1 ⎡ σd ⎤ 2 2 ⎢ ⎥ exp – ψ o 2σ n π ⎣σn ⎦
[
(
)]
2.1-7
RFP is the mean rate vn crosses ψo with positive slope. The rate of false negatives (RFN) is given by RFN = λ Pr{(vn + A) ð ψo} + λ Pr{vn > ψo}
2.1-8
Again, by integrating probability densities,
{
[
(
RFN = (λ 2) 1 + erf ( ψ o – A) σ n 2
)]} + (λ 2){1 – erf[(ψ ) (σ o
n
2
)]} 2.1-9
Now an operating cost is defined for the receptor that will be minimized: COST = C ≡ RFP + RFN – RTP
[
(
)]
[
2.1-10
(
C = (1 π)(σ d σ n ) exp – ψ 2o 2σ 2n + λ + λerf ( ψ o – A) σ n 2
[ (
– λerf ψ o σ n 2
)]}
)]
2.1-11
Figure 2.1-3 shows that for typical values, C(ψo) has a minimum. Use of the derivative of C with respect to ψo leads to frustration. The optimum value of threshold that minimizes the cost, ψopt, must be found numerically from the solution of a transcendental equation.
FIGURE 2.1-3 Plot of a typical cost function, as given by Equations 2.1-10 and 2.1-11, vs. the firing threshold. Note that there is a threshold value, ψo, where C has a true minimum.
2.1.5
SIMULATION OF A MODEL RECEPTOR WITH A CONTINUOUSLY VARIABLE FIRING THRESHOLD
Imagine a sensory receptor that must sense infrequent, random inputs of the quantity under measurement. The QUM events are considered to be a random, point process. Their times of arrival can be described by a Poisson random process. All the QUM events have the same amplitude. (For example, the events could be photons absorbed by a photoreceptor.) Ideally, the model receptor should fire 1:1 with the input events. Such output pulses are called true positives = TPs. However, because of random noise associated with the transmembrane (generator) potential, the receptor SGL
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can produce output spikes in the absence of input (false positives = FPs) and may not fire when a true input is present (false negatives = FNs). The biological cost to the animal of FPs and FNs can be varied. If a sensory neuron output occurs, it may trigger an energy-intensive behavior, such as escape swimming. If no predator was actually present (an FP event), the animal is not eaten, but expends energy it must eventually replace. When an FN event occurs, the animal does not respond, and a predator may eat it, an extreme cost. In another scenario, the receptor may detect single pheromone molecules in the air. Now FN events can lead to the animal sitting still and not finding a mate, hence the failure to reproduce. FP events can send the animal on a high-energy-cost, random search for the apparent pheromone source that may lead it to a mate by chance, or no mate. Thus, the operating cost, C, of the receptor depends on the sensory modality to be detected and its importance in the animal species’ survival. Thus C = RFP + RFN – RTP may be a good starting point in evaluating neural receptor operation, but the cost function eventually used should reflect the animal’s breeding success as governed by the receptor. Figure 2.1-4 illustrates a model for spike generation in which the output spikes act to raise the firing threshold, ϕ2, causing a reduction in the RFP and the rate of true positives (RTP). Raising ϕ2 raises the RFN. To examine the behavior of this model, which involves nonlinear dynamics and nonstationary statistical behavior, the author has written a Simnon program, ADTHRESH.t. Two independent Gaussian noise sources are used. One drives an IPFM, voltage-to-frequency converter, the output of which is random impulses representing the events the sensory neuron is to detect. The impulses are passed through a single-time constant low-pass filter to generate exponential pulses, ein, representing generator potential transients. Also added to ein is bandwidth-limited Gaussian noise, Vmn. (Vmn + ein) are acted on by a second low-pass filter representing the membrane low-pass characteristic on the neuron cell body (soma). The output of this filter, v2, is the input to a simple threshold pulse generator, which generates a sensory neuron output pulse when v2 > ϕ2, and dv2/dt > 0. This model for neural spike generation is called an RPFM (leaky integrator) neuron; it is described in detail in Section 4.3.2. The firing threshold for the RPFM neuron, ϕ2, is given by ϕ2 = ϕ20 + q2
2.1-12
q2 is the output of a two-pole low-pass (“ballistic”) filter whose input is the sensory neuron output spikes, y2. Thus, every time the RPFM neuron fires, either from sensory input or Vmn, the threshold is raised, making it less sensitive to noise and sensory input. What will be shown is that this manipulation of ϕ2 reduces RFP at the expense of raising RFN. The Simnon program follows: continuous system ADTHRESH “ 7/02/99. System uses NFB to raise phi2 of “ RPFM sensory neuron to reduce false positives. STATE v1 v2 r1 r2 p2 q2 Vmn in ein DER dv1 dv2 dr1 dr2 dp2 dq2 dVmn din dein TIME t “ Use Euler Integration w/ dt = tau. “
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FIGURE 2.1-4 Block diagram of a nonlinear, time-variable system in which the input is a random point process, y1. y1 is weighted and passed through a simple, one-pole low-pass filter. After having “membrane noise” added to it, the signal u2 is the input to an RPFM spike generator with a variable firing threshold, ϕ2. ϕ2 is the sum of a dc level, ϕ20, and q2, the output of a two-pole low-pass filter whose input is the RPFM system output pulses. Thus, the faster the RPFM spike generator fires, the larger the ϕ2. Thus, if the system is firing on the peaks of Vmn alone, ϕ2 increases to minimize noise-induced firing. “ IPFM VPC GENERATES RANDOM QUM EVENTS: dv1 = rin – z1 z1 = y1*phi1/tau w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0 e1 = z1*Do1 “ “ THE RPFM SGL dv2 = –c2*v2 + c2*ein + vmn – z2 “ Vmn added to v2 = Vm. z2 = y2*phi2/tau w2 = IF v2 > phi2 THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 – s2 y2 = IF x2 > 0 THEN x2 ELSE 0 “ “ NOISE GENERATION: dr1 = –b1*r1 + SD1*NORM(t) “ Membrane noise, 2-pole filtered. dVmn = –b2*Vmn + r1 “
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dr2 = –b3*r2 + SD2*NORM(t + Td) “ Input noise = in drive input events. din = –b4*in + r2 rin = IF in > 0 THEN in ELSE 0 “ Rectified input noise to IPFM random “ event generator. dp2 = –a2*p2 + z2*K2 “ 2-pole ballistic LPF for threshold feedback dq2 = –a3*q2 + p2 “ dein = –a1*ein + e1 “ LPF to condition random event impulses. “ PHI2 = phi20 + Kf*q2 “ Sensory neuron SGL threshold is sum of membrane noise + slow “ NFB term. plty1 = 0.3*y1 + O1 plty2 = O2 + 0.3*y2 “ “ CONSTANTS: phi1:2.5 phi20:0.75 Do1:15. K2:.2 “ Adjusts gain of BF. Kf:1“ Kf = 0 turns off NFB. a1:1. a2:.1“ a,b,c units are radians/s. a3:.05 b1:1 b2:2 b3:1. b4:1. c2:.05 Td:100 tau:.001 SD1:50 SD2:8. O1:-1.5 O2:-1 zero:0 “ END
Figure 2.1-5 illustrates some of the waveforms in the model for one simulation. (The noise waveforms are different each time a simulation is run.) Trace 1 shows the (random) input events of the QUM. Trace 2 gives the output pulses of the model, y2. Trace 3 is the generator potential of the RPFM neuron, v2. The variable threshold, ϕ2, is shown in trace 4. Whenever v2 crosses ϕ2 with positive slope, an output pulse is produced. Note that in this 400-s run, the first two output pulses are FPs. The first and fourth input pulses do not give output pulses, and are thus FNs. Five output pulses follow input pulses, and are TPs. Seven runs were made using the feedback adjustment of ϕ2, and seven runs were made with ϕ2 = ϕ20, the open-loop condition. The results are summarized in Tables 2.1-1 and 2.1-2.
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TABLE 2.1.1 Results with Constant ϕ2 Run No.
TPs
FNs
FPs
Σ Input Pulses
1 2 3 4 5 6 7 Totals
10 6 10 8 3 9 6 52
0 0 0 0 2 0 0 2
4 17 9 4 8 15 7 64
10 6 10 8 5 9 6 54
TABLE 2.1.2 Results with Feedback Run No.
TPs
FNs
FPs
Σ Input Pulses
1 2 3 4 5 6 7 Totals
6 6 6 4 6 6 5 39
1 2 2 2 3 1 2 13
1 2 1 2 1 1 2 10
8 8 8 6 9 7 7 53
Clearly, the use of feedback to raise the firing threshold following output pulses reduces the rate of FP pulses. If one uses the simple cost formula introduced in Section 2.1.4 above, feedback control of ϕ2 does indeed reduce C over the nonfeedback case. Calculations follow for the feedback case:
CF =
13 + 10 39 = –0.302 – 53 53
2.1-13
CN =
2 + 64 52 = +0.219 – 54 54
2.1-14
And without feedback,
Clearly, the use of feedback to raise ϕ2 gives a reduced cost, as simply defined. A topic for further study is to explore the effect of other system natural frequencies and gains on the degree of cost reduction.
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FIGURE 2.1-5 Results of a Simnon simulation with the program, ADTHRESH.T. Ideally, y2 should fire every time y1 occurs. In the plot, the horizontal axis is time in milliseconds, the vertical axis is in arbitrary voltage units. Traces: (1) y1 (input) pulses; (2) y2 (output) pulses; (3) V2 (input to RPFM SG); (4) the variable threshold, ϕ2. See text for more details and tabulated results for the system with fixed ϕ2 vs. the variable ϕ2.
The nonlinear and nonstationary nature of the speculative sensory neuron model above defies analysis; one must necessarily use simulation to study its signal-processing properties. Obviously, there are many system parameters that can be manipulated in seeking low operating cost. The results given above were obtained with just one combination of parameters that appeared to be “reasonable” to the author.
2.1.6
DISCUSSION
This section examined some of the general properties of single sensory receptor neurons, including their dynamic response (to a step input of stimulus), their linearity (they are generally nonlinear, perhaps due to the transduction process and the spike generation dynamics), and factors affecting their sensitivity. A heuristic, dynamic, neural model for automatic adjustment of the firing threshold of a spiking sensory receptor having noise on its generator potential was shown to minimize detection cost.
2.2
CHEMORECEPTORS
Two major classes of chemoreceptors exist in animals: external and internal. Internal chemosensors sense quantities such as pH and pO2 in the blood or extracellular fluid.
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External chemoreceptors sense a wide spectrum of molecules that figure in an animal’s life. External chemoreceptors are examined first. External chemoreceptors are sensory neurons that respond to specific molecules in the air (as gases, liquids, or solids in vapor phase, or solids suspended as aerosols). Aquatic and marine animals also have chemoreceptors that respond to specific molecules dissolved or suspended in water. The chemical senses include olfaction (smell) of airborne substances and gustation (detection of waterborne substances). Flavor is a complex sensation involving both gustation and olfaction when food is eaten. Substances sensed by olfaction generally affect animal behavior. Many animals are known to release pheromones, which act as sexual attractants and stimulate courting and mating behavior. The best-known pheromones are certain insect attractants. Synthetic, female gypsy moth pheromone, for example, is used to bait traps that attract and kill the male moths, who follow the scent gradient upwind to the source. Most other insects make pheromones to attract mates, as do crustaceans and vertebrates. Chemical signaling to attract mates and stimulate mating is universally distributed in the animal kingdom. The sensing and identification of complex odorants involves an animal’s entire olfactory system, which is generally composed of many thousands of olfactory cells. Each olfactory cell has surface protein receptors that bind specifically to a unique odorant molecule. The binding generally triggers a complex internal sequence of biochemical events that lead to depolarization of the olfactory cell, and spike generation on its axon. The spike frequency increases monotonically with the odorant concentration. The olfactory systems of vertebrates can generally be considered to be massively parallel arrays of chemosensory neurons. Their axons project into the olfactory bulb in the brain, which in turn sends interneurons to the amygdala. Comparative anatomy teaches that behavior governed by the sense of smell evolved long before mammals evolved a cortex and the ability to reason. Air- and waterborne molecules evoke complex behavioral modes, ranging from “keep out” (animals mark territory with urine, feces, scent glands), to sexual receptivity (pheromones), and in directions a food source is near (blood, amniotic fluid, etc.) Scent can also alter mood (infant smells mother), leading to nursing behavior and the release of endorphins, or elicit fear (animal smells predator), leading to the release of adrenaline. Many animals have specialized glands they use for territorial marking, usually in conjunction with urine. Odorants have been extracted from the feces of predators such as African lions and marketed for use in keeping deer out of gardens. The study of odorant molecules that affect animal behavior is called semiochemistry (Albone, 1997). Much research today is being directed toward understanding how certain animal olfactory systems can detect (as evidenced by behavior) nanomole concentrations of certain odorants. The ability of bloodhounds to track a specific person through a crowd or to follow a days-old trail when searching for lost persons or fugitives is legendary. The noses of trained dogs are also the most effective system for locating buried land mines, explosives, or drugs in luggage and automobiles. What is not known exactly is to what molecules the dogs are responding. Is it the vapor phase of the analyte (e.g., C4, TNT, cocaine, etc), or is it some less obvious scent, such as the case or envelope containing the analyte, or is it human scent, or some
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combination? Because of the commercial importance of odors in the food and cosmetic industries, and in detecting drugs and explosives, much research is being invested in “artificial noses,” using a variety of physical, immunochemical, electronic, and photonic means. It appears that humans have a long way to go to make a sensor with the threshold sensitivity of certain insect pheromonal chemoreceptors, or the bloodhound’s nose.
2.2.1
THE VERTEBRATE OLFACTORY CHEMORECEPTOR
Vertebrate chemoreceptor neurons are unique in that they undergo apoptosis (programmed self-destruction) about every 40 to 60 days and are replaced by new cells. The new cells send their axons into the olfactory bulb (first synaptic relay point for olfactory information), where they synapse with appropriate target mitral cells. The mitral cells, like all other neurons in the CNS, do not divide. Figure 2.2-1 illustrates a schematic section through a mammalian olfactory epithelium. Each olfactory cell has from 8 to 20 cilia at its end immersed in mucus. The cilia are from 30 to 200 µm in length, and their surfaces contain the odorant receptor molecules that initiate receptor spike generation. A mouse may have 500 to 1000 different odorant receptor proteins (Leffingwell, 1999). It is estimated that about 1% of the total genes specifying an entire mammal (mouse) are used to code its olfactory receptor proteins. Airborne odorant molecules must dissolve in the mucus before they can reach receptor molecules on the olfactory cilia. Also floating in the mucus are odorant binding proteins (OBPs), which have been postulated to act as carriers that transfer odorants to their specific receptors on the cilia. The OBPs may act as cofactors that facilitate odorant binding, and they may also participate in the destruction of a bound odorant, freeing the receptor to bind again. OBPs are made by the lateral nasal gland at the tip of the nasal cavity (Kandel et al., 1991, Ch. 34). The exact role of the OBPs is unknown. The mechanism of olfactory transduction is as follows. An odorant molecule dissolves in the 60-µm-thick mucus layer overlying the receptor cells and their cilia. It combines with a specific OBP and then collides with and binds to a surface membrane receptor protein on a cilium that has a high affinity for that odorant. The presence of the OBP may be necessary to trigger the next step, which is the activation of a second messenger biochemical pathway that leads to a specific ionic current and receptor depolarization. Odorant binding to the receptor protein activates a G-protein on the intracellular surface of the cilium. The α-subunit of the G-protein activates a molecule of the enzyme, adenylate cyclase, which in turn catalyzes the formation of a cyclic nucleic acid, cyclic-3′,5′-adenosyl monophosphate (cAMP) from ATP. cAMP then binds to and opens a cAMP-gated Ca++ channel, causing an inward JCa++. The increase in intracellular Ca++ appears to activate an outward chloride current that depolarizes the receptor (Leffingwell, 1999). The more of a given odorant that binds to a specific chemoreceptor cell, the larger the total JCa++ and JCl–, and the greater the depolarization, and the higher the spike frequency on the axon of the receptor. cAMP then degrades by hydrolysis to AMP; the G-protein also returns to its resting state.
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FIGURE 2.2-1 Schematic section through a vertebrate olfactory epithelium. Note that there are receptors, and basal and supporting cells. (There is a total of about 105 olfactory receptor neurons in the rabbit.) (From Kandel, E.R. et al., Principles of Neural Science, 3rd ed., Appleton & Lange, Norwalk, CT, 1991. With permission from the McGraw-Hill Companies.)
The receptors on a given chemoreceptor cilia can have affinity to more than one odorant molecule. Also, more than one receptor can be activated (in varying degrees) by a given odorant. Hence, the very large chemoreceptor array must have a mechanism for eliminating odor ambiguity, since several receptors may respond to a given odorant in different degrees. A further complication to understanding the physiology of olfaction is the recent evidence that there may be a second, second-messenger pathway in certain olfactory chemoreceptors. A different odorant binds to its receptor, leading to the production of the messenger, inositol-1,4,5-triphosphate (IP3). IP3 opens calcium ion channels, allowing a JCa++ to flow inward, hyperpolarizing the cell (Breer, 1997). In lobster olfactory neurons, apparently the internal role of the IP3 and cAMP messengers is reversed; odorants that release the cAMP messenger cause hyperpolarization, while IP3 causes depolarization (Breer, 1997). Olfaction is a complex process! How are specific odorant receptors recycled? A problem in any sensory communication system is to preserve sensitivity to a new stimulus. It is known that when an odorant binds to its receptor, the second messenger cascade is initiated; and has been recently discovered that the second messenger cascade is inhibited by the second messengers activating a protein kinase, which phosphorylates the receptor protein, interrupting the transduction process (Breer, 1997). It is also
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reasonable to assume that the odorants are slowly broken down enzymatically. Olfactory systems are extremely sensitive, but (perhaps mercifully) adapt profoundly to a sustained, external odorant concentration to the point where one is not conscious of the odor. Bundles of 10 to 100 axons from the ciliated olfactory chemoreceptors project through the pores in the ethmoidal cribiform plate on the roof of the nasal cavity into the olfactory bulbs of the brain, where the first level of signal processing takes place. In structure, the olfactory bulb has several layers of interneurons, not unlike a retina. Figure 2.2-2 illustrates schematically the basic, five-layered structure of a mammalian olfactory lobe. Not shown is the tremendous convergence of information in the olfactory bulb. For example, in the rabbit, about 26,000 receptors converge into ~200 glomeruli which then converge at 25:1 on each mitral cell (Leffingwell, 1999). Humans have about 5 million olfactory receptors; those having one specific receptor appear to be distributed randomly in the nasal epithelium. However, their axons sort themselves out and converge on the same area of the optic bulb. Thus, there appears to be order out of chaos and a basis for some sort of fuzzy combinatorial logic to identify complex scents containing several odorants (Berrie, 1997). A
B
Olfactory mucosA
- Receptor cell
Olfactory fibers
Cribriform plate
Periglomerular cell
Periglomerular cell
Olfactory nerves Glomeruli
Tufted cell
1o
External plexiform layer 2o
Mitral body layer
Mitral cell
Mitral cell Granule layer
Granule cell
Centrifugal fibers
Recurrent collateral Granule cell
[ 100µm To lateral olfactory tract
To olfactory tract
From ipsilateral anterior olfactory From contralateral nuclei olfactory nuclei
FIGURE 2.2-2 (A) Anatomical schematic of the connections of various cells within the olfactory bulb. Anatomically, there are five layers in the olfactory bulb (OB). Shepherd (1970) observed that there is congruence between cells of the vertebrate retina and olfactory lobe cells. Retinal horizontal cells are analogous to OB periglomerular cells, and retinal amacrine cells are analogous to OB granule cells. There are no cells analogous to retinal bipolar cells in the OB, but the long primary dendrite of the mitral cell fills that position. Mitral cell axons also are functionally analogous to retinal ganglion cells. Granule cells and periglomerular cells are inhibitory interneurons. (B) Schematic interneuron connections in the OB. The olfactory receptor cell axons synapse with mitral cells, tufted cells, and periglomerular cells. The mitral cell dendrites also receive inhibitory inputs from the periglomerular cells. The granule cells make inhibitory synapses on the secondary dendrites of the mitral cells. Interesting, efferent control fibers from the CNS synapse on the granule and periglomerular cells. (From Kandel, E.R. et al., Principles of Neural Science, 3rd ed., Appleton & Lange, Norwalk, CT, 1991. With permission from the McGraw-Hill Companies.)
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The periglomerular cells and granule cells in the olfactory lobe are known to be inhibitory on the mitral cells. Centrifugal (efferent) control fibers from the anterior olfactory nuclei stimulate these inhibitory cells. The receptor cell axons synapse excitatorily on the mitral, tufted, and periglomerular cells. The mitral and tufted interneurons send their (afferent) axons to the lateral olfactory tract. These axons synapse further in five different areas of the olfactory cortex. It is well known that olfactory sensitivity can vary more than a factor of 103 between normal individuals. Some individuals are totally unable to smell certain odors, while others can. As persons age, there is a loss of threshold sensitivity for certain odors. Certain odors such as the “rotten egg” scent of H2S can be sensed in concentrations of several parts per trillion. The noses of trained dogs still remain the most reliable and most sensitive for detecting drugs and explosives. As noted, the feats of bloodhounds in tracking human trails is legendary. Certainly, there is a long way to go to invent an artificial nose with the sensitivity of a dog or a bear. That olfaction in invertebrates is equally important in determining survival is discussed in the following section.
2.2.2
OLFACTION
IN
ARTHROPODS
The first part of this section addresses olfaction in insects. Insects have evolved very efficient olfactory chemoreceptors with which they locate food, congenial substrate, and other insects of the opposite sex. Insect chemoreceptors are found all over their bodies; they are principally found on their antennae, but also on their heads, labial palps, legs, and feet. They take a variety of shapes, including the sensillum placodeum (olfactory plate) found in the wasp Vespa and the honeybee; the sensillum basiconicum (olfactory cone) of Vespa, Locusta, and Necrophorus vespillio; the sensillum trichodeum (chemosensory hair) of Antheraea pernyi and Amorpha; the sensillum ampullaceum of the bee; the sensillum rhinarium from the antenna of Drepanosiphum (Homoptera); a chemosensory bristle from the wax moth, Galleria; a long sensillum basiconicum from a grasshopper. A collection of these insect chemosensors is illustrated in Figure 2.2-3. Note that some insect olfactory sensilla have only one sensory cell (e.g., sensillum ampullaceum in bees), others have two receptors (sensillum trichodeum in lepidoptera), while others have many receptors (sensillum placodeum of bees). The first insect sex attractant (pheromone) was identified by Butenandt in 1959 for the silkworm moth, Bombyx mori L., and named bombykol. It has been discovered that Bombyx females actually emit three sex attractant pheromones: bombykol [(E,Z)-10,12-hexadecadien-1-ol], bombykal [(E,Z)-10,12-hexadecadienal], and (E,E)-10,12-hexadecadien-1-ol. At present, the sex pheromones of more than 1300 insect species have been identified. These pheromones, which number in the hundreds, have significant structural homologies (Pherolist, 1999). The male Bombyx senses the pheromone with “tuned” receptors in hairs on its rather elaborate antennae. A male Bombyx may have as many as 2.5 × 104 hairs on each antenna. Each hair contains the dendrites of two bipolar chemoreceptor cells
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e
b
vacuole
pore
sheath
dendrite
cilium
cuticle
d
a
c
trichogen cell
epidermal cell
receptor cell
tormogen cell
basement membrane
axon
glia cell
FIGURE 2.2-3 Insect chemoreceptors: (a) sensillum trichodeum (chemosensory hair); (b) sensillum basiconicum; (c) sensillum coeliconicum of Locusta; (d) sensillim placodeum of Apis; (e) peg-shaped insect olfactory sensillum. (From Schneider, D. and Steinbrecht, R.A., in Invertebrate Receptors, McCarthy, J.D. and G.E. Newall, Eds., Academic Press, New York, 1968. © Academic Press. With permission from Academic Press.)
at its base; one neuron has receptors for bombykol, the other for the related compound bombykal (Yoshimura, 1996), and probably some may respond to (E,E)10,12-hexadecadien-1-ol, as well. When the female moth emits her sex pheromones, they travel downwind in an elongated plume. Airflow can be laminar or turbulent, depending on wind velocity; thus, the concentration of the pheromones as a function of distance from the female can decrease monotonically from diffusion effects (in slow, laminar airflow), or be “noisy” due to air turbulence. It appears that the male moth does not follow a simple scent gradient toward the female “emitter”; the pheromone concentration as a function of distance and time is too noisy. Instead, the male moth is stimulated by the reception of the pheromone to fly upwind (anemotaxis) in a sinusoidal, zigzag path to right and left of the wind vector. If the moth loses the odor plume, its flight behavior is modified to a search pattern of larger zigzags of up to 90° to the wind vector (the moth may even fly downwind) in an attempt to regain the scent. When it regains the plume, it again flies upwind in the smooth, zigzag pattern (Reike et al., 1997). The molecules of bombykol, bombykal, and (E,E)-10,12-hexadecadien-1-ol are shown in Figure 2.2-4. They are 16-carbon, long-chain, lipidlike molecules. It is estimated that the binding of only 20 bombykol molecules to their receptors within 100 m. will produce a behavioral response in a male B. mori. There are many insects that are important because they present human health hazards (e.g., Anopheles mosquitos → malaria; tsetse flies → sleeping sickness; certain mosquitos → Eastern equine encephalitis; deer ticks → Lyme disease, etc.), or they damage crops or trees (many, many examples). The fact that pheromones can be used to catch and kill harmful insects or to confuse their mating
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FIGURE 2.2-4 Basic structure of the silkworm moth pheromones, bombykol, bombykal, and (E,E)-10,12-hexadecadien-1-ol. Each black ball is a carbon atom.
behavior has stimulated research in isolating these molecules from the insects in question, as well as leading to investigations on the molecular mechanisms of pheromone detection and olfactory cognition. Insects are also covered with other chemoreceptors (on legs, labial palps, etc.), which sense environmental molecules that enable them to select the correct food or to lay their eggs in the correct plant (or animal). The tsetse fly (Glossina sp.) spreads sleeping sickness in central Africa (e.g., Zimbabwe). One means of poisoning these flies is to emit bait odors emulating cattle, which is their prey (other than humans). It has been discovered that tsetse flies locate their prey by optomotor-steered, upwind anemotaxis, following attractive kairomones. The main kairomones are 1-octen-3-ol, acetone, 3-methylphenol, 4methylphenol, and CO2. The methylphenols were isolated from ox urine, and CO2, octenol, and acetone are found in cattle breath (Spath, 1995). Fruitflies (Drosophila sp.) are perhaps an ideal model system in which to study olfaction. The maxillary palp of this insect contains only about 60 olfactory sensilla trichodeum, each with a pair of sensory neurons. The neurons fall into six functional classes, so, theoretically, there could be 21 different combinations of receptors in the hairs. The situation is made even more complex by the fact that a certain odorant can excite one class of receptor and inhibit another, and a particular receptor can be excited by one odor and inhibited by another. Thus, a complex odor composed of two or more odorants can produce a unique pattern or spatial distribution of receptor axon activities. The CNS of the insect must sort out this spatiotemporal pattern and generate an appropriate response (behavioral, biochemical, etc.) (de Bruyne et al., 1999). Some questions that should be asked: Do fruitflies carry a genetically determined neural template for the odors in their repertoire? Is there a multidimensional, AND operation between the odor-generated olfactory pattern and the stored template? Even olfaction in fruitflies appears very complex, in spite of their size. Perhaps researchers should try to find a simpler model system.
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Threshold chemoreception of dissolved sugar by the fly, Calliphora sp., can be determined behaviorally by observing whether the fly extends its proboscis (as if to feed) when a drop of sugar solution is presented to the labellar taste hair. Such behavioral experiments have led to the determination of a threshold sensitivity of 2.5 mM for dissolved sucrose. (The human threshold is 20 mM, while the butterfly Dania can sense an amazing 8 µM.) The trichoid sensillae on the fly’s labella also sense salt, water, and touch (Miller and Thompson, 1997). Chemoreception (underwater olfaction) has also been extensively studied in lobsters and crabs. Lobsters are addressed here. Olfactory-related behavioral and nervous responses of both male and female lobsters of two species have been studied extensively (e.g., Homarus americanus and Panulirus argus). The olfactory chemoreceptors of these lobsters appear to be located principally on their antennules. They are called aesthetasc sensillae; each sensilla is innervated by an average of about 300 chemosensory neurons. A lobster has two main antennae (left and right), which project forward from universal pivots that allow them to be oriented forward, to the side, or to the rear over the animal’s back. Medial to the main antennae, projecting forward from either side of the rostrum, are the shorter, paired antennules (two right, two left). Other parts of the lobster’s head and mouth contain other types of chemosensors that are probably used to sense food. By ablating the antennules on male and female Homarus lobsters and observing their mating behavior, Cowan (1999) was able to show that both male and female lobsters emit (perhaps in their urine) pheromones and that these chemical messengers affect behavior of the opposite sex. Male Homarus is normally agonistic toward members of its own species. That is, individuals compete for living space, food, and dominance. As with other animals of diverse species, a male Homarus strives for dominance in its territory, i.e., to become a dominant male. A dominant male evidently secretes a pheromone that advertises this fact. Female lobsters follow this pheromone trail upcurrent to the den of the dominant male. The female evidently secretes another pheromone that advertises her availability and inhibits the male’s normally agonistic behavior toward her. She lives in his den with him (cohabits) for a few days until she molts her shell. At this point, she is food for any marine predator because of her soft body. The male shows admirable restraint, and guards her from predators during this vulnerable period; he does not touch her. Once her new carapace has partially hardened, she mates with the male; she remains in his company for a few days before going on her way. Cowan showed that removal of the antennules altered this normal courtship behavior, demonstrating that it was indeed regulated by the reception of chemical messengers from the opposite sex. It is known that the California spiny lobster, P. interruptus, emits an aggregation pheromone that causes other solitary lobsters to gather in a group around it (ZimmerFaust et al., 1985). Such a pheremone, if isolated, could be used to increase the efficiency of baited traps. The chemoreceptor (aesthetasc) sensillae on lobster antennules have been shown to respond to various chemicals. Cromarty and Derby (1997) showed that isolated, individual chemoreceptor cells detected (at least): taurine, β-alanine, hypotaurine, L-glutamate, glycine, proline, cysteine, NH4Cl, and adenosine-5′-monophosphate
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(AMP). Some of the amino acids listed may be associated with damaged muscle, i.e., a food source. Cromarty and Derby concluded that individual chemoreceptor neurons from aethetasc sensillae express at least two types of receptors mediating excitation; one principal receptor class giving the strongest response and the greatest sensitivity to the principal input substance, and a second, minor class that also acts in an excitatory manner but which is present in a lower density on the receptor cell or which has a lower affinity to a second input substance. Other invertebrates such as nematodes and even flagellated bacteria exhibit chemotaxis in response to a chemical “field” gradient. Space limitation prevents describing these interesting systems. The interested reader will be impressed by the plethora of information on these topics by doing World Wide Web searches.
2.2.3
DISCUSSION
This section has discussed at the mammalian olfactory system and chemoreception in insects and crustaceans. The vertebrate olfactory system is most sensitive in certain species such as dogs (e.g., bloodhounds), bears, and predators such as wolves. Indeed, the nose of a trained dog can find hidden drugs or explosives better than any existing machine made by humans. Arthropods were seen to make extensive use of surface, airborne, or waterborne chemical messengers to attract mates, mark territory, and establish paths to food. While mammalian chemoreception of external stimuli resides in the nose and tongue, arthropods have chemoreceptors all over their bodies (antennae, labial palps, legs, feet, etc.). One of the problems in modeling chemoreception is that the exact chemical details of the processes, including by what reactions the odorant molecules are broken down is not known. Although it is known that mammalian olfactory neurons, unlike all other neurons, undergo programmed cell death (apoptosis) every month or so, we do not know whether the new receptors that grow to replace them have the same receptor specificity for odorants. If they do not, how is the neural “wiring” in the olfactory lobe reconfigured to accommodate the sensitivity of the new cell? Here is a ripe area for neural modeling to test hypotheses resulting from anatomical and neurophysiological (wet) work.
2.3
MECHANORECEPTORS
Mechanoreceptors are specialized sensory neurons that transduce the mechanical parameters of force or stretch of muscle fibers or tendons to spike frequency (muscle spindles, Golgi tendon organs, Mytilus anterior byssus retractor muscle stretch receptors). They also sense the bending of specialized hairs (trichoid hairs of arthropods, tricobothria, tricholiths) in a particular range of directions, or the angle of one limb segment with respect to the next segment (chordotonal organs). Other mechanoreceptors respond to the fluid pressure directly surrounding the sensor (mammalian pacinian corpuscles), or the stretch of an elastic walled vessel in response to internal fluid pressure (mammalian baroreceptors).
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Whether a mechanoreceptor neuron is said to respond to stretch, force, or pressure depends largely on the mechanical design of the supporting connective tissues to which the receptor is attached, or in which it is embedded. A mechanosensory neuron by itself is soft and has a great deal of compliance (little stiffness). So that the neuron will not be torn apart by relatively large forces and displacements, it is generally associated with tough, elastic tissue that is effectively in parallel with it and that protects it mechanically. For example, force sensing is generally done under relatively isometric conditions. That is, there is little compliance in the tendons that attach a muscle to its origin and insertion; the tendon behaves like a very stiff spring. The terminal arborizations of the force-sensing neurons attach to the collagen fibers of the tendon and thus experience little physical displacement themselves, enough, however, to produce a firing rate proportional to the total force on the tendon. Another strategy is used to sense muscle length. A relatively stiff fiber is embedded in the muscle in parallel with its fibers, so that when the entire muscle is passively stretched to a new length, the embedded fiber undergoes a small elongation. It is this small elongation that is sensed by a stretch mechanoreceptor. The stretch mechanoreceptor stops firing when the muscle generates internal force and shortens. In the vertebrate spindle organ, one will see that the stiff fiber is a specialized muscle fiber that can be made to shorten to “take up the slack” when the entire muscle shortens, thus maintaining the sensitivity of the length receptor over a wide range of muscle length. Muscle length receptors generally generate a spike frequency signal that has a strong derivative component in it, as well as being sensitive to the stretch, ∆L. The terminal processes of mechanoreceptors responding to force, length, or pressure have specialized, strain-gated ion channels that admit Na+ (or some other cation), which depolarizes the neuron and causes the SGL to generate a spike output. Below, some important vertebrate and invertebrate mechanoreceptors are described. Note that there are many more mechanoreceptors in nature than there is room to describe in this chapter. The examples chosen are interesting and cover a wide range of stimulus modalities.
2.3.1
INSECT TRICHOID HAIRS
One of the most plentiful and simple mechanoreceptors is the ubiquitous trichoid hair mechanoreceptor found on all insects (and often on their larvae) in great numbers. Trichoid hairs are exosensors; they are deflected by direct touch, air currents, low-frequency sound, or a nearby static electrical charge of either polarity. Patches of trichoid hairs also act as external proprioceptors, advising the animal of the relative position of its head with its thorax, and the positions of its upper legs (coxae) with respect to the body (Schwartzkopff, 1964). Trichoid hairs are located all over insect bodies, including legs, abdomen, thorax, wings, head, and antennae. Orthopteran insects such as cockroaches have twin, spikelike appendages protruding from their anal region, called cerci (plural of cercus). Each cercus is covered with trichoid hairs, and in some species, other specialized mechanoreceptors (which may have evolved from hairs) that sense the direction of the gravity vector (see Section
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2.6). Some of the trichoid hair sensillae on insect antennae also carry chemosensor cells and thus serve a dual exosensory role for the insect. A trichoid hair has a specialized, elastic socket in the body surface cuticle that in the absence of external force maintains the hair erect from the body surface. One or two sensory neurons send thin, specialized, distal processes that attach to the movable base of the hair. When the hair bends in its socket, for whatever reason, the tips of the distal processes are stretched. This stretch induces a membrane depolarization, which spreads electrotonically to and over the soma of the sensory neuron, to the SGL on the axon where nerve spikes are initiated. The firing frequency is generally proportional to the rate of hair deflection plus its deflection. Always found in association with the monopolar sensory neuron(s) of a trichoid sensilla are two types of specialized epidermal cells: trichogen cells, which embryonically secrete the specialized cuticle of the hair, and tormogen cells, which secrete the cuticle forming the flexible socket. Figure 2.3-1 illustrates a schematic cross section through a typical (generic) insect mechanosensory hair. The diagram of another sensory hair system on the caterpillar of Vanessa urticae is shown in Figure 2.3-2.
FIGURE 2.3-1 Schematic cross section through a typical insect hair plate sensillum. A single mechanosensory cell sends a thin process into the hollow core of the movable hair. The tormogen cell secretes the elastic cuticle that supports the hair and allows it to bend. The trichogen cell secretes the specialized cuticle that forms the hair. Deflection of the hair strains the distal process of the sensory cell and leads to depolarization and spikes. (Not shown are the basement membrane cells, glial cells around the sensory cell, and supporting tissues.)
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a
b l c k
d
i
e f
h
g
FIGURE 2.3-2 Schematic of cross section through the base of a tactile hair from the caterpillar of Vanessa urticae. Parts: a, base of hair; b, articular membrane; c, sensitive process with scolops; d, sense cell; e, sheath cell (neurilemma); f, cuticula; g, trichogen cell; h, basal membrane; i, tormogen cell, k, vacuole; l, hypodermis. (From Schwartzkopff, J., in The Physiology of Insecta, Vol. 1, M. Rockstein, Ed., Academic Press, New York, 1964, © Academic Press. With permission from Academic Press.)
In addition to body part proprioception, trichoid sensillae found on the heads of flying insects provide the creature with information about aerodynamic speed and air direction, and thus can probably sense flight instability such as yaw. Because an insect hair is an electret, i.e., it has permanent bound charges on its surface, the near approach of any other charged or conducting object will attract or repel the hair, producing a sensory output. This property can be of use to insects like cockroaches in avoiding predators in the dark.
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2.3.2
INSECT CAMPANIFORM SENSILLA
Like trichoid hair sensors, campaniform sensilla (CS) are widely distributed over the bodies of terrestrial insects and chelicercates (scorpions). CS are proprioceptors, however, sending information to the insect CNS about mechanical strains present in the surrounding cuticular exoskeleton. They are found in groups in various locations on an insect’s body where it is important to monitor the forces acting on certain body parts. For example, they are found in two groups on the subcosta of each of the locust’s forewings, and in one group on each subcosta of the hind wings. When the CS are destroyed or blocked electrically, reflex control of wing twisting (required for normal flight) and body orientation during flight are abnormal. The CS of the hind wings are necessary for the regulation of forewing twisting during constant-lift, and those of the forewing for the maintenance of stability of the body about its three axes (Finlayson, 1968). A more static application of CS is on the various segments of the legs of all insects. On the trochanter of the cockroach leg there are about 70 CS, located in three ventral and one dorsal group. The CS sense strain in the leg cuticle; hairs and other interoceptors monitor the positions of the parts of a leg. Finlayson (1968) states, “Probably the major function of the campaniform sensillum is to register stresses produced (a) by the weight of the insect’s body on the limb, (b) by the resistance of the cuticle to the actions of the muscles and (c) by external agencies that tend to alter the normal spatial relationship between different regions of the legs, at rest or in motion, and the body.” Since most insects lack specific gravity receptors, it is apparent that the CS figure in that important role, albeit indirectly. DiCaprio et al. (1998) examined the nonlinear transfer function of CS in the cockroach tibia using the white noise method (see Section 8.8). Another insect organ that is well populated with CS is the haltere (see Section 2.7). Halteres are organs evolved from the hind wings of dipteran insects that serve as vibrating gyroscopes. It can be shown that any departure from level flight (i.e., roll, pitch, or yaw) will produce torques on the bases of the halteres at twice the vibration frequency. It is known that the bases of each haltere are well endowed with sensors including hairs, chordotonal organs, and CS (Bullock and Horridge, 1965). It is likely that the CS measure the gyro torques on the halteres and send this information to the fly’s flight control center. A CS has one ciliary-based sensory cell located under the cuticle of the insect exoskeleton. The sensory cell sends a modified ciliary microtubule distal process through a conical opening in the cuticle to make contact with the cap, which is held in a craterlike socket on the outside of the exoskeleton. Scanning electron micrographs show the cap to be ellipsoidal, seen from the top. The ciliary distal process of the sensory neuron is covered with a thick tubule that makes contact with the cap. Thus, shear forces in the cuticle that distort the cap (presumably more on one axis because of its noncircularity), presumably impart a microdisplacement to the ciliary microtubule distal process, leading to depolarization and spikes on the axon. Figure 2.2-3 illustrates a schematic cross section through a CS. The crater diameter is about 20 to 25 µm. Similar to insect hair cells, there are supporting cells surrounding the sensory cell under the cuticle.
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FIGURE 2.3-3 Highly schematic cross section through a campaniform sensillum. The “crater” that holds the cap plate is about 20 µm in diameter. Insects use campaniform sensilla to measure shear forces in their exoskeletons.
2.3.3
MUSCLE LENGTH RECEPTORS
One of the more interesting mechanoreceptors known to physiologists and to biomedical engineers is the vertebrate muscle spindle. This organ serves to advise an animal’s motor control system of the length of an individual muscle and the rate of change of its length (rather than the force acting on the muscle). The spindle is particularly interesting because it is an enteroreceptor that operates under feedback control that preserves its sensitivity regardless of muscle length.
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Figure 2.3-4 illustrates a spindle schematically. The entire spindle is 4 to 10 mm in length, and contains from 3 to 12 specialized intrafusal muscle fibers. There are three types of intrafusal muscle fibers: (1) a dynamic nuclear bag fiber; (2) a static nuclear bag fiber; (3) one or more nuclear chain fibers (see Kandel et al., 1991, Ch. 37). The contractile states of these muscle fibers are controlled by two motor nerves: a dynamic gamma motor axon (γd) (innervates the dynamic nuclear bag fiber), and a static gamma motor axon (γs) (innervates the static nuclear bag fiber and the nuclear chain fibers). (γ refers to the diameter classification of the motor fibers; they are often called fusimotor fibers, as well.) The centers of the intrafusal muscle fibers of the spindle are noncontractile, elastic connective tissue. Thus, the γ motor innervation is applied to both ends of the intrafusal fibers.
FIGURE 2.3-4 Schematic of a muscle spindle of about 4 mm in length. It is nestled between the main bundles of fibers of the extrafusal muscle (EFM) and connected to them by fascia. Note the two γ-motoneutrons that stimulate the intrafusal muscle fibers (IMF) of the spindle, whose output nerves are the type Ia and II afferents shown. See text for more details. (From J.R. LaCourse,48 with permission.)
Wrapped around the centers of each fiber are the dendritic endings of the sensory neurons. The sensory endings of the single, primary, type Ia, afferent fiber (annulospiral endings) make contact with all three types of intrafusal muscle fibers. However, the sensory endings of the type II, secondary fiber (flower spray endings) make contact mostly with the centers of the nuclear chain fibers and also the center of the static nuclear bag fiber. Surrounding the γ motor end plates and the sensory endings of the type Ia and II spindle afferent nerves is a protective, fluid-filled capsule. The whole spindle is attached by connective tissue in parallel with the normal extrafusal fibers of a muscle. When the muscle is stretched (lengthens), so is the spindle to a lesser degree. The stretching of the intrafusal fibers stimulates the sensory nerves to spike. When the extrafusal muscle shortens (contracts actively), the spindle fibers go slack and the spike activity on the type Ia and II sensory nerves goes to zero. However, a spindle normally operates under closed-loop conditions so that the CNS activates the γ fusimotor fibers © 2001 by CRC Press LLC
when the α motoneurons fire, which causes the intrafusal muscle fibers also to shorten and again pick up some degree of tension, restoring the output of the spindle and its sensitivity to small length changes. The CNS can modulate the γ fusimotor fiber activity and thus adjust the spindle sensitivity to ∆x and dx/dt at a given muscle length (xo). The γs activity alters the sensitivity to ∆x on the group II afferent, and γd input affects the type Ia fiber response to dx/dt. More will be said of this control system below. The number of spindles per muscle volume varies considerably with the type of muscle. For the cat, Granit (1955) gives 45 spindles for the big gastrocnemius medalis, vs. 56 and 57 in the considerably smaller soleus and tibialis anterior muscles, respectively. The number of spindles may be determined by the fineness of motor control required by the muscles considered. If this is true, then muscles associated with fine hand movements and movements of the tongue ought to have a larger density of spindles than skeletal muscles associated with maintaining posture. Primate and ungulate extraocular muscles have very high densities of spindles, as do neck muscles. The role spindles in extraocular muscles and in neck muscles in controlling the tracking of visual objects has yet to be described. Two events stimulate the output of the primary, Ia, afferent fiber of a spindle: (1) an increase in the length of the extrafusal muscle, causing stretch of the sensory endings; and (2) at constant extrafusal muscle length, stimulation of the fusimotor (γ) motor efferent fibers, which causes the intrafusal muscle fibers to shorten, creating tension. Both events stretch the sensory endings in the centers of the intrafusal fibers. Spindles only fire for increasing intrafusal fiber tension. The type II afferents fire with a frequency proportional to extrafusal muscle length; type Ia fiber frequency strongly depends on stretch velocity, with only a small length component. There are many experimental scenarios possible in the study of spindle dynamics. Note that a spindle is a three-input [x(t), γd and γs activity], two output [Ia and II afferent fiber frequencies], nonlinear system. To begin with, one can do an openloop experiment with an isolated spindle.The extrafusal muscle is stretched while keeping γd and γs activity at zero and recording from the Ia and II afferent fibers. Such a scenario is shown schematically in Figure 2.3-5A and B. The stretch is repeated with only the dynamic, γd, motoneuron stimulated. Note that the increased tension on the dynamic nuclear bag fiber produces a significantly enhanced response on the Ia sensory neuron for the same stretch. There is also a slightly enhanced type II sensory fiber response as well. A second experiment on the isolated (open-loop) spindle involves stimulation of the nuclear chain fibers via the γs intrafusal motoneuron, while recording from the type Ia and type II afferents and stretching the extrafusal muscle. The results of this protocol are illustrated schematically in Figure 2.3-6A and B. The chain fibers bias the discharge of the type Ia and II afferent fibers, alter length sensitivity of the type Ia fibers, and increase the length sensitivity of the type II fibers. In lower vertebrates such as amphibians, the γ fusimotor fibers are simply branches from the α motoneurons. In this basic scheme, the intrafusal muscle fibers are stimulated and shorten along with the skeletal muscles, maintaining their sensitivity to ∆x and dx/dt; there is no feedback per se. In mammals, as has been seen,
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FIGURE 2.3-5 (A) Spindle type Ia afferent instantaneous frequency in response to extrafusal muscle length change (stretch) with and without open-loop, γd motor stimulation. (B) Spindle type II afferent instantaneous frequency in response to extrafusal muscle length change (stretch) with and without open-loop, γd motor stimulation.
spindles receive separate γ fusimotor innervation and thus their sensitivities can be modulated by the CNS independently of the α motor signals. It is interesting to examine the behavior of a theoretical mechanical model for an intrafusal muscle fiber, such as the static nuclear bag fiber. The two series polar regions of the fiber can be represented by a single polar region muscle model consisting of a force source in parallel with a linear constant internal viscosity D (a
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FIGURE 2.3-6 (A) Spindle type Ia afferent instantaneous frequency in response to extrafusal muscle length change (stretch) with and without open-loop, γs motor stimulation. (B) Spindle type II afferent instantaneous frequency in response to extrafusal muscle length change (stretch) with and without open-loop, γs motor stimulation.
dashpot symbol is used), in parallel with a linear elastic element, K1. (In reality, the viscosity and elastic element are functions of the muscle state of activity; however, for simplicity, they are considered constants.) The polar regions of the intrafusal muscle fiber are in series with the purely elastic, center nuclear bag region in which the “flower spray” endings of a type II sensory neuron are attached. It is assumed that the firing frequency, r2, of the type II fiber is proportional to the stretch of the center region. That is, r2(t) = Ko(x2 – x1). Here x2 is the actual stretch of the spindle, and x1 is the length change of the polar regions of the fiber. The system is shown in Figure 2.3-7. Below is derived a mathematical model for the mechanical behavior of the simple spindle model. The Newtonian force equilibrium is
fi + Dx˙ 1 + K1x1 = K 2 (x 2 – x1 )
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2.3-1
FIGURE 2.3-6
(Continued).
By Laplace transforming and solving for X1(s):
X1 (s) =
K 2 X 2 (s) – Fi (s) sD + (K1 + K 2 )
2.3-2
The difference, X2 – X1, is just
⎡ ⎤ Fi K 2 X1 X 2 – X1 = ⎢ X 2 – + ⎥ sD + K1 + K 2 sD + K1 + K 2 ⎦ ⎣
2.3-3
The output frequency is thus the sum of two terms, one a function of spindle stretch, x2, and the other a function of the force, fi, generated by the intrafusal fiber when activated.
R 2 (s) = K o ( X 2 – X1 ) = X 2 (s)
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K o (s + K1 D)
s + (K1 + K 2 ) D
+
K o Fi (s) D s + (K1 + K 2 ) D
2.3-4
FIGURE 2.3-7 An oversimplified, lumped-parameter mechanical model for the static nuclear bag portion of a spindle. The polar regions of the IFM are represented by a dashpot (viscosity) in parallel with a linear elastic element in parallel with a force source activated by the frequency on the γ motoneuron. These parallel elements, in turn, are in series with a linear elastic element representing the nuclear bag region of the IFM. The output instantaneous frequency on the model type II fiber is proportional to the stretch of the nuclear bag “spring.”
In the first case, let Fi be a constant, Thus,
R 2 (s) = X 2 (s)
K o (s + K1 D)
s + (K1 + K 2 ) D
+
Fi K o K1 + K 2
2.3-5
The second term is a bias firing rate or “tone.” Now let the input be a step stretch: X2(s) = X2o/s. The first term is of the form:
⎧ s+a ⎫ (a – b) – bt L–1 ⎨ e ⎬=a b– b ⎩ s(s + b) ⎭
2.3-6
Thus, the time domain response of the model to a step stretch is r2(t) = [Ko X2o K1/(K1 + K2)]{1 + (K2/K1)exp[–t(K1 + K2)/D]} + Ko Fi/(K1 + K2)
2.3-7
Note that r2(t) cannot be negative. Figure 2.3-8 illustrates the model output fiber frequency given a step stretch of the spindle followed by a step return to zero stretch. Note that the initial overshoot in r, which indicates a degree of rate sensitivity, comes from the viscoelastic properties of the intrafusal fiber model. Figure 2.3-9 illustrates the block diagram of a simple theoretical type 0 neurophysiological control loop postulated to describe the CNS feedback mechanism
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FIGURE 2.3-8 The instantaneous frequency of the type II fiber of the model of Figure 2.3-7 when subject to a step stretch, X20, followed by a return to zero stretch.
FIGURE 2.3-9 Block diagram of a theoretical type 0 control loop for a spindle designed to keep the steady-state output instantaneous firing rate, r2, constant. Note that the CNS output to the spindle γs motor fiber is proportional to some re = ro – r2, re = 0 if (ro – r2) < 0.
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driving the intrafusal, γs motoneuron that acts to keep the static basal firing rate of the spindle type II afferent constant regardless of changes in muscle length. The type II afferent output acts inhibitorily in the CNS at a “neural differencer,” which is also excited at a fixed rate, ro. The differencer output rate, re, ultimately determines the firing rate on the γs fiber to the spindle, hence the static tension of the spindle. The steady-state dc gain of the feedback system is easily shown to be
r2 SS = x 2 o
K1K o ( K1 + K 2 )
1 + K CNS K m K o (K1 + K 2 )
+ ro
K CNS K m K o (K1 + K 2 )
1 + K CNS K m K o (K1 + K 2 )
2.3-8
Assume 1 Ⰶ KCNS Km Ko/(K1 + K2). Then, r2ss ≅ x2o K1/(KCNS Km) + ro
2.3-9
In other words, ro produces a tonic bias firing rate to which signals from spindle stretch are superimposed. It allows the spindle to signal shortening of the spindle as a reduction in the tonic firing rate, as well as stretch as an increase in type II fiber frequency. Again, note that the firing frequency, r2, is non-negative. Animals such as arthropods and mollusks also need muscle length and force proprioceptors. LaCourse (1977a, b) decribed mechanoreceptors located among the smooth (i.e., nonstriated) muscle fibers of the anterior byssus retractor muscles (ABRM) of the plecypod mollusk, Mytilus edulis L. (Mytilus is the common blue mussel that can be found in the intertidal zone along the coast of New England.) A Mytilus has two ABRMs, which are used to maintain tension on a web of tough biopolymer threads spun by the animal to support itself on its habitat substrate (rocks, pilings, etc.). The muscle is unusual because it is a smooth muscle, and because it exhibits the property of “catch.” Catch is basically a controlled state of rigor whereby an ABRM contracted under load maintains its stiffness and shortened length without additional motor nerve input. There is evidence that the ABRM is unique in that it has a relaxing motor nerve (RMN) system that, when activated, releases a neurotransmitter that causes the prompt termination of the catch state (Northrop, 1964). (The relaxing nerves are not the same as inhibitory motor nerves that prevent contraction.) If no further motor stimulation is given, the ABRM relaxes upon stimulation of the RMNs. If both motor nerves and RMNs are simultaneously stimulated with a burst, the muscle produces a twitch; i.e., it contracts, then promptly relaxes without entering the catch state. Using methyene blue stain and Rowell’s silver stain (Rowell, 1963), LaCourse (1977a) identified structures in the ABRM that were similar in appearance to the annulospiral endings of vertebrate muscle spindles. Typically, a nerve fiber 0.5 to 1.1 µm in diameter forms a large spiral twisted around a smooth muscle fiber that is thinner (3.6 to 4.05 µm diameter) than a regular contractile smooth muscle fiber (4.0 to 4.5 µm diameter). Just before the axon joins the muscle, there is an apparent bipolar cell body about 3.5 to 4.1 m long by 0.75 to 1.8 µm in diameter. Figure 2.3-10 illustrates a section of the ABRMs and their associated nerves and ganglia.
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FIGURE 2.3-10 Schematic of the Mytilus two ABRMs, showing the superficial nerve trunks recorded from by LaCourse (1977a). Key to abbreviations: CG, paired cerebral ganglia; PG, paired pedal ganglia; VN, visceral nerves; CVC, cerebrovisceral connective nerve; x, point of recording with hook or suction electrode; , point where CPC nerve cut. (Courtesy of J.R. LaCourse.)
Neurophysiological recording was made from one isolated cerebropedal connective (CPC) nerve connected to an ABRM though the branches of the visceral nerves. The neurally unstimulated ABRMs were placed under tension by applying a force to the byssus threads in the direction in line with the ABRMs. The force was modulated by a perpendicular displacement of the tensioning wire imposed by a small loudspeaker. Recorded along with the sensory nerve spikes on the CPC nerve was the tension applied to the ABRMs. The muscle length followed the applied force according to the force–length curve of the unstimulated muscle (Figure 2.3-11). LaCourse (1977) found two major classes of mechanoreceptor response: those that fired for various ranges of increasing tension and those that responded for various ranges of decreasing tension. In general, ABRM mechanoreceptor responses were found to be nonhabituating. Mechanoreceptors did exhibit some rate sensitivity, however, as evidenced by adaptation to step stretch responses. LaCourse estimated the sinusoidal frequency response of ABRM mechanoreceptors to have its peak response at about 1 Hz. Response was down –40 dB at 2 Hz, and –7 dB at 0.1 Hz. It is known that externally applied 5-hydroxytryptamine (5HT) or dopamine (DA) in concentrations of 10–4 or 10–6 M are effective in causing a contracted ABRM in the catch state to relax. LaCourse found that externally applied DA at 10–4 or 10–6 M caused all mechanoreceptors to totally cease firing. After a seawater rinse, mechanoreceptor activity revived. Externally applied 5HT at 10–4 or 10–6 M temporarily depressed ABRM mechanoreceptor sensitivity for about 8 or 4 s, respectively. Sensitivity spontaneously returned without washing, suggesting that 5HT was enzymatically broken down in vivo. The anatomy of Mytilus mechanoreceptors suggests that they may behave similarly to spindles. However, unlike spindles, some are unique in firing faster for decreasing length or tension. Spindles evidently signal muscle shortening by decreasing their tonic firing rate. LaCourse (1977; 1979) proposed the theory that ABRM contraction and catch is mediated by motoneurons that release acetylcholine (ACh). The mechanoreceptors send spike signals back to the pedal ganglion regarding whether the muscle length
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FIGURE 2.3-11 Approximate shape of the force–length curve of an unstimulated ABRM.
is increasing or decreasing. He postulates that the RMNs release 5HT at their endings and that 5HT terminates catch. A third set of efferent neurons specifically innervates the stretch receptors. These “R-neurons” release the neurotransmitter DA, which decreases the sensitivity of the receptors. The activity of the R-neurons coupled excitatorily to the RMNs in the pedal ganglion, so that when R-neurons fire, 5HT is also released, producing phasic ABRM behavior. Another scenario is that the activity of the receptors that sense increasing ABRM length stimulate the contractile motoneurons locally, producing a load-resistant constant length control loop. What can be learned from the Mytilus ABRM stretch receptor story is that the simpler-appearing systems in nature turn out to have very complex physiology, challenging the modeler.
2.3.4
MUSCLE FORCE RECEPTORS
Sometimes the distinction between muscle force (tension) and length receptors is not so clear. Striated skeletal muscles generally have static force–length curves of the form shown in Figure 2.3-12. Below the peak of the stimulated tension curve, there is a monotonic relation between muscle force and length. If a mechanoreceptor is located in the relatively inelastic end of a muscle where it joins the tendon, it is generally said that the receptor is behaving like a force or tension receptor. A stretch receptor located in the more compliant body of a muscle generally behaves like a length receptor (e.g., a spindle). However, active length changes generally require the muscle to generate force internally, which must equal any external load plus internal viscous and elastic forces associated with the contraction. Because of the high number of actin–myosin bonds, active muscle not only generates tension and can shorten, but it is also stiffer and has increased viscosity. If a muscle is not stimulated (i.e., it is passive), then it behaves statically like a nonlinear spring in which receptor stretch is a function of the externally applied force. A heuristic test
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FIGURE 2.3-12 (A) The force–length curves of stimulated and unstimulated striated skeletal muscle. (B) Measurement setup to obtain the curves of A.
for a tension receptor is that it fires for an external force load with approximately the same sensitivity regardless of muscle length. The best known and most widely studied muscle force sensor is the Golgi tendon organ (GTO), found in mammals. The GTO is an encapsulated structure, about 1 mm long and about 0.2 mm in diameter. Its output is carried by a single myelinated nerve fiber about 16 µm in diameter. The naked (unmyelinated) dendritic endings of the GTO fiber arborize and twist vinelike between the twisted collagen fibers making up the fine structure of the tendon. There is one GTO for a functional group of about 10 to 15 muscle fibers. When the muscle is force-loaded in passive or activated condition, the force causes the tendon collagen fibers to elongate and squeeze together, generating shear forces on the GTO dendrite membrane. The GTO spike frequency in response to muscle force has a proportional plus derivative component (see Figure 2.3-13). GTOs generally fire more when a muscle is active at a given force than when it is unstimulated at the same force load.
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FIGURE 2.3-13 Schematic of approximate firing behavior of GTOs and spindles when a muscle is passively stretched and allowed to shorten under load (isotonic contraction).
The anatomy of a GTO is shown schematically in Figure 2.3-14. Note that the muscle end of the tendon divides to attach to small groups of muscle fibers, the sum of which comprise the entire muscle. A GTO therefore responds so its instantaneous spike frequency, r, is proportional to · r = Kp f + Kd[f]+
2.3-10
where f is the positive (tension) force on the muscle fibers in series with the GTO, · and [f]+ is the positive derivative of the force (the negative derivative is zero). The myelinated fiber from a GTO on a flexor muscle tendon is called a type Ib axon. It projects into one of the many dorsal root ganglia where its cell body is located. The Ib fiber continues into the spinal cord, where it excites a type Ib
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FIGURE 2.3-14 Schematic drawing of the sensory endings of a GTO in intimate contact with tendon collagen fibers in the GTO capsule. The GTO type Ib afferent axon fires when tension on the muscle forces the twisted collagen fibers to squeeze the GTO nerve endings. (From Kandel, E.R. et al., Principles of Neural Science, 3rd ed., Appleton & Lange, Norwalk, CT, 1991. With permission from the McGraw-Hill Companies.)
inhibitory interneuron, which in turn inhibits the flexor α-motoneuron that causes the tension on the GTO. The same GTO afferent excites a type Ib excitatory interneuron, which excites the extensor α-motoneuron. The net effect of this local GTO reflex arc is to unload the flexor muscle if the load is sensed as too high for it. This reflex system is shown in a simplified schematic in Figure 2.3-15. (Not shown in the figure are the ascending afferent fibers from the spindles and GTOs, and the sensory neuron cell bodies. Circular synapses are inhibitory; arrow synapses are excitatory.) Muscle force receptors are found throughout the animal kingdom. Eagles and Hartman (1975) described the properties of force receptors associated with the tailspine muscles of the horseshoe crab, Limulus polyphemus. Two classes of receptors were identified with light microscopy using methylene blue staining: (1) elaborately branched multipolar neurons (somas ~30 × 100 µm) associated with the flexor muscles where they insert upon the tailspine apodeme (tendon) and (2) smaller bipolar neurons (~20 µm diameter) located along the shafts of the tailspine apodemes bearing insertions of the flexor muscles. These receptors sent their axons to the
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FIGURE 2.3-15 Schematic of the role of the GTOs and spindles in the crossed, flexor/extensor reflex pathways. Key: R, Renshaw inhibitory interneuron; αF, flexor α-motoneuron; αE, extensor α-motoneuron; γaf, spindle length control motoneuron from CNS; FL, flexor load force.
animal’s CNS through a mixed nerve (motor plus sensory fibers) called hemal nerve 16 (h.n. 16). Eagles and Hartman showed indirectly that the above-mentioned neurons behaved as muscle tension receptors. They did not measure muscle force directly, but recorded from the receptor axons while stretching the muscle a known xo + ∆x. They also stimulated the motor nerves innervating the muscles having the receptors from which they recorded. In general, the larger xo, the larger the receptor response for a given ∆x. When the muscle was stimulated, there was a big increase in receptor firing, because of the greatly increased tension. Eagles and Hartman found no evidence that there were intrafusal (accessory) muscles associated with the Limulus receptors, such as are found in mammalian spindle muscle length sensors. Isometric tension generation also showed that the putative tension receptors would respond without an external length change.
2.3.5
STATOCYSTS
Statocysts are neuro-sensory organs that transduce the direction of the gravity vector relative to the animal’s body. Statocysts are found in a number of invertebrate phyla:
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Annelida (segmented worms and leeches), Arthropoda (crustaceans, but not insects), Coelenterata (jellyfish, sea anemones, etc.), and Mollusca (especially gastropods and cephalopods). They also respond to linear and angular acceleration of the animal’s body, and perhaps to very low frequency vibrations of the animal. The basic design of a statocyst is simple; it consists of a fluid-filled, roughly spherical cavity lined with closely packed mechanoreceptor cells with sensory hairs (cilia or setae). Also in the cavity is a statolith, consisting of one or more dense particles such as grains of sand, an accretion of sand grains stuck together to form a pebblelike mass, or an internally secreted calcified “stone.” When an animal is at rest, the mechanosensors in the statocyst respond to deflection of their hairs due to gravity pressing the statolith down on them and causing them to bend. If the animal undergoes roll or pitch, the statolith presses on a new region of sensory cells. The sense cells send nerve impulse data to the animal’s CNS to apprise it of its orientation in the Earth’s gravity field. Statocysts can also send data about acceleration; if the animal accelerates backward, such as a crayfish escaping a predator, the reaction force on the statoliths press them forward, presumably sending a false message that the animal is pitching head down. Angular acceleration around a statocyst produces complex forces acting on the internal hair cells. The statolith and endolymph both tend to remain at rest, so the hairs effectively move past them, deflecting. Once the fluid plus statolith gain the angular velocity of the animal, there are no further acceleration forces on the hairs until the angular velocity stops. Then the statolith and fluid tend to keep moving, again stimulating hair cells, this time in the opposite direction. Figure 2.3-16 illustrates a schematic section through generalized invertebrate statocyst. The nearly spherical cavity is lined with ciliated mechanoreceptor cells that respond to force from the statolith mass resting on them, or reaction force when the statolith is accelerated. In lobsters, the statocysts are located inside the basal joint of the left and right antennule. A crescentic sensory cushion for the statolith has four rows of hair cells, three of which are normally in contact with the statolith. In addition, there are thread hair cells located on the posteriormedial wall of the statocyst, which are apparently deflected by the flow of endocystic fluid past them as a result of angular acceleration. Lobster hair cells have been subdivided into three categories. Type I position receptors that respond to static pitch of the animal with a nonadapting spike frequency that is proportional to the angle of pitch. There is one statocyst nerve axon per hair cell, and each cell has a different range of peak sensitivity. Type I cells respond little to roll of the animal. Type II position receptors exhibit some adaptation to angle changes, and thus appear to be more angular velocity sensitive than Type I cells. Some Type II cells are sensitive to roll and others to pitch. The axons of the thin, “thread” hair cells on the medial wall of a statocyst fire when the endocystic fluid swirls due to yaw, pitch, or roll; they appear to signal the lobster angular acceleration information. Still other statocyst nerve fibers fire in response to underwater lowfrequency vibrations, or large angular accelerations (Bullock and Horridge, 1965, Ch. 18). Figure 2.3-17 illustrates the typical averaged firing rate of a Homarus (lobster) Type I statocyst fiber as the animal is pitched head down (90° is straight
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FIGURE 2.3-16 Schematic cross section through a representative simple statocyst. SC, ciliated mechanosensory cells; CAV, fluid-filled statocyst cavity; SL, statolith. This type of organ responds to any motion or force that deflects the cilia on the receptor cells. Note the canal connecting the cavity to the water outside.
down; 180° is upside down, facing to the rear). Note that the peak firing rate occurs for 85° tilt, indicating that the receptor recorded from is 5° from vertical. Coelenterates (jellyfish, comb jellies, anemones, medusae) have very primitive statocyst organs. They have few hair cells and one or a few statoliths. Annelids (segmented worms and leeches) also have elementary statocyst organs. Polychaetes (marine tube worms) have reasonably well developed statocysts with sand grain statoliths. Because these worms are sessile, and often live in turbid water, they use their statocysts for geotaxis. The phyllum Mollusca also has interesting statocysts. Wood and von Baumgarten (1972) recorded rotation and tilt responses from the 13 axons of the statocyst nerve of the marine gastropod mollusk, Pleurobranchaea californica. The two Pleurobranchaea statocycsts are located lateral to the left and right pedal ganglia within the connective tissue sheath that encloses the mollusk’s CNS. Each statocyst is only about 200 µm in diameter, and contains a single statolith about 150 µm in diameter. Figure 2.3-18 illustrates the response of a single statocyst nerve fiber to roll of the animal through 360°, both clockwise (CW, right-side down) and counterclockwise (CW, left-side down). Note that for CW rotation, there is a sharp
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FIGURE 2.3-17 A graph showing the representative firing frequency of a single Type 1 lobster statocyst position receptor. The curve shows the response of the receptor to head pitched down. At 180°, the lobster is upside down; i.e., its back is down. Peak response occurs when the lobster is pointing down at 84°. Other Type 1 statocyst receptors around the cavity will have frequency peaks at other angles, depending on their positions. This curve can be modeled by the equation: r(θ) ≅ 31.5 – 28.5 cos[2(θ + 6°)], where r(θ) is in pps and θ is the pitch down angle in degrees. (Original data from Cohen, 1955.)
peak in spike frequency at about 60°. Not surprising, the same CW unit has a minimum response at about 60° + 180° = 240°. However, CCW rotation caused the same axon to fire maximally at about 160o of roll with a broad null at 340°. Note that the maximum firing rate of the statocyst nerve is only 5 pps. Nothing happens rapidly in gastropods. In cephalopod mollusks (octopi and squids), the statocysts are well developed, each with four differentiated nerves projecting to the CNS (Bullock and Horridge, 1965, Ch. 25). One would expect the statocysts of cephalopods to be highly developed to provide the animals, which are agile swimmers, not only with gravity vector information but also neural information on linear acceleration, as well as signals on angular roll, pitch, and yaw and their derivatives (Young, 1960). Octopus statocysts are large, endolymph-filled cavities (~4 mm diameter in a 0.5-kg octopus), and are richly endowed with sense cells. The statolith is located in an ellipsoidal macula inside the statocyst structure. The macula has many hair cells. The details of their anatomy will not be described here, but it is worth noting that the sense cells in cephalopods have interneurons and lateral connections reminiscent of the neuropile underlying the Limulus compound eye and the lamina ganglionaris of compound eyes. Perhaps lateral inhibition acts here, too, to sharpen mechanical response. Also unique in the complex statocysts of the octopus is the presence of efferent nerve fibers, suggesting some sort of CNS feedback control of statocyst responses. Clearly much neurophysiology has yet to be done to understand the sense cell interactions and the role of efferent signals on the dynamic behavior of the octopus statocysts.
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Position in degrees FIGURE 2.3-18 Response of a single fiber from the statocyst nerve of the gastropod mollusk, Pleurobranchaea californica to roll around the longitudinal axis. Solid curves, average frequency in the first 10 s following position change. Dotted curves, average frequency in the period 110 to 120 s following position change. (From Wood, J. and von Baumgarten, R.J., Comp. Biochem. Physiol., 43A: 495, 1972. With permission from Elsevier Science.)
The fact that statocysts of various degrees of complexity have apparently evolved independently in a number of invertebrate phylla argues for the robustness of its design. Also, the more complex (in terms of degrees of freedom and speed) the animal’s means of locomotion, the more complex its statocysts (e.g., lobsters, octopi). Finally, compare the statocyst with the gravity receptors of the cockroach Arenivaga, in Section 2.6.
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2.3.6
PACINIAN CORPUSCLES
Pacinian corpuscles (PCs) are found in vertebrates. They are an excellent example of a phasic mechanosensory neuron that responds approximately to the absolute value of the time derivative of pressure applied to tissues surrounding the corpuscle. The exterior of the PC is an ellipsoid about 500 µm in length and 250 µm in diameter. The largest PCs are about 4 × 2 mm. The ellipsoid is made up from many thin cellular layers or lamellae, much like the structure of an onion. At the center of the corpuscle there is a cylindrical viscous fluid-filled region surrounding the tapered bare ending of the sensory axon. See Figure 2.3-19 for a schematic of a PC with its myelinated axon. Myelin beads are seen surrounding the axon at the distal end of the capsule and running down the axon to the spinal dorsal root ganglion where the soma of the receptor is located. The PC axons are from 8 to 14 µm in diameter, and conduct spikes at 50 to 85 m/s. They are classified as type Aβ fibers (Guyton, 1991). The SGL for the PC is at the first node of Ranvier. A electron micrograph crosssection through a PC perpendicular to the axon is shown in Figure 2.3-20; note the many lamellae. The axon is in the center of the lamellae.
FIGURE 2.3-19 Schematic cross section through a PC. The onionlike lamellae are thought to give this pressure its highly phasic rectifying response.
PCs are located in deep visceral tissues (mesentaries), the joints, as well as in the hands and feet. Their spatial resolution is relatively poor compared with Merkel’s receptors and Meissner’s corpuscles. They evidently function to sense abrupt changes in tissue loading, and do not respond to steady-state (dc) pressure stimuli. The PC response to a step of applied pressure to the surrounding tissues adapts completely to zero in about 0.3 s. When the step of pressure is released, the PC again fires a burst as it returns to mechanical equilibrium. If a steady-state sinusoidal displacement stimulus is used, the threshold skin indentation required to elicit spikes is seen to be a function of frequency. Maximum sensitivity (minimum displacement) is seen at 300 Hz, and sensitivity decreases as frequency is either raised above or lowered below 300 Hz (Kandel et al., 1991, Ch. 24). At 30 Hz, the sensitivity has decreased by a factor of about 0.01 (i.e., if the stimulus threshold is a skin indentation of 1 µm at 300 Hz, a 100 µm indentation is required at response threshold at 30
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FIGURE 2.3-20 Electron micrograph of a transverse section through the bulb of a PC. The axon tip is at the center. Count the lamellae. (Edge-enhanced image available from www.udel.edu/Biology/Wags/histopage/colorpage/cne/cnepc.GIF.)
Hz). At the high-frequency end of the PC threshold sensitivity frequency response curve, the response is down by 0.01 times at about 1700 Hz. If a square wave displacement stimulus is used at low frequency, the PC fires twice per cycle, generating an instantaneous frequency waveform at double the stimulus frequency. The generator potential from an isolated PC in response to sinusoidal vibrations has a nonlinear, asymmetrical, full-wave rectified form (Pietras and Bolanowski, 1995). The nonlinear, phasic response characteristics of the PC disappear if the outer lamellae are dissected away, and the nerve ending with one fluid-filled envelope is deflected. In this case, the generator potential (GP) decays slowly while the deflection is maintained. When the deflection is removed, the GP does not show a positive transient. If an intact PC is deflected, a sharp transient depolarization is seen in the GP at the beginning of deflection, and another is seen when it is removed. From this behavior it can be concluded that the lamellar cover of the PC acts like a mechanical differentiator, transmitting the rate of change of pressure to the sensory neuron tip. The mechanically activated ion gates in the tip membrane also contribute to the PC phasic response. Apparently, they deactivate at a steady rate, even when pressure is maintained on the tip. However, most of the phasic behavior of the PC appears to be derived from its lamellar coating. A block diagram modeling the nonlinear dynamics of the PC is shown in Figure 2.3-21. The SGL has RPFM dynamics. This model does not have built-in saturation; but it does exhibit a firing threshold.
2.3.7
DISCUSSION
Mechanosensory receptor neurons contain one common property: shear forces applied to the specialized dendrite or neurite membrane causes ion channels to open,
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FIGURE 2.3-21 A block diagram of a dynamic model emulating the behavior of a PC. The time-varying pressure in the tissues around the PC is differentiated and then rectified (the PC fires to both increasing and decreasing pressure). It is then low-pass-filtered to emulate electrotonic conduction to form the generator potential, Vg(t). Vg is the input to an RPFM (leaky integrator) pulse generator. The instantaneous frequency of y is the PC output.
leading to depolarization of the membrane voltage and the generation of nerve spikes. Whether or not a mechanosensory neuron responds to length changes, tension force on a tendon, tissue pressure changes, or simple deflection of one or more cilia on the cell body depends on the mechanical support of the surrounding tissues and its connection to them. The neuron itself is soft (compliant), and responds to length changes or deflections as small as a micron. (To sense length changes on the order of cm, the 1 cm length change must be divided by about 104.) Thus a force sensor neuron (e.g., a GTO neuron) must be embedded in a very stiff, elastic substrate so that micron-level displacements occur for kg-level forces. In the case of insect trichoid hairs, the hair acts as a lever so that a large angular hair deflection produces a small deflection on the tip of the sensory neuron’s dendrite (see Figure 2.3-1). Probably one of the more amazing mechanosensory organelles is the vertebrate muscle spindle. Designed to respond to muscle stretch, either passive or under load with muscle fibers active, the spindle is loosely attached in parallel with the main (extrafusal) muscle fibers. Thus a major stretch of the muscle produces a small stretch of the spindle organelle. Inside the spindle are several intrafusal muscle fibers whose lengths are under CNS control. The active endings of the mechanosensory neuron are wrapped around the elastic midregion of the intrafusal muscles. Thus, if the main muscle shortens, so will the spindle, and any tension will be taken off the receptor neuron; it will cease to fire. The CNS then stimulates the intrafusal muscles to shorten, reestablishing some tension on the receptor. The CNS stimulation acts in this case as a kind of automatic sensitivity control. If the muscle is stretched, so is the spindle and the intrafusal muscles, causing the receptor neurons to fire briskly.
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Now a CNS reflex inhibits the motor input to the intrafusal muscles, causing them to lose tension, and the firing of the receptor slows. Statocysts were shown to be found in several invertebrate phyla, e.g., crustaceans (but not insects), annelids (segmented worms), and mollusks. The design of the statocyst permits it to sense the animal’s body angle in the gravity field, and also respond to linear and angular acceleration. A statocyst is basically a fluid-filled cavity lined with ciliated, force-sensing receptor cells, and some sort of mass (statolith) that presses on the receptor cells. The animal has the job of integrating the outputs of the array of sensory cells that line the statocyst cavity to provide it with orientation information that will affect behavior.
2.4
MAGNETORECEPTORS
Scientists studying the neurophysiology underlying animal behavior (neuroethologists) have been amazed for many years at the ability of migrating animals unerringly to find their destinations in the apparent absence of obvious terrestrial or celestial orienting features. Factors such as the positions of stars, sun, and moon (when visible) and water chemical content (where relevant) have been implicated as navigational guides, and in some cases may be guides. However, behavioral evidence has accumulated showing that certain diverse animals such as yellowfin tuna, trout, sea turtles, pigeons, spiny lobsters, honeybees, and certain mollusks and bacteria use the Earth’s magnetic field to guide their migratory and systematic movements, often in conjunction with certain other geophysical factors mentioned above. The Earth’s permanent magnetic field vector, Be, varies over the surface of the Earth. It is, of course, the geophysical phenomenon that in past centuries, has permitted planetary exploration. The Earth’s magnetic field has two orthogonal vector components, one tangential to the Earth’s surface (the horizontal component) and a second, vertical component directed radially downward (or upward). For example, in Cambridge, MA, the horizontal component is about 1.7 × 10–5 Tesla directed about 15° west of true (polar) north; the departure of the horizontal component of Be from true north is called its variation or declination. The vertical component of Be is about 5.5 × 10–5 Tesla. The dip or inclination angle of Be is about 73° (from horizontal) at Cambridge. (Note that 1 Tesla = 1 w/m2 = 104 Gauss.) By definition, the dip angle and vertical component of Be are zero on the Earth’s magnetic equator. The declination, dip angle, and strength of the Earth’s Be vary from place to place on the Earth’s surface, and change slowly with time. To orient itself in the Earth’s weak dc magnetic field, an animal obviously must have neural sensors that can sense the Earth’s magnetic field vector relative to its body direction. A migrating animal must also possess the cognitive apparatus that allows it to relate its present orientation determined magnetically with its desired travel direction, i.e., it may have a stored, magnetic “map sense.” In the case of north/south migration, requirements on the magnetic sensor may be simple; the animal must follow a constant course until other physical cues tell it that it is near its destination, where a “fine approach” mode of navigation can be employed. There is also behavioral evidence that some animals use the inclination angle of Be in their
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magnetic orientation/navigation behavior. That is, they effectively sense Be and break it down into its horizontal and vertical components.
2.4.1
BEHAVIORAL EVIDENCE
FOR
MAGNETIC SENSING
Many animal ethologists studying migration have found strong direct evidence that the Earth’s magnetic field is used in conjunction with other sensory modalities for guidance. Unfortunately, the location of the neural receptors responsible for static magnetic field sensing are known in few cases (see Section 2.4.2). Some magnetoreceptors are thought to use cells containing microscopic magnetic iron oxide (magnetite) crystals that interact with the Earth’s magnetic field to create forces that are somehow transduced to nerve spikes. Other workers have implicated elements of the visual system in magnetoreception. The bottom line is that at this time there is no hard evidence for a specific mechanism for magnetoreception, even in Tritonia Pd5 neurons. Much basic neurophysiology and neuroanatomy has yet to be done. The list of animals thought to use the Earth’s magnetic field for guidance is impressive. These include but are not limited to Vertebrates, including homing pigeons, bobolinks, sea turtles, rainbow trout, sockeye salmon, yellowfin tuna Arthropods, including spiny lobsters, honeybees, the mealworm beetle, Tenebrio molitor Mollusks including the sea slug, Tritonia diomedea, and the snail, Lymnaea Bacteria exhibiting magnetotaxis have also been found. Homing pigeons have been widely studied because of their ability to find their home loft under amazing conditions of distance and weather. Some of the earliest evidence that these birds use the Earth’s magnetic field for navigation was obtained by Keeton (1971; 1974), who glued small, permanent magnets onto pigeon heads to change the magnetic field vectors experienced by the birds. Homing behavioral studies showed that birds with magnets had random “vanishing bearings,” while birds carrying brass control weights had vanishing bearings clustered around the direction of their loft. (The vanishing bearing is the direction the pigeon flies out of sight on the horizon after release.) Keeton found that the birds’ magnetic compass sense interacted with their “sun compass.” In another important set of pigeon experiments, Walcott and Green (1974) equipped their birds with solenoidal electromagnets on their heads to reverse the N–S polarity of the vertical component of Be. Birds with north up often flew 180° away from home under overcast conditions, but could navigate normally when the sun was out. Birds having south up were able to navigate normally under both overcast and sunny conditions. In another series of experiments, Walcott (1977) applied graded vertical components of 0, 0.1, 0.3, and 0.6 Gauss, N or S up, with the coils worn by the pigeons. He found that even on sunny days, the artificial field caused an increase in the scatter of vanishing bearings. These results suggested that the outputs of both the sun and magnetic compass senses were functionally integrated by the bird, and did not act autonomously. The sensitivity of
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pigeons to the magnitude of the vertical component of Be suggests that they use the complete Be vector for navigation, not just the horizontal component. Another interesting experiment on bird magnetic sensing was described by Beason et al. (1995), working with the European bobolink, a migratory bird. These workers sought to verify that the birds’ transduction of Be depended on sensors using magnetite crystals (magnetic iron oxide, Fe3O4). A bird’s head was placed inside a solenoid, and a 5-ms, 0.5-T pulse was applied to strongly and permanently magnetize magnetite crystals assumed to be in the head and associated with the bird’s migratory compass. The pulse was brief to prevent the magnetite crystal(s) from physically rotating to align themselves with the strong external field. Magnetization was done in one of three bird orientations: north anterior, north posterior, and north up. North anterior means that if the beak were iron, it would attract the south end of a compass, etc. Space does not permit a detailed description of Beason et al.’s results; however, in summary, the head magnetization did alter the birds’ ability to orient themselves correctly for migration. Birds that were magnetized N-anterior had a significantly different mean heading from that of birds magnetized S-anterior, and each group differed significantly from its control. N-up magnetized birds had two mean headings, 180° apart. The saga of the spiny lobsters is worthy of consideration here. Lohmann et al. (1995) tested spiny lobsters (Panulirus argus) for their ability to guide their migration or homing movements by using the Earth’s magnetic field. An underwater magnetic coil system was constructed with which the experimenters could independently and exactly reverse the vertical or horizontal components of Be by passing appropriate dc currents through the coils. The lobsters were tethered so they would remain in the effective field of the coils; their direction of movement under the various field conditions was noted. Control lobsters (coils unenergized) marched in diverse directions, but were consistent in those directions. When the vertical component of Be was reversed by coil, there was little change in individual march directions. When the horizontal component of Be was reversed by coil, after about 5 min, nearly all of the lobsters were marching in directions 180° from their control directions. This experiment demonstrated that spiny lobsters use the horizontal component of Be for guidance. In 1984, Lohmann using a sensitive SQUID magnetometer reported that the spiny lobster had four sites showing natural remnant magnetization (NRM), indicating the presence of ferromagnetic material in its body. Three of the four sites were located in the cephalothorax; the NRM of the one in the center was directed to the right, the NRM of one in the left posterior cephalothorax was directed posteriorly, and the NRM of the site in the right posterior cephalothorax was directed anteriorly. A fourth site in the telson-uropods region had NRM directed to the animal’s left. The most likely material providing NRM is magnetite crystals, which may lie in a yet-to-be-discovered magnetosensor cell or cell complex. Many other interesting examples of animal directional responses to the Earth’s magnetic field can be found in the literature. Of the animals examined, their responses can be put in two broad categories: those that only use the horizontal component of Be (e.g., the spiny lobster), and others that use both the vertical and horizontal components (entire) of Be (e.g., pigeons, sea turtles).
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2.4.2
THE PUTATIVE MAGNETORECEPTOR NEURONS
OF
TRITONIA
Tritonia diomedea is a large, North Pacific nudibranch mollusk (sea slug) that has been shown not only to respond behaviorally to induced magnetic fields, but to have two individually identifiable neurons believed to be magnetoreceptors (Cain et al., 1999). The neurons are located in the symmetrical left and right pedal ganglia, and have been designated LPd5 and RPd5, respectively. They have enormous cell bodies, ~500 µm in diameter, and when stimulated by action potentials, their terminal branches secrete neuropeptides that increase the beat frequency of pedal cilia on the left and right sides of the animal, respectively, producing turning. If all central nerves are intact, Pd5 neurons responded with an increased rate of firing when the horizontal component of the Earth’s magnetic field was rotated 60° clockwise The increase in firing rate was delayed for 6 to16 min following the directional change of B (Lohmann et al., 1991). Curiously, Pd5 neurons did not respond if recorded from an isolated Tritonia brain. However, when all nerves were cut except Pedal 2 and 3, response was obtained. These perplexing results suggests three possibilities: (1) nerves P2 and/or P3 transmit spikes from magnetosensors located in the animal’s peripheral tissues; or (2) P2 and/or P3 are the magnetosensors, or (3) cutting the axon of Pd5 keeps it from spiking in response to magnetic stimuli in the isolated brain. No one as yet has examined the ultrastructure of Pd5 giant neurons, or determined whether or not they contain magnetite. As will be seen below, magnetic fields can theoretically be sensed in several ways other than by the forces acting on magnetized magnetite particles.
2.4.3
MODELS
FOR
MAGNETORECEPTORS
The author can think of four reasonable hypothetical physical models for animal magnetosensor systems. Three of the models make use of the law of physics governing the force acting on a charged particle moving in a magnetic field. 2.4.3.1
The Magnetic Compass Analog
This type is based on a hypothetical, mechanosensory neuron containing microscopic, domain-sized particles of magnetic iron oxide (magnetite). To be maximally effective, the magnetite particles must have a permanent magnetic moment, i.e., be magnetized. Similar to the operation of a magnetic compass, the south pole of the particle aggregate will experience a magnetic force or torque trying to align it with the (north) magnetic vector of the Earth; conversely, the north pole of the magnetite will be attracted to the south magnetic vector. (Early magnetic compasses used a piece of magnetized lodestone, magnetite pivoted on a pin or attached to a wooden float and free to turn in a water-filled bowl.) In the case of cells with magnetized magnetite particles inside them, the tiny internal forces or torques created by nonalignment with the Earth’s magnetic field would have to be coupled to a spikegenerating mechanism that can signal by its frequency the degree of nonalignment with Be. If the magnetite crystals are attached to myosin filaments that, in turn, are attached to certain ion-gating proteins in the magnetoreceptor cell’s membrane, then
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the microscopic forces or torques produced by misalignment of the crystals with Be could provide the necessary coupling mechanism. How many sensors would be required to give unambiguous information on heading in a 360° circle about magnetic north? (See Section 2.6 for how the outputs of four sensors might be resolved to give absolute heading.) Three key questions must be answered to accept the hypothetical magnetitecontaining magnetosensor: (a) How do the magnetite crystals (insoluble in seawater) get inside the hypothetical magnetoreceptor neurons? Is it biogenic, i.e., made by the cell? (b) How do they get permanently magnetized? (c) How do forces and torques on the magnetized magnetite crystals (single domains, or small aggregates) produce changes in receptor resting potential leading to spikes? That is, what is the transducer coupling mechanism? 2.4.3.2
A Hall Effect Analog
Another possible mechanism for a neural magnetoreceptor might be based on the Hall effect. In this scenario, an animal has a cell with a special membrane Figure 2.4-1). Inside the membrane, or attached to its surface, a thin layer of densely packed macromolecules actively transports electrons parallel to the membrane surface at an average drift velocity, vn, in the –x direction. The By component of Be that lies perpendicular to vn causes a perpendicular Lorentz force, FL, to act on each moving electron according to the vector equation:
FIGURE 2.4-1 Diagram of a hypothetical Hall-effect-based biological magnetosensor membrane. Electrons are transported at velocity vn in the –x direction, giving a net current density Jx in the +x direction. vn is ⊥ to the Earth’s magnetic field, Be. A Lorentz force, FL, acts at right angles to vn and Be and pushes the moving electrons to the outside of the membrane, depolarizing it in that region. This model requires a yet-to-be-discovered array of electron transfer molecules embedded in the membrane.
FL = q vn × By = qvnBysin(θ) uz
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2.4-1
Because vn, Be, and FL are orthogonal (θ = 90°), FL = q vn By uz forces the moving electrons toward the outer cell membrane, creating a space charge that acts to depolarize the cell. Note that uz is a unit vector with the direction of the Lorentz force, i.e., uz = 1. Figure 2.4-1 illustrates schematically this process for the orthogonal case. Note that By is the vector component of Be that by definition is orthogonal to vn in the x–y plane of the membrane. If the cell orientation is such that Be is parallel with vn , θ = 0°, and there will be zero Lorentz force and no change in the cell resting potential. The Hall-induced electric field across the current-carrying volume of thickness t can be shown to be Ez = vx By V/m in the direction shown. The induced Hall potential is simply VH = t Ez = t vx By = Jx By/(q n). q is the electron charge, and n is the average density of electrons moving in the volume. (Note that if By or vn changes sign, the cell will be hyperpolarized, and will fire at a lower rate or will be inhibited.) In closing, note that the parallel current density need not be carried by electrons; any anion or cation with good mobility (and thus high vx) will do. 2.4.3.3
A Lorentz Force Mechanism in Vertebrate Photoreceptors
In the dark, the cell membrane of the outer segment of a vertebrate rod or cone has a heavy leakage current density of certain positive ions into the cell through gated cation channels embedded in the cell membrane. 80% of the current is carried by Na+, 15% by Ca++, and 5% by Mg++ ions. The inward leakage current is balanced by active outward pumping of Na+, etc., by metabolic pumps in the inner segment of the cell. When light is absorbed by the photopigment molecules on the disks inside the outer segment, a complex cascade of chemical reactions leads to the closure of the cation channels, allowing the photoreceptor cell to hyperpolarize (Kolb et al., 1999). The important thing in this model is that in the dark there is a radially directed, cation current density Jx crossing the membrane of the outer segment of the photoreceptor cell. If a magnetic field is applied perpendicular to Jx, the resulting Lorentz force, FL, given by the right-hand screw rule, will be perpendicular to both B and Jx, and tangential to the cell membrane. Thus, a moving cation will experience a force perpendicular to its direction of motion through the ion channel. This lateral force could slow or block the passage of individual leakage cations, reducing the dark Jx and causing the cell to hyperpolarize slightly, mimicking low-level light being absorbed by the receptor. Obviously, such dual-use of a rod or cone as a photoreceptor and a magnetoreceptor would only be possible in the absence of light. If such a magnetic field-sensing rod (or cone) in the retina were oriented on the animal’s anterior–posterior axis when the animal was aligned with Be, there could be maximum neural output by a special ganglion cell fiber to the CNS. Semm and Demaine (1986) showed that visual neurons in the pigeon’s brain responded to changes in the direction of a magnetic field. Their assumption was that the neural activity of magnetic origin came from the eyes, and was not from magnetite-containing magnetosensors heretofore implicated in pigeon navigation. The responses occurred only in light, which is contrary to the hypothetical scenario
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above, unless a special magnetosensing rod or cone lacked the normal biochemical machinery to close cation channels in response to light, and responded only to Be. 2.4.3.4
A Magnetosensory System Based on the Faraday Streaming Effect
Still another hypothetical model for magnetic sensing could use the Faraday streaming effect. The Faraday streaming effect generates an electromotive force across an electrically conducting fluid moving at a velocity v lying in the same plane as a magnetic field, B. The Faraday motional electromotive force (EMF) is given by the vector integral:
EF =
d
∫ (v × B) ⋅ dL
2.4-2
0
Do not be suprised that the Faraday effect also involves the Lorentz force. The conducting fluid stream necessarily must contain positive and negative ions. As these charged ions pass through the magnetic field with velocity, v, the Lorentz force causes them to separate and create an electric field in the fluid perpendicular to v and B. (See Figure 2.4-2 for a simple description of the Faraday scenario.) The conductive fluid could be blood flowing in an artery, or seawater passing though gill slits as a fish swims. If v and B are mutually perpendicular vectors, then the EMF between the electrodes is EF = vBd v, and the electric field over d is simply (vB) V/M. The EMF or E-field from the Faraday effect can then be sensed by electroreceptors. Fortunately, unlike magnetoreceptors, electroreceptor structure has been well studied and documented (see Section 2.5). Note that the Faraday effect yields a signed output EMF: EF = dv Bsin(θ)
V
2.4-3
where θ is the angle between B and v (see Figure 2.4-2). If the flow velocity vector v and B are parallel, then EF = 0. A possible model for the magnetic navigation aid for sharks makes use of the Faraday streaming effect. In this scenario, assume a shark is swimming toward magnetic east; i.e., it is swimming at right angles to the horizontal component of Be with a body velocity v. Assume that the shark’s body has a much lower conductance than the seawater, so that it is effectively an insulator. Be(hor) points from the south magnetic pole toward the north magnetic pole. Thus, the direction of the Lorentz force, qv × Be(hor), for positive ions is up. This means that the top of the shark’s head will be positive with respect to the bottom. As a result of this potential difference there will be an electric field surrounding the head whose sign depends on the swimming direction (east or west), and whose magnitude depends on the swimming velocity and the actual angle, θ, between v and Be(hor).
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FIGURE 2.4-2 A hypothetical Faraday streaming effect magnetosensor. When an ionic fluid flows at a velocity v ⊥ B, an EMF is generated in the orthogonal u direction. The EMF has to be sensed by electroreceptor cells. See text for discussion.
The magnitude and sign of the dc electric field around the head is sensed by the ampullae of Lorenzini (electroreceptors) on the shark’s head. (See Section 2.5 for details about these organs). The head field will approach zero if the shark is swimming toward either magnetic north or south, or if it stops swimming. Thus, sensitive electroreceptors can contribute to the sensing of Be for navigation in sharks.
FIGURE 2.4-3 Another Faraday magnetoreceptor scenario. Here, an insulated “fish” moves at velocity v ⊥ B in an ionic liquid (seawater). A high-conductivity channel of length L through the fish’s body at right angles to v and B develops an EMF by the Faraday induction law of E = B v L V in the mutually perpendicular case. See text for discussion.
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Another theoretical scenario than may be applicable to the sensing of the Earth’s magnetic field is illustrated in Figure 2.4-3. Here is illustrated an insulated fish or torpedo-like object moving at a uniform velocity, v. In the ideal case, v is perpendicular to the magnetic field, B. Inside the fish, body conductivity is high. Two highconductance apertures, shown by the small clear and dark circles, are opposite each other on the body, and form a conductor of length L inside the body. As this conductor moves relative to B as the fish swims, an EMF is induced in the conductor according to the differential vector form of Faraday’s induction law: dE = (B × dl) · v
V
2.4-4
If B, v, and dl are mutually perpendicular as shown in the figure, the EMF is maximum and is given by E = B vL
V
2.4-5
Using numbers, if B = 1.7 × 10–5 T, v = 10 m/s, and L = 0.1 m, E = 17 µV. (This would be for a east- or west-swimming fish. E Ý 0 for a north- or southswimming fish.) This EMF acting over 0.1 m produces an electric field of 170 µV/m across the head, which is large enough for an array of ampullary electroreceptors to sense. Finally, consider a molluscan statocyst, normally apprising the animal where “up” is relative to its static body orientation. If some magnetite crystals are placed in a statocyst, it will signal the animal the vector sum of the magnetic force plus the gravity force acting on the particles and their acceleration. Now let the particles be embedded in freely moving supporting cells (such as a cluster of magnetic bacteria, Blakemore; 1975) so that the whole mass has neutral buoyancy. Now the statocyst will respond only to Be (and acceleration).
2.4.4
DISCUSSION
There is a wealth of behavioral evidence for the ability of animals to orient themselves or navigate with the aid of the Earth’s magnetic field. One of the simplest systems of magnetotaxis is found in certain north-seeking bacteria that have been found to contain linear chains of microscopic single magnetic domain-sized magnetite crystals (Fe3O4). The magnetite evidently is of biogenic origin (Bazylinski, 1990). In more complex magnetic-responding animals, magnetite has been detected in their brains or bodies, but no specific organ or neuron containing magnetite has yet been described. Alternate models for magnetoreception (other than by magnetite-containing cells) have been suggested in which moving charged particles (ions leaking through membrane receptors, or being pumped into or out of neurons) might be affected by a dc magnetic field. The Hall effect, the Faraday streaming effect, and a Lorentz force model were described. Much work has yet to be done to isolate and describe the ultrastructure and transducer mechanism of magnetosensing neurons, where they exist.
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2.5 ELECTRORECEPTORS Electroreceptors are specialized neurons that, in an aquatic or marine environment respond to the presence of an electric eld (or , equivalently, a voltage gradient across them). They are found in the skin of certain sh and in the beak of the duck-billed platypus, the only mammal having this sensory modality. Underwater electric elds can be subdivided into two classes: those that are constant (dc) or slowly varying (up to 50 Hz), and those high-frequency elds arising from special electrogenic organs in so-called weakly electric sh that use self-generated, low-voltage ac elds for navigation. One type of electroreceptor used to detect dc and low-frequency elds is called the ampulla of Lorenzini; it is found in large numbers on the heads of sharks, skates, and rays. Low-frequency elds can be the result of a ph ysical phenomenon, e.g., the Faraday streaming effect (see Section 2.4.3), or be of biological origin, such as from the contracting muscles of a swimming sh, especially one that has been injured so that its skin is torn. Tuberous high-frequency eld sensors are called knollenorgans; they are found in weakly electric sh (Mormyrids) and are part of the comple x electric guidance system used by these sh. Mormyrid sh also ha ve smaller, less sensitive electroreceptors called mormyomasts, and some low-frequency ampullary electroreceptors. Knollenorgans respond to frequencies ranging between 300 Hz and 20 kHz. The threshold eld sensiti vity for ampullary receptors is about 0.1 to 0.2 ∝V/m (Kalmijn, 1998). Knollenorgans respond to ac elds ha ving a peak value of ~10 mV/m, far less sensitive than ampullary electroreceptors (Carr and Maler, 1986). Of signi cance is the f act that sh electroreceptors are or ganized into arrays. Mormyrid sh ha ve several hundred knollenorgans distributed over their bodies, with concentrations on the head and caudal region of the anks. Knollenor gans are integrated into a perceptual system that allows the sh to sense nearby en vironmental features on the basis of how they distort the electric eld surrounding the sh that the sh produces with its electric or gan. Such “blind guidance” is necessary for the survival of these sh, who generally li ve in muddy water where vision is useless. Ampullae of Lorenzini on elasmobranchs are found mostly on their heads and snouts, but are not restricted to these areas. Their distribution is species dependent. On skates, the ampullae are most dense on the anterior surfaces of their pectoral n “wings.”
2.5.1
AMPULLARY RECEPTORS
The ampullae of Lorenzini (AoL) found in sharks, skates, and rays were rst discovered by Marcello Malpighi in 1663; they were described in detail by Stephano Lorenzini in 1678, and bear his name. Anatomically the AoL are relatively simple structures. A jelly- lled canal projects through a pore in the skin. The surface pores of the AoL on the face of a shark give it the appearance of having acne. The canal jelly has a high conductivity due to its high content of Cl– and K+ ions; it is probably secreted by the cells that line the pore canal and that surround the sensory cells. The length of the canal is species dependent, but is on the order of 1 to 2 mm. Its diameter
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FIGURE 2.5-1 Highly schematic cross section drawing of an AoL. The gel that lls the canal and body of the ampulla has a much higher conductivity than the surrounding skin and epithelium. Thus, minute electric currents are directed into the ampulla where they may affect the sensory cells.
is about 150 ∝m. The inner end of the canal opens up into a sacklike chamber about 400 ∝m in diameter; the chamber is lined with a single layer of epithelial cells, some of which are the electrosensory cells; others are support cells. The electrosensory cells have no cilia or projections into the gel- lled ca vity (Murray, 1965). Figure 2.5-1 illustrates a schematic section through an AoL. One myelinated sensory nerve is shown for simplicity. About six myelinated nerves innervate each ampulla. Each
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nerve provides two to three synapses per sensory cell, and innervates several cells. The synapses have a characteristic tight “ribbon and gutter” morphology, not unlike the synapses of retinal rods, or cochlear hair cells (Murray, 1965). The synapses are evidently chemical in nature, and the bases of the electrosensory cells contain presynaptic vesicles. The myelinated ampullary nerve bers run together to the anterior lateral line ganglion at the base of the brain. Murray (1965) examined the electrophysiological properties of single AoL from the skate, Raja clavata. In the absence of external current passing through the canal into the AoL, a single AoL nerve would typically re at about 19 pps in the steady state with a regular rhythm. Murray (1965) gives an interval histogram of the zero-current spikes from an ampullary nerve ber , showing that its period is quite regular. If an external current on the order of nanoamps was passed in or out of the AoL canal, there was a nearly linear relation between the frequency of ring of the nerve and the current. This relation is approximately: f = fo + KI. f represents the frequency of the nerve in the rst quarter second follo wing application of the current. fo was 19 pps, and K was 16.5 pps/nA. The instantaneous frequency of the AoL nerve showed adaptation to steps of applied current, and rebound when it was switched off. The ring decreased to zero for inward current greater than 1.15 nA, and increased nearly linearly for outward current up to about 3 nA. Above 3nA outward current, nerve ring w as inhibited (Murray, 1965). From Murray’s gures, the author estimated that the threshold canal current to in uence the AoL “clock” frequency should be on the order of ±100 pA. (Note that British sign convention for current ow assumes that current is carried by negative charges; U.S. sign convention assumes that current is carried by positive charges. In copper wires in the U.S., the electrons travel in the opposite direction from current. Thus, Murray’s outward current can be visualized as being carried by Cl– ions moving out of the canal, or curiously, by positive ions [e.g., K+, Na+] moving inward.) Compared with the sh’ s skin and the canal lining, the canal, sensory cells, and sensory epithelium have a relatively high conductance, directing the minute ionic currents associated with an external eld to o w through the canal and through the apical region of the sensory cells. At the molecular level, the electrosensory cells of the AoL may release neurotransmitter because voltage-gated transmembrane proteins allow Ca++ ions to enter and depolarize them (Adair et al., 1998). Kalmijn (1998) reported that an AoL, when electrically stimulated, sources a dc current in opposition to the inward, stimulating current (U.S. sign convention). This reaction current could be carried by a variety of ions, and represents negative feedback, as far as the excitatory potential on the electrosensory cell is concerned. This opposing current could come from an electrically activated ion pump, or be from controlled outward K+ leakage or, equivalently, from controlled inward Cl– leakage. Its purpose may be to act as an automatic gain control for the AoL, giving it a log-linear sensitivity characteristic. Kalmijn (1998) claimed threshold sensitivities for shark AoLs (operating as an array in vivo) can be as low as 1 to 2 nV/cm (0.1 to 0.2 ∝V/m). This is an incredible sensitivity! Behavioral studies of shark electroreception by Dijkgraaf and Kalmijn (1962), and Kalmijn (1966; 1971; 1973; 1974) have shown that sharks use their AoL as low-frequency electric eld sensors to locate prey, and as input devices for
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geomagnetic navigation. Threshold sensitivity for AoL electroreceptors in the dogsh shark, Scyliorhinus canicula, has been estimated by behavioral experiments to be 100 ∝V/m. The skate R. clavata can sense an amazing 1 ∝V/m. This sensitivity is not for one AoL, but rather for the whole array of electrosensors on a freeswimming sh. To obtain these gures, sharks and skates were conditioned to nd and eat an injured at sh hidden on the ocean oor . They did this by using the at sh’ s electric eld from muscle potentials. Scent w as controlled for, and was not a factor. Then anthropogenic electric elds of kno wn strength were set up on the ocean bottom with buried electrodes; the sharks and skates mistook the dc electric eld for an injured sh and w ould attack the electrodes. That sharks, skates, and rays (elasmobranch sh) ha ve a real sixth sense (electroreception) that they use for prey location, and quite possibly for geomagnetic navigation, is remarkable. However, when these sh are stationary , their electroreceptive sense is useful only over a short range, probably less than a meter. Range is limited by the fact that an electric eld in v olts per meter decreases proportional to R–3, where R is the distance from the shark to the source. The electrical potential in volts of a dipole source falls off as R–2. The great sensitivity of the AoL organs may make them subject to interference from elds generated within the shark’ s own body, and by its own swimming motions. How the shark compensates for this “noise” in its CNS is just beginning to be understood. How the spike signals from elasmobranch ampullary arrays interact centrally to permit these sh to sense a weak electric eld and home in on its source promises to be a major challenge for computational neurobiologists. The inputs to the detection system are the many ampullary nerve bers that re re gularly at slightly different fos. The presence of an external electric eld will cause the ampullary nerv e ring frequencies to increase slightly or decrease, depending on its sign (which determines the direction of current o w in the ampullary canals). As a sh swims in the presence of a dc electric eld, its body motions will continuously modulate the ampullary nerve ring frequencies up and do wn, depending on the geometry. How does the sh use this information? That is, what neural circuit architectures might be involved in this system? Ampullary electroreceptors are also used in passive prey detection by cat sh (Whitehead et al., 1998), and by the paddle sh, Polydon spathula, an eater of zooplankton. Zooplankton such as Daphnia sp. were shown to emit weak, lowfrequency (dc–20 Hz) electric elds from dipole-lik e sources of their internal organs. Tested with arti cial, sinusoidal elds from electrodes, paddle sh preferred signals around 10 Hz (Wojtenek et al., 1998; Wilkens et al., 1998).
2.5.2
WEAKLY ELECTRIC FISH
AND
KNOLLENORGANS
Weakly electric sh (WEF) are one of nature’ s great curiosities. There are two major groups of WEF: mormryids from Africa and gymnotids from South America. Both groups live in muddy rivers where visual perception is impossible. Like elasmobranchs, they, too, use underwater electric eld for guidance. There is one important difference, however. Instead of only sensing electric elds passi vely in their environments, WEF have electric organs that produce a periodic (ac) electric eld of
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constant frequency. The electric organ output is called the electric organ discharge (EOD) in the literature. The electric organ is generally located on the sides of the WEF’s near the tail. WEFs of both groups of sh sense the distortion of their self-generated electric elds by nearby animate and inanimate objects with conducti vities different from the water in which they are swimming. Gymnotids sense the temporal phase shift of their EODs caused by an external object close to part of their body relative to the EOD phase sensed at a distance from the object. Mormyrids sense reactive (capacitive)-caused distortions of their ac EOD waveform caused by a nearby organic object (von der Emde, 1999). (Inanimate objects such as rocks do not have signi cant capacitance at the audiofrequencies of the EOD, and thus produce no phase shift.) It should be pointed out that the electric guidance system of WEF has a relatively short range (10s of cm) because of the way an electric eld caused by a v oltage dipole is attenuated with distance. It is easy to show that the (scalar) electric potential of a dipole attenuates as R–2, while the electric eld (v ector) magnitude drops off as R–3, where R is the distance from the observation point to the center of the dipole, d is the dipole separation, and R d (Corson and Lorrain, 1962). Several hundred knollenorgans are distributed over the body of a mormyrid sh; they are concentrated on the head and on the sides near the tail. Mormyrids also have smaller mormyromast electroreceptors, and a scattering of ampullary electroreceptors. Figure 2.5-2 illustrates a typical knollenorgan. One to eight electrosensory cells with diameters of 40 to 60 ∝m are clustered in a hemispherical mass inside the organ cavity. Each receptor cell makes electrotonic synaptic contact with the terminal branches of the single afferent nerve ber . There is a loose plug of epithelial cells lling the skin pore o ver the organ. Knollenorgan afferents travel in the lateral line nerve to the electrosensory, anterior lateral line lobe in the sh’ s brain, thence information is sent to other parts of the CNS. Much is known about the central processing of knollenorgan sensory information, and how it interacts with the brain center that controls the EOD. The interested reader is urged to see Heilingenberg (1991) to pursue this topic in further detail. Knollenorgans can be organized into two subclasses. The T units behave like tuned band-pass lters ha ving tuning peaks ranging from 100 Hz to 20 kHz, depending on the mormyrid sh species. The T nerves re on the rising edge (positi ve rst derivative) of the stimulus, one spike per EOD cycle positive zero crossing, with very little phase jitter. P units re asynchronously with the EOD; their instantaneous frequency of ring is e vidently proportional to the average peak amplitude of the stimulus (or maybe its rms value). Taken together, the P and T afferent signals from one part of the sh’ s body code the exact phase of the ac electric eld in the w ater at that point, and also its average peak amplitude. The CNS compares left and right signals from corresponding parts of the body and looks for difference-mode signals, i.e., asymmetry between left and right sides, as a signal that some nearby object is distorting the ac E eld from the EOD. Rasnow and Bower (1999) have pointed out that a weakly electric sh has the task of mapping the features of an object in real (underwater) space into its perceptual space. Object features surely include physical parameters such as the range and bearing of the object from the sh, and the object’ s size, shape, conductivity, and
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FIGURE 2.5-2 Highly schematic cross section drawing of a knollenorgan electroreceptor. Note that there are some anatomical congruences with the AoL. See text for discussion.
dielectric constant. Perceptual features are based on the “electric image” features which include the amplitude and phase distribution of the E eld along the sh’ s body, including left vs. right asymmetries in the sensed ac E eld amplitude and timing. From this sensed information the sh can mak e simple decisions whether the object is dangerous, food, a potential mate, or merely something like a rock in its immediate environment.
2.5.3
DISCUSSION
The ampullary electroreceptors of certain sh were shown to exhibit awesome sensitivity to low-frequency electric elds. It is a considerable challenge to disco ver how this ability to resolve signals out of noise is accomplished. No doubt it is due in large part to the fact that the AoLs are found in three-dimensional arrays over the heads and bodies of the sh that use them. Thus, the sh can use some form of © 2001 by CRC Press LLC
spatiotemporal averaging to achieve their great sensitivity in prey location. How the sh processes the spik e signals from individual AoL axons in its CNS remains to be described. A WEF has an electric organ that produces an ac eld of constant frequenc y around its body. Distortions in this eld caused by nearby objects having conductivities different from the water are sensed and used by the sh for guidance in murk y water. The knollenorgan receptors, like the AoL, are arranged in arrays. By using an ac eld, in principle, the sh can use some kind of synchronous detection to obtain an impro ved resolution and SNR. (Synchronous detection is used in the well-known lock-in ampliers.) Again, neuroethologists and neurophysiologists are just beginning to understand the CNS mechanisms underlying the navigation skills of WEF.
2.6 GRAVITY RECEPTORS OF THE COCKROACH, ARENIVAGA SP. This section shows that the tricholith gravity sensors of the cockroach Arenivaga sp. are an excellent example of a small receptor array. Tricholiths have been shown to respond to the position of the cockroach with respect to the Earth’s gravity vector. Only four large identi able interneurons carry spatial orientation information from the tricholith arrays to the animal’s CNS where it is processed by many interneurons. Willey (1981) recorded from single, position-sensitive neurons in the animal’s protocerebrum and demonstrated that a de nite central enhancement or sharpening of the spatial orientation response occurs. The Arenivaga spatial position sensing system is a rare example of a divergent sensory system, where a sparse sensory array projects only four bers to the CNS. Most neuro-sensory systems consist of lar ge receptor arrays whose outputs are processed convergently. That is, the sensory output of many receptors is processed in neuropile to give a few afferent outputs that are “operations” or feature extractions on the sensory input data. Insects as a class are generally thought to lack organs analogous to statoliths, otoliths, or semicircular canals used to sense an animal’s orientation in the Earth’s gravity eld. Man y insects determine their orientation by vision coupled with mechanoreceptor information provided by limb loading and joint positions (Walthall and Hartman, 1981). Certain crickets have clavate hair sensors on their cerci, which have been shown neurophysiologically to respond to changes to the cricket’s gravity orientation (Bischof, 1975). (The cerci of orthoptera are two roughly cylindrical organs projecting to the rear from the medial anal region of the insects, tapering to points. They are very noticeable in crickets and cockroaches. The cerci project at about 25° to 45° from the animal’s midline depending on the species, in the horizontal plane; they are generally covered with various hairlike sensillae; (Hoyle, 1977). A few insects have evolved arrays of specialized gravity-sensing sensilla on their cerci. Most widely studied has been the gravity response of the desert-burrowing cockroach, Arenivaga sp. (Hartman et al., 1979; Walthall and Hartman, 1981). Similar sensillae have been noted on the cerci of three genera of crickets, Achaeta, Gryllus, and Gryllotalpa, and on the cerci of several species of polyphagid cockroaches (Hartman et al., 1979).
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FIGURE 2.6-1 (A) Ventral view of an adult male burrowing cockroach, Arenivaga sp. Note the small, hornlike cerci projecting from the insect’s posterior. (B) SEM image of the ventral surface of the right cercus showing the ball-like masses of the tricholith gravity receptors. Other interesting hairlike receptors are also seen. (From Walthal, W.W. and Hartman, H.B., J. Comp. Physiol. A, 142: 359, 1981. With permission.)
The cerci of a male Arenivaga project at about 45° from the body axis, and lie in the horizontal plane. A cercus is about 1 to 1.5 mm in length, and is covered on its ventral surface with sensory hairs (trichobothria) and sensory bristles (sensilla chaetica). Figure 2.6-1A shows a ventral view of an adult, male Arenivaga sp; the cerci emerge from the rear of the abdomen. Figure 2.6-1B, a low-power, scanning electron microscope (SEM) micrograph, illustrates the hairs and tricholiths on the ventral center surface of the right cercus. Note that the tricholiths are arranged in a depression on the ventral surface of each cercus in two parallel rows. In an adult Arenivaga, each row has seven or eight tricholiths, giving a total of 14 to 16 tricholiths per cercus. The dense round ball of a tricholith is about 22 ∝m in diameter and it is suspended downward on a slender, cylindrical stalk from a socket in the cercus. Gravity pulls down on the ball so the axis of the stalk is aligned with the gravity vector. If the animal experiences roll or pitch, the stalk makes an angle less than 90° with the axis of the cercus. In males, the tricholiths are protected mechanically from above by the wings that overhang them, and ventrally by the trichobothria and the sensilla chaetica. The cerci of larval and female Arenivaga are protected in cavities in the abdomen. Thus, the tricholiths appear to be protected from mechanical perturbation from soil particles when the animal burrows.
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A mechanoreceptor neuron associated with the tricholith socket senses the relative position of the tricholith stalk with respect to the cercus, signaling the animal its orientation in the gravity eld. Because Arenivaga burrows in the sandy, quasiuid soil of its nati ve southwest Texas desert habitat during the day to escape the heat and predators, it is deprived of the normal insect orientation cues of vision, limb loading, and position and appears to rely on its tricholith system to enable it to navigate underground and return to the surface. Hartman and his graduate students were the rst neuroph ysiologists to characterize the input end of the tricholith system quantitatively (Walthall and Hartman, 1981). Sensory bers (one from each tricholith or ganelle on each cercus) pass into the sixth abdominal ganglion where they synapse with four large afferent positional interneurons that project up the paired ventral nerve cord to the insect’s protocerebrum. There are two large positional interneurons (PIs) in each of the paired ventral nerve cord tracts, facilitating electrical recording of the PI action potentials.
2.6.1
HARTMAN’S METHODS
To test the responses of the tricholith/positional interneuron system, a male Arenivaga cockroach was fastened ventral-side down to a motor-driven tilt table in the horizontal plane with pins and wax. The ventral nerve cord (VNC) was exposed by removing the wings and the cuticle on the back between terga 1 to 4. The internal organs were removed to expose the nerve cord between abdominal ganglia 1 and 2. A buffered insect saline solution was used to keep the preparation moist and neurons alive. Electrical recordings were made from the left and right VNC connectives using two suction electrodes attached to the tilt table. A variable-speed dc motor was run to generate various rates of tilt and maximum tilt angles (φ). Tilt was measured electronically using an LVDT sensor coupled to the platform. By rotating the preparation and electrodes on the tilt table, Walthall and Hartman (1981) were able to subject the insect to various degrees of roll and pitch. Tilt, using aerodynamic nomenclature, can be characterized in terms of pitch and roll (Figure 2.6-2). Pure pitch is when the animal is oriented along the 0° to 180° axis. (The platform always pivots about the 90° to 270° axis.) Pitch up is when the animal is facing 0° and its head is tipped up. The amount of pitch up can vary between 0° and 90°. Thus, a pure pitch up of 30° can be written in terms of two angular coordinates, (. = 0°, φ = +30°): the rst number is the angle the animal is facing, and the second the maximum amount of displacement. If the head is pitched down 30°, then (0°, –30°). Pure roll is when the animal is rotated around its longitudinal axis. Roll up right 45° is when the right side of the animal goes up and the left side down; it is written (90°, 45°). Roll down right is (90°, –45°). As it turns out, because the cerci and the rows of tricholiths project at approximately 135° and 225° in the horizontal plane, maximum sensitivity of the four VNC positional interneurons to roll and pitch occurs when the animal has both roll and pitch at a tilt so that the . is at 45° or 315° (Figure 2.6-3).
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FIGURE 2.6-2 Dorsal view of an Arenivaga on a horizontal tilt platform, showing tilt directional conventions.
2.6.2
HARTMAN’S RESULTS
It appears that each tricholith socket is innervated by one stretch- or positionsensitive mechanoceptor neuron. Using ablation experiments, Walthall and Hartman showed that all the mechanoreceptor neurons of the tricholiths in the medial row of the animal’s right cercus provide additive excitatory inputs to the (larger) right ipsilateral positional interneuron (RIPI). Tricholiths in the lateral row on the right cercus provide additive excitatory inputs to the left (smaller) contralateral positional interneuron (LCPI). Similarly, the medial row of tricholiths on the left cercus drive the (larger) left ipsilateral positional interneuron (LIPI), and the lateral row of tricholiths on the left cercus drive the (smaller) right contralateral positional interneuron (RCPI). It is well known that the larger spikes recorded extracellularly from a nerve bundle arise in the nerve bers with the lar ger diameters. Thus, it was easy to separate out the spikes visually from the RCPI and the RIPI, etc. on the oscilloscope. Figure 2.6-4 summarizes the results reported by Walthall and Hartman (1981). All four VNC positional interneurons were found to have approximately cosine(2. ) response patterns (number of spikes in the rst 10 s after tilt) in terms of pitch/ya w angle, given a x ed rate of tilt and maximum tilt angle. For example, when the animal was oriented so its head pointed to 45°, and was tilted so its head went down and its right side came up, the RCPI, driven from the lateral row of tricholiths on
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FIGURE 2.6-3 Schematic dorsal view of the tricholith system “wiring.” Each separate tricholith sensory neuron (from a row of tricholiths) sends an axon forward to the cell body of the giant afferent interneuron in the sixth abdominal ganglion. There are only four giant afferent interneurons: The right ipsilateral positional interneuron (RIPI) is driven by the right medial row of tricholiths. The right contralateral positional interneuron (RCPI) is driven by the left lateral row of tricholiths. Similarly, the LIPI is driven by the left medial row of tricholiths, and the LCPI is driven by the right lateral row of tricholiths. The four PI axons travel uninterrupted to the brain in the ventral nerve cord.
the left cercus, red at its maximum rate. From inspection of the polar plots obtained experimentally that describe the VNC interneuron ring rates in response to v arious . and φ, it is possible to model the RCPI directional sensitivity by fRCPI = 300(0.5){1 + cos[2(. – 45°)]},
–45° = . = 135°,
0 elsewhere
2.6-1
for a φ = –45° net tilt. The 300 is the number of spikes counted in 10 s following initiation of tilt. Note that . is the direction to which the animal’s head points, and φ is the tilt angle of the platform. Similarly, for the same animal head orientation, the LIPI’s response to the excitation from the medial row of tricholiths on the left cercus can be approximated by
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0o PIC's
315o 300
45o
150
90o
270o s
/10
kes
spi
PII's
225o
135o 180o
FIGURE 2.6-4 Polar plot of the responses of the four positional interneurons recorded from the ventral nerve cord while the animal was tilted 30° in various directions. Spikes were counted in the rst 10 s follo wing a tilt. In Walthal and Hartman’s notation, PIC stands for positional interneuron driven from tricholiths on the contralateral cercus, PII is for positional interneuron driven from tricholiths on the ipsilateral cercus. Thus, the dotted line at the upper right of the polar plot is the RCPI response to various tilt vectors, the solid line in the upper left of the polar plot is the LCPI responses, the solid line in the lower right is the RIPI responses, and the dotted line in the lower left is the LIPI responses to tilt. The little cockroach drawings with tilt arrows clarify the exact degree of roll and pitch each direction around the polar plot represents. There are three major observations one can make from this gure: The contralateral PIs have the stronger response, the nearly circular plots suggest a cosinusoidal directional response (see text for discussion), and two PIs are ring for e very tilt direction except 45°, 135°, 225°, and 315°. (From Walthal, W.W. and Hartman, H.B., J. Comp. Physiol. A, 142: 359, 1981. With permission.)
fLIPI = 140(0.5){1 + cos[2(. – 225°)]}, 135° = . = 315°, 0 elsewhere
2.6-2
for the head tipped up and right-side down. Note that the RCPI does not re for this stimulus. Next considered are responses from tricholiths on the right cercus. The RIPI is driven from the medial row of tricholiths on the right cercus. The RIPI 10-s spike count for a φ = 45° tilt is given by fRIPI = 140(0.5){1 + cos[2(. – 135°)]},
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45° = . = 225°,
0 elsewhere
2.6-3
The animal’s head points to 135°, and the maximum response occurs for tilt such that the head and the right side come up. The LCPI interneuron is driven from the lateral row of tricholiths on the right cercus. The 10-s spike count is approximated by fLCPI = 300(0.5){1 + cos[2(. – 315)]},
225° = . = 405°,
0 elsewhere.
2.6-4
The LCPI responds only to tilts that make the head and the right side go down. The RIPI does not re when the LCPI res, and vice v ersa. Figures 2.6-2 and 2.6-3 show that if the cockroach head angle is . = 90°, a standard tilt of the 0° point down will produce pure roll, right side up, and the RIPI and the RCPI neurons will both re. In summary, any tilting of an Arenivaga that causes the tricholith balls to swing away from the centerline of a cercus will stimulate the corresponding sensilla sensory neurons and cause the targeted VNC interneuron to re. Thus, in the example above, where . = 90°, the left lateral row and the right medial row of tricholith sensillae are stimulated because the corresponding tricholiths bend away from the center lines of the cerci. Figure 2.6-5A and B summarize the dynamics of the tilt responses of an RCPI and an LIPI interneuron given different tilt maxima φ and rates of tilt, dφ/dt. Walthall and Hartman (1981) showed that the four positional interneurons had a strong derivative component in their respective responses. That is, their instantaneous ring frequency was proportional not only to the tilt angle, φ, but also to dφ/dt. When dφ/dt . 0, the peak ring rate decayed slo wly toward zero. Walthall and Hartman stated, “it took many minutes during a maintained 15° displacement [tilt] for the PIC unit [LCPI] to cease ring. ” Thus, it would appear that an Arenivaga must move to have its CNS advised of its orientation relative to the gravity vector.
80
φ
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Frequency (imp/s)
Frequency (imp/s)
0
. = 225
0
. = 45
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45 0 60 0 45 0 60 0 30 0 30
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FIGURE 2.6-5 (A) Mean instantaneous frequency of the RCPI unit with . = 45°for various degrees of tilt (φ). Plots illustrate rate sensitivity and accommodation of the unit. Solid curves, tilt at 21°/s; dashed curves, tilt at 7°/s. (B). Mean instantaneous frequency of the LIPI unit with . = 225° for various degrees of tilt (φ). Plots illustrate rate sensitivity and accommodation of the unit. Solid curves, tilt at 21°/s; dashed curves, tilt at 7°/s. (From Walthal, W.W. and Hartman, H.B., J. Comp. Physiol. A, 142: 359, 1981, slightly modi ed. With permission.)
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2.6.3
CNS UNIT ACTIVITY INDUCED ROLL AND PITCH
IN
ARENIVAGA
BY
This section describes the results of Willey (1981) who located and recorded Arenivaga protocerebral positional (PP) interneurons, driven by the four VNC afferent interneurons described by Hartman et al., (1979) and Walthall and Hartman (1981). Willey and the author were motivated by the interest to discover how the animal re ned the information on the four afferent positional interneurons in its CNS. Was there a sharpening of the cosine(2. ) directional responses observed by Hartman et al.?
2.6.4
WILLEY’S METHODS
A large, robust Sears milling head was mounted vertically on a heavy steel plate bracket. The jaws of the milling head were used to grip an aluminum plate measuring 7 ⋅ 8 ⋅ 5/8 in. thick. Mounted on the plate was a 6-in.-diameter, angle-indexed disk supporting a vertical, 1/2-in.-diameter, 11.5 in. stainless steel rod. The disk and rod were free to rotate together in the plate to select . . A dc servomotor and an anglesensing potentiometer were attached to the milling head to generate known tilts of angle φ at programmed rates. The motor was controlled by a closed-loop servo system. An Arenivaga colony was maintained in the laboratory; the original animals were obtained from Dr. Hartman. A living animal was mounted ventral-side down on the disk using Cenco tackiwax, and dissected to expose its head. The head was then immobilized with a wax collar, and a trapezoidal window was cut in the frons. The antennae were removed and the antennal nerves severed. The protocerebrum was exposed, and kept moist with a special saline solution. Extracellular nerve recording was done using etched, glass-coated, 0.01-in.diameter, 70% Pt/30% Ir microelectrodes. Electrodes were similar to those used by Northrop and Guignon (1970). An electrode was spot-welded into the end of a No. 26 metal hypodermic needle, the tip was then electrolytically etched to a ne point in a cyanide bath, then insulated with solder glass down to the tip. The bare tip was coated with ne platinum black to lo wer electrode impedance and reduce noise. The hypodermic needle and microelectrode were held by a micromanipulator attached to the stainless steel post in the center of the rotatable disk. A ne AgAgCl wire electrode was placed in the moist interior of the head as a reference electrode. Once the microelectrode penetrated the outer membrane of the protocerebrum, it was slowly advanced until a candidate positional unit was found. Once a candidate PCP unit was located, the experimenter veri ed that it w as purely positional. If the unit were purely PCP, air puffs from a medicine dropper directed at the cerci would not affect the ring rate of the unit, nor w ould an ON–OFF visual stimulus, or loud low-frequency sounds; tilting the animal would, however. A mixture of mineral oil and white vaseline was then infused around the protocerebrum to prevent drying and to give electrical insulation. When the head angle . was changed, the disk, rod, animal, and microelectrode rotated as a unit. Tilt angle φ was controlled electrically by the servosystem. The entire system was extremely robust, and weighed over 50 kg. It was especially important that the relative position
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of the animal’s brain and the electrode tip not change during selection of . and tilting φ.φ used for all of Willey’s preparations was 45°. Tilts were done at 3.2°/s. An exposed microelectrode tip was typically a 45° cone, about 3 ∝m high. Thus, in dense nerve tissue such as the protocerebrum, it picked up the external potential changes caused by several neurons ring in its proximity . A relevant protocerebral positional unit (PPU) was not always the largest unit potential recorded. Thus, Willey used a pulse-height window circuit (Northrop and Grossman, 1974) to select the desired PP unit. If another, non-PP unit had an amplitude close to that of a desired PP unit, the window was useless, and a new recording site had to be found.
2.6.5
WILLEY’S RESULTS
Willey (1981) found six PP unit sites in 649 electrode penetrations done on 84 different animals’ protocerebrums. In each of the six penetration sites, several different PP units could be isolated with the pulse-height discriminator window. All discriminated PP units exhibited random steady-state ring with the animal in the horizontal plane (φ = 0°). Willey averaged the number of window-discriminated spikes in six 3-s intervals to obtain the primary orientation background level (POBL) of a unit in pps. Curiously, PP units exhibited suppression of the POBL (background) ring if tilted in certain ranges of . , generally about 180° away from the . giving the maximum response of the unit. This behavior is illustrated in Figure 2.6-6.
FIGURE 2.6-6 A polar plot of the tilt response of a protocerebral positional unit from Willey (1981). The dotted circle shows the average random ring rate of the PPU when the animal is level. There was complete suppression of the POBL ring for tilts in the sector 280° to 260°. The animal was always tilted φ = 45° at 3.2 °/s in this and subsequent plots from Willey (1981).
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Willey (1981) recorded from single PP units that showed single, very sharp directional responses. For example, see Figure 2.6-7. This unit had a strong, sharp
FIGURE 2.6-7 An Arenivaga PPU with a very sharp directional sensitivity.
peak at . = 135°. Its background ring w as strongly suppressed for tilts over . = 135° . 360° . 30°. In Figure 2.6-6, the response of the unit is not quite as sharp as the one shown in Figure 2.6-7. Its peak angle is at . = 270°, and there is strong suppression of the POBL ring for . = 285°. 360° . 255°. The fact that its peak angle is at 270° suggests that its response is in some way due to the ring of the LCPI and LIPI VNC neurons which both re for . = 270°. Other sharp unimodal response units are shown in Figures 2.6-8 and 2.6-9. Some PP units were unusual in having two or more response peaks (frequency maxima) as a function of . . One such response is shown in Figure 2.6-10. Note that the peaks are very sharp compared with the cosine(2x) response seen on the four VNC positional interneurons (RCPI, RIPI, LCPI, LIPI). They are also about 180° apart (. = 135° and 330°). A smaller, broader response peak at . = 195° is also present. Note that the peak at . = 135° is the same angle that produces maximum ring in the RIPI VNC giant ber . One possible explanation of the three-peak behavior is that three PP units having exactly the same pulse height were discriminated by the window, giving a superimposed response. Or the response may be the real (counterintuitive) behavior of a single PPU. Of special interest is the sharpening of the directional responses. Two PP units showed broad, single peak responses. The unit shown in Figure 2.6-11 had a peak at . = 195°, and showed a small suppression of ring belo w the POBL rate over 300° to 45°. Figure 2.6-12 shows the response of another broad, unimodal unit having its peak around 315°. Some small suppression was seen for . between 15° and 90°.
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FIGURE 2.6-8 An Arenivaga PPU with sharp directional sensitivity.
FIGURE 2.6-9 An Arenivaga PPU with a very sharp directional sensitivity main peak at 260°, and a smaller sharp peak at 220°.
All protocerebral positional units studied had low POBLs, and adapted rapidly following the tilt φ. Adaptation times were dif cult to measure because the increased ring rate due to tilt at the preferred . decayed toward the random POBL ring rate. Willey’s data suggest that adaptation times range from 20 to 45 s.
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FIGURE 2.6-10 A curious PPU with a trimodal response. (The trimodal response may be because three different PPUs had spikes of nearly the same size, giving a superimposed response.)
FIGURE 2.6-11 A single PPU response with a broad directional sensitivity. Note suppression of POBL ring for tilts in the 315° to 45° range.
2.6.6
A TENTATIVE MODEL
FOR
PCP UNIT NARROW SENSITIVITY
A tentative, conceptual neural model to describe the behavior of directionally sharp PP units has been devised. Such a model must necessarily involve inhibition because
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FIGURE 2.6-12 Another broadly sensitive PPU that has a sensitivity peak at about 315°, similar to the input LCPI VNC unit.
PP units generally involve the suppression of the background ring (POBL) when the animal is tipped at angles away from the peak angle (e.g., see Figure 2.6-6). Inspection of the four combined angular response plots of the VNC interneurons (see Figure 2.6-4) given by Walthall and Hartman (1981) suggests ways that PP neurons can be driven through excitatory and inhibitory synaptic inputs that may lead to directionally sharp responses. Figure 2.6-13 illustrates a possible organization of PPUs. Examine, for example, the 315° PPU. Sharpening of its directional response is possible by two mechanisms: (1) by the relative weighting of the excitatory and inhibitory synaptic inputs and (2) by use of feedback inhibition analogous to the reciprocal inhibition in arthropod compound eye systems. (Reciprocal inhibition is introduced in Section 4.2 of this book; it is shown under certain conditions to lead to a sharpening of contrast in a visual image.) Thus, if the animal is tilted away from 315°, either the RCPI or LIPI unit begins to re and reduces the 315° PP unit ring rate via inhibitory interneurons. In the case of a PP unit with a 270° peak sensitivity, sharpening is accomplished by driving the unit excitatorily with the LIPI and LCPI VNC inputs, while inhibiting it with RIPI and RCPI inputs. Further inhibition aids sharpening in the form of reciprocal inhibition from the outputs of the 315° and 225° PP units. Note that 24 model interneurons are required to simulate the PPU model shown in Figure 2.6-13. (Details of the models used for this size of neural model are discussed in Chapter 3.) To demonstrate sharpening in the model, the input frequencies on the four VNC connectives is preset according to Figure 2.6-4 for each . . At each . assumed, there will be only two connectives ring at rates determined by Figure 2.64. (At the unique angles of . = 45, 135, 225, and 315°, there will be only one connective
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FIGURE 2.6-13 A prototype neural model for directional sharpening in PP units. Note that feed-forward inhibition and lateral inhibition is hypothesized to cause directional sharpening. Four input lines diverge to eight output lines in this model, 24 neural elements and a total of 60 synapses must be simulated.
ring maximally.) Note that, for simplicity, this model is based on constant input frequency ring, rather than the dynamic adapting beha vior seen in the insect. Because the 24 neuron model contains eight output neurons (PP units assumed recorded), plus 16 inhibitory interneurons, and 60 synapses, each of which has an analog psp emulated by two coupled rst-order ODEs, there is a total of 148 states in the model, including four neurons acting as voltage-to-frequency converters for the four afferent VNC sensory giant bers.
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Rather than simulate the entire 148-state model, a reduced model consisting of only three output PPUs (the 45°, 90°, and 135° units in the 24 neuron model) has been chosen. Associated with the three output PPUs, only four inhibitory interneurons are needed, giving a total of seven neurons to model. Also, 18 synapses are needed in the reduced model; 4 inhibitory and 14 excitatory. Thus, a total of 7 + 2 ⋅ 18 + 4 = 47 states is needed for the reduced model. The model is shown in Figure 2.6-14. The reduced model is simulated using a Simnon program, ARENcns3.t, listed in Appendix 1.
FIGURE 2.6-14 Reduced PPU model. For proof of concept, the model of Figure 2.6-14 was reduced to this 7-neuron, 18-synapse model. The narrow response is expected at the 90° output neuron.
In the program above, each of the four VNC units were modeled by rst generating an analog quantity whose magnitude followed the tilt sensitivity shown in Figure 2.6-4. For example, the response of the RCPI VNC unit was given by frcpi = (maxF1/2) * {1 + cos[2(. – 45)]} To make this RCPI unit not respond for 135° < . < 315°, use the statements:
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2.6-5
rcpi1 = IF THETA > 135 THEN 1 ELSE 0 “ Supresses rcpi for theta rcpi2 = IF THETA < 315 THEN 1 ELSE 0 “ between 135 - 315°. rcpi3 = rcpi1*rcpi2 rcpi = IF rcpi3 > 0 THEN 0 ELSE frcpi
The analog signal rcpi is the input to an IPFM voltage-to-frequency converter (state r1). The frequency of uRCPI is proportional to rcpi. The steady-state pulse frequency of uRCPI is one of the inputs to the reduced neural model. The periodic pulses in uRCPI are inputs to the two-pole, ballistic lters, 7, 14, and 17, which generate epsps for neurons 2, 3, and 4, respectively (see Figure 2.6-14). Similar signal processing occurs in the other three VNC unit analogs.
FIGURE 2.6-15 Results of a simulation of the model of Figure 2.6-14. Model neuron spikes are shown. Simulation was done using the 47-state, Simnon program, ARENcns3.t found in Appendix 1. The insect orientation angle, . , was assumed to be smoothly varied over 0° to 225°using a tilt of 45°. The horizontal axis of the gure is . . (Recall that . = 90° corresponds to pure roll right up; see the insects in Figure 2.6-5.) Part of the program modeled the cosine directional responses of the four VNC PI units driving the model. The bottom spike train is the simulated activity of the RCPI axon; the trace immediately above it shows the spikes on the RIPI axon during the change of tilt direction, . . The third trace up shows the spikes on the LIPI axon; the fourth trace up is the LCPI activity. The numbered traces are 1, output of the 45° PPU; 2, output of the 90° PPU (note that it res for . in the range of 80° to 120°, giving a sharp response); 3, output of the 135° PPU.° 4 to 7 show activities of interneurons 4 to 7, respectively. It is not surprising that 1 and 3 do not have sharp responses because the model is complete only around PPU 2.
To simplify the operation of the neural model, let . vary continuously and linearly from 0° to 225°, and plot the spike outputs of the seven neurons in the model as well as the spike frequencies associated with the IPFM models of the four VNC afferent units. A typical plot is shown in Figure 2.6-15. (A more accurate way
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to use the model would be to input x ed . values and measure steady-state pulse frequencies of the seven neurons.) The model parameters used were: [ARENcns3] “ Parameters 2/27/99 AREN6.t : maxF1:2, R:57.296, maxF2:1, phi2:3, tau:0.002, Do:0.5, Doo:0.5, Doi:0.5, co:1, phi:2, g2:5, g3:1, g3e:1, g6:0.3, ci:1, ae:0.5, ai:0.5, pi:3.14159.
Note that the three output neurons re o ver a considerably narrower range of . than do the VNC units (80 to 120°); hence, the model demonstrates that sharpening of the positional response is possible, and it offers a possible mechanism responsible for this enhanced resolution.
2.6.7
DISCUSSION
It is apparent that the protocerebral positional units of Arenivaga have only four inputs, i.e., the VNC connectives RCPI, RIPI, LCPI, and LIPI. These VNC interneurons show broad, cosine-like directional sensitivities. Willey (1981) has shown that certain PP units have very sharp directional responses, at . values not restricted to the major sensitivity directions of the VNC connectives: e.g., 45°, 135°, 225°, and 315°. Furthermore, it appears that at any . , there will be at most two VNC units carrying tilt information to the protocerebrum, and only one unit if . = 45°, 135°, 225°, or 315°. Thus, the protocerebral interneurons work with a paucity of inputs to advise the animal that it is being tilted at a very speci c . with respect to the gravity vector. Why does the animal need sharp PPU directional responses? Are they involved in the motor control of burrowing, as Hartman et al. (1979) have suggested, and can they help stabilize the animal in ight? Both the VNC interneurons and the PP units habituate to zero rate. Thus, if the animal burrows deep in ne sandy soil and remains still for se veral minutes, its protocerebrum is theoretically not being sent any positional information from the tricholiths. However, if the animal is still, it does not need positional information! As soon as the animal moves, because of the dynamic nature of its gravity vector sensing system, it is advised of its orientation and can, for example, sense roll and pitch up as it burrows toward the surface. It is reasonable to assume that the PP units outputs are used to make adjustments to the leg motor system so that the animal can minimize roll and control pitch when it is burrowing in ne sandy soil. Because of the phasic nature of the tricholith sensillae response to changes in animal position in the gravity eld, it is also possible that the system could be used to stabilize ight, albeit ight is only by males, and is done infrequently.
2.7 THE DIPTERAN HALTERE Anyone who has tried to swat a house y can appreciate the agility in ight e xhibited by these creatures. Unlike ying insects ha ving two pairs of wings (dragon ies, moths, butter ies, etc.), dipteran ies ha ve one pair of wings and short, stubby bodies. That they can y in a controlled, purposeful manner must be considered remarkable. What is even more fantastic is the fact that ies use dynamic visual
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information along with inertial information to stabilize their ight. The inertia information for ight stabilization of dipteran ies comes from a unique pair of vibrating gyroscopes (angular rate sensors), which have evolved over the millennia from the hind wings. These dumbell-shaped organs, called halteres, reside on the sides on the y in back of the wings, and are vibrated at wing-beat frequenc y by modi ed ight muscles. Each haltere swings through an arc of nearly 180° (Schwartzkopff, 1964). Figure 2.7-1 shows a haltere on the left side of a (dipteran) mosquito.
Note the haltere
FIGURE 2.7-1
Side view of a mosquito (diptera), showing its left haltere behind the wing.
Where the halteres attach to the y’ s body there are many mechanoreceptor cells (campaniform sensillae) whose presumed function is to respond to the vibrating, gyroscopically induced torques on the halteres produced when the y under goes a rotational departure from level straight ight (roll, pitch, or yaw) (Bullock and Horridge, 1965). More particularly, the haltere system can theoretically respond to roll, pitch, and yaw rates to provide high-speed feedback to the wing ight muscles to stabilize ight. The bases of Dipteran halteres are endowed with a profusion of chordotonal organs and campaniform sensilla mechanoreceptors that presumably respond to the dynamic forces and torques on the haltere support as the insect undergoes roll, pitch, and yaw in its ight. The outputs from these sensory neurons at the base of the halteres is evidently integrated with visual information from the compound eyes to give complete ight control. A good illustration of haltere mechanosensors can be found in Figure 20.9 in Bullock and Horridge (1965). The next section derives expressions for the torques produced at the base of a model haltere when it is subject to angular velocity in certain planes.
2.7.1
THE TORSIONAL VIBRATING MASS GYRO
A schematic of a single, torsional vibrating mass angular rate sensor is shown in Figure 2.7-2. This structure is analogous to one haltere. This particular example (1) introduces and illustrates Lagrange’s method of analyzing complex, dynamic biomechanical systems (Cannon, 1967); (2) allows extension of the analysis of a single, torsional, vibrating-mass, angular-rate sensor to a paired structure in which the masses vibrate 180° out of phase (a crude model for the paired halteres of a complete y).
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Platform
FIGURE 2.7-2 Schematic (top view, side view) of a single torsional vibrating rate sensor. A mass M on the end of a rod of length R is caused to vibrate sinusoidally through an arc of ± . m. See analysis in text.
A haltere is attached and pivoted at its small end to the insect’s body below and behind each wing. It is basically a mass at the end of a stiff rod that vibrates in one plane at the wing-beat frequency when the insect ies. To derive a mathematical description for the torques at the base of the vibrating mass, this section uses the well-known LaGrange method (Cannon, 1967), which is based on Newtonian mechanics. For a complex mechanical system one can write N equations of the form: d . .L. .L – = Qk dt .. . q k .. . q k
2.7-1
where qk is the kth independent coordinate. The qk can be rotational or translational. Qk is the kth moment (torque) if rotational dynamics are considered, or the kth force if the dynamics involve linear translation. T is the total kinetic energy of the system, and U is the total potential energy of the system. L = T – U. L is called the system’s “LaGrangian function.”
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FIGURE 2.7-3 Schematic of ying insect sho wing the left haltere vibrating in the xz plane.
Consider Figure 2.7.3 in which the linear ight v elocity vector is along the y axis, and for simplicity, let the halteres vibrate in the XZ plane. In this example, the independent coordinates are . , the angle of the vibrating mass in the XZ plane with the z axis, and φ, the input rotation around the Z axis (pitch up is shown). The system potential energy is U = (1/2)K. . 2 J. K. is the spring constant of the torsion spring that restores the resting mass to align with the Z axis. A muscular “torque motor” (not shown) keeps the mass vibrating sinusoidally in the XZ plane so that . (t) = . m sin(. m t). There are two components to the system kinetic energy:
( )
T = (1 2)J o . 2 + (1 2)M rφ
2
2.7-2
Here Jo is the moment of inertia of the haltere (mass on the end of a rod of length R): It is well-known that Jo = MR2. r is the projection of R on the XY plane: r = R sin(. ). ( rφ ) is the tangential velocity of the mass M in the XY plane due to the input rotation rate. Thus, the LaGrangian of the system is:
( )
L = (1 2)J o . 2 + (1 2)M rφ
2
– (1 2)K.
2
= (1 2)J 0 .
+(1 2)MR (1 2)[1 – cos(2. )]φ – (1 2)K . . 2
2
2
2.7-3
2
And thus, .L = – (1 2)MR 2 (1 2)[ –2 sin(2. )]φ 2 – K. = (1 2)MR 2 sin(2. )φ 2 – K . . ..
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2.7-4
.L = J o. ..
2.7-5
d . . L. . . = J o. dt . . . .
2.7-6
· · Hence M. = Jo. + K. . – (1/2)MR2 sin(2. ) φ2 is the torque required to be produced by the muscles to maintain the haltere in simple harmonic motion. Here the interest is in the (instantaneous) torque around the pivot in the XY plane, · Mφ, due to the input pitch angular velocity, . . One can write, .L =0 .φ
2.7-7
.L = M r 2 φ = MR 2 sin 2 (. )φ = MR 2 (1 2)[1 – cos(2. )] .φ
2.7-8
d . . L. 2 2 2 . = MR φ(1 2) – MR φ(1 2) cos(2. ) + MR φ . sin(2. ) . dt . . φ .
2.7-9
M φ = φ(1 2)MR 2 – φ(1 2)MR 2 cos(2. ) + (1 2)MR 2 φ . sin(2. )
2.7-10
Thus,
Because 2. m 10° in the haltere system, one cannot use the approximations: sin(2. ) . 2. , and sin2(. ) . . 2. The Mφ moment can be written:
[
M φ = φ(1 2)MR 2 – φ(1 2)MR 2 cos 2. + φ(1 2)MR 2 . m .
m cos(.
[
m
m t ) sin 2.
sin(.
m
m sin(.
]
t)
]
m t)
2.7-11
·· Thus the Mφ moment lies in the XY plane and contains a dc term proportional to φ, · ·· and also double-frequency terms in φ and φ. The desired angular velocity term is larger by an amount . m, which can be 900 r/s in mosquitos. If it is assumed that the sensory neurons associated with the base of each haltere · ·· respond to the twisting torque, Mφ, induced by φ and φ, then the sensory output will contain a pulse frequency code proportional to pitch angular acceleration and, more interestingly, pitch angular velocity at twice the wing-beat frequency. The same haltere can be shown to respond to angular acceleration and velocity in the yaw direction.
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Refer to Figure 2.7.3. To simplify the yaw calculations, assume that there is no restoring spring torque (K. = 0). The Lagrangian is thus the kinetic energy of the system: L = (1 2)J o . 2 + (1 2)M(. α )
2
2.7-12
· where .α· is the tangential velocity of the mass M due to yaw angular velocity, α. The instantaneous effective radius of the mass (projected onto the z axis) is . = R cos(. ). The forced oscillatory displacement of the haltere can be written: . (t) = . m sin(. m t). Thus, the Lagrangian can be rewritten as L = (1 2)J o . 2 + (1 2)Mα 2 R 2 cos 2 (. ) = (1 2)J o .
2
+(1 2)MR 2 α 2 (1 2)[1 + cos(2. )]
2.7-13
.L = (1 2)MR 2 α 2 [ – sin(2. )] ..
2.7-14
.L = J o. ..
2.7-15
d . . L. . . = J o. dt . . . .
2.7-16
Thus the driving torque required on the haltere is M . = J o . + (1 2)MR 2α 2 sin(2. )
2.7-17
Due to yaw, α, there will be a moment at the base of the haltere in the YZ plane, Mα. This moment is found by .L =0 .α
2.7-18
.L = (1 2)MR 2 (1 2)[1 + cos(2. )]2α .α
2.7-19
d . . L. 2 2 2 . . = (1 2)MR α + (1 2)MR α cos(2. ) + (1 2)MR [ –2 sin(2. )]. α dt . . α . Thus,
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2.7-20
[
M α = α MR 2 cos 2 .
m
sin(.
m
]
t ) – α MR 2 . m .
m
cos(.
m
[
t ) sin 2.
m
sin(.
m
]
t ) 2.7-21
Compare Mα with Mφ:
[
M φ = φ MR 2 sin 2 .
m
+ φ (1 2)MR . m .
sin(.
2
m
m
]
t)
cos(.
m
[
t ) sin 2.
m
sin(.
m
]
t)
2.7-22
Note that the pitch and yaw velocity moment responses differ by (1) an algebraic sign and (2) lying in orthogonal planes (Mφ in the XY plane; Mα in the YZ plane). No wonder the sensory cells at the base of an haltere have complex morphology. Note that if a haltere were mounted so that it vibrated in the animal’s XY plane, then it would appear to be responsive to angular acceleration and angular velocity due to roll and pitch. If a haltere were to vibrate in a plane at 45° to the ZY plane, the neural sensors at its base may possibly extract information simultaneously about pitch, roll, and yaw acceleration and velocity.
2.7.2
DISCUSSION
The results of this analysis of the torques at the base of a vibrating mass rate sensor ·· produces show that they are complex to interpret. For example, yaw acceleration, α, a dc torque component, as well as a time-variable one proportional to cos[2. m sin(. m · produces a time-variable torque proportional to cos(. t) sin[2. t)]. Yaw velocity, α, m m sin(. m t)]. This term gives a torque at double the haltere vibration frequency, . m. Mechanosensory cells in the bases of the halteres presumably re synchronously with the haltere vibration. Because of the high frequency of vibration, amplitude information for α· is probably coded by recruiting more and more cells to re. Thus, yaw and pitch rate information could be coded by the number of mechanosensors ring e very cycle or half-cycle, giving a high data sampling rate. One can conjecture from an engineering point of view how this information might be demodulated by the y , but it boggles the mind how it might be done with real neurons. Dipteran insect ight stabilization in the daytime is probably dominated by inputs from the compound eyes and aerodynamic hair patches and antennae. In the dark, or under low-light conditions, it is reasonable to expect roll, pitch, and yaw signals from the haltere sensors to play a signi cant role in stabilizing ight. That the ght stabilization system has three sensory input modalities (inertial, visual, and aerodynamic) argues for its complexity. How a y decodes this information and uses it to stabilize its ight is going to remain a mystery for some time.
2.8 THE SIMPLE “EYE” OF MYTILUS EDULIS Mytilus edulis L. is the common, blue marine mussel found in the intertidal regions of northeast North America. It is a bivalve, plecypod mollusk that in its adult form attaches itself to rocks and pilings by a network of tough byssus threads, which it
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spins. (Byssus is a natural biopolymer spun by the animal for the express purpose of attaching itself to a supporting substrate.) The eyes of Mytilus lack any dioptric apparatus (a lens or pinhole) with which to form an image on its primitive photoreceptor cell array. Mytilus has two eyes; each eye is a simple invaginated cup, containing about 37 photoreceptor cells (PRCs), and is about 50 to 60 ∝m in diameter, regardless of animal size. Figure 2.8-1 shows a light micrograph of a transverse section through an eye. A larger magni cation of an e ye is shown in Figure 2.8-2. Note the granules in the pigment cells. The eyes are located at the base of the left and right inner gill laments, between the inner and outer gills. Each PRC sends an axon directly in the optic nerve to the cerebral ganglion. LaCourse (1981), observed that there was one optic nerve ber per PRC. That is, this simple visual system has no intervening synapses between the PRCs and the optic nerve bers.
FIGURE 2.8-1 Artist’s line drawing of a transverse section through an eye of M. edulis (from a light micrograph). Object in the eyecup is an artifact; a shrunken ball of mucus and cilia. Scale line is 20 ∝m. (From LaCourse, J.R., Ph.D. dissertation, University of Connecticut, Storrs, 1981. With permission.)
2.8.1
EYE MORPHOLOGY
LaCourse (1981) examined the structure and ultrastructure of Mytilus in detail using light and transmission electron microscopy. He found that there were pigment cells in close membrane association with the PRCs. The arrangement between PRCs and pigment cells was irregular. The PRCs were unpigmented. Their distal ends were covered with many microvilli and one or more cilia with the 9 + 2 arrangement of microtubules in their axonemes. Figure 2.8-3 is an artist’s schematic of a slice through the eyecup based on electron microscope photomicrographs. The optic nerve
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FIGURE 2.8-2 Transmission electron micrograph of pigment cells (PC) and ciliated sensory cells (SC) on the margin of the eyecup. (⋅ 3000). (From LaCourse, J.R., Ph.D. dissertation, University of Connecticut, Storrs, 1981. With permission.)
axons are about 1 to 2 ∝m in diameter. Note the slender region between the inner nuclear part of the PRCs and the outer section, which contains the cilia and microvilli that project into the eyecup. Each microvillus is about 12 ∝m long and 0.15 ∝m in diameter. Its interior is lled with granular material and small v esicles about 0.03 ∝m in diameter. The microvilli presumably contain the photoreceptor pigments and are the primary transduction sites where captured photons trigger photochemical reactions leading to PRC hyperpolarization. Of the PRCs, LaCourse (1981) comments: Slender cell processes, passing through the pigmented region contain microtubules and numerous cytoplasmic vesicles as supportive or transportive elements. These processes are without pigment granules and therefore do not function as shading elements. The large nuclear regions of the sensory cells house rich accumulations of glycogen granules, Golgi apparatus, mitochondria, and both smooth and rough endoplasmic reticulum. The presence of these organelles indicates the high level of synthetic activity common to visual cells.
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C
SC
MV
MV
PC Z MT PG
Z
CV
SER PN
SN M
LP
NT
BL
FIGURE 2.8-3 Artist’s reconstruction of the cells in the Mytilus eyecup based on many TEM micrographs. Key to abbreviations: BL, basal lamella; C, cilia; CV, cytoplasmic vesicles; LP, lipid droplets; M, mitochondria; MT, microtubules; MV, microvilli; NT, axons; PC, pigment cells; PG, pigment granules; PN, pigment cell nucleus; SC, photosensor cell; SER, smooth endoplasmic reticulum; SN, sensory cell nucleus; Z, zonules. (From LaCourse, J.R., Ph.D. dissertation, University of Connecticut, Storrs, 1981. With permission.)
2.8.2
PHYSIOLOGY
OF THE
EYE
The rst interesting property of the e ye discovered by LaCourse (1981) and described by LaCourse and Northrop (1983) was that it is an entirely OFF system. That is, the eye produced spikes on its optic nerve bers only at OFF of general illumination. The number of spikes and their instantaneous frequency depends on the intensity, wavelength, and duration of the light exposure before the light is turned off. LaCourse found that there was an optimum light level (other factors held constant) to elicit a maximum number of spikes at OFF. The peak response occurred for a white (tungsten) light stimulus of 2 ∝W/cm2; it fell to zero for intensities less than
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about 2 nW/cm2, and above about 1 mW/cm2. LaCourse also examined the wavelength action spectrum of the OFF response using nearly monochromatic light from a single-grating monochromator adjusted for constant intensity. Response was strongest between 460 to 550 nm wavelengths; there were two peaks in the action spectrum, however: one at 490 nm (blue) and the other at 520 nm (green). LaCourse also found that an intensity-dependent latency to the OFF response existed. At threshold intensity, the latency was very large, about 2 s; it steadily decreased to about 100 ∝ at the maximum response intensity, and again slowly increased as the intensity was increased past that for the peak response. LaCourse did not do intracellular recording from the Mytilus PRCs. He did measure the extracellular “ERG” voltage for the eye, however, using a glass microelectrode lled with sea water. The reference electrode was placed far from the eye in the gills. The microelectrode was advanced from the eyecup into the middle of the PRCs and thence to the area on the optic nerve. A ash of light produced a positive ERG potential, suggesting that the light caused the intracellular potential (Vm) of the PRCs to go negative, hyperpolarizing them. At OFF, when spikes occurred on the optic nerve bers, the ERG went ne gative, suggesting that Vm went positive, depolarizing the cells and leading to spikes. A putative scenario describing the Mytilus’ PRC/optic nerve behavior can be borrowed from what is known about vertebrate rods and cones. In their case, light (photons) is captured by a photoreceptor pigment (a rhodopsin). An enzyme cascade occurs in which rhodopsin . activated rhodopsin (metarhodopsin II) . a GTP-binding protein (transducin) . an enzyme hydrolyzing cGMP (cGMP-phosphodiesterase) . closure of membrane-bound, cGMP-gated cation channels. When the channels close, the normal (dark) rate of cation inward leakage (e.g., Na+) is exceeded by the rate of outward cation pumping, so Vm goes negative, hyperpolarizing the cell (Kolb et al., 1996). In the dark, the membrane cation channels gradually open, allowing Vm to again go positive, depolarizing the rod. A similar series of events may occur in Mytilus eyes. That the these eyes only re at OFF is reasonable because this is when Vm rebounds positively toward what might be the normal ring threshold for these PRCs.
2.8.3
DISCUSSION
In sensory neurophysiology, things are seldom simple. What is really occurring in the Mytilus PRCs will be better understood when intracellular recordings are made simultaneously from a PRC and a pigment cell, and optic nerve spikes are observed. The fact that PRC and pigment cell membranes are in intimate contact is no coincidence; function follows form, and vice versa. Unfortunately, Mytilus eye cells are small and soft, and are not easily recorded from. In closing, one might speculate why Mytilus has evolved an OFF photoreceptor with 37 PRCs. Why so many PRCs? Redundancy? Certainly no image is possible in the absence of a focusing apparatus, and the eye deep in the tissues between the gill tissues. Are they the vestigial remnant of a divergent evolutionary pathway taken by the Mytilidae? LaCourse tested the theory that Mytilus eyes served to warn the mollusk that danger is near (a shadow passing over the eye causing dimming), so
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the valves close for protection. He destroyed the eyes, then found that the shells still closed when the light was dimmed. He hypothesized that Mytilus has other singlecell PRs located on the gills, mantle, or foot that mediate this action. Then, what are the eyes used for? Do they sense photoperiod and thereby regulate feeding and/or the reproductive cycle? Perhaps increased light indicates low tide, and triggers appropriate behavior, such as spinning more byssus threads. The moral is that even apparently simple organisms are far more complex at the physiological level than one can imagine. Good research always raises new questions, and suggests new models.
2.9 CHAPTER SUMMARY This chapter has tried to provide the reader with an interesting sampling of basic sensory receptors found in vertebrates and in invertebrates. Some examples of unusual and little-known receptors (magnetoreceptors, sh electroreceptors, in vertebrate gravity sensors, and the haltere, a vibrating, angular-rate sensor of dipteran ies) ha ve been included. Most receptors signal their sensed quantity by generating nerve spikes that are sent to the animal’s CNS. The spike “code” used is seen to be generally nonlinear; the steady-state spike frequency is proportional to the logarithm of the sensed quantity, or is described by a power-law where the sensed quantity is raised to some power < 1. Receptors generally exhibit rate sensitivity, where a step of sensed input produces a high-frequency burst of spikes at rst, and then slo ws to a steady-state ring rate. In some receptors, such as the mammalian spindle, the steady-state spik e rate is zero, and the receptor res a b urst when pressure is suddenly applied, and again when it is removed. Most receptors show unidirectional rate sensitivity, ring a burst only at the onset of their stimulus, and simply stopping ring when the stimulus is removed. Some receptors exhibit amazing sensitivities to threshold levels of their stimulus. A hypothetical model has been illustrated, where the ring threshold of the receptor is adjusted by a feedback mechanism so there are a few random spikes produced (false positives) by to noise in the spike generator potential. By maintaining such an optimum, low threshold, the receptor can minimize its number of false positives, while true positives are sensed with very few false negatives (missed input stimuli). Certain photoreceptors, low-frequency electric eld receptors, and chemoreceptors are good examples of receptors that have enormous sensitivities to low-level stimuli. In some cases, such great sensitivity may be in part due to nonlinear interactions between receptors (lateral inhibition or multiplicative processing). A challenge in sensory neurophysiology that should be met in the next decade is the identi cation of speci c magnetoreceptor neurons and the elucidation of ho w they work at the molecular level. It is known from magnetic bacteria that living cells can assemble chains of single-domain-sized magnetite (Fe2O3) particles inside themselves by biochemical means, as well as other iron/oxygen and iron/sulfur ferrimagnatic crystals. Most mechanoreceptor neurons sense distortions of their cell membranes when an external displacement, force, or pressure causes a depolarizing generator potential to occur. A model for a magnetoneuron would have the minute © 2001 by CRC Press LLC
forces generated by the Earth’s magnetic eld on biogenic magnetite crystals inside the sensor couple to the membrane where ion-gating proteins could be activated, causing depolarization and spike generation. Section 2.4.3 examined some speculative models for magnetoreception that do not involve internal or external magnetite crystals. Noise reduction in neuro-sensory systems having threshold sensitivities might be carried out in several ways. Multiplicative signal processing, synaptic averaging, and low-pass ltering by electrotonic conduction on dendrites and nonspiking axons may all gure in noise reduction.
PROBLEMS 2.1. A chemical kinetic model for photoreceptor transduction has been proposed* in which the depolarization voltage of the photoreceptor cell is proportional to the concentration, c, of product C in the cell; that is, vm = k6c. Product C is made according to the reaction shown in Figure P2.1: The conversion of A to B proceeds at a rate proportional to the log of the light intensity, k1 log[1 + I/Io]. The rate of conversion of B to C contains an autocatalytic term: (k2 + k3 c). In the absence of light, C is converted to A at rate k4. From chemical mass-action kinetics, one can write the three ODEs (a is the concentration of molecule A in the cell, and b is the concentration of B):
[
a = k 4 c – a k 1 log 1 + l l o
[
]
]
b = a k 1 log 1 + l l o + k 5 c – b( k 2 + k 3 c) c = b( k 2 + k 3 c) – ( k 4 + k 5 )c a. Simulate the three nonlinear chemical kinetic equations above: Use a(0) = 1, other ICs = 0, k1 = 4, k2 = 0.3, k3 = 40, k4 = 10, k5 = 0.1, k6 = 80, Io = 1. Let I = 0.1, 1, 10, 102, 103, 104 for 10 ms. Plot the depolarization, vm(t). Does the system saturate? Plot the initial peak and vm at 10 ms as a function of intensity. b. *Let I = 2. Plot a(t), b(t), and c(t) over 20 ms. Use the parameters in (a). 2.2. a. Consider a putative, Hall-effect, magnetosensor based on Figure 2.4-1. Protons, instead of electrons, are actively pumped and form Jx. Find the mean transport velocity, vx, required to produce a Hall voltage of 100 ∝V, given that By = Be = 5.8 ⋅ 10–5 T (maximum Earth’s magnetic eld), and t = 10 –7 m. b. Describe several known biological systems that actively transport protons or electrons. * Jones, R.W., D.G. Green, and R.B. Pinter.1962. Mathematical simulation of certain receptor and effector organs, Fed. Proc., 21(1): 97.
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FIGURE P2.1
2.3. Sharks and other elasmobranch sh use their AoL electrosensors to locate wounded prey sh lying on the ocean oor . This problem will model a wounded sh and calculate the electric eld in the w ater at some distance R and angle . from the sh. The wounded sh is represented by an electric dipole. Assume that the sh has major w ounds on its head and tail, and thus may be represented electrically as two conductors separated by a thin insulator (its body) of length L. The wounds have a potential difference between them of VF V, and an electrical capacitance between them of CF F. Thus, at a distance R L, the wounded sh can be represented by an electrical charge dipole. See Figure P2.3. The equivalent charges on the ends of the sh are Q = C FVF C, and its dipole moment is p = QL = CFVF L C meters. Consider the sea bottom to have the same dielectric constant as water (80), and not to introduce a boundary condition on the electric eld. Ne glect conductivity effects. A well-considered problem in electrostatics is the computation of the electric eld produced by a char ge dipole at a point (L, . ). In cylindrical coordinates, the electric eld has tw o components given by*: E = Er + E. =
C F VF L a cos(. ) + a. 2 π . εoR3 r
[
1
2
sin(.
)]
where ar and a. are unit vectors, and . is the angle the radial line makes with the dipole axis. a. Calculate E at the shark, given: L = 0.2 m, CF = 80 pF, VF = 50 mV, R = 2 m, . = 30°, εo = 8.85 ⋅ 10–12 F/m, and . (dielectric constant of water) 80. b. *Using the numbers above, nd the R where E = 0.2 ∝V/m. 2.4. The stalk and mass of a tricholith of an Arenivaga cockroach are restrained from moving by a stiff torsion spring when the animal is tilted. Tilting the cercus, as shown in Figure P2.4, generates a torque, M. , which is sensed by a strain-sensitive, mechanoreceptor cell. The output spike frequency of the cell is given by: fo = KM. . (There is no spike output for any tilt in the other direction.) Give an expression for the ring frequenc y as a function of . . * Kraus, J.D. 1953. Electromagnetics, McGraw-Hill, New York, Ch. 2.
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FIGURE P2.3
FIGURE P2.4
2.5. The feedback model for spindle control of Figure 2.3-9 has been modi ed to include a hypothetical CNS integrator (see Figure P2.5). a. Give expressions for R2(s)/X2(s) and R2(s)/Ro(s) in Laplace form. b. Let ro(t) = RoU(t + T) (ro is a step of height Ro applied long in the past so that the system is in the steady state. Also, let x2(t) = X2 U(t). Write expressions for r2(t). Plot and dimension r2(t). 2.6. Refer to Figure P2.6, a model for the dynamic behavior of a PC. Simulate this system using Simnon or Simulink. Let k = 4, . n = 0.5 r/ms, Q = 1/(2. ) = 0.707, c = 1 r/ms, . = 1. Use Euler integration with dT = 0.001 ms. The derivative of the applied pressure, pr, can be approximated by the analog lter:
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FIGURE P2.5
FIGURE P2.6
p = – b * p + b * pr q = a * q + a * pr dpr = a * b * ( p – q ) ( b – a )
(dpr estimates pr over 0
= . = a 2 r ms)
Give the spike responses of the RPFM spike generator for: a. A pulse of pressure: p(t) = Po [U(t) – U(t – 20)]. The pulse is 20 ms in duration; Po = 3 psi. Let a = 2, b = 3 r/ms.
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b. A pressure sine wave of varying frequency: p(t) = Po sin(2πf t). Let Po = 3 psi and let 0.05 = f = 5 cycles/ms. Plot the steady-state spike frequency vs. f. 2.7. It is known that the primary response of some animals’ photoreceptor cells is hyperpolarization upon illumination, while absorption of light in others causes depolarization. Describe what anatomical and physiological features these two classes of receptors have in common, and also the unique differences. Consider photoreceptors in mollusks, arthropod compound eyes, arthropod ocelli, annelids, and vertebrates. 2.8. Section 2.4.3 suggested that the Faraday streaming effect could induce an electric eld in and around a shark’ s gill slits as seawater is forcefully expelled through them in the Earth’s magnetic eld. The Earth’s magnetic eld has, in general, a vertical and a horizontal component. (At the “magnetic equator,” the vertical component is zero, and over the north and south magnetic poles, the horizontal component is zero.) As a rst approximation to modeling a hypothetical Faraday streaming effect magnetic sensor, consider the shark to be swimming at some low velocity that is small compared with the peak velocity of water expelled from the gill slits. Assume that water is expelled from the gill slits perpendicular to their (vertical) plane with peak velocity, vw. High peak water velocities can be obtained by muscular contraction of the pharynx. In general, the Faraday streaming effect produces an electric eld gi ven by: E = vw ⋅ Be. The direction of E is normal to the plane containing vw and Be, and points in a direction a right-hand screw would advance when vw is rotated into Be. E = ve Be sin(. ); is the angle between vw and Be. E is maximum when vw and Be are orthogonal. Assume a model shark with one left and one right rectangular gill slit. Figure P2.8A shows a top view and a right side view of the right gill slit of a north-swimming shark. Note that for a shark pointing toward magnetic north anywhere in the oceans other than the magnetic equator or the magnetic poles, the dip angle, . , of Be will, in general, generate a Faraday E- eld in the plane of the gill aperture at an angle of (90° – . ). The magnitude of this E eld will be vw Be because of orthogonality. Note that in the left gill aperture, the E eld will be 180° re versed from the right eld. That is, EL = –ER. In Figure P2.8B, the model shark is swimming due east. Again, the right gill aperture is examined. (The geometry in this case is more complex.) Here it helps to break Be into its vertical and horizontal components. Because vwR is 180° from Bh, their cross-product is zero. The crossproduct of vwR with Bv lies in the plane of the aperture and points anteriorly, producing a E eld with the orientation sho wn. (This approach is valid because the cross-product is distributive, i.e., A ⋅ (B + C) = A ⋅ B + A ⋅ C.) a. Find the maximum E eld in micro volts/meter in the right gill aperture when vwR = 10 m/s, Bh = 17 ∝T, and Bv = 55 ∝T. Is this eld lar ge
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FIGURE P2.8A
enough for ampullary electroreceptors located on the edges of the aperture to sense? b. Find the magnitude and direction of E in the plane of the right aperture for a shark swimming northeast, and for a shark swimming southeast. c. Describe what happens to E in the right aperture when a shark swims south, west, northwest, and southwest. (Hint: De ne three orthogonal axes: x points north (direction of Bh), y points up, and z points east. i is a unit vector pointing in the +x direction, j is a unit vector pointing in the +y direction (up), and k is a unit vector pointing in the +z direction (east). The cross-product of two vectors described in rectangular coordinates can be written in general as i A ⋅ B= a x bx
j ay by
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k a z = i a y b z – b y a z + j(a x b z – b x a z ) + k a x b y – b x a y bz
(
)
(
)
FIGURE P2.8B
Be = i Bh + j (–Bv) + k 0, north of the magnetic equator. The vw components will depend on the shark’s orientation.) d. Comment on the practicality of magnetic orientation by the gill slit method. Are there ambiguities? That is, is there a unique E- eld v ector associated with each of the eight directions (S, N, E, W, NE, SW, NW, SE)? 2.9. Repeat Problem 2.9 for a shark swimming on the magnetic equator (Bv = 0).
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3
Electronic Models of Neurons: A Historical Perspective
INTRODUCTION Early neural modelers (in the 1960s) were faced with a dilemma. Digital computers of that era were not user-friendly as tools for interactive modeling, and analog computers were also unwieldy. Also, far less was known about the behavior of specific ionic conductances and transmembrane proteins in determining the observable electrical phenomena of neuronal transmembrane voltage. Thus, many neural modelers in the 1960s developed dedicated, compact transistor circuits to emulate spike generation, and various nonlinear RC low-pass networks to model the generation of epsps and ipsps, and signal conduction on dendrites. Such electronic neuromimes were relatively crude phenomenological models. They did offer the experimenter two advantages, however; they ran in real time, and they could be easily interconnected with patch cords. Also, the modeler could listen to their spike outputs on headphones or loudspeakers as the familiar “pop, pop” heard when recording from biological neurons. Thus, the experimenter’s hearing could detect subtle changes in phase between two spike outputs, frequency changes, bursting, etc. A definitely qualitative approach, but fun to do. The following sections examine some of the criteria used by the neuromime modelers, and neuromime circuits developed by the author.
3.1
NECESSARY ATTRIBUTES OF SMALL- AND MEDIUM-SCALE NEURAL MODELS
An important question to ask when one is contemplating modeling the behavior of a small assembly of neurons is what level of detail to pursue. Common sense tells one to use as little detail as possible to still obtain an accurate emulation of known biological behavior. Because an action potential travels at constant velocity and has a constant shape [Vm(x, t)] as it propagates along an axon, it can be described in summary by a unit impulse delayed by a simple transport lag between the spike initiation event and the MEPP, epsp, or ipsp. One does not need to model the axon by a linked series of HH modules, or be concerned with sodium, potassium, and calcium ion currents. Detail is needed, however, to accurately model the spike generation process and absolute and relative refractory periods. The origin of a spike generator potential
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(SGP) should reflect the low-pass filtering propagation delays (if relevant) and summation inherent in the electrotonic conduction of epsps and ipsps over various dendritic branches to the soma and the SGL. It is also necessary to model the effect of axosomatic inhibitory chloride synapses if they exist. Their effect is to clamp the SGP to the chloride Nernst potential through a high conductance that effectively attenuates the pooled epsp and ipsp inputs from the dendrites. Thus, the axosomatic inhibitory synapse has a powerful action in preventing the SGP from reaching the firing threshold. In an analog neuromime simulation of dendritic inputs, it is easy to emulate the epsp ballistic potentials, attenuate and low-pass filter them, then delay them (if appropriate) before summing to form the SGP. The generation of the epsp or ipsp, its attenuation, and low-pass filtering can all be combined and modeled with two or three nonlinear low-pass filters (see Figure 3.2-2). There are several ways to model the spike generation process electronically: 1. The simplest model for spike generation is integral pulse frequency modulation (IPFM) (see Section 4.3.1 for details). In IPFM, the positive SGP is integrated. When the integrator output voltage reaches a threshold voltage with positive slope, an output impulse is produced. This output impulse causes the integrator to be reset to zero. The process repeats, and it is easy to show that the IPFM process is an ideal, linear voltage-tofrequency converter for constant positive inputs. 2. The relaxation pulse frequency modulator (RPFM) is very similar to the IPFM system except that, instead of an integrator with infinite “memory,” the RPFM system inputs the SGP into a simple low-pass filter (LPF) with the transfer function, H(s) = a/(s + a). The output of the LPF then must exceed a firing threshold voltage, Vϕ, with positive slope to initiate an output spike. As in the case of IPFM, the output spike resets the LPF output to zero. The RPFM system is not a linear voltage-to-frequency converter, and it has finite memory to transient SGP inputs. The RPFM system is generally more realistic neurobiologically than is the IPFM spike generator (see Section 4.3.2 for details). Early neuromime circuits generally used IPFM or RPFM spike generation methods. IPFM was used in the neuromime of Figure 3.2-1. Transport lags are a feature useful for realism in modeling any neural system in which the conduction time (δ = axon length /conduction velocity) is an appreciable fraction of the epsp or ipsp time constants. Thus, if a peripheral sensory neuron in the foot has a 1.5 m axon and a conduction velocity of 20 m/s, it will take 75 ms for information to reach the spinal cord. Thus, a transport lag of 75 ms should be put in the path between the sensory transduction process (including spike generation) model and a spinal reflex model. Many interneuron axons in the CNS are sufficiently short that their axonal delays are negligible compared with their dendritic integration times and psp time constants. Thus, in dense, compact neuropile such as the retina or olfactory system, axonal transport lags probably do not need to be included in a modeling scenario.
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One way to realize transport lags with analog neuromimes is to use an acoustic delay line, or a spinning magnetic drum with write, read, and erase heads. (For example, if the drum is spinning at 300 rpm, this is equivalent to 1800°/s. If the read head is spaced 36° from the write head, then the output pulse will occur 20 ms after the input pulse.) The delay drum was used by the author with his early neuromime simulations.
3.2
ELECTRONIC NEURAL MODELS (NEUROMIMES)
Neural modeling, or computational neurobiology as it is now called, began in the early 1960s with the creation of real-time, analog, electronic models of single neurons, synapses, and spike generation. Leon Harmon at Bell Labs was one of the first workers to design an analog neural model using PNP bipolar junction transistors (BJTs) that exhibited summation, threshold (for firing), excitation, inhibition, refractoriness, and delay (van Bergeijk and Harmon, 1960). This circuit contained a voltage-controlled astable multivibrator (VCO) that simulated spike generation in response to generator potential. Harmon coined the term neuromime to describe this class of analog neural model. Harmon and co-workers applied their neuromimes to elementary modeling studies of visual and auditory systems, and later to describing the behavior of two reciprocally inhibited neurons (Harmon, 1964). A seminal paper in the era of analog neuromimes was written by Edwin R. Lewis. Lewis (1963) formalized the concept of synaptic loci, devising simple, RC diode circuits to model the behavior of synapses in generating “ballistic potentials” in the subsynaptic membrane. Circuits were given to model: simple epsps, epsps with antifacilitation, epsps with facilitation (this circuit also contained a PNP BJT, as well as diodes, Rs and Cs), adaptation (the adaptation circuit also contained a PNP BJT), ipsps, and local response loci. A three-PNP transistor, dc excitable, monostable multivibrator (retriggerable) was given for the spike generator locus (SGL) (site of action potential origination). This SGL had an adjustable dc threshold and also exhibited absolute and relative refractoriness. The author and graduate students J.-M. Wu and E.F. Guignon, inspired by the work of Harmon and Lewis, designed and built their own neuromime computer in 1966. We followed Lewis’s designs for synaptic loci, but developed our own SGL circuit using a unijunction transistor (UJT) relaxation oscillator that closely followed the IPFM architecture (Li, 1961). Our SGL circuit is shown in Figure 3.2.1. Note that the SGL is a current-controlled oscillator where the UJT firing frequency is nearly linear with the dc current leaving the input node. The capacitor C1 charges under constant-current conditions from the collector of the PNP BJT, Q1. When VC1 reaches the firing threshold for the 2N489 UJT, its gate conductance abruptly increases, discharging C1. A 6 V negative pulse about 100 s in duration is seen at the UJT B2. This pulse is reshaped by a one-shot (monostable multivibrator) circuit and given a low output impedance at the SGL output. Note that we used negative signals, i.e., a more negative input to Vin caused the UJT firing frequency to increase. An ipsp was therefore positive going in this system. Figure 3.2-2A illustrates a simple epsp synaptic ballistic filter that we used; an antifacilitating synaptic filter is shown in Figure 3.2-2B.
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FIGURE 3.2-1 Schematic circuit of an analog SGL circuit developed by the author. Design is based on integral pulse frequency modulation (IPFM) voltage-to-frequency conversion. When a negative voltage (equivalent to depolarization) is applied to the input node, the 2N3905 PNP transistor is turned on and current flows from its collector into C1, charging it so VC1 goes positive. When VC1 exceeds the firing theshold for the 2N489 unijunction transistor, it abruptly conducts, discharging C1 and also producing a negative pulse at its base 1. This negative pulse triggers another UJT connected as a one-shot multivibrator, producing a negative rectangular pulse at the SGL output. The 1N657 diode connected between output and the VC1 node clamps VC1 to the minimum pulse level, preventing C1 from charging over the duration of the output pulse (100 µs), creating an absolute refractory period. A charge put on C1 from a transient input to the circuit slowly leaks off in the absence of further inputs.
Another approach to constructing analog, real-time neuromimes was taken by Hiltz (1962). Instead of using RC diode low-pass filters to model synaptic potentials, Hiltz used linear op amp, low-pass filter circuits and summers to model the generator potential. A retriggerable one-shot multivibrator was used to generate output spikes. The output spikes were fed back to create an absolute and relative refractory period for the SGL. In 1964, Lewis described an approach to analog electronic neural modeling that used BJT active circuits to model the specific ionic conductances for sodium and potassium ions as described by the nonlinear ODEs in Hodgkin and Huxley’s famous 1952 paper. Lewis put his gK and gNa circuits in parallel with a (linear) leakage conductance and a membrane capacitance and found that it did indeed produce realistic action potentials when appropriately “depolarized.” By manipulation of certain RC parameters, Lewis’s conductance model could exhibit a number of features observed when recording transmembrane potential in real neurons. Clearly, this was a flexible analog neuromime with realistic behavior based on the known transmembrane ionic events at the time. To obtain these effects, however, circuit parameters had to be adjusted, often by trial and error, to obtain the desired emulations. Unfortunately, the analog circuit approach is subject to the demons of analog electronics, which in a nonlinear regenerative circuit, such as Lewis’s, make troubleshooting difficult. Certainly, Lewis’s approach was the most biological of the
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FIGURE 3.2-2 (A) A two-pole, ballistic filter designed to emulate an epsp (albeit negative). The diode prevents the charge on the input capacitor from leaking off back into the SGL circuit. (B) An antifacilitating filter for emulating antifacilitation in an epsp. Voltage on the diodeisolated, 0.66 µF capacitor from an input pulse slowly leaks off. Any voltage remaining on this capacitor when subsequent input pulses are applied effectively subtracts from the transients driving Vantif through the output ballistic filter.
analog neuromimes. It also foreshadowed the approach used in purely digital simulation applications such as Neuron and Genesis, where many detailed ionic conductances enter the picture when simulating at the membrane level.
3.3
DISCUSSION
Analog neuromimes, in their time, were fun to experiment with. A multichannel strip-chart recorder was needed to obtain permanent quantitative data from their outputs. Neuromimes are now obsolete because they were temperamental to use, and they were limited to a phenomenological locus approach. Programs for digital simulation of neural behavior are now so flexible and easy to use that it is unlikely that anyone would ever again attempt an analog neuromime approach.
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4
Simulation of the Behavior of Small Assemblies of Neurons
INTRODUCTION There are two main ways to approach the dynamic modeling of small biological neural networks (BNNs). The first is at the so-called compartmental level; the second, the locus method, uses a more heuristic, phenomenological approach. Both approaches have advantages and disadvantages. In the compartmental method, the membrane of each of the neurons to be modeled is subdivided into passive, local response, and active areas. The passive areas generally cover most dendrites and parts of the soma. Passive membrane is described by constant-parameter, linear RCG modules that resemble the elements approximating an RC transmission line when connected together. Each dendrite is subdivided into short cylinders of membrane of known length and diameter. Each cylinder is a “compartment” with a total shunting conductance in siemens and a capacitance and farads, derived from the per-unit area Cm and Gm. Each compartment is connected to its nearest neighbors by resistances representing the internal and external axial (longitudinal) resistance based on the compartment diameter and the resistivities of the axoplasm and the extracellular fluid. Refer back to Figure 1.2-1 for the compartmental circuit model of three dendrite sections. Dendrite taper is handled by changing the section diameters, hence the numerical values of cm, gm, ro, and ri. Branches in dendrites are handled by joining three (or more) ris and ros at common junction nodes (see Figure 9.0-1). Modeling local response membrane and active (spike-generating) membrane in the compartmental context allows the modeler to add specific, voltage-dependent conductances (i.e., for Na+, K+, Ca++) in parallel with the constant general leakage conductance and cm for each compartment. This approach allows the Hodgkin–Huxley (HH) (1952) model for spike generation to be put in the model, and also local response transients and spike propagation on an axon to be emulated. The action of chemical synapses in a compartmental model is generally done by using a comparator operator to detect the arrival of the propagated spike at the boutons or motor end plate at the end of the axon. This event is used to trigger a transient sodium conductance increase in the subsynaptic membrane (SSM, on an otherwise passive dendrite compartment). This conductance increase may be of the form: gNa(t) = GNamax a2 t exp(–at). This sodium conductance transient causes an epsp to be generated and propagate in the dendritic tree model. The compartmental approach allows the insertion of a great deal of detail about voltage- and chemical-dependent conductances to be included in the model. Such
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detail may be useful if long-term, plastic changes in neuron behavior are of interest. However, in many cases this amount of detail is unwarranted when one is only interested in the input/output behavior of a BNN. In the locus approach, a neuron is again subdivided into regions: synapses, passive dendrites and soma, spike generation, and axon. Instead of directly modeling neurotransmitter-induced conductance changes in the SSM, the locus approach directly models the epsp and ipsp at the SSM. Nerve axons are not modeled, but the spike propagation delay from the spike generator locus (SGL) to the synapse is simulated. The psp voltage transients at synapses are passed through high-order, low-pass filters to model their propagation from SSM to SGL on dendrites. All propagated psps are summed at the SGL. In the locus approach, spike generation is modeled phenomenologically. No consideration is given to the HH formalism or voltage-gated conductances. Instead, the relaxation pulse frequency modulation (RPFM) model (also known by some as the leaky integrator spike generator) is generally used. In RPFM, the voltage at the SGL node is acted on by a simple RC low-pass filter (LPF). When the LPF output reaches a preset firing threshold voltage, Vφ, the RPFM spike generator puts out a unit impulse, and simultaneously resets its output voltage to zero. The locus approach is computationally simpler than the compartmental method. It is not necessary to simulate the dendritic tree, only its net effect in conditioning psps as they impinge on the SGL node. No conductances are required in a locus model; however, the HH spike generation process can be substituted for the RPFM SGL, if so desired. The following sections examine the details of synaptic behavior, dendrite behavior, and RPFM spike generation in the locus context, and examine the behavior of simple BNNs simulated with locus model neurons.
4.1
SIMULATION OF SYNAPTIC LOCI
This section considers various linear and nonlinear dynamic models for chemical synapses described in Section 1.3, and the generation of simulated “ballistic potentials” associated with epsps and ipsps. As has been seen, the epsps and ipsps are summed both in time and in space (over the dendrites of a neuron and/or its soma where synapses occur) and the resultant summed depolarization seen at the SGL is what the neuron responds to in producing its output spike train. Characteristic of all chemical synapses is a small delay between the arrival of the presynaptic spike and the onset of the psp. This delay has four significant components: (1) a delay associated with the presynaptic release of neurotransmitter; (2) a time for the neurotransmitter to diffuse across the 20 to 40-nm synaptic cleft; (3) a time for the transmitter molecules to bind to the ion channel receptor proteins; and (4) a time required for ion channels to be gated open. The total delay between the arrival time of the presynaptic action potential peak at the bouton and the onset of the corresponding epsp or ipsp can range from 0.1 to 1 ms, depending on the preparation, the temperature, etc. (In most simulations, it is acceptable to ignore this small synaptic delay because it can be included in the model by increasing the rise time constant of the psp.)
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The sections below illustrate simple Simnon computer models for (1) a linear, two-pole ballistic filter to generate psps, (2) a nonlinear model of a synapse showing facilitation in the generation of a series of psps, and (3) another nonlinear model showing antifacilitation of synaptic fatigue in the generation of a series of psps. Facilitation is a phenomenon where each succeeding psp is larger than the previous one, providing the presynaptic (input) pulses are closely spaced in time. The physiological reasons for this phenomenon are complex, and will not be covered here. If the times between the successive input pulses are long enough, the facilitory effect from each previous input pulse dies out, and the “basic response” of the synapse to each input pulse is seen. Antifacilitation is a behavior also seen in the generation of psps from an input pulse train. In this case, however, if the time between input pulses is short enough, each successive input pulse causes a progressively smaller psp than the basic response. Antifacilitation can be viewed as a fatiguing of the synapse (i.e., one or more of its components).
4.1.1
A LINEAR MODEL
FOR PSP
GENERATION
To model the generation of an epsp or ipsp at a chemical synapse, the presynaptic nerve spikes are treated as a point process characterized by delta functions; that is,
p(t ) =
k =∞
∑ δ( t – t )
4.1-1
k
k =1
The so-called ballistic filter that operates on p(t) can be described by a pair of firstorder, linear ODEs:
x˙ 1 = – a x1 + p( t )
4.1-2A
x˙ 2 = – b x 2 + x1
4.1-2B
By Laplace transforming, the transfer function is
H(s) =
1
(s + a )(s + b)
=
X2 (s) P
4.1-3
The impulse response of the synaptic ballistic filter is
h( t ) =
1
(b – a)
[e
– bt
– e – at
]
4.1-4
The total area under the h( t ) curve can be shown to be 1/ab. Thus if H(s) is multiplied by ab, the area under h(t) = 1, regardless of a and b. Thus, for time-domain simulation purposes, one can write the ODEs for constant-area psps:
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x˙ 1 = – a x1 + p( t )
4.1-5A
x˙ 2 = – b x 2 + ab x1
4.1-5B
In the special case where a = b, the transfer function will be
H(s) =
a2 (s + a )2
4.1-6
The impulse response of this filter is sometimes called the alpha response; it is: h(t) = a2 t exp(–at) = x2(t)
4.1-7
A Simnon program to generate linear psps is given: continuous system LinSynBF “ v. 3/10/99 A Linear ballistic filter STATE x1 x2 DER dx1 dx2 TIME t “ dx1 = -a*x1 + Vs dx2 = -b*x2 + x1*a*b “ ab factor causes all h(t) to “ have same area under curve. “ Vs1 = IF t > dt THEN 0 ELSE Vso “ Makes three input impulses. Vs2 = IF t > t2 THEN Vso ELSE 0 Vs2n = IF t > (t2 + dt) THEN -Vso ELSE 0 Vs3 = IF t > t3 THEN Vso ELSE 0 Vs3n = IF t > (t3 + dt) THEN -Vso ELSE 0 Vs = Vs1 + Vs2 + Vs2n + Vs3 + Vs3n “ “ CONSTANTS: a:1 b:1 Vso:100 dt:.01 t2:3 t3:6 “ END
The basic response of the linear BF is shown in Figure 4.1-1. The natural frequency b is varied with a = 0.5 r/ms constant: Trace 1, b = 8 r/ms; 2, b = 2 r/ms; 3, b = 0.5 r/ms; and 4, b = 0.125 r/ms. Vso was increased as b was decreased to have the peak values of each psp equal 1. Note that the psps simulated in Figure 4.1-1 represent the behavior of one synapse; there may be one or two to hundreds of synapses between a pre- and postsynaptic neuron. Furthermore, the epsps from each synapse must be combined
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STATE : x1
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0.
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PAR
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dt
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t2
21.
t3
23 .
0.
Vs1
0.
Vs2
0.
0.
Vs3
0.
Vs3n
0.
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: Vs Vs2n
>
FIGURE 4.1-1 Simulated epsps using Simnon program, LinSynBF.t. Normalized vertical scale, horizontal scale in milliseconds. Parameters: a = 0.5 r/ms; b = 8, 2, 0.5, and 0.125 r/ms in traces 1, 2, 3, and 4, respectively. Two linear ODEs were used.
additively with weights dependent on their locations. Some subsynaptic potentials may have to propagate along passive dendrites, over the surface of the soma, to the SGL. Because of the lossy RC cable properties of dendrites, the potentials are effectively low-pass-filtered, delayed, and attenuated before they sum at the SGL. psps arising on the surface of the soma fare better; they are attenuated, but far less than those arising peripherally on dendrites. Thus, they have more weight relative to spike generation.
4.1.2
A MODEL KINETICS
FOR EPSP
PRODUCTION BASED
ON
CHEMICAL
The “ballistic shape” of the epsp can also be modeled by a simple, nonlinear, chemical kinetic model. In this case, consider a nicotinic acetylcholine (ACh) synapse in which two molecules of ACh must bind to a subsynaptic receptor site in order for the ion channel to open, depolarizing the SSM. Stated in terms of a chemical reaction, a bolus of ACh is released by the presynaptic action potential, and it diffuses across the cleft to the receptors. There, at every receptor, two transmitter molecules must bind to open the ion channels. In the reactions below, T is an (active) transmitter
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molecule; R is an unbound receptor site; RT2 is two transmitters bound to a receptor, – opening the ion channel; T is an enzymatically inactivated transmitter molecule; the k are reaction rate constants. Thus, k3 k2 k1 2 T + R ←⎯⎯ ⎯ → R * T2 ⎯⎯ → 2T + R y
( N−x )
x
α ⎯ ⎯ → T ⎯⎯→ * k4
y
By mass-action kinetics (Northrop, 1999), one can write
y˙ = α(t ) – k 4 y + k 2 2 x – k 1y n ( N – x)
4.1-8A
x˙ = k 1y n ( N – x) – ( k 2 + k 3 )x
4.1-8B
where x is the fraction of the total receptors each bound to two ACh molecules that are conducting ions that depolarize the SSM. Thus it is assumed that the depolarization voltage is proportional to x; y is the concentration of free transmitter in the cleft and the exponent n = 2. To simulate this synaptic epsp, the Simnon program, nicochan.t, is used: Continuous System nicochan “ 3/11/99 System to emulate nicotinic ACh STATE x y “ channel dynamics. EPSP V proportional to x. DER dx dy “ Note 2 ACh molecules required/channel to TIME t “ t in ms. open it and cause epsp. “ y = ACh conc in synaptic cleft. “ dy = alphad - k4*y + k2*2*x - k1*(y^2)*(N - x) dx = k1*(y^2)*(N -x) - (k2 + k3)*x “ x = Density of open channels on SSM. “ vm epsp is proportional to x. alpha = IF t < to THEN ao ELSE 0 “ Pulse input rate of ACh. alphad = DELAY(alpha/to, D) “ “ CONSTANTS: D:.3 k1:10 k2:1 k3:1 k4:1 ao:1 N:1“ Max fraction of ion channels. (N - x)=fraction of unbound channels. to:.1 “ END
Figure 4.1-2 illustrates the response of the system to a narrow (100 µs) pulse of presynaptic ACh release; 300 µs is allowed for the diffusion time. Note that the modeled psp is proportional to x(t); x(t) differs from vm(t) by a scale factor.
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FIGURE 4.1-2 Results of a nonlinear chemical kinetic model for epsp generation. The Simnon program, nicochan.t, was used. See text for details. Vertical scale, arbitrary units; horizontal scale, ms. Parameters: K1 = 10; K2 = K3 = K4 = 1. Trace 1, y (ACh conc. in cleft); 2, x (2ACh+ion channel complex conc.); 3, α(t).
4.1.3
A MODEL
FOR A
FACILTATING SYNAPSE
A facilitating synapse (Bullock, 1958) is one where one presynaptic pulse (pp), applied to the synapse after a long quiescent time, causes a single epsp called the “basic response.” If the first input pp is then followed a few milliseconds later by a second pp, the epsp in response to the second pp is larger than the basic response. The same behavior is true for a third pp, etc. The facilitation phenomenon is seen to die out in time following the last pp. If the pps are far enough apart in time, no significant facilitation will be observed in the corresponding epsps. Facilitation is clearly a time-dependent, nonlinear phenomenon. Several early neural modelers devised nonlinear analog electronic models (Lewis, 1964; Harmon, 1964) for facilitation; i.e., the models produce simulated epsps with facilitation. Mathematically, a facilitating synapse can be simulated by multiplying each linear basic epsp response by a factor dependent on the instantaneous frequency of the input pulses. This technique is illustrated in the Simnon model below, FACILBF.t: continuous system FACILBF “ 3/10/99 Use Euler integration w/ DT = .001. STATE x1 x2 f g DER dx1 dx2 df dg TIME t “ dx1 = -a*x1 + Vs “ Primary BF dx2 = -b*x2 + a*b*x1 “ Vsd = DELAY(Vs, DEL) df = -c*f + Vsd “ Facilitating BF dg = -d*g + c*d*f ff = (1 + g*k) “ Facilitation factor
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xo = x2*ff “ Facilitated epsp “ Vs1 = IF t > dt THEN 0 ELSE Vso “ 3 input pulses. Vs2 = IF t > t2 THEN Vso ELSE 0 Vs2n = IF t > (t2 + dt) THEN -Vso ELSE 0 Vs3 = IF t > t3 THEN Vso ELSE 0 Vs3n = IF t > (t3 + dt) THEN -Vso ELSE 0 Vs = Vs1 + Vs2 + Vs2n + Vs3 + Vs3n “ a:1 b:1 c:.3 d:.3 k:7 “ Facilitation gain DEL:.2 dt:.001 t2:3 t3:6 Vso:1000 “ END
In this program, xo is the facilitating epsp, and x2 is the basic response (unfacilitated). The time course of facilitation is determined by the (f, g) LPF with natural frequencies c and d r/ms. Figure 4.1-3 illustrates the response of the model to three impulses spaced 4 ms. Trace 1 = x1, 2 = x2 (basic response), 3 = xo (facilitated epsp), 4 = ff (the facilitation factor). If the input pulses are spaced farther apart, there is less facilitory effect. This is shown in Figure 4.1-4, where the pulse spacing is 10 ms; each xo(t) response is affected by ff, but in 10 ms, the g(t) response has almost died out, so the facilitation is minimal. Facilitation is interesting because it provides a mechanism whereby a jump in the instantaneous pulse frequency in the presynaptic signal (e.g., by inserting an extra spike) produces a disproportionate postsynaptic response.
4.1.4
A MODEL
FOR AN
ANTIFACILITATING SYNAPSE
Antifaciltation, as the name suggests, is where successive epsps in response to an input pulse train grow progressively smaller than the basic response epsp. It is easy to interpret antifacilitation as a fatigue phenomenon, such as might be caused by temporary depletion of the neurotransmitter caused by its release rate exceeding its replacement rate in the bouton. As in the case of facilitation, if the input pulses occur with a long period, the antifacilitating response has time to decay, and the epsps evoked are close to the basic response. A Simnon program, ANTIFAC.t, modeling antifacilitation is show below: continuous system ANTIFAC “ 3/10/99 Use Euler integration w/ DT = .001. STATE x1 x2 f g “ Antifacilitating psp. DER dx1 dx2 df dg TIME t “
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4 4
2 4 4 4
1
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2
2
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FIGURE 4.1-3 The simulated response of a facilitating ballistic filter using the Simnon program, FACILBF.t. Vertical scale, arbitrary units; horizontal scale, ms. Traces: 1, x1 (state); 2, x2 (unfacilitated ballistic filter output); 3, xo (facilitated epsp); 4, ff. Note that each successive xo pulse (3) grows in size. Parameters: a = b = 1; c = d = 0.3; k = 7. See text for details.
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FIGURE 4.1-4 The simulated response of a facilitating ballistic filter using the Simnon program, FACILBF.t. Traces: 1, x1 (state); 2, x2 (unfacilitated ballistic filter output); 3, xo (facilitated epsp); 4, ff. Note that each successive xo pulse (3) has almost the same size because the input pulses are more widely spaced. Parameters: a = b = 1; c = d = 0.3; k = 7. Same scales. See text for details.
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dx1 = -a*x1 + Vs “ Primary BF dx2 = -b*x2 + a*b*x1 “ df = -c*f + Vs “ Anti-Facilitating BF dg = -d*g + c*d*f ff = exp(-k*g) “ Anti-Facilitation factor xo = x2*ff “ Anti-Facilitated psp “ Vs1 = IF t > dt THEN 0 ELSE Vso Vs2 = IF t > t2 THEN Vso ELSE 0 Vs2n = IF t > (t2 + dt) THEN -Vso ELSE 0 Vs3 = IF t > t3 THEN Vso ELSE 0 Vs3n = IF t > (t3 + dt) THEN -Vso ELSE 0 Vs = Vs1 + Vs2 + Vs2n + Vs3 + Vs3n “ “ PARAMETERS a:1 “ radians/ms b:1 c:.3 d:.3 k:1“ Antifacilitating factor. dt:.001 t2:3 t3:6 Vso:1000 “ END
Figure 4.1-5 illustrates the response of the antifacilitating synaptic model to three pulses. Trace 1 = x1, 2 = x2 (basic response), 3 = xo (antifacilitated epsps), #4 = ff (antifacilitation factor). Simulation parameters are the same as in the program. Note that an exponential factor was used to effect antifacilitation. A Hill function can also be used; thus,
ff =
1 1+ k g
4.1-9
Figure 4.1-6 shows the antifacilitation response of the model to six pulses with a period of 3 ms. In this case: a = b = 1, c = 0.1, d = 0.3, k = 2. Note that the antifacilitated output, 3, grows steadily less than the basic responses, 2.
4.1.5
INHIBITORY SYNAPSES
There are several types of inhibitory synapses. Those that produce a strong negative (hyperpolarizing) psp generally gate potassium ions. The Nernst equilibruim potential for K+ across a neuronal membrane is given by EK = (RT/F)ln[Ko/Ki] = (0.025 V) ln[20 mM/400 mM] = –75 mV
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(at 20°C for squid axon)
4.1-10
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1
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0
3 0
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10
15
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FIGURE 4.1-5 The modeled response of an antifacilitating ballistic filter simulation using the Simnon program, ANTIFAC.t. Scales: vertical, arbitrary units; horizontal milliseconds. Traces: 1, x1 (state); 2, x2 (simple linear BF output); 3, xo (antifacilitated epsp); 4, ff (the antifacilitation factor). Parameters: a = b = 1; c = d = 0.3; k = 1. Note that the epsp pulses, 3, grow progressively smaller.
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FIGURE 4.1-6 The modeled response of an antifacilitating ballistic filter simulation using the Simnon program, ANTIFAC.t. Scales: vertical, arbitrary units; horizontal milliseconds. Traces: 1, x1 (state); 2, x2 (simple linear BF output); 3, xo (antifacilitated epsp); 4, ff (the antifacilitation factor). Different parameters: a = b = 1; c = 0.1; d = 0.3; k = 2. Note that the epsp pulses, 3, grow progressively smaller with each successive presynaptic input.
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Thus, as more and more SSM K+ channels open in a potassium inhibitory synapse, the inhibitory subsynaptic membrane potential tends to go toward –75 mV, that is, to hyperpolarize, giving a definite “ballistic” ipsp. Summed ipsps force the generator potential away from the firing threshold of the SGL. Chloride ion–mediated inhibition is found in spinal motoneurons and in CNS interneurons. A typical Nernst equilibrium potential for Cl– is –69 mV. If for some reason the resting potential across the SSM were –69 mV, upon stimulation of the chloride-gated inhibitory synapse, one would see no ipsp, nor would there be any Cl– flux through the gated channels. Although no visible ipsp is present, the transient high Cl– conductance effectively clamps the SSM to the –69 mV potential, preventing epsps from moving the SGL generator potential toward the firing threshold. If the nerve membrane potential is artificially hyperpolarized to –80 mV by a voltage clamp apparatus, stimulation of a chloride inhibitory synapse will produce a positivegoing ipsp! This ipsp is positive because the membrane voltage tries to reach ECl = –69 mV. If the membrane potential is clamped at –60 mV, the Cl– ipsp will be negative-going as it tries to reach –69 mV. One means of modeling the clamping effect of gated chloride inhibition is to assume that the inhibitory synapse makes contact with the postsynaptic neuron (PSN) soma (axosomatic synapse) near the SGL. In this position, one inhibitory synapse can have a greater effect in attenuating the net epsp excitation at the SGL. If the excitatory generator potential is Vex, Vex = (Ve1*k1 + Ve2*k2 + … + VeN*kN)*Hi
4.1-11
where Vek = the kth epsp generated either on the soma or dendrites of the PSN. kn = nth synaptic weighting function. (In general, the farther from the SGL, the smaller kn.) Hi is an attenuating Hill function emulating increased chloride conductance in the SSM when the inhibitory synapse is activated. The Simnon program, ClInh1.t, illustrates the dynamics of a chloride, axosomatic, inhibitory synapse: continuous system ClInh1 “ 3/12/99 STATE x1 x2 x3 x4 v1 f g DER dx1 dx2 dx3 dx4 dv1 df dg TIME t “ ms. “ dx1 = –a*x1 + Vs1 “ BF for axodendritic synapse 1. dx2 = –b*x2 + a*b*x1 Ve1 = DELAY(x2, D1) “ dx3 = –a*x3 + Vs2 “ BF for axodendritic synapse 2. dx4 = –b*x4 + a*b*x3 Ve2 = DELAY(x4, D2) “ df = –c*f + Vi “ BF for axosomatic inhibitory synapse dg = –d*g + c*d*f “ Hi = 1/(1 + g*Ki) “ Hill function for Cl- inhibitory synapse. Vex = (Ve1*k1 + Ve2*k2)*Hi “ Sum 2 epsps; inhib with Cl- synapse. “
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dv1 = –c1*v1 + c1*Vex - z1 “RPFM SGL w1 = IF v1 > phi THEN 1 ELSE 0 s1 = DELAY(w1, tau) q1 = w1 – s1 y1 = IF q1 > 0 THEN q1 ELSE 0 z1 = y1*phi/tau “ “ INPUTS: Vs1 = IF t < dt THEN Vso ELSE 0 Vs2 = IF t < dt THEN Vso ELSE 0 vi1 = IF t > ti THEN Vso ELSE 0 vi2 = IF t > (ti + dt) THEN -Vso ELSE 0 Vi = vi1 + vi2 “ PARAMETERS: tau:.001 “ ms. ti:1 dt:.001 Vso:1000 phi:.1 c1:.3 a:1 b:1 c:.5 d:.5 D1:0.3 D2:0.1 k1:.5 k2:.9 Ki:5 “ END
Figure 4.1-7 illustrates the uninhibited PSN model response to two nearly coincidental excitatory epsps with different weights. Three output pulses are produced by the SGL. Note that the generator potential of the SGL, v1, is reset each time the RPFM SGL model generates a pulse. Figure 4.1-8 shows that one pulse to the inhibitory synapse at t = 1 ms prevents postsynaptic firing. Model parameters are listed in the program above. The traces are 1 = Ve1, 2 = Ve2, 3 = Hi, 4 = v1, 5 = y1 (SGL output), 6 = Vex.
4.1.6
DISCUSSION
Neural inputs to neurons can be inhibitory as well as excitatory, permitting gating or downward modulation of information flow. An example of such downward modulation by inhibition is seen in the neuronal system carrying pain information to the brain. Potassium-gating inhibitory synapses generate negatively going ipsps which can sum with positively going epsps in passive dendrites and membrane. A dominance of ipsps over epsps will slow or stop the firing of a neuron. A more powerful form of inhibition is found in the chloride-gating axosomatic synapses that make contact with the PSN near its SGL. Activation of a chloride-gated
© 2001 by CRC Press LLC
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FIGURE 4.1-7 Results of simulating a chloride axosomatic inhibitory synapse with the Simnon program, ClInh1.t. Response of RPFM SGL with no inhibitory input. Vertical scale, arbitrary units; horizontal scale, milliseconds. Traces: 1, Ve1 (first epsp); 2, Ve2 (second epsp); 3, Hi ≡ 1 (Hill function emulating Cl– inhibition); 4, V1 (RPFM state; note resets at pulse occurrence times); 5, y1 (spike output of RPFM); 6, Vex (RPFM input voltage). Parameters: a = b = 1; c = d = 0.5; c1 = 0.3 r/ms; Vso = 1E3; dt = 0.001, Ki = 50; ϕ = 0.1.
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FIGURE 4.1-8 Results of simulating a chloride axosomatic inhibitory synapse with the Simnon program, ClInh1.t. Response of RPFM SGL with one inhibitory input at t = 1 ms. Vertical scale, arbitrary units; horizontal scale, milliseconds. Traces: 1, Ve1 (first epsp); 2, Ve2 (second epsp); 3, Hi (Hill function emulating Cl– inhibition); 4, V1 (RPFM state; note resets at pulse occurrence times); 5, y1 (output of RPFM); 6, Vex (RPFM input voltage). Parameters: a = b = 1; c = d = 0.5; c1 = 0.3 r/ms; Vso = 1E3; dt = 0.001, Ki = 50; ϕ = 0.1. No output pulses occur. Note that Vex is suppressed by Hi for t > 1 ms.
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axosomatic synapse will clamp the membrane potential in its vicinity to ECl and strongly inhibit firing of the SGL. The Simnon models above do not follow the detailed, physiologically oriented, ionic conductance approach used by some workers (Koch and Segev, 1998). Rather, they use a phenomenological approach to approximate the input/output behavior of small assemblies of real neurons. Facilitation, antifacilitation, fatigue, and inhibition can all be modeled using simple ODEs and nonlinear functions.
4.2
DENDRITES AND LOCAL RESPONSE LOCI
As shown in Section 1.1, there are many different morphologies for neurons, depending on their location and function. However, neurosecretory cells, spiking interneurons with chemical synapses, and motor neurons generally follow a common plan: 1. They have a cell body or soma that contains the nucleus. 2. There is an axon that propagates nerve spikes regeneratively to the next site in the communications chain. 3. The axon joins the cell body at the axon hillock. The axon hillock is generally the site of nerve impulse generation, i.e., the SGL. 4. Also attached to the soma are branching, treelike structures called dendrites. The tubular cross section of dendrites becomes smaller as the number of branchings increases and the farther the branch is from the soma. Dendrites can provide an extensive, diverse contact surface for the synapses of other neurons. Some neurons have small “dendritic fields,” while others, especially in the CNS, have huge dendritic trees, implying that there are many input synapses from a number of neurons. Some CNS interneurons have dendrites and axon terminal branches that are symmetrical with respect to the soma, and it is easy under the light microscope to confuse dendrites with terminal branches (see Kandel et al., 1991, Figure 50-6]. 5. At the end of the axons of interneurons are terminal branches ending in synaptic boutons that make contact with the cell membrane of the soma of the next neuron, its axon hillock, or its dendrites. Or, if the neuron is a motoneuron, at the end if its axon motor end plates make intimate contact with muscle membrane. The properties of the neuron membrane vary from passive (ionic conductances in the membrane are not voltage sensitive until an unrealistic 30 mV of depolarization is reached) to active, where sufficient depolarization (e.g., 10 mV) initiates a nerve action potential that can propagate over the active membrane. Dendrite membrane is generally thought to be passive. Axon hillock and axon membrane is active, and can propagate spikes. The cell body and terminal branches of the axon may be partially active; that is, there is a local response region where there is partial regenerative depolarization due to a low density of voltage-sensitive Na+ channels that reach a threshold depolarization voltage caused by summed epsps. This partial regeneration causes the transient membrane depolarization voltage to spread rapidly
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over the local response membrane and not be attenuated as the electrical activity spreads; a spike is not generated, however. In the purely passive membrane case, a local psp generated under a synapse is attenuated and delayed as it propagates down a dendritic branch, and over the soma. The sections below consider some mathematical models that can be used to describe psp propagation in a dendritic tree toward the SGL. First considered are the electrical properties of a uniform tube of passive membrane. This tube is in effect, an electrical transmission line without inductance. It is, however, a linear circuit, so superposition applies, and Fourier and Laplace transforms can be used to describe its behavior. In general, the mathematical models for transmission lines are very complex, and detailed analysis that includes their taper and branching properties is left for computer simulation. The basics of dendrite behavior are examined below.
4.2.1
THE CORE-CONDUCTOR TRANSMISSION LINE
If one assumes that a tube of constant-diameter passive membrane is immersed in a conductive saline solution, the bulk properties of the tube compose what is called a core-conductor, illustrated in Figure 4.2-1. The specific parallel transmembrane ionic conductances (for Na+, K+, Cl–, Ca++, Mg++, etc.) are assumed to be constant and are summed to form a net gm, in S/cm tube length. Similarly, the lipid bilayers in the membrane form a transmembrane capacitance, cm, in F/cm tube length. (Passive nerve membrane has a capacitance of about 1 µF/ cm2, which must be changed to cm in F/cm by using the tube diameter. See Section 1.2.1) There is an external spreading resistance, ro ohm/cm. ro is determined by the resistivity of the external medium and the tube diameter. The homogeneous gel that fills the dendrite tube (the axoplasm) also has a conductivity ρi ohm cm that determines an internal resistance, ri ohm/cm. The entire dendrite can be modeled by linking a large number of incremental RCR sections in series, each of length ∆x, and writing a set of partial differential equations that describes the electrical behavior of the core-conductor. The next step is to write voltage and current relations for the model based on Kirchoff current and voltage laws. Then, let ∆x → dx, and differential equations are obtained that permit general solutions. Note that each section of the discrete model has a transmembrane depolarization voltage, vm, a transmembrane current, im (x, t), through cm in parallel with gm, and internal and external axial (longitudinal) currents, iL(x, t). (Although these derivations were introduced in Section 1.2.2, they are repeated here for clarity.) The first relation formed on Kirchoff voltage law (KVL):
[
]
v m (x + ∆x) = v m (x) – i(x) ro + ri ∆x
4.2-1A
↓
v m (x + ∆x) – v m (x) = – i L (x) ro + ri ∆x
[
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]
4.2-1B
FIGURE 4.2-1 Lumped-parameter core-conductor model for a dendrite with a passive membrane. Per-unit area parameters are put in per-unit length form for greater ease of solution. See text for details.
Let ∆x → 0. Then lim
∂v m = – i L (x) ro + ri ∂x
[
]
4.2-1C
Now by Kirchoff’s current law (KCL):
i L (x + ∆x) = i L (x) – i m (x) = i L (x) – c m ∆x
∂v m (x + ∆x) – g m ∆ x v m (x + ∆x) ∂t
4.2-2
Use a Taylor’s series to expand
v m (x + ∆x) ≅ v m (x) +
∂v m ∂ 2 v m ∆x 2 … ∆x + + ∂x ∂x 2 2!
Only the first term is used. Thus,
∂v (x) i L (x + ∆x) – i L (x) = –cm m – v m (x)g m ∆x ∂t which reduces to
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4.2-3A
∂i L ⎡ ∂v ⎤ = – ⎢c m m + v m g m ⎥ = – i m ∂x ∂t ⎣ ⎦
4.2-3B
Substituting Equation 4.2-1C into Equation 4.2-3B yields
∂2 vm ⎡ ∂v ⎤ = ( ro + ri )⎢c m m + v m g m ⎥ = i m ( ro + ri ) ∂x 2 ∂ t ⎣ ⎦
4.2-4
and by algebraic manipulation:
∂2i L ⎤ ⎡ ∂i = ( ro + ri )⎢c m L + g m i L ⎥ 2 ∂x ∂t ⎣ ⎦
FIGURE 4.2-2 constant, λ.
4.2-5
The infinitely long dendrite example used in the derivation of the space
Equations 4.2-4 and 4.2-5 are called the “telegrapher’s equations,” or “telephone equations” (Lathi 1965); they describe the spatial and temporal distribution of transmembrane voltage and longitudinal current on an idealized dendrite (RC transmission line). To see how they can be used, examine the simplest case of steadystate (in time) behavior of an infinitely long dendrite (0 ð x ð ×) given a dc voltage source at one end (see Figure 4.2-2). The voltage distribution in x is given by
∂2 vm = ( ro + ri )[v m g m ] = i m ( ro + ri ) ∂x 2
4.2-6
Laplace transforming Equation 4.2-6 yields Vm [s2 – (ro + ri)gm] = 0 Equation 4.2-7 has roots at s = ±
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4.2-7
(ro + ri )g m , so the general solution is of the form:
v m (x) = A exp ⎡ – x ⎣⎢
(ro + ri )g m + B exp[+ x (ro + ri )g m ⎤⎦⎥
4.2-8
Considering the boundary conditions, vm(0) = Vo, vm(×) = 0, B must = 0, and A = Vo. Therefore:
[
v m (x) = Vo exp – x
(ro + ri )g m ] = Vo e – x λ
4.2-9
The parameter λ is called the dendrite space constant (analogous to the time constant in an RC circuit):
rm
λ=
(ro + ri )
cm
4.2-10
Example 4.2-1 As an example, calculate the cable parameters and space constant of a “typical” dendrite of length L in which C = 1 µF/cm2, D = 0.6 µm or 6 × 10–5 cm (diameter), axoplasm conductivity σi = 1.333 × 10–2 S/cm, assume ri Ⰷ ro so neglect ro → 0, Gm = 4 × 10–4 S/cm2, line length L = 103 µm. Am/L is the dendrite membrane area /length in cm2/cm. First, calculate ri:
ri = [L (Aσ )][1 L ] =
(
π 3 × 10
)
1
–5 2
1.333 × 10 –2
= 2.653 × 1010 ohm cm
4.2-11
The shunt conductance /length, gm, is next considered: gm = Gm (Am/L) = Gm (πDL/L) = 4 × 10–4 (π 6 × 10–5) = 7.540 × 10–8 S/cm 4.2-12 Thus rm = 1.326 × 107 Ω cm. Similarly, cm = C (Am/L) = 10–6 (π 6 × 10–5) = 1.885 × 10–10 F/cm
4.2-13
Thus, the dendrite space constant is
λ =1
g m ri = 1
7.540 × 10 –8 S cm × 2.653 × 1010 ohm cm = 2.236 × 10 –2 cm = 223.6 µm 4.2-14
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Next examined are the dynamic properties of a uniform diameter, core-conductor (dendrite), using the notation of Lathi (1965). The passive, cylindrical dendrite is characterized by the functions:
Characteristic impedance : Z o (s) = Propagation function :
γ (s) =
Reflection coefficient :
ρ(s) =
(ro + ri ) (g m + sc m )
4.2-15
(ro + ri )(g m + sc m )
4.2-16
Z L (s) – Z o (s) Z L (s) + Z o (s)
4.2-17
Lathi (1965) derives voltage and current transfer functions for a uniform, finite transmission line:
H v (x, s) =
H i (x, s) =
e – γ ( x−L ) + ρ e γ ( x−L ) eγ L + ρ e–γ L
1 Z o (s)
[e
– γ ( x−L )
[e
γL
4.2-18
– ρ e γ ( x−L )
+ ρe
–γ L
]
]
4.2-19
The characteristic impedance of the line, Zo, can be written:
Z o (s) =
(ro + ri ) (g m + sc m ) = (ro + ri ) λ
1 (1 + sτ)
4.2-20
where λ is the space constant derived above, and the time constant of Zo is τ = cm/gm s. If an ideal voltage source, vin, sets vm at x = 0, then in general: vm(x, t) = L–1{Vin(s) Hv(x, s)],
0ðxðL
i(x, t) = L–1{Vin(s) Hi(x, s)],
4.2-21
0ðxðL
4.2-22
Now let vin(t) = δ(t) at the left end (x = 0) of the line, terminate the line in its characteristic impedance, Zo(s), so ρ = 0, and examine vm(t) at x = L/2. Needed is
⎧ eγ L 2 + 0 ⎫ –γ L 2 –1 v m ( L 2, t ) = L–1 ⎨1 γ L = L–1 exp[ – a ⎬= L e ⎩ e +0 ⎭
{
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}
{
(s + c)
}
4.2-23
(note that the same result will be obtained if the line is half-infinite, i.e., it ends at L → ×) where a = (L/2)
(ri + ro )c m
, and c = gm/cm r/s. The inverse Laplace
transform for Equation 4.3-23 is not known. One means of finding vm(L/2, t) is to let s = jω in Equation 4.2-23, and compute the inverse discrete Fourier transform of {Vi (jω)exp[–a
( jω + c) }. (Note that in polar form, the vector, v = a ( jω + c) ,
can be written v = a
(ω
2
)
+ c 2 ∠ {[tan–1(ω/c)]/2}.)
Many interesting, complex dendritic architectures have been analyzed by Rall (in Koch and Segev, 1989). Several principles of dendritic behavior emerge from Rall’s calculations. (1) The farther a psp site is out along the dendritic “tree,” the smaller and more delayed the corresponding psp will be at the SGL. (2) There is also a loss of high frequencies from the psp as it propagates toward the soma and SGL. Thus, the peak of the psp at the soma is lower and delayed with respect to the psp at the synapse; it is also more rounded due to the high frequency attenuation (see Figure 4.2-3).
FIGURE 4.2-3 Figure illustrating how an impulse input at the tip of a dendrite propagates toward the soma and thence out a neurite (an axon is not shown). Arrows mark the peaks of the wave; note the progressive delay and attenuation with distance from the source.
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4.2.2
DISCUSSION
The analysis above has considered the psp stimulus to be a voltage source applied at some point on the x axis of a dendrite model. More realistically, the input to the dendrite can be considered to be a local conductance increase at the SSM for a specific ion (Na+ for epsps) caused by the arrival of neurotransmitter molecules at receptor sites in the SSM. The inrush of Na+ locally depolarizes the SSM, generating the epsp considered to be a voltage source. In a more realistic simulation of dendrite function, one might apply a transient increase of gm at a point. For this parametric input to be effective, the core-conductor dendrite model must include a dc bias voltage to represent the –65 mV resting potential. Still another approach is to inject some charge (a current pulse) at the SSM site to represent the transient Na+ inrush. The bottom line about dendrites is that their analysis in terms of neuron function is exceedingly complex, and any meaningful detailed description of the function of dendrites with complex geometry necessarily must rely on tedious computational means. Medium-scale neural modeling can avoid the detailed simulation of dendrite effects by passing a unit impulse representing the action potential of a presynaptic neuron’s through a two-time-constant, low-pass (“ballistic”) filter to generate the psp, then pass the psp through an attenuating low-pass filter followed by a pure delay operation to emulate the psp popagation along the dendrite to the SGL. The resulting processed epsps (or ipsps) are literally summed at the SGL to determine whether the neural model will generate a spike.
4.3
INTEGRAL AND RELAXATION PULSE FREQUENCY MODULATION MODELS FOR THE SPIKE GENERATOR LOCUS
How the SGL of a spiking neuron behaves approximately as a voltage-to-frequency converter (VFC), driven by a (positive) generator potential (GP) that is the spatiotemporal sum of epsps and ipsps has been described. As seen from the discussion of the HH model equations in Section 1.4, spike generation is a nonlinear process. There is a minimum GP below which a neuron will not fire, and the steady-state spike frequency is a nonlinear saturating function of the GP. To model spike generation heuristically on a computer, there are two mathematical models that are considerably simpler than the HH equations. The simpler model for action potential generation as a function of a GP is called integral pulse frequency modulation (IPFM) (Li, 1961; Meyer, 1961; Pavlidis, 1964). A more realistic SGL model uses relaxation pulse frequency modulation (RPFM), in which the integrator is replaced with a simple, one-pole low-pass filter (Meyer, 1961). The properties and simulation of these simple SGL models are described below.
4.3.1
IPFM
The IPFM model for SGL action is an ideal linear VFC that generates a pulse train output whose average frequency is equal to the average input voltage. Figure 4.3-1 illustrates a block diagram of an IPFM system. The input to an ideal integrator is a
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continuous positive analog voltage, e. e is integrated until the integrator output v exceeds the pulse generation threshold, ϕ, at t = tk. As ϕ is exceeded, an impulse (delta function) of area Do is produced at tk at the IPFM generator output, and, simultaneously, an impulse of area –ϕ is added to e at the integrator input. This feedback pulse resets the integrator output to v = 0 at tk+, and the integration of e
FIGURE 4.3-1 Block diagram of a system producing IPFM. When an output pulse occurs, it is fed back to reset the integrator output voltage to zero.
continues. This process may be described mathematically by the set of integral equations:
ϕ=
∫
tk
K i e(t ) dt,
k = 2, 3, …, ∞
4.3-1
t k −1
where Ki is the integrator gain. e(t) is the input; e(t) = 0 for t < 0. tk is the time the kth pulse is emitted. Equation 4.3-1 can be rewritten as
rk =
K 1 1 = i t k – t k −1 ϕ t k − t k −1
∫
tk
e(t ) dt,
k = 2, 3, …
4.3-2
t k −1
Here rk is the kth element of instantaneous frequency, defined as the reciprocal of the interval between the kth and (k – 1)th output pulses, τk = (tk – tk–1). So the kth element of instantaneous frequency is given by
rk ≡ 1 τ k = (K i ϕ){e} τ
k
4.3-3
Thus if e is constant and e Š 0, r = (Ki/ϕ) e pps. Simulating an IPFM SGL using Simnon can use the following subroutine: dv = e - z “ Integrator with dv is its derivative. w = IF v > phi THEN 1 ELSE 0 s = DELAY(w, tau) x = w - s
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Ki = 1. v is a state, “} “} ← Pulse generator. “}
y = IF x > 0 THEN x ELSE 0 “ Pulses are > 0. z = y*phi/tau “ Pulse resets integrator from phi to 0. u = y*Do/tau “ Pulse train output, pulse areas = Do.
Note that Euler (or rectangular) integration must be used with ∆T = τ. By using Simnon, the actual, simulated “unit” pulses, b, are triangular with height 1 and base width 2τ. Thus pulses y(t) have areas A = 1(2τ)/2 = τ. To reset the integrator from v = ϕ to v = 0, the unit impulse y is given area ϕ by multiplying it by (ϕ/τ), and then subtracting this pulse from the input, e. The IPFM model is a simple linear VFC, responding only to v > 0. It provides an “α-emulation” of a neural SGL. The IPFM SGL has infinite memory for past subthreshold inputs, which is decidedly unbiological. A relative refractory period can be easily added to the Simnon program for IPFM voltage-to-frequency conversion by manipulating the firing threshold, ϕ. Define: phi = phi0 + fphi
4.3-4
phi is the instantaneous firing threshold. phi0 is the steady-state value of phi when the IPFM system has not produced a spike in a long time, and the relative refractory period is made by inputting the output pulse, z, occurring at t = tk, into a low-pass filter. Thus, the first-order ODE makes a low-pass filter with time constant 1/a: dfphi = – a * fphi + Kphi * (y/tau)
4.3-5
(y/tau) is a unit impulse, so the output of the low-pass filter is the simple exponential decay, which is added to ϕ(t) at t = tk: fphi(t) = Kphi exp[–a(t – tk)]
4.3-6
Note that the RPFM system below can use the same scheme to give its SGL a refractory period.
4.3.2
RPFM
The RPFM SGL model differs from the IPFM model only by the inclusion of a realpole low-pass filter instead of a pure integrator acting on e. Thus the “memory” of v decays exponentially. The RPFM spike generation model is often called the “leaky integrator” SGL. A property of RPFM is that if a constant E < ϕr occurs, the model will not fire as v never reaches ϕ, creating a dead zone in the system e vs. frequency characteristic. Figure 4.3-2 shows the block diagram of an RPFM SGL model. Note that 1/c is the time constant of the low-pass filter. The general solution for an RPFM system is given by a set of real convolution integrals for n = 1, 2, … N, where t0 = 0. µ is the variable of integration.
ϕr =
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∫
tn
[
]
e(µ )c exp – c(t n – µ ) dµ
t n −1
4.3-7
FIGURE 4.3-2 Block diagram of a system producing RPFM. It is the same as the IPFM system except that the ideal integrator is replaced by a single real-pole, low-pass filter. When an output pulse occurs, it is fed back to reset the LPF output voltage to zero. The RPFM pulse generator is also called a “leaky integrator” pulse generator by some computational neurobiologists.
3
2
1
0 0
2
4
6
8
10
FIGURE 4.3-3 A Simnon simulation of an RPFM pulse generator given a step input of Eo = 3.5 V dc. The rising exponential trace is the LPF output when the firing threshold, ϕ > Eo = 3.5 V. When the threshold is set to ϕ = 2 V, output spikes occur, and the LPF output is reset to zero at each spike. Euler integration was used with dt = 0.001, c = 0.5 r/ms.
Example 4.3-1 As a first example, examine the steady-state firing rate of an RPFM SGL model for a constant e = Eo U(t), where Eo > ϕr. The RPFM low-pass filter output (if not reset) can be shown to be v(t) = Eo (1 – e–ct ), 0 ð t ð ×. The first output pulse occurs at t1 when v(t1) = ϕr. v(t) is reset to 0 by the feedback pulse, and again begins to charge exponentially toward Eo. The second pulse occurs at t2 such that (t2 – t1) = t1 = τ, the third pulse will be at t3 so that (t3 – t2) = τ, etc. This behavior is illustrated in Figure 4.3-3. It is easy to see that the RPFM output pulse period will be constant and equal to τ = t1 when the dc input is Eo > ϕr. One
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can write an algebraic expression for the RPFM SGL steady-state output frequency as r(Eo) = 1/t1. First, v(t1) = ϕr = Eo (1 – e–ct1)
4.3-8
By algebraic manipulation, the steady-state period can be written as: t1 = 1/r = (1/c) ln [Eo/(Eo – ϕr)]
4.3-9
Hence,
r=
[
c
]
ln E o ( E o – ϕ r )
>0
4.3-10
For example, inspection of Equation 4.3-10 reveals that for E = 2ϕr, the firing rate of the RPFM SGL equals c/ln(2) = 1.443c pps; for E = 10ϕ, r = c/ln(10/9) = 9.49c pps. Of course, for Eo < ϕr, the system does not fire (here c = 1, ϕr = 2). A Simnon model for an RPFM SGL is given below. Note that only the first line of code differs from the IPFM model. dv = –c*v + c*e – z “ Analog LPF with inputs e and z. w = IF v > phir THEN 1 ELSE 0 s = DELAY(w, tau) x = w – s y = IF x > 0 THEN x ELSE 0 “ y(t) is triangular: height =1, base = 2tau. z = y*phir/tau “ The pulse z resets v to 0. u = y*Do/tau “ RPFM SGL output pulses
In summary, the RPFM model for a SGL is “more biological.” It possesses a dead zone, mimicking subthreshold stimulation, and also exhibits lossy memory, where the effect of a singular epsp input decays toward zero with time constant, 1/c s. As noted in the preceding section, it can easily be given a relative refractory period. Figure 4.3-4 illustrates the RPFM response of the neural model to impulse inputs to its LPF. It fires on the ninth input pulse, which causes vr to Š ϕr, then resets its LPF to zero. As an example of subthreshold behavior of an RPFM neuron, calculate the minimum frequency of a periodic input pulse train that will not cause the RPFM neuron to fire. Let each periodic input pulse have unit area; set the firing threshold to ϕ = 15, and let c = 10 r/s. The LPF output v(t) will have a sawtooth appearance as shown in Figure 4.3-5. It is easy to see that the first peak is at v1pk = Do c V. By using superposition, the second peak can be shown to be at v2pk = Do c (1 + e–cT); the third peak will be at v3pk = Do c (1 + e–cT + e–2cT). In general the nth peak can be written as
v( n ) pk = D o c
k=n
∑e k =1
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– cT ( k −1)
4.3-11
3
2
1
0 0
5
10
15
20
FIGURE 4.3-4 A Simnon simulation of an RPFM pulse generator given a constant frequency train of impulses as an input. The RPFM SGL fires at every ninth input pulse. c = 0.5 r/ms; ϕ = 2 V; Euler integration was used with dt = 0.0001. The sawtooth y waveform is the LPF output. Note that reset of the LPF is not perfect; this is a computational artifact.
3
2
3
3
3
2
2
2
1
1
1
3
3
3
2
2
2
1
1
1
1
1
0 0
5
10
15
20
FIGURE 4.3-5 A Simnon simulation of an RPFM pulse generator given a constant frequency train of impulses as an input. The input frequency has been adjusted so that the RPFM SGL does not fire. Trace 1, Vr (low-pass filter output); 2, ϕ = 2 V; 3, output (no pulses); c = 0. 5 r/ms. See discussion in text.
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T is the period of the input impulses of area Do c. Note that the series: Sn = 1 + x + x2 + … + xn-1 can be written in closed form as
Sn =
1 – xn , 1– x
x <1
4.3-12
In the limit, as n → ×, S× = 1/(1 – x). Thus as t → ×, the peaks of v(t) → Do c/(1 – e–c/t) = v×pk. For the RPFM neuron to fire after a very long time, fmin is needed.
v ∞pk = ϕ r =
Doc 1 – e – c fmin
4.3-13
Solving,
fmin =
c = 0.5 ln (1.5) = 1.74 pps ln(ϕ r ϕ r − D o c)
4.3-14
Example 4.3-2 In a second example, find the conditions whereby two subthreshold area Do input pulses elicit an RPFM SGL output pulse. How far apart in time T can these pulses occur and still give an output pulse at T? The peak amplitude of v(t) when the second pulse occurs is v2pk = Doc(1 + e–cT)
4.3-15
v2pk must equal ϕ in the limiting case. Solve for T as in the previous example: T = – (1/c) ln [ϕ/(Doc) 1] = 1.386 ms
4.3-16
Thus, if the time T between the two pulses exceeds 1.386 ms, there will be no output pulse.
4.3.3
MODELING ADAPTATION
Given a constant input stimulus, a spiking receptor cell such as the Pacinian corpuscle pressure-sensing neuron in the abdomen will gradually decrease its firing rate to zero. This inability to respond to a steady-state stimulus is called adaptation by neurophysiologists. Adaptation can arise from several causes; it may be due to the primary mechanical structure of the transducer or mechanism, or it may involve time dependence of certain ion channels that give rise to depolarization (Kandel et al., 1991). Functionally, adaptation of a pacinian corpuscle can be modeled by placing a high-pass filter of the form:
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H(s) = s/(s + a)
4.3-17
followed by an absolute value (absval) operation between the positive sensory stimulus and the RPFM SGL. Thus, a broad pulse input of pressure of height Po and of duration, δ, P(t) = Po [U(t) – U(t – δ)], will give rise to a generator potential, VG(t), of the form: VG(t) = kPo e–at + kPo e–a(t-δ) tŠ0
4.3-18
tŠδ
This VG(t) will give a burst of spikes when pressure is applied, and another when it is removed. Other spiking receptors may adapt more slowly (e.g., the gravity receptors of Arenivaga, see Section 2.6). In this case, the absval operation is replaced with a half-wave rectifier.
4.3.4
MODELING NEURAL FATIGUE
Many interneurons, driven by a single excitatory presynaptic neuron having a constant pulse frequency, will fire more slowly as time progresses. This gradual loss of response is called neural fatigue. Fatigue can be due to exhaustion of neurotransmitter, exhaustion of ATP reserves, failure of ion pumps in the membrane to keep up with Na+, K+, and Ca++ pumping, etc. Functionally, fatigue can be modeled by passing the SGL output pulses through a low-pass filter having two or more real poles, and adding the LPF output to the firing threshold. This is illustrated below: dv = –c*v + c*e – z “ Analog LPF with inputs e and z. w = IF v > phi THEN 1 ELSE 0 s = DELAY(w, tau) x = w – s y = IF x > 0 THEN x ELSE 0 “ y(t) is triangular: height =1, base = 2tau. z = y*phi/tau “ The pulse z resets v to 0. u = y*Do/tau “ RPFM SGL output pulses dp = –a*p + k*u “ Feedback LPF dq = –b*q + p phi = phi0 + q “ Phi is raised prop.to firing rate.
The time constants 1/a and 1/b should be long compared with the initial firing period of the SGL. The negative feedback from the LPF output raises phi, lowering the output frequency.
4.3.5
DISCUSSION
It is clear from the discussion above that the RPFM SGL model is more realistic in applications such as the simulation of small assemblies of neurons. The input to an
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RPFM SGL model is the spatiotemporal sum of epsps and ipsps arriving at synapses on dendrites or the cell body (soma). e is the depolarization (the positive voltage added to the negative resting potential, Vm0, that moves the net membrane potential, Vm, toward the firing threshold). Thus, Vm = Vm0 + e
4.3-19
If the synaptic inputs to the neural model are purely inhibitory, e can go negative, forcing Vm < Vm0 (hyperpolarization of the GP). As has been seen, there is a practical, asymptotic negative value of e set by the Nernst potential for the gated ion participating in the ipsp generation. Thus, it adds a dimension of reality to the overall neural model to define a lower bound, or Nernst clamping level, for each ipsp. Several examples in this text use the RPFM SGL model to simulate various hypothetical interactions between small assemblies of neurons.
4.4
THEORETICAL MODELS FOR NEURAL SIGNAL CONDITIONING
This section examines several simple, theoretical neural models for conditioning and selecting “features” from a train of spikes from a receptor or interneuron. Such mechanisms may, in fact, exist in nature. Whether they can be identified in natural neurophysiological systems depends on future research in neural signal processing and neuroanatomy. They are considered biomedical engineers because they offer useful signal processing paradigms with simple structures. Their actual existence in nature would not be surprising. In discussing the fine structure of a neural spike sequence, the sequence is treated as a point process characterized by unit impulses occurring at the times of the peaks of the nerve spikes. That is,
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k
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Note that the index k starts at 2 because one needs two pulses to define an interpulse interval, hence instantaneous frequency element. The original work on “resonant networks” was done by Reiss (1964). This section illustrates some of Reiss’s models using the Simnon RPFM neuron model developed above. Reiss described a 1:1 neuron model, a T-neuron, and several types of frequency-selective neural models called band detectors. The 1:1 neuron fires a single output pulse every time it receives an (excitatory) input pulse. There is a synaptic plus axonal transport delay associated with the T-neuron. The other models are described below.
4.4.1
THE T-NEURON
A T-neuron has two (or more) excitatory inputs, P and Q. If a second (Q) input pulse occurs more than T seconds after the first (P) pulse, the T-neuron will not fire. If the Q input pulse occurs less that T seconds following the P pulse, the T-neuron will produce a single output pulse at t = T. The T-neuron thus behaves like an AND gate with a memory, T. It does not matter which pulse is first; an output pulse will occur if P follows Q in less than T seconds, as well. A Simnon model for a T-neuron is given below. The two input delta function “spikes” are conditioned by low-pass filters with two real poles, generating epsps. The epsps are summed and form the generator potential, Vex, for an RPFM model for the SGL. continuous system TNeuron “ 4/22/99 Use EULER integration with Dt = .001. “ STATE x1 x2 x3 x4 v DER dx1 dx2 dx3 dx4 dv TIME t “ msec. “ dx1 = –a*x1 + Vs1 “ 2-pole, synaptic LPF dx2 = –b*x2 + a*b*x1 “ x2 is output epsp “ dx3 = –a*x3 + Vs2 “ 2-pole, synaptic LPF dx4 = –b*x4 + a*b*x3 “ Vex = x2 + x4 “ Two epsps summed to form generator potential. “ “ RPFM T-NEURON dv = –c*v + c*Vex – z w = IF v > phir THEN 1 ELSE 0 d = DELAY(w, tau) q = w - d y = IF q > 0 THEN q ELSE 0 z = y*phir/tau “ “ INPUT PULSES: Vs1 = IF t < dt THEN Vso/dt ELSE 0 Vs21 = IF t > t2 THEN Vso/dt ELSE 0 Vs22 = IF t > (t2 + dt) THEN -Vso/dt ELSE 0 Vs2 = Vs21 + Vs22
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“ RPFMout = y/8 + .5 “ “ PARAMETERS: tau:.001 dt:.001 Vso:1. phir:.2 a:1 “ Units radians/ms. b:1 “ “ “ c:1 “ “ “ t2:1 “ END
Figure 4.4-1A illustrates the response of the T-neuron when the two input pulses are separated by T = t2 = 3 ms. No output spike occurs. Trace 1 = v, 2 = x2 = epsp1, 3 = x4 = epsp2. In Figure 4.4-1B, T = t2 = 2.05 ms. Again, no output spike. In Figure 4.4-1C, T = t2 = 1.0 ms. The peak v reaches the threshold phi = 0.355, and the RPFM neuron fires. Note that v resets to 0, then climbs again and falls back to zero. Trial-and-error simulation determines that this T-neuron model will fire when v(0) = 0 and T ð 2.0 ms. Note that this RPFM model is not a strict T-neuron; it has “memory” because of the behavior of the LPF that conditions Vex to give v. The larger c is, the shorter the memory of the LPF. In general, if one considers the single-pole memory of the LPF, there will be a minimum instantantaneous frequency of a single input pulse train (the others being zero) below which there will be no output pulses. Thus, for a single input channel, the T-neuron model above behaves like a highpass filter. 0.6
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C FIGURE 4.4-1 (A) Computed responses of a simulated T-neuron. Traces: 1, Vex (input to RPFM SGL; 2, x2 (input 1 epsp); 3, x4 (input 2 epsp); 4, ϕr (RPFM firing threshold); 5, SGL output. Simulation parameters in all scenarios in this figure: a = b = 1; c = 0.5; ϕr = 0.355, pulse 2 delay, T = 3 ms; input pulse areas = 1. SGL obviously does not fire. (B) Same conditions except input pulse spacing, T = 2.05 ms. SGL on threshold of firing. (C) Same conditions except input pulse spacing, T = 0.3 ms. SGL fires one pulse at about t = 1.8 ms. Simulation indicates that the SGL will fire for T ð 2.0 ms.
4.4.2
A THEORETICAL BAND-PASS STRUCTURE: THE BAND DETECTOR
Reiss (1964) proposed a theoretical neural structure that displayed selectivity for a range of input frequencies. He called this structure a band-detector. It produced output pulses for a steady-state periodic input pulse train with a frequencies between
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rmin1 and rmax1 , and also outputs in harmonic bands, 2rmin1 to 2rmax1, 3rmin1 to 3rmax1, etc. Eventually the bands overlap, and there is a continuous output at high input frequencies. Figure 4.4-2 illustrates Reiss’s band detector. N1 is the input neuron (source of periodic pulses), N2 is a 1:1 neuron that delays the N1 pulses by D before they synapse on N3, a T-neuron. Both inputs to N3 are excitatory and have equal weight. Each excitatory input has a duration T after the pulse arrives at N3. The input from N1 is not delayed. It is possible to show, assuming strict interpretation of the 1:1 and T-neuron function, that the first pass band has: rmin1 = 1/(D + T), rmax1 = 1/(D – T), and the arithmetic center of the band 1 is rcenter1 = D/(D2 – T2). Thus, the “Q” of the first pass band is
FIGURE 4.4-2
Schematic of a band detector neural system.
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4.4-4
The second passband is rmin 2 = 2/(D + T), rmax2 = 2/(D – T), etc. One of the problems with the basic band detector is that it passes harmonics of the fundamental passband. To circumvent this problem and to make a pure, single band-pass system, Reiss developed a band-detector with harmonic suppression. The output is active only in the primary passband. Figure 4.4-3 illustrates a harmonicsuppressing band detector. The input pulse train inputs without delay to synapse x on the T-neuron. The output of a 1:1 neuron, driven by the source, is delayed D2 then inputs to the T-neuron through excitatory synapse y. Another 1:1 neuron driven by the source, is delayed D1, then inputs to the T-neuron through inhibitory synapse w. D2 > D1. The dwell of each excitatory input is T, as in the case of the band detector, and the inhibitory input has dwell I. That is, no output of N3 can occur,
FIGURE 4.4-3 Schematic of a band detector with harmonic suppression. An RPFM T-neuron is used.
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regardless of the enabling condition, if the enabling condition occurs within I seconds following the inhibitory input. Reiss shows that second- and higher-order harmonic bands will not appear in the output if the conditions, D1 > T, (D1 + I) + (n – 1)/r > (D2 + T), n Š 2, are met. The primary passband of N3 is given by rmin1 = 1/(D2 + T), and rmax1 = 1/(D2 – T), as in the case of the simple band detector. Determining the conditions on the steady-state input pulse period, 1/r, that will produce an N3 output is best done by trial and error, sketching various cases for the three input waveforms. To examine the performance of a more realistic neural model band detector with harmonic suppression (BDHS), the author has written a Simnon neural model using an RPFM SGL for N3. The program is given below: Continuous system BDsupr1 “ V. 3/04/99 “ Use EULER integration with delT = tau. There are 9 states. “ This system is Reiss’ BD with harmonic suppression. STATE noise v1 v3 p1 q1 p2 q2 p3 q3 DER dnoise dv1 dv3 dp1 dq1 dp2 dq2 dp3 dq3 TIME t “ t is in ms. “ dnoise = -wo*noise + SD*NORM(t) “ BW limiting ODE for noise. Vin = noise + Kr*t + Vo “ Noise + ramp + const. drive for N1 “ IPFM VFC. “ THE IPFM SPIKE SOURCE: “ dv1 = Vin – z1 “ IPFM neuron N1: Integrator Ki = 1. w1 = IF v1 > phi1 THEN 1 ELSE 0 “ N1 behaves as an ideal VFC. s1 = DELAY(w1, tau) “ Pulse generator. x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0 “ Pulses are > 0. z1 = y1*phi1/tau “ Pulse resets integrator. u1 = y1*Do1/tau “ Pulse train output, pulse areas = Do1. “ y2 = DELAY(y1, D2) “ Excit. delay D2 ms. to N3 synapse. u2 = Do2*y2/tau“ Thru excit. 1:1 interneuron (xparent). “ yi = DELAY(y1, D1)“ Inhibitory delay (thru xparent 1:1 neuron). ui = Do3*yi/tau “ “ SYNAPTIC BALLISIC FILTER ODEs. (There are 3 synapses on N3) dp1 = –a1*p1 + a1*u1 dq1 = –a1*q1 + a1*p1 “ dp2 = –a2*p2 + a2*u2 dq2 = –a2*q2 + a2*p2 “ dp3 = –a3*p3 + a3*ui “ BF for ipsp. Determines effective delay, I. dq3 = –a3*q3 + a3*p3 “ “ OUTPUT RPFM T-NEURON, N3: dv3 = –c3*v3 + c3*e3 – z3 z3 = y3*phi3/tau w3 = IF v3 > phi3 THEN 1 ELSE 0
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s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 e3 = g1*q1 + g2*q2 – g3*q3 “ “ OFFSET SCALED SPIKE OUTPUTS FOR PLOTTING: b0 = y1/4 + .1 b1 = y2/4 + .4 b2 = yi/4 + .7 b3 = y3/4 + 1.0 “ “ CONSTANTS: a1:1 “ r/ms. a2:1 a3:.55 g1:1 g2:1 g3:.5 c3:7 phi1:3 phi3:.22 tau:0.001 “(millisec) Do1:1 Do2:1 Do3:1.2 wo:1.5 SD:0 Kr:.0125 Vo:0 D1:0.333 D2:2 “ END
To examine the performance of the neural model with a linearly increasing input frequency, an IPFM SGL model is used as a linear VFC. The IPFM SGL is given a linearly increasing analog input, Vin = Kr * t (no noise or offset is used at first). Figure 4.4-4 illustrates the performance of the band detector with harmonic suppression. The input frequency is swept up linearly. Parameters used are listed in the program above. Trace 1 = input spikes from IPFM VFC, 2 = y2 = excitatory input spikes with delay D2, 3 = y3 = inhibitory input spikes with delay D1, 4 = y3 = N3 output spikes, 5 = v3 (RPFM output neuron state). Note that the output fires over a narrow range of input frequencies. Also note that the v3 peaks in the output range of frequencies. This more detailed simulation of a band detector with harmonic suppression illustrates that this frequency-selective behavior indeed exists for a neural model that includes the sum of two ballistic epsps minus a ballistic ipsp as the drive for an RPFM SGL “T-neuron.” The interested reader will find it instructive to introduce random noise into the IPFM VFC input voltage, Vin, causing a noisy instantaneous frequency, rk , of the input pulse train. This BDHS cannot detect single rks lying in the passband because of the “memory” of the RPFM SGL low-pass filter.
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FIGURE 4.4-4 Results of simulating the BDHS using Simnon program, BDsupr1.T. The input frequency is increased linearly. Traces: 1, swept-frequency input spikes; 2, y2 (exitatory input spikes with delay, D2); 3, y3 (inhibitory input spikes with delay D1); 4, y3 (N3 output spikes); 5, ϕ3 = 0.355 (N3 firing threshold); 6, v3 (N3 state). Vertical scale, arbitrary; horizontal scale, ms. Note that the BDHS system fires over a narrow range of the input neuron instantaneous spike frequency.
Where might BDs or BDHSs be found in nature? An obvious application might be in frequency analysis in the auditory system. Another might be in the central processing of electrosensory signals used for guidance in weakly electric fish (Heiligenberg, 1991) (see Section 2.5).
4.4.3
DISCUSSION
This section has examined the results of modeling two theoretical neural operations with signal-processing implications; the T-neuron and the band detector. Modeling was carried out using the locus approach and the RPFM spike generator algorithm. psps were modeled using the so-called alpha function so that a presynaptic spike produced a psp of the form: q(t) = K t exp(–at). Postsynaptic potentials were simply summed to form the spike generator potential that was the input to the RPFM SGL of the output neuron. It is interesting to note that simulation with the more realistic neural models with synaptic ballistic potentials and RPFM SGL showed that the T-neuron did
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indeed act as an AND gate with a memory. The band detector with harmonic suppression behaved as a band-pass filter for an input pulse train with slowly changing frequency. Reiss’s (1964) original T-neuron and band detector only used AND logic, delays, and one-shot multivibrator elements giving gating dwells in their representations. Reiss’s systems were closer to logic than neurons.
4.5
RECURRENT INHIBITION AND SPIKE TRAIN PATTERN GENERATION
Recurrent or reciprocal inhibition and feed-forward inhibition are well-documented neural architectures that have been proposed to describe the generation of burst firing in an output neuron, and alternate burst firing in a pair of motor neurons innervating antagonistic muscles (in biomedical engineering speak, this is a two-phase motor output). Recurrent inhibition has been discussed by Eccles (1964) as a possible CNS mechanism to generate phased bursts of neural firing observed in the CNS. Kleinfeld and Sopolinsky (1989) discuss central pattern generators (CPGs) that drive motoneurons that effect rhythmic activities such as swimming, scratching, chewing, walking, and breathing. These authors state, “The smallest [neural] circuit that can produce a rhythmic output consists of two neurons coupled by reciprocal inhibitory synaptic connections.” Figure 4.5-1 illustrates the basic reciprocally inhibited pair as traditionally configured in the literature. Two output neurons (N1 and N2) each receive some excitatory input. Each output neuron also excites an inhibitory interneuron (N3 and N4), which in turn sends its axon to the opposite output neuron, where inhibitory synaptic contact is made. Conventional wisdom says that this circuit should be a two-phase, burst generator. One output neuron is supposed to fire several pulses, which prevent the other output neuron from firing. Then, the second output neuron fires, inhibiting the first, etc. The problem is that this circuit, as proposed, is not a negative feedback circuit; rather, it has positive feedback. (There are two effective sign inversions around the four-neuron loop. Their product gives an effective positive loop gain.) As discussed below, the basic RI pair is unstable in that once an output neuron fires, its output continually inhibits the other output neuron, and there are no alternating spike bursts generated. If the basic RI pair is run with low loop gain, both output neurons fire continuously at a slower rate, but nearly synchronously; there are no bursts. Wide changes in the synaptic ballistic filter time constants do not change this behavior. (Note that no transport delays were used in this model.) The following section, examines the behavior of the basic reciprocal inhibitory neural model for its effectiveness in producing two-phase burst outputs, given a common input.
4.5.1
THE BASIC RI SYSTEM
Figure 4.5-2 shows a more-detailed architecture of the basic reciprocal inhibition system discussed in the introduction to this section. A Simnon program, RECIPIN3.t, to simulate this model follows:
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FIGURE 4.5-1 Schematic of a basic reciprocally inhibited pair of neurons. Note that they have a common input. Continuous system RECIPIN3“ V. 3/16/99 “ The Basic 4 Neuron RI model. “ STATE v1 v2 v3 v4 p1 p2 p3 p4 p5 p6 va DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dp5 dp6 dva TIME t“ t is in ms. “ “ IPFM VFC TO GENERATE INPUT SPIKES: dva = Ea – za“ Ea is analog input to VFC. wa = IF va > phia THEN 1 ELSE 0 sa = DELAY(wa, tau) xa = wa - sa ya = IF xa > 0 THEN xa ELSE 0 za = ya*phia/tau ua = ya*Doa/tau “ “ THE RPFM RI PAIR: dv1 = -c1*v1 + c1*E1 - z1 “ Output Neuron 1. RPFM model. w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0
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FIGURE 4.5-2 A more-detailed model of the RI pair, showing synapses and critical parameters. This model is simulated with the program, RECIPIN3.T. An RPFM model is used for each of the four neurons. A variable-frequency input is generated for the output neurons, N1 and N2. Single-time-constant, low-pass filters are used to simulate epsps and ipsps. z1 = y1*phi1/tau u1 = y1*Do1/tau “ dv2 = –c2*v2 + c2*E2 – z2 “ Output Neuron 2. RPFM model. w2 = IF v2 > phi2 THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 – s2 y2 = IF x2 > 0 THEN x2 ELSE 0 z2 = y2*phi2/tau u2 = y2*Do2/tau “ “ 2 RPFM INHIBITORY INTERNEURONS: dv3 = –c3*v3 + c3*e3 – z3 “ Inhibitory interneuron 3. w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 z3 = y3*phi3/tau u3 = y3*Do3/tau “
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dv4 = –c4*v4 + c4*e4 – z4 “ Inhibitory interneuron 4. w4 = IF v4 > phi4 THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 – s4 y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau “ “ 1 TIME-CONSTANT, SYNAPTIC BFs: “ dp1 = –a1*p1 + ua “ Synapse 1 ballistic filter (ipsp) “ dp2 = –a2*p2 + ua “ Synapse 2 ballistic filter. q2 is output. “ dp3 = –a3*p3 + u1 “ dp4 = –a4*p4 + u2 “ dp5 = –a5*p5 + u3 “ dp6 = –a6*p6 + u4 “ “ Inputs to RPFM neurons E1 = p1 – p6 “ epsp – ipsp input to output neuron 1. E2 = p2 – p5 “ epsp – ipsp input to output neuron 2. E3 = p3 “ epsp input to inhibitory interneuron 3 E4 = p4 “ epsp input to inhibitory interneuron 4. “ “ Offset outputs for plotting. oa = ya/5 + 1 o1 = y1/5 + 1.3 o2 = y2/5 + 1.6 o3 = y3/5 + 1.9 o4 = y4/5 + 2.2 “ “ INPUTS TO IPFM VFCs: Ea = A“ Nonzero input to IPFM VFCa. “ “ PARAMETERS: A:2. tau:0.001“ ms a1:1 “ All natural frequencies are in r/ms. a2:1 a3:.5 a4:.5 a5:.2 a6:.2 c1:1. c2:1 c3:1 c4:1 phia:1 phi1:1 phi2:1 phi3:1
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phi4:1 Doa:1. Do1:0.75 Do2:0.75 Do3:1 Do4:1 “ END
Figure 4.5-3 illustrates the system response with no recurrent feedback (Do3 = Do4 = 0). With feedback, and with symmetrical parameters as listed in the program above, all four neurons fire more slowly, but in phase. This is shown in Figure 4.5-4. If one makes the circuit asymmetrical by letting Do4 = 1.2 (instead of 1), it is clear from Figure 4.5-5 that N2 dominates the output. From extensive investigation of this model by varying the input frequency and all its synaptic and RPFM parameters, it is clear that this basic RI model is incapable of generating a chopped, two-phase output. However, if symmetrical delays are inserted in the outputs of the two inhibitory interneurons (see Problem 4.10), the basic RI system will generate in-phase bursts on N1 and N2. The next section examines another RI neural model, attributed to Szentagothai by Bullock and Horridge (1965). 2.5
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FIGURE 4.5-3 Results of a simulation of the RI system when there is no inhibition, i.e., Do3 = Do4 = 0. Traces: 1, constant-frequency input pulse train; 2, N1 output; 3, N2 output; 4, interneuron N3 output; 5, interneuron N4 output. See program in text for simulation parameters used.
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FIGURE 4.5-4 Results of a simulation of the RI system when symmetrical inhibitory feedback is used. Same trace numbering as in Figure 4.5-3. Note that the output neurons fire two pulse bursts, in phase. Same simulation parameters except Do3 = Do4 = 1. 2.5
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FIGURE 4.5-5 Results of a simulation of the RI system when asymmetrical inhibitory feedback is used. Same trace numbering as in Figure 4.5-3. The N1 firing is completely inhibited by N2, which becomes dominant. Same simulation parameters except Do3 = 1; Do4 = 1.2.
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4.5.2
SZENTAGOTHAI’S RI CIRCUIT
Figure 4.5-6 illustrates a variation on the basic RI architecture. Note that the reciprocal inhibition acts around an inhibitory interneuron pair and they gate the output neurons. The output neurons do not have feedback around them in this case.
FIGURE 4.5-6 Schematic of Szentagothai’s four-neuron RI system. The listing of the Simnon program, SZGRIsys.t, used to simulate this system can be found in Appendix 2.
If a common constant frequency drive is applied to N3 and N4, they behave precisely like the RI pair described above in Section 4.5.1. No patterned bursts are generated, and if a slight asymmetry is introduced into the N3, N4 RI system, one neuron dominates and holds the other off. On the other hand, the circuit will sharpen patterned inputs delivered as two-phase spike trains. Two IPFM, VFC “neurons” Na and Nb, are used to convert the nonnegative analog inputs: Ea = A * [1 + cos(ωo t)]
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FIGURE 4.5-7 “Burst-sharpening” behavior of the Szentagothai system. Traces: 1, Na (input); 2, Nb (input); 3, N1 (output); 4, N2 (output); 5, N3 (interneuron): 6, N4 (interneuron). The instantaneous frequency of the inputs is sinusoidally modulated 180° out of phase. The system generates short, six-pulse output bursts, 180° out of phase. It behaves as an input burst sharpener, rather than a true pattern generator.
and Eb = B * [1 – cos(ωo t)] to FM spike trains. The Simnon program, SZGRIsys.t, to simulate the Szentagothai RI circuit is listed in Appendix 2. Figure 4.5-7 shows the spikes elicited from the four model neurons, and the input spike trains. Curiously, this RI system makes the bursts from N1 and N2 (traces 3 and 4) shorter and more regular, and it preserves the 180° phase shift in the FM inputs.
4.5.3
A
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BURST GENERATOR
It is possible to use the recurrent feedback concept to generate alternate output spike bursts from a constant activation input spike train. However, symmetrical positive feedback RI, as described above is not used. Figure 4.5-8 illustrates such a system, proposed by the author. This system works because the output of neuron N1 is chopped by the delayed, local, high-gain, negative feedback from inhibitory interneuron, N2. Inhibitory interneuron N4, driven from the N1 output, keeps N2 from firing, thus producing alternate bursts of spikes from N1 and N2. (In linear control theory, a delay in a high-gain, negative feedback loop can destabilize the system, causing limit cycle oscillations, Ogata, 1990.) The Simnon program, BURSTNM1.t, for this bursting system is listed in Appendix 3. Figure 4.5-9 illustrates the ability of this neural model to generate a two-phase, patterned output from N1 and N2, given a common, constant frequency source of spike excitation to N1 and N2. This architecture is a candidate for a central pattern generator (CPG) (Kleinfeld and Sompolinsky, 1989).
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FIGURE 4.5-8 Model of the author’s CPG system. Again, two output neurons and two interneurons are used. However, this system uses a single, delayed, negative feedback loop to obtain a 180° bursting output. The Simnon program, BURSTNM1.T , that simulates this system is given in Appendix 3.
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FIGURE 4.5-9 Results of a simulation with BURSTNM1.T. Traces: 1, input clock; 2, N1 output; 3, N2 output; 4, interneuron N3; 5, interneuron N4. Note that this system generated a genuine, two-phase burst output, given a common clock input to N1 and N2. Simulation parameters are listed in Appendix 3.
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4.5.4
A RING CPG MODEL
WITH
NEGATIVE FEEDBACK
Kleinfeld and Sompolinsky also suggested ring oscillators as pattern generators. As in the case of the RI pair, a ring oscillator with positive feedback (all excitatory synapses around the ring) will be unstable. The net frequency around the ring rises to a saturation value set by the refractory periods of the neurons; no stable pattern is seen. If the ring uses negative feedback, as shown in Figure 4.5-10, and the epsp time constants, RPFM time constants, and pulse weights are “tuned,” a stable, patterned burst oscillation around the ring will occur.
FIGURE 4.5-10 A ring oscillator model that uses negative feedback. A depolarizing dc drive, Vin, is applied to neuron N1. Sustained circulating bursts of firing are produced.
Four neurons are connected in a ring. N1 excites N2, N2 excites N3, N3 excites N4, and N4 inhibits N1, which is also excited with dc. The ring uses negative feedback and acts as a stable oscillator. The Simnon model for this candidate CPG is given below: continuous system RINGOSC1 “ 3/16/99 “ Model for a neural ring of 4 oscillator using NFB. “ STATE v1 v2 v3 v4 p1 p2 p3 p4 q1 q2 q3 q4 DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dq1 dq2 dq3 dq4 TIME t “
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“ RPFM NEURONS (4). dv1 = –c1*v1 + c1*E1 – z1 “ Neuron 1. RPFM model. w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 - s1 y1 = IF x1 > 0 THEN x1 ELSE 0 z1 = y1*phi1/tau u1 = y1*Do1/tau “ dv2 = –c2*v2 + c2*E2 – z2 “ Neuron 2. RPFM model. w2 = IF v2 > phi2 THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 – s2 y2 = IF x2 > 0 THEN x2 ELSE 0 z2 = y2*phi2/tau u2 = y2*Do2/tau “ dv3 = –c3*v3 + c3*E3 – z3 “ N3 RPFM model. w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 z3 = y3*phi3/tau u3 = y3*Do3/tau “ dv4 = –c4*v4 + c4*E4 – z4 “ N4 RPFM model inhibits N1. w4 = IF v4 > phi4 THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 – s4 y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau “ “ 2 TIME CONSTANT SYNAPSES: dp1 = –a1*p1 + u1 dq1 = –b1*q1 + p1*a1*b1 “ dp2 = –a2*p2 + u2 dq2 = –b2*q2 + p2*a2*b2 “ dp3 = –a3*p3 + u3 dq3 = –b3*q3 + p3*a3*b3 “ dp4 = –a4*p4 + u4 “ Inhibitory synapse dq4 = –b4*q4 + p4*a4*b4 “ “ NEURON INPUTS: E1 = Vin – q4“ N1 has a dc clock drive input, Vin E2 = q1 E3 = q2 E4 = q3 “ “ PLOTTING VARIABLES: o1 = y1/5 + .1 o2 = y2/5 + .4
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o3 = y3/5 + .7 o4 = y4/5 + 1.0 “ “ CONSTANTS: tau:.001 Vin:1.5 c1:.7 c2:.75 c3:.75 c4:.75 Do1:0.8 Do2:1.2 Do3:0.7 Do4:3. phi1:1 phi2:0.3 phi3:0.35 phi4:0.4 a1:.3 b1:.3 a2:.3 b2:.3 a3:.3 b3:.3 a4:.3 b4:.3 “ END
Figure 4.5-11 illustrates the patterned bursts produced by RINGOSC1.t. Note that the N4 burst is not a full 180° from the N1 burst. This is essentially 4/5 of a five phase oscillator. By adding one RPFM neuron outside the ring, driven from N4, it is possible to realize the missing phase lag. Neurons 1 and 5 become 180° out of phase (Figure 4.5-12). The program is the same as above with the addition of one RPFM neuron and one synapse. (The additional parameters are a5 = 0.25, b5 = 0.25, c5 = 0.3, and phi5 = 1.) Another ring oscillator is shown in Figure 4.5-13. In this case, the four neurons are modeled more simply with IPFM SGLs, and the synapses are represented by single time-constant low-pass filters generating psps. This alpha model is used to test the hypothesis that a ring oscillator must have negative feedback to produce stable bursts. It is instructive to examine the equivalent loop gain of the four-neuron system. Each synaptic ballistic filter (including the inhibitory one) is of the single time-constant form:
p˙ k = – a k * p k + u k * a k
4.5-1
Laplace transforming, yields the transfer function:
Pk a (s) = k = H k (s) Uk s + ak
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4.5-2
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FIGURE 4.5-11 Steady-state bursting activity around the four-neuron ring oscillator. Neurons from the bottom: N1, N2, N3, and N4 at the top. Time in ms. Parameter used in program listing in text.
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FIGURE 4.5-12 When RPFM neuron N5 is added to the system outside the ring with the simulation parameters given in the text, the outputs of N1 and N5 are almost 180° out of phase.
It is easy to see that the dc gain of each synapse is unity. That is, when the input to a synapse is a constant-frequency train of delta functions of area Doj the average output will equal the average input. That is,
Pk = 1* U k = D oj rj
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4.5-3
FIGURE 4.5-13 Another four-neuron ring oscillator. IPFM spike generaters are used instead of RPFM, and all psps are the outputs of single time-constant, low-pass filters. A spike train input to N1 is used instead of dc.
(rj is the frequency of the jth input to the kth synapse.) As has been shown, the steady-state firing frequency of an IPFM SGL model is
rj =
Kj phi j
Uj
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The integrator gain, Kj, in this case, is unity and phij is the firing threshold. Thus, the dc loop gain of the system can be approximated by
A L (0) = −
k=4
∏ (D k =0
ok
phi k )
4.5-5
The Simnon program used to simulate the four-neuron burst generator, RING4.t, is listed below:
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Continuous system RING4 “ V. 3/21/99 “ A Basic 4 Neuron RING model with NFB. 1 input to N1. “ Uses IPFM neurons and single TC synaptic BFs. System unstable for “ hi gain; does burst oscs, but not 2-phase. “ STATE v1 v2 v3 v4 p1 p3 p4 p5 p6 va DER dv1 dv2 dv3 dv4 dp1 dp3 dp4 dp5 dp6 dva TIME t “ “ IPFM VFCa TO GENERATE INPUT SPIKES: dva = Ea – za wa = IF va > phia THEN 1 ELSE 0 sa = DELAY(wa, tau) xa = wa – sa ya = IF xa > 0 THEN xa ELSE 0 za = ya*phia/tau ua = ya*Doa/tau “ “ THE IPFM SGLs: “ dv1 = E1 – z1 “ Output Neuron 1. w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0 z1 = y1*phi1/tau u1 = y1*Do1/tau “ dv2 = E2 – z2 “ Interneuron 2. w2 = IF v2 > phi2 THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 – s2 y2 = IF x2 > 0 THEN x2 ELSE 0 z2 = y2*phi2/tau u2 = y2*Do2/tau “ dv3 = E3 – z3 “ Interneuron 3. w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 z3 = y3*phi3/tau u3 = y3*Do3/tau “ dv4 = E4 – z4 “ Inhibitory interneuron 4. w4 = IF v4 > phi4 THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 – s4 y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau “ “ 1 TC SYNAPTIC BFs: dp1 = –a1*p1 + a1*ua “ Synapse 1 BF (epsp). dc gain = 1 “
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dp3 = –a3*p3 + a3*u1 “ dp4 = –a4*p4 + A4*u2 “ dp5 = –a5*p5 + a5*u3 “ dp6 = –a6*p6 + a6*u4 “ Synapse 6 BF. Generates ipsp. “ “ Inputs to IPFM neurons E1 = p1 – p6 “ epsp – ipsp input to neuron 1. E2 = p5 “ epsp input to neuron 2. E3 = p3 “ epsp input to interneuron 3 E4 = p4 “ epsp input to inhibitory interneuron 4. “ “ Offset outputs for plotting. oa = ya/5 + 1 o1 = y1/5 + 1.3 o2 = y2/5 + 1.6 o3 = y3/5 + 1.9 o4 = y4/5 + 2.2 “ “ INPUT TO IPFM VFC: Ea = A “*(1 – cos(wo*t)) “ Nonzero input to IPFM VFCa. wo = 6.28*fo “ “ PARAMETERS: A:0.75 fo:.025 tau:0.001 a1:1. a3:.2 a4:.2 a5:.2 a6:.2 phia:1 phi1:1 phi2:1 phi3:1 phi4:1 Doa:1. Do1:1 Do2:2 Do3:2 Do4:1 “ END _
Figure 4.5-14 illustrates a stable burst behavior when AL(0) = –4.0. (The parameters are listed above.) Note that although four neurons are involved, none shows a marked phase shift from N1. If AL(0) is reduced to –1.0, the system quickly settles into a nonbursting, steady-state behavior shown in Figure 4.5-15. By experimenting with the magnitudes of the {Dok}, it was found that the borderline between sustained,
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FIGURE 4.5-14 The model, RING4.T, when given a dc loop gain of –4 (see Equation 4.5-5) exhibits steady-state bursting activity. Traces: 1, input to N1; 2, N1; 3, N2; 4, N3; 5, N4 (inhibitory on N1). Note relative lack of phase shift between the bursts.
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FIGURE 4.5-15 Simulation of the RING4.T model with AL(0) = –1. There is a progressive phase shift between the spikes of the neurons around the ring in the steady state.
steady-state oscillations occurred between AL(0) = –3.5 and AL(0) = –4.0. Figure 4.5-16 illustrates the system settling into stable firing with AL(0) = –2.5. Note that there is a prolonged settling transient, and no bursts in the steady state.
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FIGURE 4.5-16 Simulation of the RING4.T model with AL(0) = –2.5. At the higher loop gain, the system exhibits an underdamped, second-order-type response in terms of instantaneous frequency before settling down to steady-state, constant-frequency behavior.
4.5.5
DISCUSSION
Probably one of the more significant lessons learned by examining the basic RI pair (modeled as RPFM spike generators connected by single-time-constant, ballistic filters emulating epsp and ipsp) is that this neural circuit, as configured here, will not generate patterned firing unless symmetrical delays are introduced into the inhibitory feedback paths. In the absence of delays, or with asymmetrical delays, one output tends to dominate (the other is silent) when the system is given a common input pulse train. The cross-inhibition turns off the other output neuron, so no bursting is seen. Other circuit architectures based on negative feedback will, however, produce patterned burst firing, such as the four-neuron ring oscillator.
4.6
CHAPTER SUMMARY
Chapter 4 has examined in detail the functional and dynamic elements required to model neurons and small BNNs using the locus approach. These have included pure delays to emulate spike propagation down axons, one- and two-time-constant lowpass filters to emulate the production of epsps and ipsps, low-pass attenuation and delays to emulate psp propagation on passive dendrites and soma membranes, and finally IPFM and RPFM models for spike generation. The locus models were applied in studies of hypothetical neural signal processing operations (the T-neuron and band detector) and also central pattern generation, including reciprocal inhibition (RI), ring oscillators, and other “resonant” neural
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structures. The simple RI pair, as modeled here, was shown to be a positive feedback system incapable of firing alternate, two-phase bursts.
PROBLEMS 4.1. The behavior of an IPFM pulse generator can be described by the equations:
p˙ = Vg (t ) – ϕ δ( p – ϕ) y = δ( p – ϕ ) where Vg (t) is the generator potential, y is the unit impulse output, and ϕ is the firing threshold = 0.025 V. Let Vg (t) = 4e–7t , t Š 0. a. Find a general expression for tk, the time of the kth output pulse. b. How many output pulses occur? c. Find the maximum instantaneous pulse frequency output. d. Find the range of peak input voltage at t = 0 such that only one output pulse occurs. 4.2. An RPFM (leaky integrator) spike generator is described by the equations:
p˙ + p T = Vg T – ϕ δ( p – ϕ) y = δ( p – ϕ ) where Vg (t) = 4e–7t, ϕ = 0.025, and T = 0.01 s. Repeat (a), (b), (c), and (d) of Problem 4.1. A simulation can be used to obtain solutions. 4.3. An RPFM spike generator system is described by the system: ∞
x=A
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τ p˙ + p = x – ϕ τ δ( p – ϕ) y = δ( p – ϕ ) where the SGL input is a train of impulses of area A and occurring at period T. The SGL output is y, and its time constant is τ. A = 0.5, ϕ = 1.3, τ = 0.667 s. a. Draw a functional block diagram of the system. b. Find the T value above which the SGL will never fire.
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c. Now assume the SGL input is only two impulses: x = Aδ(t) + Aδ(t – θ). Find the range of instantaneous frequency, r = θ–1, over which a single output pulse is produced. Note: Parts b and c may be solved analytically, or by simulation. 4.4. The analog outputs of two adjacent photoreceptors act on two nonspiking interneurons, (N1 and N2), whose outputs, in turn, control the generator potential of a third, nonspiking interneuron (N3). A spiking interneuron (N4) is modeled by an RPFM system. Receptor A generates a depolarizing · (positive) potential, Va, at N3 according to the ODE: Va = –a*Va + K*I*a. Receptor B generates an inhibitory, hyperpolarizing potential, given by: · Vb = –b*Vb – K*I*b. The input to N4 is simply Ve = Va – Vb. N4 can generate spikes only if Ve is positive and large enough. In this system, a > b. This system is illustrated in Figure P4.4.
FIGURE P4.4
a. Sketch and dimension the RPFM SGL input, Ve (t), when both photoreceptors are given a long pulse of light: I(t) = Io [U(t) – U(t – T)]. That is, T Ⰷ a–1, b–1. U(t) is a unit step function. b. Simulate the system using a Simnon neural model, and observe the spike output of N3. The three ODEs required are:
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˙ = a*V + K*l*a V a a ˙ = b*V – K*l*b V b b v˙ = – v * c + c * Ve – z Ve = Va – Vb
(Excitatory LPF) (Inhibitory LPF) (RPFM LPF)
(SGL generator potential)
Complete simulation of the RPFM SGL with Simnon requires the auxiliary equations: · dv = – v*c + c*Ve – z “ z is reset pulse; dv = v. w = IF v > phi THEN 1 ELSE 0 s = DELAY(w, dT) x = w – s “ pulse former y = IF x > 0 THEN x ELSE 0 “ half-wave rectification. y are unit output pulses. z = y*phi/dT
In the simulation, let a = 2, b = 1, phi = 1, c = 1, dT = 0.001 (same as Euler integration ∆t), K*Io = 1. Plot I(t), Ve(t), v(t), and y.
FIGURE P4.5
4.5. This problem illustrates a theoretical model for detection of weak sensory signals received by noisy receptors. The system is illustrated in Figure P4.5. The sensory input stimulus causes a transient, analog depolarization, s(t), that is added to bandwidth-limited Gaussian noises, n1 and n2. n1 is statistically independent of n2; that is, uncorrelated noise sources are used to make n1(t) and n2(t). The rectified voltages, rVin1(t) = [n1(t) + s(t)]+ and
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rVin2(t) = [n2(t) + s(t)]+ are inputs to the two, IPFM spike generator models for the sensory neurons. The sensory neuron output spikes, y1 and y2, are passed through “α-function,” two-equal-pole, low-pass filters to form epsp inputs to a T-neuron that acts as a coincidence detector. In Simnon notation, the noises are made by: du1 = -a*u1 dn1 = -b*n1 “ du2 = -a*u2 uncorrelated dn2 = -b*n2
+ a*SD*NORM(t) + b*u1 “ n1 is BW limited Gaussian noise. + a*SD*NORM(t + To) for large To. + b*u2
“ n1 and n2 are
One of the IPFM SGLs is: dv1 = rVin1 – z1 z1 = y1*phi/tau“ IPFM reset. w1 = IF v1 > phi THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0 “ Unit output spikes.
The epsp1 is formed: dp1 = –c*p1 + Do*z1 dq1 = –c*q1 + c*p1
“ q1 is epsp1.
The RPFM T-neuron is simulated: dv3 = –d*v3 + d*(q1 + q2) – z3 z3 = y3*phi3/tau “ z3 is RPFM reset pulse. w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 “ T-neuron output pulses
The input, s(t) is to be a 5 ms pulse of height So mV starting at t1 = 20 ms. Write the complete Simnon program for the coincidence detector. Use the following parameters: a = b = 4, c = 3, d = 2, phi = 0.5, phi3 = 0.55, tau = 0.001, SD = 2, So = 0.25, Do = 3, To = 100, t1 = 20, t2 = 25. Run the program using Euler integration with delT = 0.001. See how small one can make So and still see a y3 pulse (or pulses) correlated with s(t). Simulate over at least 100 ms. Note that this is a statistical detector; on some runs false positive y3 pulses, or false negatives (no y3 output for the input pulse) may be observed. Try adjusting the system parameters, phi, phi3, Do, and d to improve detection performance. 4.6. This problem involves modeling a simplified insect “ear”; that is, an airbacked, tympanal membrane, the center of which is connected to a stretchsensitive mechanoreceptor neuron. Assume that the membrane vibrates in response to sound pressure impinging on it. Upward (outward) deflec-
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tion of the membrane causes stretch of the mechanoreceptor neuron neurite, and a consequent transient depolarization (positive generator potential). The neuron does not respond to downward (inward) deflection of the tympanum. The system is shown schematically in Figure P4.6. Because the membrane has mass, elasticity, and damping, it behaves like a linear second-order, low-pass system. That is,
FIGURE P4.6
Kω 2n δx (s) = 2 P s + s(ω n Q) + ω 2n where δx is the deflection of the membrane in microns, and P is the sound pressure level at the membrane in dyn/cm2. Assume that the neurite is stretched by a sinusoidal δx, and only the positive δx leads to membrane depolarization. Thus, the generator potential can be expressed as a Simnon statement: rVg = IF dx > 0 THEN Kc*dx ELSE 0 “ dx is x, rVg is in mV.
The half-wave rectified rVg(t) is the input to the neuron RPFM SGL. As has been written before: “ RPFM SGL: dv = –c*v + c*rVg - z z = y*phi/dT “ z resets RPFM LPF. w = IF v > phi THEN 1 ELSE 0 s = DELAY(w, dT) x = w – s y = IF x > 0 THEN x ELSE 0 “ Receptor output pulses.
a. Write a program to simulate the receptor’s output pulses for various sound input frequencies. Let p(t) = Po sin(2πft), Po = 0.5 dyn/cm2. Let f range from 0.1 to 3 kHz. Let K = 1, Kc = 5, c = 0.2 r/ms, Q = 2.5, ωn = 6.2832 r/ms, t in ms, dT = 0.001 ms (RPFM delay same as Euler integrator interval), phi = 1 mV. Plot p, rVg, v, and y. Make a plot of the average, steady-state firing frequency of the sensor model vs f. b. Examine the system response to “chirps,” i.e., bursts of sound of different lengths and frequencies.
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4.7. A major problem in auditory neurophysiology is how animals locate sounds binaurally. The pressure waves from a point source of sound arrive at the two ears with different phases and amplitudes. Spikes on the two auditory nerves carry this amplitude and phase information to the brain where the cognitive function of source location takes place. As a first step in attempting to model this complex process, examine the properties of a simple model neural phase detector. The system architecture is shown in Figure P4.7. Two spike sources of the same frequency impinge on two RPFM SGLs, N1 and N2. N1 fires if fR > fL, and N2 fires when fL > fR.
[
]
The generator potential for N1 is Vg1 ≅ (epsp R ) – (ipsp L ) K. Because the average psps are proportional to the presynaptic frequencies, Vg1 will be near zero for fR ≅ fL, and will go positive toward the firing threshold of N1 when fR > fL. Simulate each synapse with so-called alpha function (two equal, real-pole) dynamics. For example, in Simnon notation,
FIGURE P4.7
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dpLi = –a*pLi + a*yL/dT dqLi = –a*qLi + a*KLi*pLi
Here, qLi is the ipsp for the left spike source, a is the filter natural frequency in radians/ms, yL are the actual unit input impulses to the synaptic filter on the left input axon, and KLi adjusts the “gain” of the synaptic LPF. To make two equal, synchronous input spike sources for the system, will require two IPFM SGLs with dc inputs, one of which is shown below: “ LEFT IPFM Freq. source: dvL = eL – zL zL = yL*phi/dT wL = IF vL > phi THEN 1 ELSE 0 sL = DELAY(wL, dT) xL = wL – sL yL = IF xL > 0 THEN xL ELSE 0
The firing threshold is set to phi = 1, and dT = 0.001, the Euler integrator delT. eL is a dc level so that the unit impulses, yL(t), will have constant frequency. The right-hand IPFM frequency source is made to increase slightly in frequency by adding a pulse of height dE to the dc input; thus, eR = Eo + dE[U(t – t1) – U(t – t2)]. Simulate the phase/frequency difference detector system. The model will have two IPFM frequency sources, two RPFM SGLs, two excitatory and two inhibitory synapses. The synaptic dynamics are all equal, as are all the SGLs. Let K = 600, phi = 1, Eo = 0.333, dE = 3.33E – 3, a = 0.2 r/ms (synaptic poles), c = 0.5 r/ms (RPFM poles). Observe over what steady-state range of fR = fL the system will reliably detect a 1% increase in fR (Eo adjusts the SS frequency). 4.8. Section 4.4.2 we examined the architecture of a BDHS. Figure 4.4-4 illustrates the response of a BDHS to a swept frequency input. Figure P4.8 illustrates a BDHS system. a. This problem will determine this system’s steady-state band-pass characteristic, i.e., fmax, fmin, and Q of the passband. Let Vin = Vo, (no ramp). Use the Simnon program, BDsupr1.t in the text with the following parameters: SD = 0, Kr = 0 (no noise, no swept frequency), phi1 = 1, phi3 = 0.30, c3 = 7 r/ms, a1 = 1, a2 = 2, a3 = 1, g1 = 0.8, g2 = 1, g3 = 1.50, D1 = 0.333 ms, D2 = 2 ms, Do1 = 1, Do2 = 1, Do3 = 1.2. b. By manipulating D1, D3, and other system parameters, see how narrow a passband can be created. The center of the passband should be at about 30 pps. 4.9. This problem will examine the properties of a hypothetical model neural notch filter that blocks the transmission of incoming spikes having a certain range of frequency. The system is based on Reiss’s (1964) “band suppressor” architecture. Figure P4.9 illustrates the neural notch system. The program is listed below with desired parameters.
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FIGURE P4.8 Continuous system HP49 “ v. 3/13/99. rev. 01/25/00 “ Use EULER integration with delT = tau. There are 7 states. “ This system is Reiss’ Band Suppressor. STATE noise v1 v3 p1 p2 q1 q2 DER dnoise dv1 dv3 dp1 dp2 dq1 dq2 TIME t “ t is in ms. “ dnoise = –wo*noise + SD*NORM(t) “ BW limiting ODE for noise. Vin = noise + Kr*t + Vo “ Analog noise drive for N1 “ “ THE IPFM INPUT VFC, N1: dv1 = Vin – z1 w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) “ (Pulse generator.) x1 = w1 – s1 y1 = IF x1 > 0 THEN x1 ELSE 0 “ Pulses are > 0. z1 = y1*phi1/tau “ Pulse resets integrator. u1 = y1*Do1/tau “ Pulse train output, pulse areas = Do1. “ y2 = DELAY(y1, D) “ Inhib. delay D ms to N3 synapse. u2 = Do2*y2/tau“ Thru excit. 1:1 interneuron (xparent). “
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FIGURE P4.9 “ 2-POLE, SYNAPTIC BALLISTIC FILTERs: (There are 2 synapses on N3) dq1 = –a1*q1 + a1*u1 dp1 = –a1*p1 + a1*q1“ Direct excitatory synapse BF. “ dq2 = –a2*q2 + a2*u2 dp2 = –a2*p2 + a2*q2“ Inhibitory synapse BF. “ e3 = p1 – p2“ Generator potential for N3: “ “ RPFM T-NEURON, N3: dv3 = –c3*v3 + c3*e3 – z3 w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 – s3 y3 = IF x3 > 0 THEN x3 ELSE 0 z3 = y3*phi3/tau “ “ OFFSET SCALED SPIKE OUTPUTS FOR PLOTTING: yo1 = y1/5 – .75 yo2 = y2/5 – .5 yo3 = y3/5 – .25 “ “ CONSTANTS:
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a1:0.9 a2:3 c3:3.33 phi1:1 phi3:0.30 tau:0.001 Do1:1 Do2:1.1 wo:1.5 SD:0
“(millisec)
“ Cutoff freq. of noise LPF in r/ms. “ Standard deviation of broadband Gaussian noise.
Kr:6.5E-3 Vo:.20 D:2.5 “ ms. “ END
a. Run the simulation and find the stop-band range of frequencies for the model neural notch filter. Describe what happens for high frequencies. b. Investigate the effect of noise on the neural notch filter performance. 4.10. This problem investigate the effect of delays in the behavior of the reciprocally inhibited pair of neurons having common excitation shown in text Figure 4.5-2. Delays are added to u3 and u4 in the Simnon program, RECIPIN3.t. That is, the lines, u3 = y3*Do3/tau u4 = y4*Do4/tau
are placed with u3 = (Do3/tau)*DELAY(y3, D3) u4 = (Do4/tau)*DELAY(y4, D4).
a. Use the listed parameters of the program, RECIPIN3.t, in Section 4.5.1 with D3 = D4 = 4 ms to examine the behavior of the model at different input frequencies (vary Ea = A). Does the system generate bursts? Are they in phase? Vary D3 = D4 from 0 to 4 ms, and observe the results. b. Let Ea = A = 1.5. Investigate the effect of asymmetric delays on the generation of patterned firing on N1 and N2. Specifically, what happens when D3 = 0, D4 = 4, and vice versa? 4.11. In this problem, an RPFM leaky integrator SGL is given a short refractory period by causing each output pulse to raise the firing threshold exponentially. The Simnon program is CONTINUOUS SYSTEM HP411 “02/01/00. Use Euler integration with delT =.001. STATE v phi DER dv dphi TIME t “ t in ms. “ “ RPFM SGL WITH EXPONENTIAL REFRACTORY PERIOD:
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dv = –c*v + c*Vin – z w = IF v > phir THEN 1 ELSE 0 s = DELAY(w, dT) x = w – s y = IF x > 0 THEN x ELSE 0“ RPFM output. z = y*phir/dT dphi = -a*phi + a*Do*z “ Refractory increase in firing threshold. phir = phio + phi “ Firing threshold, mV. Vin = Vo + Kr*t “ SGL generator potential, mV. “ yo = y/1.25 – 1 “ Offset y for plotting “ “ PARAMETERS: phio:10 “ mV. c:0.2 “ r/ms. a:0.5 “ r/ms. Vo:7.5 “ mV. Kr:.5 dT:.001 “ ms. Do:2 “ END
a. Run the program with the parameters given. The ramp input sweeps the frequency of y upward. Plot yo, v, Vin and phir vs. t over 60 ms. Let the vertical scale range from –1 to 50 mV. From the plot, graph the instantaneous frequency of yo vs. Vin. Is it linear? [Note: Simnon allows one to expand the timescale in local regions to the pulse period, hence rk = 1/Tk, can be resolved for the kth pulse interval.] Is there a maximum r? b. Now set Do = 0 to make phir ≡ 10 mV (no refractory period). Repeat (a). Describe the net effects of the refractory period.
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5
Large Arrays of Interacting Receptors: The Compound Eye
INTRODUCTION A receptor array is considered to be a two-dimensional, spatial distribution of sensory neurons and their closely associated interneurons in a bounded area. There can be hundreds to tens of thousands of receptors. The sensory neurons are in close proximity so that their output signals can interact locally in underlying interneuronal ganglia before the processed information is sent to the CNS. Examples of large sensory arrays include the vertebrate retina, the arthropod compound eye (CE) and optic lobes (OL), the vertebrate olfactory system, and the vertebrate cochlear system. In small sensory arrays, the sensory neurons generally send their axons directly to the CNS (e.g., ampullary electroreceptors in sharks, skates, and rays), or send their axons to a small numbers of interneurons, which in turn send axons to the CNS (e.g., gravity receptors in the cockroach, Arenivaga). This chapter describes the signal processing that occurs in the arthropod compound eyes and OL. These receptor arrays have many interesting properties, including interneuron interactions that permit enhanced optical resolution and “feature extraction” from visual objects. Some visual feature extraction operations may be associated with dynamic flight stabilization; others may have to do with finding food, or a mate, or sensing danger. Unlike higher vertebrates, where sophisticated visual information takes place in the visual cortex of the brain, insects and crustaceans appear to do most of their visual feature extraction in their OLs, which lie directly under the receptors. The OL ganglia then send the processed information to the animal’s brain and ventral cord ganglia. In this chapter, the first topic considered is the anatomy of CEs and optic ganglia. The ommatidia are the functional subunits of the receptor array of the CE. Each ommatidium consists of a dioptric apparatus (corneal lens, lens, and lightpipe-like structures) and a group of light-sensing retinula cells, arranged about a central rhabdom core like the sections of a lemon. The ommatidia of the CE are modeled as a two-dimensional spatial sampling array. Each ommatidium is also characterized by a directional sensitivity function, which describes how effectively a point source of light is converted to a depolarizing voltage in each retinula cell as the light is flashed ON at different angles from the centerline of the ommatidium. Further mathematical treatment of CE optics develops equations to calculate intensity contrast as a black/white object is moved over an
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ommatidium. Intensity contrast is shown to be a measure of the resolving power of the ommatidia treated as imaging elements. A multiplicative signal processing mathematical model is offered to describe “anomalous resolution” in CE systems. The effective product of the outputs of the six retinula cells in a single ommatidium is shown to lead to improved intensity contrast over that for a single retinula cell. To further describe the signal processing properties of the ommatidia and OLs, inhibitory signal interaction between adjacent ommatidia is next modeled. Such interaction, long known as lateral inhibition, is shown to act as a spatial frequency high-pass filter, effectively enhancing edges and boundaries in the visual object. Lateral inhibition was first observed in the CE system of the horseshoe crab, Limulus polyphemus; there is evidence for it in many other CE systems, and in vertebrate visual systems. Finally, feature extraction operations in compound eye systems are reviewed. Feature extraction is defined as an OL neural response to a particular feature of a visual object. Examples include directionally sensitive neurons that fire when a long, contrasting object moves in a preferred direction, units that fire for a small dark object that is jittered anywhere over an eye, units that fire for dimming of general illumination (no object), etc. Feature extraction operations in insects tend to be simple, and most involve object motion. It is easy to hypothesize that the outputs of such operations are used for flight stabilization, or alerting the animal to potential danger. While evolution has stuck arthropods with a relatively low resolution optical system (the ommatidia of the CEs), remarkably, it has allowed these animals to develop neural systems (the OLs) that make the most of the relatively low spatial input information from the retinula cells. Insects that hunt by vision (mantisses, dragonfles) have huge numbers of ommatidia (~104/eye), and the highest spatial resolution found in CE vision. Not only do dragonflies use their eyes to locate prey (e.g., mosquitos) on the wing, but they also use visual information to control their flight; they probably have the most complex CE visual systems of all the many insects and crustaceans. Not unexpectedly, very few workers have investigated dragonfly CE vision. Unlike flies and grasshoppers, these beautiful creatures are hard to catch in the wild, and very difficult to rear in captivity.
5.1
ANATOMY OF THE ARTHROPOD COMPOUND EYE VISUAL SYSTEM
This section examines the structure of an interesting, ubiquitous, neuro-sensory array, that of the arthropod CE. The most important arthropod classes include insects, crustaceans, and arachnids. Other animals either have primitive photoreceptor arrays, or in the case of vertebrates and certain mollusks (including the octopi and the squids) have two sophisticated “camera” eyes, each with a single lens that focuses an image on a neuro-sensory array, the retina. Compound eyes, as the name suggests, are made from many fixed, subunits, each consisting of a cluster of photoreceptor cells (retinula cells) arranged radially around a common center. Each such cluster of receptors has its individual lens. CE lenses are not classic, convex, spherical lenses with surfaces defined by fixed radii,
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but rather more like cones attached to light pipes; their purpose is not to image an object on a sensory array as is done in the vertebrate eye, but rather to collect light over a narrow acceptance angle over the lens, and to conduct it to the receptor cells. The whole assembly of dioptric apparatus and receptor cells is called an ommatidium. The CE as a whole may be considered to be a discrete, spatial sampling array. Figure 5.1-1 shows a schematic frontal view of the face of the dragonfly, Libellula quadrimaculata. The individual corneal facets define the ommatidia. There are about 104 ommatidia per eye in this insect, which has excellent vision; it catches its flying insect prey on the wing. Its eyes wrap around the front of its head to give it the equivalent of binocular vision, in which vergence theoretically can be used to estimate object range. Dragonfly eyes are interesting in that they have an “equator” or horizontal meridian that divides each CE into two anatomically different halves. The ommatidia in the upper half are about 1.3 times larger than those in the lower half, suggesting that visual resolution “below the equator” is higher than above it. The dragonfly normally attacks its prey from above.
FIGURE 5.1-1 Drawing of a frontal view of the head of the dragonfly, Libellula quadrimacultata. Key: a, compound eye; b, three dorsal ocelli; c, antennae; d, larger facets in upper half of eyes; e, smaller facets in the lower half of eyes; f, buccal apparatus (mouthparts). (From Mazokhin-Porshnyakov, G.A., Insect Vision, Plenum Press, New York, 1969. With permission.)
In general, insects that rely on chemosensors and touch more than vision appear to have fewer ommatidia per eye than do highly visual insects. Also, because ommatidial size is relatively fixed, smaller insects have fewer ommatidia per eye. The cockroach Periplaneta americana has about 2000, the housefly Musca domestica has 3500 to 4000, the worker bee Apis mellifera has 4000 to 5000 ommatidia per eye (Mazokhin-Porshnyakov, 1969). Figure. 5.1-2 illustrates a coronal section through a “composite” CE, showing the different types of dioptric apparatus and retinula cells making up various types of ommatidia found in different arthropods. The first neural synaptic layer in the OLs, the lamina ganglionaris, is shown at the base of these ommatidia.
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FIGURE 5.1-2 Schematic slice through the ommatidia of a exemplary CE, illustrating various CE types. At the outer radius are shown the crystalline corneal lenslets, one for each ommatidium. Types of ommatidia: 1 to 4, pseudocone ommatidia with soft crystalline cone; 5 to 8, acone ommatidia; 9 to 10, eucone ommatidia with terminal cone; 11 to 17, ommatidia of a superposition eye; 11, light-adapted ommatidium; 11 to 14, eucone ommatidia with central cone; 15 to 17, pseudocone ommatidia with cuticular cone. Key to anatomy: AK, eye capsule; C, corneal lenslets; KK, cystaline cone; KZ, Semper cells; HPZ, primary (corneagenous) pigment cells; NPZ, secondary iris (accessory) pigment cells; BM, basement membrane; Lg, axons of neurons in the lamina ganglionaris; Psc, pseudocone; SZ, visual (retinula) cells; Tr, trachea; TrT, tracheal tapetum; Rh, rhabdom; SN, retinula cell axon. (From MazokhinPorshnyakov, G.A., Insect Vision, Plenum Press, New York, 1969. With permission.)
Note that many CEs undergo dark/light adaptation by the migration of pigment in cells that surround the retinula cells in each ommatidium, or in the retinula cells themselves. In the dark, visual resolution is traded off for sensitivity because screening, light-absorbing pigment is moved radially inward, away from the eye’s surface. This unshielding of ommatidia allows oblique light rays to excite retinula cells in more than one ommatidium. Thus, the angular sensitivity is broadened and visual resolution is lost. In the daylight, the pigment migrates outward, shielding individual ommatidia from oblique light. This shielding results in an increase in visual resolution (the directional sensitivity function is sharpened), but a loss of light sensitivity. In a light-adapted CE, it is generally possible to see a pseudopupil. The pseudopupil is a small, dark area on the surface of the CE facing the observer where light enters the eye nearly perpendicular to a small group of ommatidia, and is absorbed. © 2001 by CRC Press LLC
Insects that literally live by their vision such as dragonflies and praying mantises have very small pseudopupils, indicating high visual resolution. In a dark-adapted CE, the pseudopupil is significantly larger than that of the light-adapted eye.
5.1.1
RETINULA CELLS
AND
RHABDOMS
The number of retinula cells in an ommatidium vary among arthropod species. Damselflies (Zygoptera) have only four retinula cells per ommatidium (MazokhinPorshnyakov, 1969), as do beetles (coleoptera) (Meyer-Rochow, 1975). There are eight retinula cells in each ommatidium of dragonflies (odonata) and in flies (diptera). However, bees generally have nine, one of which is a smaller, eccentric cell (Laughlin, 1975; Gribakin, 1975). The horseshoe crab, Limulus, a chelicercate, has ten retinula cells plus a large eccentric cell per ommatidium (Bullock and Horridge, 1965). The silkworm moth, Bombyx mori, also has ten retinula cells plus an eccentric cell (Mazokhin-Porshnyakov, 1969). Figure 5.1-3 shows schematically a longitudinal section and corresponding cross sections of an ommatidium of a worker bee. Structures are identified in the figure caption. There are three major anatomical types of retinula cell defined in this figure by Gribakin (1975). Note the dense cross hatching at the center of the ommatidium where the retinula cells meet. This is the rhabdom region of the retinula cells, where photon trapping and transduction leading to retinula cell depolarization occurs. Figure 5.1-4 illustrates schematically cross sections through a Romalea (grasshopper) ommatidium. Distally, near the lens, there are six ommatidia. A section near the base of the ommatidium shows that two eccentric retinula cells without rhabdoms are present. All eight retinula cells send axons centrally. In electron micrographs, rhabdoms appear as closely packed, dense, parallel tubules, about 50 to 60 nm in diameter. Figure 5.15 shows an electron micrograph of a transverse section of a bee’s ommatidium showing rhabdoms. In this proximal section, only seven retinula cells are seen; the eighth cell lies deeper in the ommatidium (Gribakin, 1975). Grouped rhabdom tubules have been shown to act as dielectric waveguides, similar to modern optical fibers used in telecommunications. The rhabdoms contain visual pigments that trap photons, which, in turn, initiate chemical reactions that lead to retinula cell depolarization. In a given ommatidium, certain retinula cells have visual pigments that have peak sensitivities at different wavelengths, giving the animal “color vision.” Snyder (1975) has shown that in diptera (flies), there is both physical and electrophysiological evidence for retinula cells with three distinct spectral sensitivities. He claims that rhabdoms of cells 1 to 6 have green peak sensitivity, the rhabdom of cell 7 peaks in the violet or near ultraviolet (UV), and the rhabdom of cell 8 may have a yellow-green peak. The UV sensitivity of retinula cell 7 is mostly the result of the physical dimensions of its rhabdom tubules. The rhabdom of retinula cell 8 lies under that of cell 7, so blue light is presumably filtered out before impinging on the cell 8 rhabdom (Snyder, 1975). A retinula cell is a nonlinear analog sensor for light. It responds to increases in light intensity by a graded depolarization of its resting potential that propagates down its axon to its lamina cartridge. Figure 5.1-6 shows a montage of typical, direct-coupled, intracellularly recorded retinula cell depolarization responses to light flashes of increasing intensity. The eye of a bee was used. Note that dynamics of © 2001 by CRC Press LLC
FIGURE 5.1-3 Schematic of a radial cross section of an ommatidium of a worker bee. Perpendicular cross sections through the retinula cells on the right. Key: CL, corneal lens; CC, crystalline cone; SPC, secondary pigment cell; PPC, primary pigment cell; VC, retinula cell; Rh, rhabdom; 9VC, the ninth visual (eccentric) cell; OC, the optical stopper formed by pigmented extensions of the four cone cells. Nature has designed this ommatidium to trap and deliver as much light as possible to the rhabdoms where transduction occurs. (From Gribakin, F.G., in The Compound Eye and Vision in Insects, G.A. Horridge, ed., Clarendon Press, Oxford, 1975. With permission.)
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FIGURE 5.1-4 (A) A cross sectional schematic of an ommatidium from the grasshopper Romalea. This upper or distal cross section shows six retinula cells. Key: RC, retinula cell; Rh, rhabdom; Lp, “light-pipe” extensions of cone lens, 0.5 to 2 µm diameter. (B) A cross sectional schematic of an ommatidium from Romalea. This lower or proximal cross section shows six retinula cells plus two eccentric cells. Key: RC, retinula cell; Rh, rhabdom; Lp, “light-pipe” extensions of cone lens, 0.5 to 2 µm diameter; eight axons leave the ommatidium through the basement membrane.
the depolarization change with increasing light intensity, as well as their amplitude. Figure 5.1-7 illustrates the quasi-logarithmic vm vs. I response of Apis retinula cells. The normalized, peak vm from retinula cells exhibits nonlog behavior at low intensities, is log-linear over about 1.5 decades of intensity, then abruptly shows saturation at very high light intensities. Naka and Kishida (1966) give an empirical relation to describe the peak depolarization of retinula cells due to light flashes, with or without background illumination.
⎛ l +l ⎞ v m = k log⎜1 + f b ⎟ ld ⎠ ⎝
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5.1-1
FIGURE 5.1-5 Transmission electron micrograph of a cross section of a bee ventral ommatidium showing the rhabdom in the proximal part of the dark-adapted ommatidium. Key: ES, extracellular space; PG, pigment granule; PEC, principal endoplasmic cysternae; SLP, secondary pigment cell; M, mitochondria; CB, cytoplasmic bridges. Arrows show the directions of diffusion between SLP and the cytoplasm of the visual cell, and between ES and PEC. (From Gribakin, F.G., in The Compound Eye and Vision in Insects, G.A. Horridge, ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)
where vm is the peak depolarization, k is a positive constant, Ib is the steady background illumination intensity, If is the flash intensity, and Id is the “intrinsic light” of the eye.
5.1.2
THE OPTIC LOBES
Every CE has directly beneath it a complex nervous network called the optic lobe. It is tempting to view the OL as analogous to the vertebrate retina, but, in fact, it is far more complex in structure and function than a retina. One might argue that the arthropod OL fulfills the role of the retina and most of the visual parts of the CNS in vertebrates. Inside the OL there are three major dense, highly organized ganglionic layers of neurons. Figure 5.1-8 shows an overview of the relation of the OLs to the eyes and the protocerebrum in the lubber grasshopper, Romalea microptera. Figure 5.1-9 shows an artist’s composite drawing of vertical sections through a silverstained, Romalea OL. Figure 5.1-10 shows a light photomicrograph of a coronal (horizontal) section of a silver-stained Romalea OL. Note the dense neuropile in the
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FIGURE 5.1-6 Graphs of typical intracellular recordings of retinula cell depolarizations to progressively more intense flashes of 440 nm light to the compound eye of drone bees (Apis m.). Flashes are 200 ms in duration. Note the development of a peak in the retinula cell depolarization waveform as the flash intensity increases. A curious double peak in E evolves into an initial peak plus a sustained depolarization phase. This evolution suggests that two independent kinetic processes may be involved. (Figure drawn from data in Naka and Eguchi, 1962.)
lobula and medulla and the giant nerve cells and neural tracts outside these neural networks. Figure 5.1-11 is a schematic drawing of the OL of the larva of the dragonfly Aeschna, showing typical interneuron pathways seen by light microscopy on silverstained sections. Figure 5.1-12A is a another schematic of neurons in the OL of the fly Calliphora. Again, note the tracts and the very complex interconnections in the medulla. In Figure 5.1-12B, the tracts between neuropile masses in the OL of the butterfly Celerio euphorbiae are illustrated. (Both figures from Mazokhin-Porshnyakov, 1969.) The figures show that nerve fibers from the retinula cells pass in bundles through a basement membrane of the CE to the first ganglionic mass in the OL, the lamina ganglionaris. The lamina, in turn, projects fibers to the most complex ganglion layer, the medulla. From the medulla, some fibers go directly to the protocerebrum; others pass to the third ganglion, the lobula, thence to the protocerebrum. Interestingly, efferent fibers carrying nonvisual sensory information also run from the protocerebrum back to the medulla and lamina, where their signals interact with visual information.
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FIGURE 5.1-7 Graph of normalized, peak retinula cell depolarization vs. log flash intensity. (Light from a xenon flash lamp was attenuated with neutral density filters.) Note that the assymptote follows the model, Vrpk = k log(1 + I). The depolarization curve exhibits hard saturation at high light levels. (Figure drawn from data in Naka and Kishida, 1966.)
The structure and function of the lamina has been most widely investigated, and is best known among the three ganglion masses in an OL. The lamina is a thin, curved plate lying at the outer boundary of an OL. It is the region where the nonspiking retinula cells synapse with various types of interneurons. In each lamina cartridge of the fly Calliphora are found six types of neurons: 1. All but one or two retinula cells from the kth ommatidium send their axons to the kth lamina cartridge (Osorio et al., 1997). 2. There are two nonspiking large monopolar cells (LMCs) that have their cell bodies (somata) distal to the lamina, lying between the outer margin of the lamina cartridges and the outer OL membrane. They send signals to the medulla. Their axons are large, from 3 to 5 µm in diameter. 3. There are also two small monopolar cells (SMCs); their somata are also found with the LMC somata. SMC axons also go to the medulla. 4. A single T cell has its soma between the lamina and medulla, and sends an axon to the medulla and to the lamina. 5. In analogy with the vertebrate retina, amacrine cells, whose somata lie on the inner margin of the lamina, send fibers horizontally between neighboring cartridges. They also synapse with a small bipolar cell (SBC) and a T cell (Osorio et al., 1997). 6. One to three (centrifugal) C cells, whose somata lie between the medulla and the lobula, send efferent feedback along their axons to a lamina cartridge.
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FIGURE 5.1-8 Phantom drawing of the head of the lubber grasshopper, Romalea microptera, showing the outlines of the surfaces of the CEs and the underlying OLs and the protocerebrum. The “fingerlike” projections from the OLs toward the eyes are bundled retinula cell axons. They are enclosed by a regular system of hemolymph channels lying between the pigmented layer and the lamina. (From Northrop, R.B. and E.F. Guignon, J. Insect Physiol., 691, 1970. With permission.)
Intracellular recordings from the large axons of the LMC neurons reveals that these cells hyperpolarize in response to the retinula cell depolarization from flashes of light to the eye. Figure 5.1-13 shows a series of these responses in a dragonfly lamina in response to flashes of increasing intensity. Figure 5.1-14 shows a vm vs. log(I) plot for the LMC responses to flashes. The log-linear range covers about 2.5 log units (Laughlin, 1975.) There is a 1:1 mapping of the retinula cell axons of an ommatidium’s to a lamina cartridge, which then projects LMC and SMC axons centrally to the medulla. The medulla has nine or ten anatomically distinct layers of densely packed neuropil. Interneuron fibers interconnect the medullary layers both horizontally and perpendicularly. There is also anatomic and electrophysiological evidence that complex visual feature extraction takes place in the medullar neuropile (Northrop, 1975).
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FIGURE 5.1-9 Schematic vertical cross section through an OL of Romalea. (drawn from silver-stained sections). Figure is intended to give an overview of the neuroanatomy. Key: O, ommatidial layer of the CE; B, basement membrane; P, pigment layer; H, hemolymph channel; LG, lamina ganglionaris; OC, outer chiasma; M, medulla externa; IC, inner chiasma; L, lobula. (From Northrop, R.B. and E.F. Guignon, J. Insect Physiol., 16: 691, 1970. With permission.)
5.1.3
THE OPTICS
OF THE
COMPOUND EYE
No anatomic description of CEs is complete without a consideration of how light reaches the photosensory rhabdoms of the retinula cells. The outer surface of the CE is covered with transparent cuticle subdivided into many hexagonally packed corneal convex lenslets, each over an ommatidium. The cornea is from 30 to 50 µm thick, and serves to protect the soft visual cells beneath it. Directly under each corneal lenslet, there is a crystalline conical lens that acts as a “light funnel” to channel light energy down the center of the ommatidium to the retinula cell rhabdoms. The purpose of this conical lens is to concentrate photon energy on the transducer region of each retinula cell. The conical lens arises from four specialized
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FIGURE 5.1-10 Low-magnification light micrograph of silver-stained, coronal section through a Romalea OL. Note the many different layers of neurons in the medulla (M), and in the lobula (L) neuropile, the large neuron cell bodies outside the neuropile, and the tracts running between neuropile masses. (T) trachea, (H) hemolymph channel. (From Northrop, R.B. and E.F. Guignon, J. Insect Physiol., 16: 691, 1970. With permission.)
Semper cells lying under the cornea. In the CEs of primitive arthropods (e.g., Machilis), the cone is formed directly from the Semper cells, or by special structures secreted by them. In many CEs, the cone is shielded on the sides by pigment cells (Mazokhin-Porshnyakov, 1969). (Figure 5.1-2 above illustrates schematically the anatomic variations of cone lens design between arthropod species.) The rhabdom region of retinula cells is known to have a higher refractive index than do the surrounding cells. This means that light entering the rhabdoms from the cone is trapped in them (a fiber-optic effect), contributing to the efficiency of the transduction process. A retinula cell of a CE ommatidium can be characterized by a directional sensitivity function (DSF), s(θ, φ). It will be seen in the next section that, in general, the narrower the DSF, the higher the resolution of the CE. The DSF is a normalized function; i.e., s(0, 0) ≡ 1. Also, s(θm/2, 0) ≡ 0.5. To measure a DSF, the back of the insect’s head is removed, and a glass micropipette microelectrode with a very small tip is inserted into a retinula cell with minimum physical disturbance. A movable, bright point-source of white light is suspended on a semicircular track centered over the eye. The amplitude of the retinula cell depolarization is recorded as a function of the angular position of the light, which is flashed on and off. When the light is centered over the ommatidium under study (generally near a line perpendicular to
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FIGURE 5.1-11 Scheme of basic neural connections in the OL of an Aeschna (dragonfly) larva. Key: I, lamina ganglionaris; II, medulla; III, lobula; HX and BX, external and internal chiasmata, respectively; OT, optic tract. (The medulla actually has more than three anatomical layers.) Types of neurons: 1, fibers from retinula cells (swelling is not soma); 2 and 4, small and large lamina monopolar cells; 3, recurrent monopolar cell; bipolar interneurons 12, 14, 15, 16, and 17 connect individual ganglion layers directly to the brain. Other interneurons interconnect ganglia; some apparently provide centrifugal feedback. (From Mazokhin-Porshnyakov, G.A., Insect Vision, Plenum Press, New York, 1969. With permission.)
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FIGURE 5.1-12 (A) Schematic structure of the OL of the fly Calliphora vomitoria, based on the pioneering neuroanatomical work of Cajal and Sanchez, 1915. (Compare this to the author’s light micrograph, Figure 5.1-11a and b.) Key: BM, basement membrane; OK, retinula cell axons (nonspiking); other symbols as in Figure 5.1-12. Note that three, distinct afferent tracts are shown. (B) Schematic of the optic ganglia of the butterfly Celerio euphorbiae. Same notation is used. Note that fibers from the retinula cells are organized into bundles, between which are found hemolymph channels and tracheae. (From Mazokhin-Porshnyakov, G.A., Insect Vision, Plenum Press, New York, 1969. With permission.)
Monopolar
-4.3
-3.5
-2.9
-2.3
-0.6
retinula
FIGURE 5.1-13 Comparison between the intracellulary recorded potentials of LMCs and their associated retinula cells in the dragonfly Hemicordula tau. Stimulus is a 500 ms flash from a point source of white light. Light intensity is attenuated by neutral density filters. (Thus –4.3 means the source is attenuated by 4.3 log10 units.) Horizontal bars are all 500 ms; vertical bars are all 10 mV. Note the retinula cells depolarize while the LMCs hyperpolarize; neither types spike. (From Laughlin, S.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)
© 2001 by CRC Press LLC
50
30
20
Response mV
40
10
-5
-4
-3
-2
-1
0
log intensity
FIGURE 5.1-14 Graphs of response/intensity for retinula cells and LMCs of the dragonfly Hemicordula tau. —o—, magnitude of the peak response of LMCs to flashes. —•—, plateau magnitude response of LMCs. —∆—, retinula cell response. Note that retinula cell responses are log-linear over about two decades of intensity. Curiously, at high intensities, the plateau magnitude response of the LMCs decreases. (From Laughlin, S.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)
a tangent plane touching the corneal facet of the ommatidium under study), a maximum response is noted. As the light source is traversed away from the maximum axis, the response falls off to zero. The shape of the DSF curve depends on the state of light adaptation of the CE, and on absolute intensity of the light, so this must be standardized. Recall that the retinula cell depolarization responds in a logarithmic manner to light intensity (see Equation 5.1-1). The DSF in a light-adapted (LA) eye is narrower than that of a dark-adapted (DA) eye because of pigment shielding in the LA ommatidia. Figure 5.1-15 shows a large difference in the DSFs of the eye of the cockroach Periplaneta americana for a LA vs. a DA eye. Circles are electrophysiological data; solid curves are fits by a mathematical model for the DSFs. Another set of DSFs for DA and LA locust retinula cells is shown in Figure 5.1-16. These investigators found that the locust DSFs were in fact slightly elliptical in twodimensional shape, rather than circular. They found that the mean θm/2 for the intensity DSF for an LA locust retinula cell was about 1.7°. The DSF peak was pointy, which means that trying to model it with a Gaussian function is not as accurate as a Hill hyperbolic function, e.g., s(x) = 1/(1 + (x/θm/2)2). Wilson (1975) (cited in Northrop, 1975) measured DSFs in LA locust eyes, and found mean θm/2 = 0.73° in the horizontal plane and θm/2 = 0.685° in the vertical plane for ten animals. That Wilson’s θm/2 values were significantly smaller is probably due to extreme light adaptation and careful microelectrode technique. As will be seen, Wilson’s smaller θm/2 values make it easier to make a model for anomalous resolution in the CEs of locusts.
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Sensitivity (per cent) 60
50
40
30
20
10
80 LA
60
DA
40 20 00 θ
10
20
30
40
50
60
FIGURE 5.1-15 Normalized, percent angular sensitivity as a function of the angle θ of light incident upon a facet of the eye of the cockroach Periplaneta americana. Solid curves represent theoretical calculations. Circles are from electrophysiological measurements. DA, dark-adapted eye; LA, light-adapted eye. Note the large increase in θm/2 for the dark-adapted eye. (From Snyder, A.W., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, 1975. With permission of Oxford University Press.)
100%
50%
0
DA 0 LA -10
0
-5
0
0
0
+5
0
+10
0
FIGURE 5.1-16 The averaged DSFs for 50 dark-adapted and 50 light-adapted locust retinula cells. Vertical scale, linear % sensitivity; horizontal scale, linear angle between point source and ommatidial center axis. Note that the dark-adapted eye trades off resolution for sensitivity; it has about double the θm/2 as the light-adapted eye. These DSFs have sharp peaks. (From Tunstall, J. and Horridge, G.A., Z. Veral. Physiol., 55: 167, 1967. With permission from Springer-Verlag.)
5.1.4
DISCUSSION
This section has described the organization and structure of the CE optical system. The CE is basically a visual sampling array, rather than a camera with a single lens and film. The two CEs are generally built on the curved surface of an insect’s head. The maximum response axis of each ommatidium diverges from each of its neighbors
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so that a whole eye responds to objects lying in about a hemisphere of visually sampled space (2π steradians of solid angle). Each ommatidium is seen to be effectively a light-gathering and transducer assembly of retinula photoreceptor cells, rather than an imaging system. The entire CE with its thousands of ommatidia is the imaging system. The directional sensitivity function of each ommatidium is analogous to a spatial low-pass filter. Thus, the entire ommatidial array samples an analog low-pass-filtered image and converts the intensity information to retinula cell depolarization voltages. Interneurons in the OLs process these analog voltages and perform spatial filtering (edge enhancement) and feature extraction operations. The outputs of OL neurons are “coded” in terms of spike frequency.
5.2
SPATIAL RESOLUTION OF THE COMPOUND EYE
Resolution can be thought of as the ability of a visual system to resolve small, lowcontrast objects as separate entities without error. For example, to resolve two adjacent black spots as two spots, rather than one big fuzzy spot. This type of test is analogous to the time-domain resolution of two closely spaced pulses as two separate pulses after they have propagated through a low-pass filter or a transmission line. It is also possible to test the resolution of a visual system in the frequency domain, i.e., its steady-state, sinusoidal spatial frequency response. It is possible to generate an object having a one-dimensional, spatial, sinusoidal intensity variation given by p(x) = Io + Im sin(2πf x),
Io Š Im
5.2-1
Such an object allows interpretation of visual resolution in terms of the spatial frequency response in one direction, i.e., x. At very high spatial frequencies, the pattern disappears, and only Io is perceived. This loss of high spatial frequency information indicates that all visual systems are low-pass in nature. Resolution tests are generally carried out on the responses single cells, e.g., ganglion cells, retinula cells, or on the eye as a whole. Arthropod CEs are not noted for their high resolution and spatial frequency response. Their resolution is usually tested behaviorally, or by neurophysiological recording, as will be seen. To derive a quantitative model describing the resolution of CEs, it is necessary to define first a coordinate system (object space). Most CEs view over 2π steradians of solid angle (over a half of a hollow sphere, viewed from the inside). The “view” of individual retinula cells is rather narrow, however, and is characterized by the DSF. The surface of a CE is convex, so that the optical axis of the DSF of each ommatidium diverges slightly from its neighbors. Consider Figure 5.2-1. A CE views a point source of light on a concave spherical surface of radius R. The angle subtended by the light source at the eye is much smaller than the halfmax angle, θm/2, of an ommatidium DSF. Thus, the point source of light behaves like a two-dimensional spatial impulse function of intensity Io. That is, p(φ, θ) = Ioδ(φ – φo, θ – θo), where the spot is located at spherical coordinates, (R, φo, θo). In
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general, any object viewed by the eye can be described by its intensity as a function of φ and θ. For purposes of demonstration, it is mathematically more convenient to abandon spherical coordinates to describe the object intensity and instead use twodimensional rectangular coordinates (x, y). The arc lengths x and y are shown in Figure 5.2-1. If θ and φ are small, then the arc lengths approach the linear dimensions, x and y, given φ and θ. That is, x ≅ Rθ and y ≅ Rφ, θ and φ in radians.
FIGURE 5.2-1
The curved object plane of a compound eye. See text for description.
To establish the relationship between the two-dimensional, spatial distribution of intensity of an object, p(x, y), and a retinula cell depolarization voltage, vm, assume that the superposition of spatial impulse responses occurs, which leads to the following linear relation: Ie = ko [p(x, y)**s(x, y)]
5.2-2
where Ie is the effective intensity causing the photoreaction in the rhabdom of a retinula cell. (Ie is unobservable.) ko is an “un-normalizing” scale factor required to get the “true” value of Ie, noting that the DSF, s(x, y), is normalized; i.e., s(0, 0) ≡ 1. The ** denotes the operation of two-dimensional, real convolution. Once computed, vm can be found from the log relation:
⎛ l +l ⎞ v m = k 1 log⎜1 + e b ⎟ ld ⎠ ⎝
5.2-3
where Ib is the background intensity of illumination, and k1 and Id are positive constants. It is mathematically convenient to work in the spatial frequency domain to find the effective absorbed intensity, Ie, for various objects. That is, one takes the twodimensional Fourier transforms of p(x, y) and s(x, y). u and v are spatial frequencies with the dimensions of radian/mm. Thus, F[ko s(x, y)] = ko S(u, v)
DSF
5.2-4A
F[p(x, y)] = P(u, v)
Object
5.2-4B
F{ko[p(x, y)**s(x, y)]} ≡ ko P(u, v) S(u, v)
Real convolution theorem
5.2-4C
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To find Ie, take the inverse Fourier transform of Equation 5.2.4C with x = y = 0. (x = y = 0 corresponds to the center of the coordinate system for the one retinula cell under study. It is tacitly assumed that because of axial symmetry, all retinula cells in a given ommatidium have the same DSF.) The effective intensity is found by
l e = F –1[ k o P( u, v) S( u, v)] = x ,y = 0
ko 4π 2
∞
∫∫
∞
P( u, v) S( u, v) du dv
5.2-5
−∞ −∞
Still further mathematical simplification occurs if it is assumed that s(x, y) is symmetrical and independent in x and y, i.e., s(x, y) → s(x) s(y), and s(x) = s(y). Now analysis can be carried out in one-dimensional with little loss of generality. Two models frequently used for one-dimensional DSFs are
s(x) =
(
1
1 + x θm 2
)
n
,
n≥2
Hill function
5.2-6
Gaussian function
5.2-7
and
⎤ ⎡ ln( 4)x 2 ⎥ ⎢ s(x) = exp – 2 ⎥ ⎢ 2θ m2 ⎥⎦ ⎢⎣
( )
In both models, s(± θm/2) = 0.5.
5.2.1
THE COMPOUND EYE SAMPLING ARRAY
AS A
TWO-DIMENSIONAL, SPATIAL
This section analyzes the properties of the CE as whole as a spatial sampling array. Each ommatidium gathers light intensity from a corresponding point on a visual object, weighted by its DSF. Intensity information from each ommatidium can theoretically be used to reconstruct an “image” of the object in a sampling space lying in the animal’s CNS. To permit easy mathematical analysis of spatial sampling, assume that the surface of a model CE is a perfect hemisphere of radius r in which each ommatidium has an optical axis that diverges from its nearest neighbors by a uniform interommatidial angle, λ. Further, assume that the ommatidia are arranged on the surface of the eye in an equally spaced, rectangular grid rather than a hexagonally close-packed array, which is the usual case. If a model CE has 180 ommatidia on an “equator,” then the interommatidial angle is 1.0°. The object intensity is assumed to be displayed on the inside surface of a hemisphere of radius R Ⰷ r, concentric with the CE model. To examine spatial sampling mathematically, we will first consider the case of ideal, one-dimensional, spatial sampling. The ideal sampling process is mathematically equivalent to impulse modulation, in which the object intensity, g(x), is multiplied by an infinite impulse array formed by the receptor array. The onedimensional impulse array can be written as
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Pλ (x) =
∞
∑ δ ( x – nλ )
5.2-8
n =−∞
The periodic function, Pλ(x), can also be represented in the space domain by a Fourier series in complex form:
Pλ (x) =
∞
∑C
n
n =−∞
exp( – jnu s x)
5.2-9
where the fundamental spatial frequency is us = 2π/λ
r /mm
5.2-10
The complex-form Fourier coefficients are given by
Cn =
1 λ
∫
λ2
Pλ (x) exp( + ju s x) dx =
−λ 2
1 λ
all n
5.2-11
Thus the complex Fourier series for the pulse train is
Pλ (x) =
∞
∑
1 exp( – jnu s x) λ n =−∞
5.2-12
The sampler output, g*(x), is the space-domain product of Equation 5.2-12 and g(x):
g * (x) =
∞
∑
1 g(x) exp( – jnu s x) λ n =−∞
5.2-13
Now the Fourier transform theorem for complex exponentiation is F{y(x) exp(–jax)} ≡ Y(ju – ja)
5.2-14
Using this theorem, one can write the Fourier transform of the ideally sampled image as
G * ( ju) =
∞
∑
1 G( ju – jnu s ) λ n =−∞
5.2-15
Equation 5.2-15 for G*(ju) is the result of the complex convolution of G(ju) and Pλ(ju); it is in the so-called Poisson sum form, which helps visualize the effects of
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the ideal sampling process in the spatial frequency domain. Figure 5.2-2A illustrates the spectrum of an ideally sampled, one-dimensional intensity, g(x), which is bandwidth limited so that it has no spectral energy above uN = us/2 r/mm, the Nyquist frequency (Northrop 1997). When 冨G(ju)冨 contains significant spectral energy above the Nyquist frequency, a phenomenon known as aliasing occurs, shown schematically in Figure 5.2-2B. The overlap of the high-frequency spectral components of the baseband of 冨G*(ju)冨 with the high-frequency portions of the first harmonic terms of 冨G*(ju)冨 generate an aliased, high-frequency portion of the baseband of 冨G*(ju)冨 that represents unrecoverable or lost information. Note that the more the spectrum of 冨G(ju)冨 extends past the spatial Nyquist frequency, the more spectral energy in the baseband of 冨G*(ju)冨 is unrecoverable. One way to avoid aliasing, whether sampling in the time or space domains, is to precede the sampler with a low-pass filter that attenuates high frequency energy in 冨G(ju)冨 beyond the spatial Nyquist frequency, uN = π/λ r/mm. Such a low-pass filter is appropriately called an antialiasing filter. In the CE, each point in the sampling array (i.e., each ommatidium) has a builtin low-pass filter that attenuates high spatial frequencies of the object and decreases aliasing. This filter is, of course, the DSF. To be effective, S(uN)/S(0) < 0.01. For the Hill DSF, given above,
FIGURE 5.2-2 (A) Spectrum of a spatially sampled, one-dimensional intensity distribution, g(x). g(x) is bandwidth limited so no spectral energy exists for 冨u冨 > π/γ r/mm. (λ is the angular separation between ommatidial axes.) G(ju) is the baseband spectum; G*(ju) is the (repeated) sampled spectrum. Note that there is no overlap of adjacent G*(ju) elements. (B) Aliased spectrum of an object whose baseband spectrum has energy at frequencies exceeding π/λ r/mm. Note that the adjacent spectra of G*(ju) overlap and add together, creating a region of high frequencies above which G(ju) is not recoverable by an ideal low-pass filter with passband between ±π/λ r/mm acting on G*(ju).
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s(x) =
θ 2m 2
5.2-16
θ 2m 2 + x 2
The Fourier transform of this DSF is well known: S(u) = (θm/2π) exp[– θm/2冨u冨]
5.2-17
So θm/2 must be chosen so that the attenuation criterion is met. That is,
(
) [ ( )
]
θ m 2 π exp – θ m 2 π λ S( π λ ) = = exp – θ m 2 π λ < 0.01 S(0) θm 2 π
[
]
5.2-18
For this attenuation to occur, and for this DSF, the ratio of DSF half-angle to interommatidial angle, θm/2/λ, must be > 1.465. The mathematical development above was done for an ideal, one-dimensional, infinite, spatial sampling array for object intensity. In practice, interommatidial angles vary with the position of the ommatidium on the eye surface, and can differ in the x and y directions as well. It seems that the arthropod lives in a far less perfect visual world than do vertebrates with camera eyes. Still, they have survived through the ages. A behavioral test for aliasing in CEs is to see if the optomotor response reverses sign when the period of a striped object moved past the animal reaches a small value where aliasing occurs. Ideally, this would be where the fundamental spatial frequency of the stripe square wave is greater than the Nyquist frequency set by interommatidial angle, λ; i.e., 2π/Xo > π/λ, or the stripe period, Xo , is less than 2λ. (A time-domain analog of aliasing is when spoked wheels are perceived turning backward in the movies when they are moving a vehicle forward. In this case, the temporal sampling rate is the frames per second of the camera.) Bishop and Keehn (1967) have reported the phenomenon of optomotor reversal in the housefly Musca domestica. The crossover stripe period was 4 to 4.5°. Interommatidial angles in the Musca eye are variable, depending on position in the eye, and range from 1.0 to 5.4° with a mean λ = 3.9°. Bishop and Keehn (1967) noted: The orientation of the array of ommatidia with respect to the rotating pattern, results in a spectrum of interommatidial angles, some of which could contribute positively, and some negatively to the neural (optomotor) response.
Northrop (1974) also noted a loss of directional selectivity in the firing rates of the third cervical nerve (CN3) (see Section 5.4.2 below) of the locust, Schistocerca gregaria, when presented with moving stripes having a critical period. (N3 innervates muscles that move the animal’s head.) In this experiment, the animal still sensed
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stripe motion, but the neural response was unable to distinguish between the preferred and null directions of motion. The loss of directional discrimination occurred for stripe periods less than 5 to 6°. See Figure 5.2-3 for an example. Presumably this loss was also due to aliasing. A general property of CEs is that when they become DA, the DSFs of the ommatidia become broader (θm/2 increases). This broadening of s(x) means the spatial frequency response S(u) decreases, making aliasing less likely. Thus, one would not expect to see optomotor reversal for DA CEs. NR 1.00
PD
AV
0.10
0.01 3
4
5 6
10 Ao
20
30
FIGURE 5.2-3 Log–log plot of the normalized response of the cervical motor nerve N3 of the locust when a restrained animal is shown moving rectilinear stripes. Stripes were viewed in a square window 2.5 periods on a side with its sides parallel with the stripes. Horizontal bar is the background firing rate for the “C” unit; circles: C unit responses to stripes moved in a PD direction; squares: C unit responses to anterio-ventral (AV) stripe velocity; hexagons: D-unit response to PD stripe movement. The D unit was silent for no stripe motion, and did not respond to AV stripe motion. Stripes were moved two periods in 1.4 s. A 20-s i.s.i. was used to avoid habituation. Note curious “knee” in the C unit responses. N3 C unit responses ceased for stripe period less than about 5°. (From Northrop, R.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)
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5.2.2
CALCULATION
OF INTENSITY
CONTRAST
As an object moves relative to the eye, the intensity of light on the rhabdoms, Ie, will change in time, producing temporal changes in the retinula cell resting potential, ∆vm(t). One theoretical measure of visual resolution is to calculate the intensity contrast, CIe, in a retinula cell.
l max – l min l max + l min
C ie ≡
5.2-19
Simple test objects themselves can be described by a contrast function, Cobj:
C obj ≡
l o max – l o min l o max + l o min
5.2-20
Example 5.2-1 A first example examines the linear Fourier model for intensity contrast when the object is a single black spot of width, 2a. Assume that the spot contrast is unity (Cobj = 1). For simplicity, work in one dimension, x. Let the DSF be modeled by the hyperbolic Hill function:
s(x) =
θ 2m 2 θ 2m 2 + x 2
5.2-21
Figure 5.2-4 illustrates this system. The Fourier transform of this DSF is well known: S(u) = (θm/2π) exp[– θm/2冨u冨]
5.2-22
The one-dimensional, “black spot” object is centered over the ommatidium. It is described by p(x) = Io {1 – [U(x + a) – U(x – a)]}
5.2-23
The one-dimensional Fourier transform of the object can be shown to be
sin(au) ⎫ ⎧ P( u) = l o ⎨2 π δ( u) − 2a ⎬ = l {2 π δ( u) − (2a ) sinc(au π)} au ⎭ o ⎩
5.2-24
Of interest is the intensity contrast when the spot is at x = ×, and when it is directly over the ommatidium (x = 0),
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FIGURE 5.2-4 Diagram relevant to the calculation of the intensity contrast of a retinula cell. A one-dimensional, black spot object is “shown” to an ommatidium whose retinula cells absorb light with a hyberbolic, Hill-type DSF.
l e max =
ko 2π
l e min =
ko 2π
∫
∞
[(
) (
)
l o 2 π δ( u) θ m 2 π exp – θ m 2 u e 0 du = k o l o θ m 2 π
−∞
∞
∫ l [2π δ(u) − sinc(au π)] exp(–θ u )e du −∞
0
o
m2
5.2-25
5.2-26
↓
l e min = l e max –
ko 2π
∫
∞
−∞
2a
sin(au) l o θ m 2 π exp – θ m 2 u e 0 du au
[
(
)]
5.2-27
The integral above can be rewritten as a well-known definite integral, noting that the integrand is an even function in u:
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l e min = l e max – θ m 2 k o l o 2
∫
∞
0
sin(au) exp – θ m 2 u du u
[
]
5.2-28
Evaluation of this definite integral yields: Iemin = Iemax – θm/2 2 ko Io tan–1 [a/θm/2]
5.2-29
Thus, the intensity contrast is
C ie =
{ π + l {k θ
[ ]} = tan [a θ ] [a θ ]} π − tan [a θ ]
k o l o θ m 2 π – l o k o θ m 2 π − k o θ m 2 2 tan –1 a θ m 2 k o l oθ m 2
o
o m2
π − k o θ m 2 2 tan
–1
–1
m2
–1
m2
m2
↓ C ie =
5.2-30
1
π tan
–1
[a θ ] − 1 m2
The Cie function is plotted in Figure 5.2-5; note that Cie → 0 as a → 0 and/or θm/2 → ×. 1
0.8
0.6
0.4
0.2
0 0 >axes >simu >
2
4
6
8
10
h 0 10 0 10
FIGURE 5.2-5 Plot of the theoretical intensity contrast of a retinula cell having a Hill DSF. Horizontal axis, a/θm/2; vertical axis normalized to 1.
Example 5.2-2 As a second example, let the test object be an infinite, two-dimensional array of black and white stripes of equal widths. The stripes are infinitely long in the y dimension and have a period A in x. The white areas have intensity pIo and the black areas have intensity qIo, where 1 > p Ⰷ q > 0. In the x dimension, p(x, y) consists of an average (dc) intensity level, Io (p + q)/2, plus an additive square wave with
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FIGURE 5.2-6 Diagram relevant to the calculation of the intensity contrast of a retinula cell in the second example. A one-dimensional, black white, square wave object is “shown” to an ommatidium whose retinula cells absorb light with a Gaussian DSF. The square wave period is A. The DSF half intensity angle is θm/2 = σ
ln( 4) .
zero mean, f(x, y), having a period A and peak height, (p – q)/2. Note that the object itself has a contrast, Cobj = (p – q)/(p + q). Figure 5.2-6 illustrates this system. Because f(x, y) is periodic in x, one can write f(x, y) as a Fourier series in complex form:
f (x, y) = f (x + A, y) =
∞
∑ C (y) exp(+ jnu x) n
o
5.2-31
n =−∞
where uo = 2π/A rad/mm. The complex Fourier coefficient Cn(y) is found from
C n (y) = (1 A)
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∫
A2
[
]
f (x, y) exp – jnu o x dx
−A 2
5.2-32
Cn(y) is found to be
C n ( y) =
( p − q ) sinc n 2 ( )
5.2-33
2
Evaluating Cn(y) one can write the complete Fourier series for the object as
p(x, y) − l o
(p + q) + l 2
⎡ (p − q) ⎤ sinc( n 2)⎥ exp( + jnu o x) 2 ⎦ n =−∞ ∞
o
∑ ⎢⎣
5.2-34
n≠0
where sinc(z) ≡ sin(πz)/(πz) and Io (p + q)/2 is the dc intensity level. Next, take the Fourier transform of p(x, y) so Equation 5.2.5 can be used to find CIe for the striped object. Note that in general
[
]
P( u, v) = F p(x, y) ≡
∞
∞
∫ ∫ p(x, y) exp[− j(ux + vy)] dx dy
5.2-35
−∞ ∞
The parameters u and v are spatial frequencies in rad/mm. Also, the theorem: F[exp(+ juo x)] ≡ 2π δ(u – uo)
5.2-36
is used. Hence the two-dimensional Fourier transform of the object can be shown to be
p( u, v) =
(p + q) + l 2
o
4 π 2 δ( u, v) + l o 4 π 2 δ( v)
⎡ (p − q) ⎤ sinc( n 2)⎥ δ( u – nu 0 ) exp − jnu o x o ⎢ 2 ⎦ n =−∞ ⎣ ∞
∑
[
]
5.2-37
n≠0
The complex exponential term on the right of Equation 5.2.25 allows the square wave to be shifted in x by an amount xo so a black stripe can be centered over the receptor. For this example, assume the DSF of the receptor is modeled by a two-dimensional Gaussian form with circular symmetry. Thus,
(
)
⎡ x2 + y2 ⎤ ⎡ x2 ⎤ ⎡ y2 ⎤ s(x, y) = exp ⎢ – ⎥ = exp ⎢ – 2 ⎥ exp ⎢ – 2 ⎥ 2 2σ ⎢⎣ ⎥⎦ ⎣ 2σ ⎦ ⎣ 2σ ⎦
5.2-38
where σ ≡ θm/2 / ln( 4) . This Gaussian function has the well-known two-dimensional Fourier transform:
© 2001 by CRC Press LLC
S(u, v) = [2π σ2] exp[– σ2(u2 + v2 )/2]
5.2-39
Now, knowing P(u, v) and S(u, v), Equation 5.2.5 can be used to find Ie for the case where the white stripe is centered over the ommatidium DSF (xo = 0) and also for the case where the black stripe is centered over the DSF (xo = A/2). After some complex algebra, one can write for the general case:
l e (x o ) = k o 2 πσ 2 l o ⎫ 5.2-40 ⎧ ∞ ⎡ (p − q) ⎤ ⎪ ⎪ (p + q) sinc( n 2)⎥ exp − σ 2 n 2 u o2 2 exp( − jnu o x o )⎬ + ⎨ ⎢ 2 ⎦ ⎪ ⎪ 2 n =−∞ ⎣ n≠0 ⎭ ⎩
∑
[
]
where, again, uo = 2π/A, and σ = θm/2/ ln( 4) . At limiting resolution, only the dc and first harmonic (n = ±1) of the Fourier series is important. So, by using the expression for Ie(xo) above,
⎧ (p + q) (p − q) ⎫ l e (x o ) ≅ k o 2 πσ 2 l o ⎨ + (4 π) exp −σ 2 u o2 2 cos(u o x o )⎬ 2 ⎩ 2 ⎭
[
]
5.2-41
When xo = A/2, a black stripe is centered over the ommatidium, and Ie is given by
⎧ (p + q) (p − q) ⎫ l e (A 2) = k o 2 πσ 2 l o ⎨ − (4 π) exp −σ 2 u o2 2 ⎬ 2 2 ⎩ ⎭
[
]
5.2-42
(Note that cos[1(A/2)2π/A] = –1.) By using Ie(xo = 0) and Ie(xo = A/2), the absorbed light contrast, CIe, is
C ie =
[
] [ 2[( p + q ) 2]
2 ( p − 2) 2 ( 4 π) exp − σ 2 u o2 2
]
5.2-43
By canceling the 2s and dividing the top and bottom by (p + q),
[
]
[
C ie = C obj ( 4 π) exp − σ 2 u o2 2 = C obj ( 4 π) exp −14.239 θ 2m 2 A 2
]
5.2-44
From the plot of Cle/Cobj shown in Figure 5.2-7, the absorbed light contrast drops off very rapidly as (θm/2/A) increases.
© 2001 by CRC Press LLC
1
0.5
0 0
FIGURE 5.2-7 are (θm/2/A).
5.2.3
0.2
0.4
0.6
0.8
1
Plot of Cie/Cobj for the second example. Peak is at 4/π. Horizontal axis units
“ANOMALOUS RESOLUTION”
The limiting resolution of the CE-to-descending contralateral movement detector (DCMD) neuron system in the locust Schistocerca gregaria and in the lubber grasshopper Romalea microptera has been widely investigated, both optically and electrophysiologically. The DCMD neuron axons (one in each side of the ventral nerve cord) are large, produce large spikes, and are easily recorded from with simple hook or suction electrodes. The operation performed by the CE-to-DCMD neuron system is the detection of small, novel movements of small, dark test objects anywhere over the contralateral eye. DCMD units habituate quickly, and need several minutes to recover their maximum response to novel object movements. The DCMD neuron has been traced electrophysiologicaly from its origin in the lateral, ipsilateral midbrain, down and across the contralateral esophageal connective to the contralateral ventral nerve cord (CVNC), where it descends as far as the third thoracic ganglion. The DCMD neuron is driven through a rectifying electrical synapse in a 1:1 manner from the lobular giant movement detector (LGMD), neuron which is located in the lobula of the OL. The LGMD neuron also drives by a chemical synapse a descending ipsilateral movement detector (DIMD) neuron in the ipsilateral VNC. (Details of the locust movement detector system can be found in a series of definitive papers by Rowell and O’Shea, 1976a; 1980); Rowell et al., 1977, O’Shea and Williams, 1974; and O’Shea and Rowell, 1975, 1976). The resolution of the DCMD system has been called “anomalous” because a significant increase in the firing rate of a DCMD unit will occur for novel movements of a dark test object described by spatial frequencies apparently higher than the cutoff frequency that ommatidial optics and DSFs predict. Any test of the spatial resolution of DCMD neurons is made more difficult by the fact that these units respond to novel movements of the test object in their visual field (the entire contralateral CE), and soon habituate for repeated object motion. Thus, tests of
© 2001 by CRC Press LLC
DCMD unit visual resolution must be spaced far apart enough in time for the DCMD system to “unhabituate,” i.e., recover its maximum sensitivity to novel object motion (Grossman and Northrop, 1976). Interstimulus intervals used by the author in DCMD system resolution tests with Romalea were never less than 2 min. It was observed by Horn and Rowell (1968) for Schistocerca, that a 2-min recovery time was necessary to avoid response desensitization by habituation. In testing DCMD unit resolution using small, “jittery spot objects,” the author has found that while the entire eye is capable of responding to the jittery spot, it is the group of ommatidia directly under the spot that apparently habituate. If the spot is moved over an unstimulated part of the eye, and jittered without delay, little habituation is found at first. If the spot is jittered over the whole eye, then a 2-min rest is required to restore sensitivity to the whole eye. The first report of anomalous resolution was by Burtt and Catton (1962). They showed that the DCMD unit would give a significantly increased number of spikes when the eye was presented with moving black and white stripes, seen through a rectangular window. Responses were seen down to stripe periods of ~0.3°. The 0.3° limit was considered “anomalous” by other workers because it exceeded the Nyquist limit for spatial sampling by a factor of about 6.67 in the vertical plane of the locust eye, and, also, it is beyond the “resolving power” of the apertures of the individual ommatidia as determined by the Rayleigh criterion (classical optical theory). To try to explain Burtt and Catton’s “anomalous” results, other workers suggested that the anomalous response was due to imperfections in the pattern (i.e., spatial subharmonics; McCann and MacGinitie, 1965) or was due to a subharmonic generated as stripes emerged from and then passed behind the straight edges of the mask (Palka, 1965). Burtt and Catton (1966, 1969) countered these criticisms by making a precision, radially striped, “wheel” pattern, viewed by the insect in an annular window. This pattern obviously had no edge effects. Burtt and Catton again found a limiting resolution of ~0.3°. The author has also examined the threshold resolution of DCMD units in Schistocerca and in Romalea using the radial striped wheel pattern used by Burtt and Catton, as well as a fine, rotating checkerboard pattern in an annular window, and a single, black, jittery spot. The jittery black spot was held by a magnet on the gray inside surface of a thin fiberglass hemisphere, 50 cm in diameter. The hemisphere with spot was centered over the CE under test, and illuminated from the sides by diffuse white light. The spot was jittered manually by moving a corresponding magnet on the outside of the hemisphere by hand in a manner (position, direction, speed) to maximize the number of spikes elicited on the DCMD fiber over the test period (generally 1 min). DCMD units have a size preference for jittered spots; a spot of ~5° appears to give the strongest response. Smaller spots obviously give a reduced response, as do spots larger than 5°. White spots also give about half the response of black spots, other factors being equal. Figure 5.2-8 illustrates the fact that the DCMD system can confidently resolve a jittered spot as small as 0.4°. A rotating, black/white, checkerboard object viewed through an annular window (8.2 cm inside diameter; 14.5 cm outside diameter) was found to be the most potent test object for the DCMD system (Northrop, 1974). Rotation of this pattern could elicit
© 2001 by CRC Press LLC
NR 1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4 0.6 0.8 Spot diameter,°
1.0
1.2
FIGURE 5.2-8 Normalized DCMD unit responses to a single, jittered, black spot inside a large, gray hemisphere. See text for description of the experimental protocol. The confidence that the DCMD unit responded to the jitter of a 0.46° diameter spot was better than 99.5% (one-sided t-test, 9° freedom). Bar denotes the normalized, random firing rate of the unstimulated DCMD system. (From Northrop, R.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, New York, 1975. With permission of Oxford University Press.)
a 95% confident DCMD response down to a checker repeat distance of 0.21°. (A Student t-test was used to validate the responses statistically.) After demonstrating that anomalous resolution does in fact occur in the DCMD system for a number of different types of test objects, the focus turns to a theoretical model that may describe its mechanism. One strategy that the animal might use is to organize the ommatidia of an eye into a synthetic aperture system, such as used to enhance resolution in radar and radiotelescope systems having more than one antenna (an antenna array). The synthetic aperture approach has also been employed with optical telescopes (Reynolds et al., 1989). Although it has been suggested that one ommatidium is all that is necessary to sense object movement (Kirschfeld, 1972), higher resolution may be obtained by using two or more closely spaced ommatidia, and processing the rate of change of each vmk(t) so an output is obtained if one vmk(t) is increasing while another vmj(t) is decreasing. No output occurs if both vmk(t) and vmj(t) are changing with the same sign. The following section describes a hypothetical neural model proposed by the author to account for anomalous resolution.
5.2.4
A MODEL FOR CONTRAST ENHANCEMENT VISUAL SYSTEM
IN THE INSECT
The author proposed a hypothetical model of neural interaction in the lamina ganglionaris of grasshoppers (Northrop, 1975) that may account for the anomalous resolution observed in the grasshopper’s multimodal and DCMD units. The model makes use of an engineering technique called multiplicative signal processing
© 2001 by CRC Press LLC
(MSP). The model assumes that an ommatidium has six identical retinula cells that send their axons to synapse on one monopolar cell in the lamina cartridge serving that ommatidium (Horridge, 1966). Assume that the light intensity on all the rhabdoms is equal, and large enough so that each retinula cell depolarization is in its logarithmic region. Thus, for the kth depolarization, Vrk ≅ D ln(Ie/B) mV,
Ie > B > 0
5.2-45
The constants D and B are chosen to give the best fit to the Vr vs. I curve. Next, assume that all six of the retinula cell axons converge on one monopolar cell, inducing a hyperpolarizing postsynaptic potential, given by 6
Vm = K
∑V
5.2-46
rk
k =1
Assuming that all Vrk are equal, Vm = – (6KD) ln(Ie/B) = – ln(Ie/B)6KD,
Ie > B,
Vm < 0
5.2-47
The next step in formulating this speculative model is to assume an electrotonic coupling of the axon of the monopolar cell to a target interneuron in the medulla. The graded potential change in this interneuron, Vi, is also assumed to be hyperpolarizing, and exponential in shape. Vi is assumed to be directly related to the DCMD spike production rate:
(
ˆ Vi = – F exp − GV m
)
5.2-48
ˆ is V conditioned by propagating electrotonically down the nonspiking axon of V m m the monopolar cell. For steady-state or average absorbed intensities, ˆ =β V ≤0 V m o m
5.2-49
where βo is the dc cable attenuation of the axon. Combining the relations above, define the product, βo 6KGD ≡ m:
Vi = − F(l e B)
βo 6 KGD
(
)
= − F B – m l em
5.2-50
It will be shown that m must be > 1 for an improvement in the contrast seen in the hyperpolarizing interneuron voltage, CVi. A tacit assumption has been made that an improved contrast in Vi is required for the threshold detection of moving, highspatial-frequency objects. The contrast in Vi is
© 2001 by CRC Press LLC
C vi =
( ) –(F B ) l
[ + [– F B
– F B– m l emmax − – F B – m l emmin –m
m e max
–m m e min
l
]=l ] l
m e max m e max
– l emmin + l emmin
5.2-51
The maximum effective intensity in the rhabdoms was shown to be
⎧ (p + q) (p − q) ⎫ l e max = k o 2 πσ 2 l o ⎨ + (4 π) exp −σ 2 u o2 2 ⎬ 2 2 ⎩ ⎭
[
]
5.2-52
]}
5.2-53
If the (p + q)/2 term is factored out,
{
[
l e max = k o 2 πσ 2 l o ( p + q ) 2 1 + C obj ( 4 π) exp − σ 2 u 2o
The exponential term in Equation 5.2.53 is Ⰶ 1, so when Iemax is raised to the mth power, it becomes possible to use the approximation, (1 + ε)m ≅ 1 + mε. Thus,
[
] {1 + mC
[
] {1 − mC
l emmax ≅ k o 2 πσ 2 l o ( p + q ) 2
m
obj
(4 π) exp[−σ 2 u o2 ]}
5.2-54
(4 π) exp[−σ 2 u o2 ]}
5.2-55
Similarly,
l emmin ≅ k o 2 πσ 2 l o ( p + q ) 2
m
obj
Hence by using Equation 5.2.51, on Equations 5.2.54 and 5.2.55,
[
]
C vi ≅ mC obj ( 4 π) exp − σ 2 u o2 = mC ie
5.2-56
As long as m = βo 6KGD > 1, there will be contrast improvement in the membrane voltage of the medullar interneuron, Vi, as the result of moving a pattern with contrast Cobj over the eye. That is, CVi > CIe > Cobj for m > 1. It is also of interest to evaluate the contrast function for the retinula cell depolarization voltage, given by
{
}
Vmax = D ln(l e max B) = D − ln(B) + ln(l e max )
{
[(
)[
{
(
)
(
= D − ln(B) + ln k o 2 πσ 2 l o ( p + q ) 2 1 + C obj ( 4 π) exp − σ 2 u 2o 2
[
(
)]}
= D − ln(B) + ln k o 2 πσ 2 l o ( p + q ) 2 + ln 1 + C obj ( 4 π) exp − σ 2 u o2 2
)]}
5.2-57
© 2001 by CRC Press LLC
Note that the Cobj (4/π) exp(–σ2 uo2/2) term is generally Ⰶ 1, so one can use the approximation, ln(1 + ε) ≅ ε. Thus,
{
(
)
(
)]}
5.2-58
{
(
)
(
)]}
5.2-59
Vmax ≅ D − ln(B) + ln k o 2 πσ 2 l o ( p + q ) 2 + C obj ( 4 π) exp − σ 2 u o2 2 and
Vmin ≅ D − ln(B) + ln k o 2 πσ 2 l o ( p + q ) 2 − C obj ( 4 π) exp − σ 2 u o2 2 Hence,
C vr ≡
(
) ]
2 2 Vmax – Vmin C obj ( 4 π) exp − σ u o 2 ≅ = C ie ln k o 2 πσ 2 l o ( p + q ) 2 B Vmax + Vmin ln k o 2 πσ 2 l o ( p + q ) 2 B
[
[
]
5.2-60 Under the conditions that m > 1 and ln(ko2πσ2Io (p + q)/2B) > 1, it follows that CVi > Cie > CVr. The MSP model based on the six retinula cells from one ommatidium has been shown to produce enhancement of the voltage contrast function in a corresponding medullary, nonspiking interneuron. Northrop (1975) also showed that it is theoretically possible to realize signal-to-noise ratio enhancement using the same nonlinear MSP model.
5.2.5
A HYPOTHETICAL MODEL FOR SPATIAL RESOLUTION IMPROVEMENT IN THE COMPOUND EYE BY SYNTHETIC APERTURE
The synthetic aperture (SA) architecture theoretically allows the generation of equivalent, sharper, high-resolution, directional sensitivity functions in a CE system. As a general rule, the sharper and narrower the DSF of an ommatidium, the larger the intensity contrast will be for a given contrasting object (spot, stripes) moved over that ommatidium, and the higher the spatial frequency bandwidth of the system. The sharper DSFs generated by SA architecture still have the same angular spacing in the CE as do individual ommatidia. Figure 5.2-9 illustrates a two-receptor SA system. The outputs from two retinula cells, each from one of two adjacent ommatidia, are multiplied together to form an SA output. (Two retinula cells from nearest-neighbor ommatidia are used here for simplicity. Analysis is also carried out in one dimension for simplicity.) Both retinula cells have a Hill or “Cauchy” type of DSF, whose spatial impulse response is given by
s(x) =
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1 ←⎯F→ S( u) = απ exp( – α u ) 1 + x2 α2
5.2-61
FIGURE 5.2-9 Schematic of a two-ommatidia, one-dimensional, synthetic aperture system, given a single black spot object (in one dimension). The black spot is 2r° in diameter. The axes of the two ommatidia are separated by 2b°, and the DSF half-intensity angles are θm/2 = α°.
The frequency variable, u, has the dimensions of rad/mm, α is the acceptance angle of the DSF; that is, s(± α) ≡ 0.5, s(0) ≡ 1. Now since vsa(x) = vr1(x) vr2(x), p(x) = A δ(x), A = 1, P(u) = 1; hence, Vsa(u) = Vr1(u) * Vr2(u) and Vsa for the unit impulse input is Ssa(u), the Fourier transform of the synthetic aperture equivalent DSF. * denotes one-dimensional complex convolution. (Multiplication in the spatial (x) domain is equivalent to convolution in the spatial frequency (u) domain.) Thus, Ssa(u) can be written
S1 ( γ ) Ssa ( u) =
1 2π
S2 ( u − γ )
∞
∫ [πα exp(−α γ ) exp(– jbγ )][πα exp(−α u − γ ) exp(+ jb(u − γ ))] dγ −∞
5.2-62 Note that the complex convolution is carried out with respect to the spatial frequency γ. The integral above can be rewritten
© 2001 by CRC Press LLC
(
)
Ssa ( u) = π α 2 2 exp( + jbu)
∫
∞
−∞
exp( − j2 bγ ) exp( −α γ ) exp( −α u − γ ) dγ
5.2-63
Equation 5.2.63 can be rewritten as the sum of three integrals for u > 0:
⎧ Ssa ( u) = π α 2 2 exp( + jbu)⎨ ⎩
(
)
∫
0
∫
u
∫
∞
−∞
0
u
l1
exp( − j2 bγ ) exp(αγ ) exp(α( γ − u)) dγ + l2
exp( − j2 bγ ) exp( −αγ ) exp(α( γ − u)) dγ +
5.2-64
l3 ⎫ exp( − j2 bγ ) exp( −αγ ) exp( −α( γ − u)) dγ ⎬ ⎭
Evaluating the definite integrals:
l1 =
l2 =
l3 =
⎡ 1− 0 ⎤ exp( – αu) exp ( − j2 b + 2α )γ dγ = exp( −αu)⎢ ⎥ −∞ ⎣ − j2 b + 2α ⎦
5.2.65A
⎡ exp( − j2 bu) − 1 ⎤ exp( – αu) exp( − j2 bγ ) dγ = exp( −αu)⎢ ⎥ − j2 b ⎣ ⎦
5.2.65B
⎡ exp( − j2 bu) ⎤ exp(αu) exp − γ ( j2 b + 2α ) dγ = exp( −αu)⎢ ⎥ ⎣ j2 b + 2α ⎦
]
5.2.65C
⎡ 1 1 − e − j2 bu e − j2 bu ⎤ l Σ = exp( −αu 2)⎢ + + jb α + jb ⎥⎦ ⎣ α − jb
5.2-66
∫
0
∫
u
∫
∞
0
u
[
]
[
Now IΣ = I1 + I2 + I3, so
Through a maze of complex algebra, the Fourier transform of the equivalent SA DSF is finally reached:
Ssa ( u) =
πα 3 exp( −α u )
(
2 α 2 + b2
(a b) sin(b u ) + cos(bu)], ) [
for all u
5.2-67
Remember, α = θm/2 = the half-intensity (acceptance) angle of the primary DSF. b is the angle between the two adjacent DSFs. If b = α is chosen, a considerable algebraic simplification results. The Fourier transform of the synthetic aperture DSF is
© 2001 by CRC Press LLC
Ssa ( u) =
απ 2e − αu sin(αu + π 4), 4
for u ≥ 0
5.2-68
[Note that sin(αu + π/4) reaches its maximum for αu = π/4 rad; this broadens the low-frequency peak of Ssa(u).] Compare the expression for Ssa(u) above with the Fourier transform of the DSF of a single ommatidium: S1(u) = απ exp(–αu), for u Š 0. Figure 5.2-10 plots S1(u) and Ssa(u) vs. u. For simplicity, let α = 1° = b. Note that Ssa(u) has less low-frequency attenuation than a single ommatidium DSF, but has a zero at about u = 2.3.
FIGURE 5.2-10 Plot of the Fourier transform of a single receptor DSF (trace 2) and the two-receptor, synthetic aperture DSF (trace 1) vs. spatial frequency, u. Trace 3 is the zero reference.
To verify that the SA architecture leads to increased intensity contrast, first calculate the intensity contrast for a one-dimensional black spot of “diameter” 2r, centered on x = 0. p(x) for the spot is p(x) = Io {1 [U(x + r) – U(x – r)]}
5.2-69
The Fourier transform of this spot is
sin( ru) ⎤ ⎡ P ( u ) = l o ⎢2 π δ ( u ) − 2 r ru ⎥⎦ ⎣ The effective intensity of the SA system is found from
© 2001 by CRC Press LLC
5.2-70
l eSA =
lo 2π
3 ⎫⎪ sin( ru) ⎫ ⎧⎪ π α exp( – α u ) ⎧ a b b u bu + sin cos ( ) ( ) ( ) ⎬⎨ ⎨2 π δ( u) − 2 r ⎬ du ru ⎭ ⎪ 2 α 2 + b 2 −∞ ⎩ ⎪⎭ ⎩
∫
∞
) [
(
]
5.2-71 When the black spot is at x = ×, IeSAmax is found to be
l eSA max =
(
l o πα 3
2 α 2 + b2
5.2-72
)
When the spot is centered over x = 0, IeSA is minimum, and is given by IeSAmin = IeSAmax – ∆IeSA
∆l eSA =
lo 2π
5.2-73
3 ⎫ ⎧ sin( ru) ⎫ ⎪⎧ π α exp( – α u ) (a b) sin(b u ) + cos(bu) ⎪⎬ du 5.2-74 ⎬⎨ ⎨2 r 2 2 ru ⎭ ⎪ 2 α + b −∞ ⎩ ⎪⎭ ⎩
∫
∞
(
) [
]
Evaluation of this rather sinister looking integral is made easier when note is made that an even function is being integrated. After some algebra,
∆l eSA =
(α
l oα 3 2
+b
2
)
⎫⎪ ⎧⎪ ⎡ α 2 + ( r + b) 2 ⎤ –1 ⎨(α 4) ln ⎢ 2 2 ⎥ + (1 2 ) tan [( r + b ) α ]⎬ ⎪⎭ ⎣ α + ( r − b) ⎦ ⎩⎪
5.2-75
The equivalent intensity contrast, CIeSA , is found:
C leSA =
∆l eSA 2 l eSA max – ∆l eSA
5.2-76
Let α = r = b ≡ 1°. These reasonable values give CIeSA = 0.437. If the contrast for one receptor alone directly under the spot is examined, CIe1 = 0.333. Thus, it appears that the SA architecture can improve equivalent intensity contrast. A more realistic model of the synthetic aperture/MSP architecture that will improve neural signal contrast in a CE is shown in Figure 5.2-11. Here three adjacent retinula cells lying in three adjacent ommatidia with intensity DSFs s1(x), s2(x), and s3(x) have depolarizations due to effective absorbed intensities, Ie1, Ie2, and Ie3, respectively. The depolarizations Vr1, Vr2, and Vr3 are assumed to be summed at a lamina monopolar cell (LMC). The LMC hyperpolarizes by voltage Vm. Vm propagates electrotonically down the LMC axon, where it synapses with a medullary interneuron (MI), causing a hyperpolarization, Vi. It is the contrast in Vi caused by a moving object that is assumed critical in limiting spatial resolution.
© 2001 by CRC Press LLC
FIGURE 5.2-11 Schematic of a three-ommatidia, one-dimensional, synthetic aperture system. The black spot is 2r° in diameter; The axes of the three ommatidia are separated by b°, and the DSF half-intensity angles are θm/2 = α°.
Let the object be a black spot of radius, r. The DSF relations are
© 2001 by CRC Press LLC
s1(x) = 1/(1 + x2/α2)
5.2-77A
s2(x) = 1/[1 + (x + b)2/α2]
5.2-77B
s3(x) = 1/[1 + (x – b)2 /α2]
5.2-77C
The object is p(x) = Io {1 – [U(x + r) – U(x – r)]}
5.2-78
↓ F{*}
sin( ru) ⎫ ⎧ P( u) = l o ⎨2π δ( u) − ⎬ ru ⎭ ⎩
5.2-79
The effective intensity at the kth retinula cell is
l ek =
ko 2π
∞
∫ S (u) P(u) du, −∞
1, 2, 3
k
5.2-80
Thus, the voltages can be calculated Vrk = A ln(Iek/B),
k = 1, 2, 3
5.2-81
3
Vm = K
∑V
5.2-82
rj
j=3
Vi = F exp(GVm)
5.2-83
When the spot is at ×, the maximum intensity is absorbed by all three retinula cells, and Vi is maximum. That is, Ie1 = Ie2 = Ie3 = Iemax:
l e max =
k olo 2π
∫
∞
−∞
[
]
2 π δ( u) α π exp( −α u ) e 0 du = k o l o α π
5.2-84
Thus, Vr1 = Vr2 = Vr3 = A ln[(koIo απ)/B] = Vrmax
5.2-85
Vi = F exp{GKA ln[(koIo απ)3/B3] = F[(ko Io α π)/B]3GKA = Vimax
5.2-86
and
Next, center the black spot of diameter 2r over the center receptor at x = 0. The effective intensity at retinula cell 1 is reduced:
© 2001 by CRC Press LLC
k olo 2π
l e1 = l e1max – ∆l e1 = k o l o α π −
2 r sin( ru) α π exp( −α u ) du 5.2-87 ru −∞
∫
∞
[
]
Because the integrand is an even function, the integral can be written as
k olo 2 2π
l e1 = l e1max – ∆l e1 = k o l o α π −
∫
∞
0
2 sin( ru) [α π exp(−αu)] du 5.2-88 u
The definite integral has the well-known solution: Ie1 = αkoIo [π – 2 tan–1(r/α)]
5.2-89
Now the offset DSFs have the Fourier transforms:
[
]
5.2-90
]
5.2-91
F s 2 ( x ) = 1 1 + ( x + b) α 2 ⎯ ⎯ → S2 ( u) = α π exp( −α u ) e + jub 2
[
F s 3 ( x ) = 1 1 + ( x − b) α 2 ⎯ ⎯ → S3 ( u) = α π exp( −α u ) e − jub 2
The Euler relation, e±jub ≡ cos(ub) ± j sin(ub), is used to finally obtain: S2(u) = απ exp(–α 冨u冨) [cos(ub) + j sin(ub)] S3(u) = απ exp(–α 冨u冨) [cos(ub) – j sin(ub)]
5.2-92 5.2-93
Now Ie2 can be found:
l e2 = k o l oα π −
k olo 2π
sin( ru) α π exp( −α u ) [cos( bu) + j sin( bu)] du u −∞
∫
∞
[
5.2-94
The (odd) j sin(bu) term integrates to zero, so
l e2 = k o l oα π −
k olo 2 2π
∫
∞
0
sin( ru) [α π exp(−αu)] cos(bu) du u
5.2-95
Because the –j sin(bu) term integrates to zero, it is clear that Ie2 = Ie3. When the trigonometric identity, sin(A) cos(B) ≡ (1/2)[sin(A + B) + sin(A B)], is used, the definite integrals become Ie2 = Ie3 = αkoIo {π – tan–1 [(r + b)/α] – tan–1 [(r – b)/α]}
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5.2-96
Substituting Equations 5.2.89 and 5.2.96 into 5.2.81, 5.2.82, and 5.2.83 yields Vimin = F exp{GKA ln {(αkoIo/B)3 [π – 2 tan–1(r/α)][π – tan–1[(r + b)/α] – tan–1[(r – b)/α]2}
5.2-97A
↓ Vimin = F exp{ ln {(αkoIo/B)3GKA [π – 2 tan–1(r/α)]GKA[π – tan–1[(r + b)/α] – tan–1[(r – b)/α]}2GKA
5.2-97B
↓ –1 GKA –1 Vimin = F (αkoIo/B)3GKA · [π – 2 tan (r/α)] · {[π – tan [(r + b)/α]
– tan–1[(r – b)/α]}2GKA
5.2-97C
and Vimax = F[(koIo απ)/B)3GKA
5.2-98
At this point in this example, it is useful to assume reasonable numerical values for the parameters, and to calculate CVi ≡ (Vimax – Vimin)/(Vimax + Vimin) numerically and compare it to CV1 for just one receptor. Let K = 3, A = 0.01, G = 100, r = 0.1°, b = 0.2°, α = 1°. The other parameters (ko, Io, F, B) cancel in calculating Cvi. CVi is found:
F(αk o l o B) π 9 − F(αk o l o B) [π − 0.4] [π − 0.4] 9
C Vi =
9
3
6
F(αk o l o B) π + F(αk o l o B) [π − 0.4] [π − 0.4] 9
9
9
3
6
=
1.3318 = 0.2877 5.2-99 4.6300
But for just one receptor:
C V1
[ ]} { = + F{( k l α B)[π − 2 tan − 1( r α )]} F( k l α π B) F( k o l o α π B)
GKA
− F ( k o l o α B) π − 2 tan − 1( r α )
GKA
o o
GKA
5.2.100
π − ( π − 0.2) 3 = 0.09834 π 3 + ( π − 0.2) 3
=
o o
GKA
3
The ratio of contrast improvement is R = CVi/CV1 = 2.93. Thus, it is clear that this simple mathematical model of the synthetic aperture/MSP system indicates that there will be a significant improvement in contrast in the hyperpolarizing response of the medullar interneuron to a moving, contrasting object.
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Example 5.2-3 As a third example consider a local cluster of seven ommatidia with one retinula cell from each ommatidium in the cluster. The seven retinula cells each synapse with an LMC, which, in turn, has an exponential electrotonic synapse with a medullary interneuron, following the theoretical architecture of the previous example (see Figure 5.2-12). Again, assume
FIGURE 5.2-12 Schematic of a seven-ommatidia cluster, synthetic aperture system. The optical axes of the ommatidia are separated from their neighbors by ρ°. The black spot diameter is d°.
S1(u) = απ exp(–α 冨u冨)
5.2-101
For each retinula cell in an ommatidium surrounding the central cell ommatidium, the radial displacement between ommatidial centers is ρ degrees. The object is a circular black spot, as before, either at x = × or centered over the 1 ommatidium at
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x = 0. The diameter of the spot is d°. When centered at x = 0, the spot has the onedimensional Fourier transform, P(u):
⎡ sin( ud 2) ⎤ P ( u ) = l o ⎢2 π δ ( u ) − d ⎥ ud 2 ⎦ ⎣
5.2-102
It is easy to see that if the spot is at x = ×, Vi = Vimax, which can be shown to be Vimax = F(koIo απ)7GKA
5.2-103
Finding Vimin is more complex; one must start by finding the effective intensities with the black spot centered over the array of seven ommatidia. The intensity at the central retinula cell is found in the manner used in the second example: Ie1 = koIo α[π – 2 tan–1(d/2α)]
5.2-104
Because of radial symmetry, the intensities at retinula cells 2 through 7 are equal, and can be found in one dimension: Iek = koIo α{π – tan–1[(d/2 + ρ)/α] – tan–1[(d/2 – ρ)/α]},
k = 2, 3, … 7
5.2-105
Now Vimin is found: Vimin = (koIo α/B)7GKA [π – 2 tan–1(d/2α)]GKA {π – tan–1[(d/2 + ρ)/α] – tan–1[(d/2 – ρ)/α]}6GKA
5.2-106
As before, the desired contrast is
C Vi =
Vi max − Vi min Vi max + Vi min
5.2.107
Considerable simplification occurs if the approximation, tan–1(y) ≅ y, y in radians, can be used. Note that the ρ/α terms cancel. Thus, Cvi can be written:
C Vi =
π 7GKA − [π − d α ]
π 7GKA + [π − d α ]
7 GKA 7 GKA
5.2-108
If α = 1°, ρ = 0.1°, d = 0.2°, GKA = 3, then CVi = 0.598 vs. CV1 = 0.0983 for a single receptor. A contrast improvement ratio of R = 6.1 is noted.
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It has been shown theoretically that, under certain conditions, synthetic aperture architecture using as few as two receptors from adjacent ommatidia can yield improved voltage contrast in the axon of the medullary interneuron that presumably drives a spiking output neuron whose function is to sense novel movement of small contrasting objects in the visual field. Note that the success of the synthetic aperture architecture in improving contrast depends on the overall light level making the retinula cell depolarization obey the logarithmic approximation Vr = A ln(Ie/B). The actual existence of such neural connections in arthropod OLs remains to be demonstrated.
5.2.6
DISCUSSION
This section has set forth two speculative mathematical models that provide a rational hypothesis to explain anomalous resolution (AR) in insect vision. AR was seen to be a statistically significant increase in firing of certain neurons in the OLs or ventral nerve cord when stripes of very fine spatial period or small black spots were moved over a CE. The limiting stripe period and spot diameter of 0.3o appears impossible to detect if only a single ommatidium and a single retinula cell in the CE are considered. Significant contrast enhancement (as the spot object or a single stripe moves from a distant location to directly over the ommatidium optical axis was shown to occur in a multiplicative signal processing model, where the depolarization outputs of all six retinula cells in an ommatidium are multiplied together. In the synthetic aperture model, the depolarization outputs of single retinula cells from adjacent ommatidia are multiplied together to achieve increased contrast. Models with two, three, and seven adjacent ommatidia were considered. Thus, the power of the array over the single visual receptor is demonstrated.
5.3
LATERAL INHIBITION IN THE EYE OF LIMULUS
Much of the early work on CE vision was done by Hartline (1949), Hartline and Ratliff (1957; 1958), Ratliff et al., 1963; 1966, and Tomita (1958) on the lateral eyes of the horseshoe crab, L. polyphemus. Limulus, a chelicercate arthropod, has two relatively simple, CEs and is an ideal subject to study the basic anatomy, organization, and neural signal processing of a primitive CE. Limulus eyes are elipsoidal bulges on the sides of the animal’s dorsal carapace, about 12 mm long by 6 mm wide. Each ommatidium has an optical aperture about 0.1 mm in diameter; the facets are spaced about 0.3 mm center-to-center on the surface of the eye. The optical axes of the ommatidia diverge from one another, so that the combined visual fields of the ommatidia view about a hemisphere (2π sterradians). A vertical section through a Limulus ommatidium is shown in Figure 5.3-1. Note the eccentric cell (E-cell); more will be said about this cell below. Note that in crosssections of ommatidia, 11 retinula cells (from 7 to 15 have been reported; Wulff and Mueller, 1975) are arranged around the center like the sections of an orange. The striated parts of the retinula cells at the center of the ommatidium are the rhabdoms, where photon trapping and light-to-depolarization transduction take
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place. Rhabdoms are composed of closely packed microvilli, at right angles to the cell surface and continuous with the cytoplasmic reticulum. The E-cell dendrite at the center of the retinula cell cluster also has a layer of microvilli arranged radially around its circumference. The microvilli (tubules) are about 140 µm in diameter (Bullock and Horridge, 1965).
FIGURE 5.3-1 Schematic drawing of a section through an ommatidium in a Limulus CE. Note the eccentric cell, which sends a dendrite into the center of the rhabdom. Pigment cells are omitted from surrounding spaces. Cell nucleus; CL, crystaline lens; XC, glassy cells, RC, retinula cells; EC, eccentric cell; Dt, dendrite; Rh, rhabdoms; LP lateral plexus.
The number of ommatidia in a CE depends on the animal and its lifestyle. Limulus has from 700 to 1000 ommatidia per eye, while a dragonfly has about 9000 to 10,000 ommatidia per eye. There is neurophysiological and anatomical evidence that Limulus CEs are very primitive; they may have evolved earlier and separately from those of insects and crustaceans (Bullock and Horridge, 1965). Directly beneath the ommatidia of each CE, Limulus has a lateral plexus of nerve fibers emanating from the approximately 11 retinula cells and the E-cell of each ommatidium. These fibers spread laterally in all directions at the base of the eye to form the lateral plexus; the lateral fibers evidently synapse in an inhibitory manner with the spiking eccentric cell of each neighboring ommatidium. Intracellular recording
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with glass micropipette electrodes revealed that the retinula cells depolarize in a logarithmic manner when subjected to a flash of light, but do not spike Figure 5.3-2 illustrates a depolarizing receptor potential from a Limulus retinula cell. Figure 5.3-3 illustrates the logarithmic response of the instantaneous spike frequency of one E-cell at ON of light to one ommatidium. The frequency log-linearity at ON is exists over about three decades of source intensity before it saturates. The E-cell spike frequency shown in curve B, taken 3.5 s after ON, shows little log-linearity. The E-cell is evidently a true neuron rather than a photoreceptor (Ratliff, 1964).
FIGURE 5.3-2 Transmembrane potential change of a Limulus retinula cell given a 0.5-second flash of light. Note sharp overshoot at ON. (From Wulff & Meuller, 1975, in The Compound Eye and Vision in Insects. G.A. Horridge, Ed., Oxford University Press, with permission.
FIGURE 5.3-3 Graph of typical instantaneous frequency (IF), r, vs. log light intensity ratio for a Limulus eccentric cell. Open circles: the IF at ON; dark circles: the IF 4 s after ON. A single ommatidium was illuminated.
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5.3.1
EVIDENCE
FOR
LATERAL INHIBITION
Early in the study of Limulus CE, it was discovered that when the eye was stimulated with beams of light that could be focused on individual ommatidia, the firing frequency of a given E-cell axon in the optic nerve could be reduced by focusing a second beam of light on a neighboring ommatidium. The relation between the firing frequencies of the two separate, stimulated E-cells was found to be described by a pair of simultaneous, piecewise-linear algebraic equations (Ratliff et al., 1963): r1 = e1 – K12(r2 – ϕ12)
5.3-1A
r2 = e2 – K21(r1 – ϕ21)
5.3-1B
where r1 and r2 are the steady-state firing frequencies of the E-cell axons from the two stimulated ommatidia; e1 and e2 are the frequencies at which the E-cells would fire if stimulated one at a time (the other ommatidium being dark); K12 and K21 are the reciprocal inhibition constants; ϕ12 and ϕ21 are threshold firing rates, above which inhibition takes place. For example, if (r2 – ϕ12) < 0, then K12 = 0, etc. Also, if [e1 – K12(r2 – ϕ12)] < 0, r1 = 0, etc. (Clearly, there are no negative frequencies in nature.) The process described by the two equations above was called lateral inhibition (LI), because the lateral plexus evidently carried information about retinula cell depolarization from a stimulated ommatidium to its neighbors, causing a reduction in their firing rates. Figure 5.3-4 shows a systems block diagram describing the two LI equations above. Note that the LI dyad is an overall, positive feedback system when operating in its linear range. Neglecting the dc offsets, the dc loop gain of the system is: AL = + K12 K21.
FIGURE 5.3-4 Systems block diagram describing the two lateral inhibition (LI) equations (5.3-1A and B) from Ratliff, et al., 1963.
The linearity of the dyadic model was demonstrated by Ratliff (1964). Ratliff et al., (1963) were able to extend the dyadic relation above to a more general N ommatidium model:
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N
rk = e k −
∑ K (r − ϕ ), kj
j
kj
k = 1, 2, … N
5.3-2
j=1 j≠ k
The same restrictions on negative frequency and self-inhibition (Kjj = 0) apply as in the dyadic case. Ratliff et al. (1963) observed that the Kkj decreases as the physical distance between the kth and jth ommatidium increases. This decrease is reasonable both from a functional and anatomical point of view. The terminal arborizations of nerve fibers branch as they become more distant from their source; thus their density and their effectiveness in causing inhibition of E-cell firing decreases with the radial distance from their source ommatidium. Figure 5.3-5 illustrates schematically the instantaneous frequencies of three optic nerve fibers corresponding to the E-cells in three separate ommatidia stimulated individually with ON/OFF illumination (N = 3). From 0 ð t ð t1, I3 is ON and I1 and !2 are OFF; r1 = r2 = 0 and r3 = r3max = e3. At t1, I1 goes ON. Note the decrement in r3max caused by r1 > 0, also the overshoot in r1 and the corresponding undershoot in r3. When I2 comes ON at t2, there are corresponding decrements in both r1 and r3, again with undershoots caused by the overshoot in r2. Note that for t > t4, I1 = I3 = 0, and r2 → r2max = e2.
5.3.2
MODELING LATERAL INHIBITION OBJECTS
AS A
SPATIAL FILTER
FOR
This section derives a linear model for the spatial filtering that occurs in lateral LI. First consider the surface of a Limulus CE as an oval, defining an area in x,y angle space. Assume that each ommatidium is represented by a differential area (solid angle) of ∆x∆y located at j∆x, k∆y. The frequency of its E-cell will be affected, in general, by the frequency of a spiking E-cell located at p∆x, q∆y. To simplify interpretation, set the firing thresholds, φ → 0. Thus, by superposition, N
rk = e k −
∑ K (r − ϕ ), kj
j
kj
k = 1, 2, … N
j=1 j≠ k
5.3-3 or, more simply,
r( j, k ) = e( j, k ) −
N
N
∑ ∑ K( j − p, k − q) r(p, q)
5.3-4
p =1 q =1 p≠ j q ≠ k
Note that K is a function of the distances in x, y space between the ommatidium being examined and the affecting (input) ommatidium at p, q.
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FIGURE 5.3-5 (A) Diagram of three mutually inhibited eccentric cells from three separate ommatidia. The LI follows the form of Equation 5.3-2. (B) Light input stimuli vs. time for the three ommatidia. (C) Approximate instantaneous frequency vs. time for the three stimulated eccentric cells. Note that stimulation of adjacent ommatidia reduces the firing rate of the stimulated eccentric cell being recorded from.
Rather than work with a two-dimensional, discrete model using two-dimensional z-transforms, the properties of a continuous, linearized, one-dimensional, LI model using Fourier transforms will be examined. The discrete, one-dimensional, linearized LI model is
r ( x ) = e( x ) –
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∫
∞
K ( x – σ ) r ( σ ) dσ
–∞
5.3-5
The summation term is interpreted as a one-dimensional discrete convolution of the inhibition weighting function, K(j∆x) with the one-dimensional distribution of frequencies, r(j∆x). To make the model continuous, assume that N → ×, ∆x → 0, and j∆x → x, and p∆x → σ in Equation 5.3-5. Thus, the discrete, finite LI model becomes ∞
∫
r ( x ) = e( x ) –
K ( x – σ ) r ( σ ) dσ
5.3-6
–∞
When Equation 5.3-6 is Fourier transformed, R(u) = E(u) – K(u)R(u)
5.3-7
u is the spatial frequency in rad/mm. Collecting the terms, the transfer function is
R( u ) 1 = = L( u ) E( u ) 1 + K( u )
5.3-8
Example 5.5-1 As a first example of calculating the effects of LI predicted by the model, assume k(x) = ko exp(–α 冨x冨). (As noted above, experimental workers have found that K decreases with the interommatidial distance parameter, x.) Thus, K(u) = 2 α ko/(u2 + α2), and from Equation 5.3-8, L(u) = (u2 + α2)/(u2 + 2αko + α2). This L(u) attenuates the E-cell rate of firing, r(x), for low spatial frequencies; thus, for u <
2αk o + α 2 ,
L = α2/(2αko + α2) < 1, and at high u, L = 1. To examine the behavior of this model further, let the input be a general dc level of illumination, B, plus a bright spot over x = 0. That is, e(x) = B + Aδ(x). Fourier transforming yields
E( u ) =
∫
∞
B exp( − jux) dx +
−∞
∫
∞
A δ(x) exp( − jux) dx = 2 π B δ( u) + A
5.3-9
−∞
Now one can use the inverse Fourier transform to find r(x). Clearly, R(u) = L(u) E(u).
r(x) =
1 2π
⎡ u2 + α2 ⎤
∞
∫−∞[2π B δ(u) + A] ⎢ u 2 + γ 2 ⎥ exp(+ jux) du, ⎣
⎦
γ 2 ≡ 2α k o + α 2 5.3-10
The first integral is simply r1(x) = B α2/γ2. The second integral can be broken into two parts:
r2 (x) =
A 2π
⎡ α2 − γ 2 ⎤ ⎢1 + u 2 + γ 2 ⎥ exp( + jux) du −∞ ⎣ ⎦
∫
∞
[ (
= A δ( x ) + A α 2 − γ 2
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) (2γ )] exp(− γ x )
5.3-11
Note that (α2 – γ2) = –2αko. Thus, r(x) = B α2/γ2 + A δ(x) – [A αko/γ]) exp(–γ冨x冨)
5.3-12
Figure 5.3-6 illustrates the general form of this r(x). Clearly, r(x), because it is a frequency, is non-negative. For this to occur, it is possible to show that B must satisfy B > A ko 1 + 2 k o α . Note that the effect of the exponential, spatial distribution of LI is to suppress the frequencies r(x) in the vicinity of the origin with an exponential shape. Curiously, the space constant of the exponential is γ rather than α.
FIGURE 5.3-6 (A) Stimulus for a one-dimensional, linear, continuous LI system. A bright point source of intensity A is superimposed on a background intensity of B. An exponential falloff of inhibition is assumed [k(x)]. (B) Calculated output firing frequency as a function of x for the one-dimensional LI system. Note that the LI supresses the output instantaneous frequency in an exponentially weighted region around the origin.
Example 5.5-2 As a second example, assume that the object is such that e(x) = B + AU(x) (overall dc illumination B plus a step of height A for x Š 0). The Fourier transform of e(x) is E(u) = 2π B δ(u) + A[πδ(u) + 1/ju]
5.3-13
The k(x) = ko exp(–α冨x冨), as in the first example. Thus, r(x) is found:
r(x) =
1 2π
1 + 2π
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∞
⎡ u2 + α2 ⎤
∫−∞[2πB δ(u)] ⎢ u 2 + γ 2 ⎥ exp(+ jux) du ⎣
⎦
⎡ u2 + α2 ⎤ Aπ δ( u) + A ju] ⎢ 2 exp( + jux) du [ 2 ⎥ −∞ ⎣u + γ ⎦
∫
∞
5.3-14
These integrals are solved using conventional techniques. The result can be written
⎤ ⎡ r(x) = B α 2 γ 2 + ⎢A α 2 γ 2 + Ak o α γ 2 exp( − γx)⎥ U(x) ⎥⎦ ⎢⎣ ( all x ) ( x≥0 )
(
)
(
) (
)
5.3-15
⎤ ⎡ − ⎢ Ak o α γ 2 exp( + γx)⎥ U(x) ⎥⎦ ⎢⎣ ( x≤0 )
(
)
A dimensioned graph of Equation 5.5-15 is shown in Figure 5.3-7. The peak r(x) occurs at x = 0+: r(0+) = (α2/γ2)(A + B + Ako/α). The minimum r(x) occurs at x = 0–: r(0–) = (α2/γ2)(B – Ako/α). In order for r(0–) > 0, it is clear that B > Ako/α. The shape of r(x) suggests that LI performs an edge enhancement function, as well as boosting high spatial frequencies. Just to the left of the light step, the resultant frequencies are reduced, making the dark edge of the step appear darker. Just to the right of the step, the resultant frequencies are raised, making the light edge appear lighter, accentuating the edge.
FIGURE 5.3-7 (A) Stimulus for a one-dimensional, linear, continuous LI system. A uniform object intensity of B is applied for –× ð x < 0, and a uniform object intensity of A > B is applied for 0 ð x ð ×. (B) Calculated output firing frequency as a function of x for the onedimensional LI system. The output IF is higher just to the right of the origin, and lower just to its left. If output IF codes perceived intensity, then the perceived boundary will be enhanced, a darker band lying to its left, and a lighter band to its right. Such boundary enhancement is a direct result of LI.
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Example 5.5-3 As a third and final example of the one-dimensional model for LI, consider a rectangular K(x) = ko for 冨x冨 ð xo/2, and K(x) = 0 for 冨x冨 Š xo/2. It is easily shown that K(u) = koxo sin(uxo/2)/(uxo/2). Thus, L(u) = 1/[1 + K(u)] can be unstable (i.e., L(u) → ×) if its first (negative) minimum → –1. To see at what value of its argument the first minimum of sin(x)/x occurs, one differentiates it and sets the derivative to zero. 1.5
1
0.5
0 0
10
20
30
>axes h 0 30 > simu -10 30 >
FIGURE 5.3-8 The calculated spatial frequency response, L(u), of the linear, continuous, one-dimensional model of LI when the inhibition weighting function, k(x), is rectangular with width ±xo. Parameters: xo = 2, ko = 0.5, L(u) = 1/[1 + sin(u)/u]. See text for details.
x cos(x) − sin(x) d =0 sin (x) x] = [ dx x2
5.3-16
From this relation, it is evident that the argument for the first minimum occurs when x = tan(x). Solution of x = tan(x) occurs for x = 4.49341 rad, or 257.45°. Substituting this value back into sin(x)/x gives the first minimum = –0.217234. So ko < 1.6116/xo for LI system stability. The first peak in the frequency response, L(u), occurs for uxo/2 = 4.49341, or u = 8.9868/xo. Figure 5.3-8 illustrates the frequency response, L(u), of the LI system. Note that the ripples die out to a steady-state, high-frequency level of L = 1, and L(0) = 1/(1 + koxo). There is much evidence that LI occurs in certain vertebrate visual systems. Neurophysiological evidence taken from the ganglion cells (optic nerve fibers) of the eyes of cats, frogs, pigeons, etc. indicates that basic high-pass spatial filtering takes place in vertebrate retinas before high-frequency cutoff. As has been seen, spatial high-pass filtering enhances contrasting edges. Other “operations” on visual information, called feature extraction, also take place at the retinal level in vertebrate eyes (Lettvin et al., 1959), as well as in insect OLs. Feature extraction is described in detail in Section 5.4. Indirect evidence for LI can also be taken from psychophysical studies of human visual perception. Our visual system definitely sharpens
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contrasting boundaries. For example, stare at a sheet of paper, half-white, half-black. At the white/black boundary, people perceive the white whiter, and the black blacker. In some cases, it is even possible to see one or two bands at the boundary, illustrating an underdamped spatial frequency response of L(u).
5.3.3
DISCUSSION
LI was first described in the CEs of the horseshoe crab Limulus polyphemus by Hartline (1949). Subsequent neurophysiological studies and mathematical models revealed that LI had the overall effect on the spatial frequency response of the CE of boosting high spatial frequencies over low, before cutoff. This was shown in terms of image processing to accentuate image boundaries and edges. This section has examined the anatomy of Limulus CEs, and has mathematically derived an expression for an equivalent LI spatial high-boost filter in terms of the LI spread function. (The LI spread function describes how the effective inhibition of one retinula cell decreases with distance to neighboring ommatidia.) Several psychophysical visual responses in human visual experience can be attributed to LI in human vision.
5.4
FEATURE EXTRACTION BY THE COMPOUND EYE SYSTEM
It has been observed that certain qualitative and quantitative features of visual objects presented to both arthropod and vertebrate visual systems give rise to neural signals that are specific for those features. Features in this context are simple, contrasting, geometric properties of the object, which can be moving or stationary. That is, the feature of an object can involve shape, contrast, and a velocity vector. For example, a type of neuron in the medulla of the OL of the lubber grasshopper will significantly increase its firing rate if a long black stripe is moved from front to back over the ipsilateral CE. Motion from back to front silences the unit, and a black or white spot given the same motion has a negligible effect on the random, background firing of the neuron. Also, motion of the stripe at right angles to its preferred direction produces little response. Such units that respond only to long stripes moving in a preferred direction have been called “vector edge units” (Northrop, 1974). Prewired feature extraction operations offer neural economy in simple nervous systems. It will be seen that the features extracted have survival value for the animal, particularly in the detection of prey, avoidance of predators, stabilization of flight, and perhaps even in the location of a mate. Examples of a visual neural array are the retina in the case of vertebrate eyes, and the OLs of a CE, including the lamina ganglionaris, the medulla, and the lobula when arthropod eyes are considered. Both retinas and OLs are highly organized neural networks dedicated to visual data preprocessing and feature extraction.. It is supposed that most of the visual feature extraction used by arthropods occurs in their OLs; the OLs probably serve most of the functions of the vertebrate retina, tectum, and visual cortex. A number of well-defined nerve tracts carry information
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to and from the OLs. Bullock and Horridge (1965) describe such tracts in the bee and locust. For example: 1. There is a tract of fibers leaving the back of the lamina ganglionaris that runs tangential to it toward the front of the lobe, thence to a primary optic association center in the dorsal posterior part of the protocerebrum. 2. Two tracts arise in the medulla; one leaves from the dorsal edge, and the other from the ventral edge. Both tracts run to the calyces of the corpora pedunculata. 3. The anterior optic tract runs from the lobula to the optic tubercle on the anterior brain, thence to the ipsilateral corpora pedunculata. 4. The superior medial optic tracts are short and long tracts from the lobula to the α and β lobes of the corpora pedunculata. 5. The inferior medial optical tract has fibers running from the lobula to the ventral center of the brain. 6. Decussating tracts between left and right OLs include two tracts between dorsal and ventral medullas, and one between left and right lobulas. 7. Giant descending neurons from the lobula project into giant ventral nerve cord fibers. (These are the DCMD and DIMD VNC neurons described by Rowell, et al., 1977.) It is apparent that the arthropod OLs are “well-connected” to the other components of the animal’s CNS. Thus, it is not surprising to find signals in OL neurons that respond to visual stimuli, as well as to other sensory modalities (touch, sound, airflow). These are the multimodal units described in Locusta by Horridge et al. (1965), in moths by Blest and Collett (1965), and in Romalea by Northrop and Guignon (1970). Other visual units (N3 vector units) described by Northrop (1974) responded to moving edge visual stimuli and stimulation of aerodynamic hair patches (wind velocity sensors) on the head. More will be said about these units below. Early workers examining CE, OL unit responses explored a variety of arthropod visual systems. Burtt and Catton (1960), Ishikawa (1962), Schiff (1963) and Dingle and Fox (1966) examined single unit OL unit responses to simple changes in illumination in locusts, silkworm moths, mantis shrimp, and crickets, respectively. Not surprisingly, units were found responding to ON, OFF, and ON/OFF of general illumination, as well as sustaining units that fired steadily in steady-state light or dark. Because an eye operates in a complex visual environment where many object features can be found (spatial frequency, contrast, color, relative velocity, etc.), some workers have explored how visual systems respond to certain object features, as well as to changes in overall and local illumination. Both approaches, i.e., the study of responses to simple changes in illumination, and of responses to certain object features are necessary to provide a complete understanding of how the CE visual system works. The seminal work on visual feature extraction was done by Lettvin et al. (1959) on the frog retina. What was new in their work were three important findings: (1) Object size (hence spatial frequency) was a factor in determining the strength of response of some ganglion cells. (2) Object motion, and direction of motion were
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also response factors for some ganglion cell units. (3) Both size and motion were factors, as well. Other workers (see Section 6.3), have extended their approach to a variety of other vertebrate retinas and CNSs. Horridge et al. (1965) were the first workers to explore systematically feature extraction in the CE system of Locusta, in which they ambitiously classified some 20 types of OL units. The feature extraction approach has also been applied to flies (diptera) by Mimura (1970; 1971; 1972; 1974), McCann and Dill (1969), Bishop et al. (1968), and DeVoe (1980), and to lepidoptera (moths and butterflies) by Swihart (1968) and Collett (1971). Wiersma and colleagues (254, 257) have also examined the CE visual system of crabs and other crustaceans from a feature extraction point of view. These works will be discussed below. By combining the properties of responses to changes in illumination with the detailed responses to moving objects, several of the workers mentioned above have identified large numbers of units classes (20 for Horridge et al., 1965 in Locusta; 12 OL classes by McCann and Dill, 1969, in flies). The author, working with the grasshopper Romalea microptera, adopted a more parsimonious approach and designated eight OL units with distinct properties based on form and motion, as well as ON and OFF responses These are described in detail below.
5.4.1
FEATURE EXTRACTION
BY
OPTIC LOBE UNITS
OF
ROMALEA
The grasshopper Romalea microptera is a large, flightless insect native to the Everglades of Florida and other swamps and marshes in the southeast U.S. It is a pleasure to work with because it is large (adult females can be 7.5 to 10 cm in length), and consequently its head, CEs, and OLs are proportionally large. Inspired by the work of Lettvin et al. (1959) on the frog’s retina and Horridge et al. (1965) on the locust’s OLs, Northrop and Guignon (1970) and, later, Northrop (1974) examined single units from the OLs of Romalea for feature extraction operations on simple, contrasting visual objects, moving and stationary, presented on the inside surface of a hemisphere concentric with the eye under test. Hemispheres 50 and 28 cm in diameter were used. The insides of the hemispheres, viewed by the insect, were painted light gray so either white or black test objects could be used. Black and white disk and square objects were 14, 28, and 56 mm in diameter, or on a side; these subtended 3.2, 6.4, and 12.8° on the 50-cmdiameter hemisphere, and 5.7, 14.5, and 22.9° on the 28-cm hemisphere, respectively. Spot objects were moved manually by magnets from outside the hemispheres, or by a Ledex rotary solenoid outside the large hemisphere. A flat spring arm was connected to the solenoid; a magnet was glued to the end of the arm, which moved a spot in an arc inside the hemisphere. One 28-cm hemisphere was constructed like a giant ice cream scoop; it had a 2.5 cm wide (10.2°), flat band pivoted at the rim by two bearings. The band was moved manually; its angle could be read on a protractor, and also sensed electrically by a potentiometer that moved with the band. The band could be made black or white, and its width could be made greater than 10.2°. Objects inside the hemispheres were illuminated by a 6.3 V dc, 24 W, highintensity tungsten lamp. Illumination was nearly uniform on the object surface.
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A modified ophthalmoscope was used to project a microbeam of intense white light directly on the eye’s surface; as few as seven ommatidia could be illuminated at a time with this device to test for simple, local ON and OFF responses Neurophysiological recordings from OL units were extracellular, made with electrolytically sharpened Pt-30% Ir microelectrodes. About 3 to 5 µm of Pt-blackcoated tip was exposed; the rest of the electrode shank was glass-insulated. Conventional low-noise amplification was used to condition spike signals. Northrop and Guignon (1970) approached the problem of unit classification from a parsimonious point of view. They were primarily interested in finding if the insect visual system could extract object size information (i.e., responded selectively to objects containing a restricted range of spatial frequencies), as well as to the direction of moving objects. Unit responses to simple changes in general and local illumination were not considered to be as important as any potential, spatiotemporal filtering operations. On the basis of tests used, Northrop and Guignon (1970) were able to claim five OL unit classes, described below: 1.
Multimodal Units
MM units were the most frequently encountered unit class (46% of the 181 units studied). These neurons fired erratically in both steady-state light (LSS) and dark (DSS) in the absence of a moving object, their average rate slightly higher in LSS. There was always an MM burst at ON and at OFF of general illumination. The receptive field of a MM unit was the entire ipsilateral eye. Maximum response was elicited by small, jerky object motions in random directions. MM units responded maximally to motions of a black disk, square, or triangle from 3° to 5° in diameter, although good responses could be obtained from white test objects of the same size range. Continuous, smooth motion of a spot in one direction did not give a sustained response. If a spot were jittered in a restricted region of the hemisphere, an MM unit would quickly adapt; if the same spot were moved to an area viewed by an unstimulated area of the eye, the MM response would again return, then adapt. The moving stripe of the “ice cream scoop” object would cause an MM unit burst only at the onset of motion. There was no motion directional sensitivity for any object to which an MM unit responded. MM units behaved similarly in their responses to visual objects as the descending contralateral movement detector (DCMD) neuron axons in the ventral nerve cord. However, MM units were unique in that they could be caused to fire faster when the animal was given tactile stimuli (e.g., stroke the legs and abdomen with a paintbrush or fine wire), or acoustic stimuli (e.g., loud clicks). Tactile and acoustic stimuli adapted far more slowly than responses to jittered spots. Mechanical and visual stimuli given close in time generally revealed no particular response potentiation of one vs. the other. Northrop and Guignon (1970) used radio-frequency lesioning to estimate the recording sites of OL units. Of 30 MM units lesioned, 8 were found on the proximal, distal, and lateral surfaces of the medulla, 3 were in the outer chiasma (between the lamina and the medulla), 2 were in the lamina, 5 were in the volume of the medulla,
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7 were in the lobula, and 5 were in the inner chiasma (between the medulla and lobula). Receptive fields of the MM unit were both CEs (ipsilateral and contralateral). MM units were also found in Locusta by Horridge et al. (1965) (type CE) and in moths by Blest and Collett (1966). Over 30 years have past, and the role of these ubiquitous units is still not known. A good guess is arousal, a sort of sensitivity control triggered by high-spatial-frequency object’s new movement anywhere in the animal’s visual field, or by a novel sound or touch. 2.
Directionally Sensitive, Contrasting Edge Detectors
Also called “vector edge” (VE) units by Northrop and Guignon, these units fired at the same average rate in LSS and DSS. However, the firing in DSS was more regular, less random. They constituted 7% of the 181 OL units studied. They gave a sharp burst at OFF of general illumination, followed by a long period of reduced firing before the unit regained its DSS rate. There was no burst at ON. The receptive field of the VE units appeared to be the entire ipsilateral eye, with the posterior half generally more sensitive. VE units were not responsive to mechanical or acoustic stimuli. VE units were caused to fire above their base rate by the motion of a contrasting, linear (black/white) boundary in the preferred direction. The preferred direction for Romalea VE units was generally distributed around the anterior + dorsal direction, that is, forward and up. Object velocity in the opposite, or null, direction would suppress the LSS firing rate under certain conditions. If a single, long, black/white boundary was used as the test object, there would be no firing rate suppression when the motion was in the null direction if the white field was anterior. Suppression occurred when the black field was anterior. VE units gave little or no response to small contrasting spots moved in the preferred direction; the object must have a linear contrasting boundary subtending at least 30°, perpendicular to the velocity vector. VE units respond over a very wide range of velocities in the preferred direction. Response drops off as the cosine of the angle between the preferred direction and the velocity vector of the object. When a multiple striped object was used (square-wave pattern), VE unit response to continuous motion in the preferred direction adapts. The suppression effect for continuous motion in the null direction also adapts. VE units are extremely sensitive to small incremental movements of ð0.33° in the preferred direction, giving a large burst for such incremental motion. VE unit responses fall off for square-wave striped patterns with periods less than ~4°. Clearly, there appears to be a trade-off between spatial resolution (object size) and incremental motion sensing. Northrop (1974) reported having found another class of directionally sensitive units giving maximum response to small, contrasting spots rather than to long contrasting boundaries. These vector spot (VS) units responded similarly to the VE units except they showed a burst at ON followed by a relatively long period of suppressed firing. The VS units firing was suppressed by spot motion in the null direction. VS units showed little adaptation to continuous spot motion. They also responded to stripes moving in the preferred direction. Lesioning of VS units showed that recording sites were in the lobula or in the tracts of large fibers running from the lobula to the protocerebrum.
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Romalea VE units are probably the same as the type BG units described by Horridge et al. (1965) in the OLs of Locusta. Many other workers have identified directionally sensitive units in insect eyes. Collett and Blest (1966) found VE units in the hawk moth OLs. Swihart (1968) reported directionally-sensitive (DS) units in the butterfly Heliconius. Eight subtypes of DS units were specified for flies by McCann and Dill (1968); more will be said on fly units below. DeVoe et al. (1982) has reported on DS units in the OLs of bees. 3.
Light Units
Light units (LU) constituted 37% of the 181 units studied. They had monocular RFs, about 20° to 40° in diameter. These are ipsilateral units, insensitive to patterns or moving spots, except that moving a large white spot over an LU RF would cause an increase in firing rate. Similarly, moving a large black spot over the RF would suppress firing. LUs can be subdivided into two subclasses: tonic LUs and phasic or ON-type LUs. The firing rate of tonic LUs increased smoothly at ON; there was no burst. ON-type LUs gave a burst at ON then slowed to a steady, lower rate under LSS conditions. A few of the LUs exhibited an ON-gated mechanoresponse. Such an LU would fire a burst in response to stroking a leg with a wire only immediately following its ON burst. Stroking the leg in the same manner in the dark, or several seconds following ON, produced no extra burst in this curious type of LU. In several instances, the microelectrode would pick up both a tonic and a phasic LU at the same time, indicating that their axons ran in intimate contact. Horridge et al. (1965) called Locusta tonic LUs type AD; phasic LUs were called type AC and BC (the BCs had small RFs, down to 7°). 4.
Net Dimming Units
Northrop and Guignon (1970) only found three dimming units (DUs) out of 181 units classified. They were monocular, ipsilateral units with RFs evidently involving the whole eye. As in the case of LUs, DUs have two subtypes: Those that give phasic responses to OFF, and those that give tonic responses. A tonic DU would fire slowly in LSS, then accelerate its rate smoothly at OFF. All DUs fired regularly in DSS. A phasic DU bursts at OFF, then slows to a steady DSS rate. Horridge et al. (1965) called the tonic DUs in Locusta type AF; phasic DUs were called type BD. Horridge et al. were able to measure the RFs of type BD at about 15° to 25°. 5.
Unmodulatable and Weakly Sensitive Units
This class of unit comprised 3.9% of the 181 unit total. These units were very perplexing because their steady firing appeared unchanged by ON and OFF of general illumination, spot and stripe motion, sound and mechanical stimulation. Some of these units could be “aroused” by repeated mechanostimulation, such as stroking the animal’s feet with a paint brush. An “aroused” WSU would develop a very weak response to light in the manner of a tonic LU. No one had any idea what role these units played in the animal’s visual responses.
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6.
Anomalous Units
Some 4.5% of the units examined had visual and mechanical responses that did not fit any of the five consistent categories above. Because this sample population was small and was diversified, Northrop and Guignon felt conditions did not warrant the establishment of new unit categories. If 1810 units had been studied instead if 181, these workers might have had more confidence to generate new unit response categories.
5.4.2
FEATURE EXTRACTION
BY
OPTIC LOBE UNITS
IN
FLIES
Much basic early work (anatomy, neurophysiology, optomotor studies) was been done on the visual systems of flies by DeVoe and Ockleford (1976), DeVoe (1980), Mimura (1970; 1972), Bishop et al. (1968), etc. As will be seen, rapidly flying insects face special challenges to their visual systems. In addition to having to avoid obstacles, a flying insect faces the additional burden of having to provide its CNS with visually driven, flight stabilization information in order that the flight motor system can compensate for yaw, pitch, and roll. These are extremely complex operations to understand in terms of muscle forces and instantaneous aerodynamic forces and torques. It stands to reason that the more agile the flier, the more sophisticated will be the control. Thus, one should not expect sessile or crawling insects to have the visual motion-sensing sophistication that can be found in agile fliers such as dragonflies, moths, and flies. Several classes of flying insects have additional, nonvisual aids for flight stabilization. For example, the diptera (flies, mosquitos, etc.) have vibrating gyroscopes called halteres (see Section 2.7) to sense roll and yaw, and locusts have hair patches on their heads that sense wind velocity; their antennae have the same role and may sense yaw and roll from differential wind forces (Gewecke, 1970). Flying insects also face the problem of separating the visual background motion due to flight from the (relative) motion of other flying insects. Wide-field DS units have been found in fly OLs that have a rear-to-front (anteriad) preferred direction (PD); others have a posteriad PD. A wide-field DS unit responds well to stripes, as well as to spots and square-wave gratings. When an object is moved in its RF in the PD, it fires at a rate given approximately by r(v) = k log(1 + v/vo), where k and vo are positive constants and v is the speed of the object (Collett and King, 1974). From an engineering viewpoint, it is easy to see how a flying fly might use the difference between the firing rates of left and right DS units with posteriad PDs to correct for yaw in flight (assume stationary, vertical contrasting objects on both sides of the fly’s flight path). Anteriad DS units could also be stimulated if a flying fly, for example, turns sharply to the left. The posteriad DS units of the right eye would be stimulated strongly, but if the object of the left eye has a relative, forward tangential velocity that exceeds the forward flight velocity, the object will appear to move anteriad, and stimulate the left eye DS units with anteriad PDs. (See Collett and King, 1974, for an excellent review of the problem of directional vision during flight.) Note that DS units with anteriad PDs would be stimulated if the fly is sitting, and some object, e.g., another fly, flies past it from the rear.
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Collett and King (1974) have recorded from the so-called small-field DS (SFDS) units in the OLs of the hoverfly with microelectrodes located in the external chiasma, the medulla, and the lobula. Mimura (1974a) reported that SFDS units were most commonly found in the region between the medulla and the lobula (inner chiasma). The SFDS unit RF was ipsilateral, contralateral, or bilateral to the recording site. The RFs were ~20° in diameter. SFDS units gave little or no response at ON and OFF of general illumination. Many but not all SFDS units were directionally sensitive with posteriad-moving object preferred directions. SFDS units were not spontaneously active; they responded best to high-spatial-frequency objects (2 to 4° diameter black spots) and not at all to moving long bars, gratings, or single contrasting edges. A curious property of the SFDS units was that their optimum spot object size was velocity dependent. For example, a 2.5° diameter black spot moving in the preferred direction (posteriad) at 70°/s gave a strong response, but a 7° diameter spot moving at the same velocity in the PD gave little response. When the 7° spot was moved at 430°/s in the PD, it gave a strong burst while in the RF; the 2.5° spot given the same motion gave no response. Collett and King (1974) found in the hoverfly that the surround of an SFDS unit strongly inhibited the center response to a moving spot when it was stimulated by a complex pattern moving at a lower or equal velocity than the spot, in the spot PD. Thus, it is probable that most of the SFDS units would be “turned off” (inhibited by background motion) during flight. Mimura (1970) reported the finding of LUs in the fly Boettscherisca peregrina that gave complex directional responses. One such unit studied responded to a 1.15° diameter light spot moved on the left front of the fly in a plane perpendicular to the center (flight) axis of the fly. (See Figure 5.4-1.) The unit gave a positive response to a linear spot motion moving to the left in the fly’s upper left quadrant, also to a linear downward motion in the left upper and lower quadrants, and to a left to right linear motion in the lower left quadrant. It was evident that the unit responded maximally to a counterclockwise rotation of the spot in the left hemifield. Another LU responded to counterclockwise light spot rotation over the entire frontal field; a clockwise rotation unit was also found. In the same paper, Mimura reports having found the more common, linear DS units, and units that responded similarly to the LGMD/DCMD units in grasshoppers. No control was made to find the exact recording sites. It is tempting to speculate that the “feature” extracted by Mimura’s rotation units is in fact roll (while the animal is flying), and that such units could have direct inputs to the fly’s flight stabilization control system. A comprehensive study of fly OL units was reported by Bishop et al. (1968) in which they classified units according to their response properties. Nine classes (including subclasses) of OL units were described, as well as three subclasses of visual unit recorded in the brain corpora pedunculata region. Bishop et al. worked with the flies Calliphora phaenicia and Musca domestica. Metal extracellular microelectrodes were used. Their Class I unit was nondirectionally sensitive. It was found in the anterior medulla tracts; the RF was ipsilateral and relatively small for a CE, 15 to 40° in diameter. Class I units responded to:
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FIGURE 5.4-1 Schematic of a fly viewing a light spot rotated counterclockwise on its left side. Mimura (1970) found a lobula unit giving such a complex response. See Section 5.4.2 for discussion. “patterns fixed in space with transient illumination, or to moving patterns. Their response to moving patterns appeared independent of the direction of motion.”
Information was not given whether the response to transient illumination was at ON or OFF, or both. Also, no tests for object size and contrast optimality or adaptation to object motion were reported. The Class II units were true directionally sensitive units, recorded from the lobula–lobula plate region. Class II units responded to ON of general illumination with an ON burst; there was a weak or no OFF burst. In the SSD, Class II units fired randomly at 1 to 3 spikes/s. In SSL with no pattern motion, the firing rate increased to 5 to 20 spikes/s. Stimuli were circular spots with contrasting stripes appearing within them. The stripes could be moved within the stationary spot, or the spot with fixed stripes could be moved as a whole. Spot diameter, d, and stripe period, λ, could be varied independently. Stripes were always moved perpendicular to their long dimension. For spots about 22° in diameter and less, the measured DS response fell off as the cosine of the angle α between the PD and the spot/stripe vector (up to ±90°). The Class IIa group had four subclasses, dependent on the PD: Class IIa units had contralateral, full-eye receptive fields. Type IIa1 had an approximately horizontal PD directed toward the anterior of the animal (anteriad), the type IIa2 PD was vertical downward (ventrad), the type IIa3 PD was horizontal toward the posterior (posteriad), and the type IIa4 PD was vertical upward (dorsad). Class IIb units had ipsilateral, full-eye RFs. They had horizontal PDs; the type IIb1 PD was anteriad, and the type IIb2 PD was posteriad. Class III units were found in the brain corpora pedunculata region. Class IIIa units had monocular contralateral RFs, and had similar properties to Class IIa; Class IIIb units had monocular ipsilateral RFs, and had properties similar to Class IIb. Class IIIc units were binocular with contralateral response dominance. Their firing appeared to be the summations of paired combinations of Class IIa and b unit responses. DeVoe and Ockleford (1976) recorded intracellulary from single units in the optic medulla of the fly Calliphora erythrocephala. Because intracellular recording was used, slow potential shifts could be seen as well as spikes when units fired.
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Thus, more pieces of the structure/function puzzle were available to gain understanding of how the various types of DS units in the dipteran visual system work. In addition to several types of DS units, DeVoe and Ockelford found a new type of unit that they named the change of direction (CoD) unit. The CoD unit fired as long as the test object was moving, regardless of its direction. When the object changed direction, there was a pause in the firing rate of the CoD unit. Firing resumed when object speed resumed in the null direction. Unlike grasshopper multimodal units (Northrop, 1974), and LGMD units (O’Shea and Rowell, 1976), CoD units did not adapt. The purpose of the CoD unit is unknown.
5.4.3
EYE MOVEMENTS
AND
VISUAL TRACKING
IN
FLIES
The preceding section demonstrated that the visual systems of flies are richly endowed with at least two types of DS visual neuron, and several preferred directions. It is reasonable to assume that these neurons carry information to the complex motor centers that stabilize flight. In the visual systems of other arthropods, e.g., crustaceans, which have movable eyes on eyestalks, visual DS units control eye movements (Burrows and Horridge, 1968; Horridge and Burrows, 1968). Insect eyes are generally considered to be fixed to their heads, and only move when the head moves, so that rapid scanning or fixation saccades are not possible. The exception to the fixed eye occurs in the eyes of flies and mosquitoes (the Diptera). Figure 5.4-2 shows a schematic of a horizontal section through the middle of a fly’s head; the section passing through the eye’s “equator.” A thin muscle (M. orbito-tentoralis, or MOT) has its origin at the heavy chitin of the tentorium (TT) at the rear of the head. The tentorium lies in line with the mass of the optic ganglia and the center of the CE. The muscle inserts on the inner margin of the orbital ridge, an elastic ring that surrounds the base of the retinular structure and gives it support. The muscle has 14 to 20 tubular fibers, each 7 to 10 µm in diameter, innervated by multiple motor end plates from a single motor nerve fiber, about 6 µm in diameter (Hengtstenberg, 1972). There is one muscle and nerve for each eye; i.e., it is a paired structure. The nerves arise from the sides of the subesophageal ganglion. The mechanics of the fly’s eye muscle system are not clearly understood. Because the muscle inserts at the central medial margin of the orbital ridge, increased muscle tension and shortening presumably move the proximal ends of the medial central ommatidia medially, while the distal ends of the ommatidia remain relatively fixed, anchored by the attachment of the cuticle on the outside of the eye to the heavy head cuticle at the margin of the eye. This medial movement of the proximal ends of the ommatidia appears to swing their optical axes laterally, toward the animal’s rear, providing what is in effect a scanning eye movement toward the rear. This eye movement affects the most anterior or medial ommatidia the most, and the lateral ommatidia (on the rear margin of the eye) probably do not move at all. Early workers on fly vision observed so-called clock spikes from the OLs of flies (Leutscher-Hazelhoff and Kuiper, 1966). In the absence of visual stimuli, or in the dark, these spikes fired very regularly at a mean rate determined by the fly’s temperature. From a graph given by Leutscher-Hazelhoff and Kuiper (their Figure 2), the author found that the clock spike frequency in Calliphora was approximately given
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o OBJ.
R OA
θ
OA'
e =Rθ
ANT
OL
NA
RET NMOT
LAM
OE MED
MOT SOG
TT OL
MRH HK
HL HT
FIGURE 5.4-2 Artist’s summary of a horizontal section through the middle of a fly’s head. A visual object is moving from left to right in front of the fly. When the MOT contracts, the visual axes of the medial ommatidia swing to the right, tracking the object. Ideally, vo = ve · = Rθ. Key: MOT, muscle orbitotentorialis; NMOT, motor nerve to MOT; RET, ommatidia; LAM, lamina ganglionaris; MED, medulla of OL; TT, tentorium; ANT, antenna base; NA, antennal nerve; OE, esophagus; SOG, subesophageal ganglion. (Modified from Qi, 1989. With permission.)
by f ≅ –40 + 6.25 T pps. T is the Celcius temperature, and no spikes occur for T > 36° or T < 15°. At room temperature, f is about 85 pps. Subsequent workers found that the clock spike was related to (if not actually) the MOT nerve (NMOT) firing in order to maintain a constant state of tension in the MOT. Thus, any increase or decrease in the firing frequency will increase or decrease muscle tension, respectively, and cause the anterior (medial) ommatidia to scan to the rear or forward, respectively. Burtt and Patterson (1970) and Patterson (1973a,b) reported that the NMOT frequency underwent a transient increase in frequency for OFF or dimming of general illumination. Conversely, ON or brightening caused a transient decrease in the NMOT firing rate. Burtt and Patterson (1970) also reported that when a vertical dark stripe was moved from front to rear around the head, the ipsilateral NMOT firing rate increased during the motion, and decreased when the stripe was moved from rear to front. This suggested that DS visual units may play a role in modulating the frequency of the NMOT mean rate. Burtt and Patterson also observed that puffs of air directed at the head caused transient changes in the frequency of the NMOT. This multimodal behavior suggests that there is a functional connection between the
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yaw stimulation of aerodynamic mechanoreceptor hairs and the need to scan the medial retinula cells. The author and graduate student Xiaofeng Qi decided to investigate the dynamics of the clock spike and eye muscle (CSEM) system of the fly Calliphora erythrocephala. Qi (1989a, b), using fixed, nonflying Calliphora flies, recorded MOT action potentials from both left and right MOTs while presenting the animal with various controlled moving visual stimuli. To characterize better the rapid changes in frequency on the left and right CSEM systems, Qi used two instantaneous pulse frequency demodulators (IPFDs) to convert the instantaneous frequency (IF) of the spike trains to voltage in the following manner: The kth interspike interval (defined by two adjacent spikes), Tk, is by definition, Tk = (tk – tk-1). tk is the time the kth spike in the sequence occurs, tk-1 is the time the previous spike occurs. Since two spikes are needed to define an interval, k = 2, 3, … ×. The kth element of instantaneous frequency is defined as rk ≡ 1/Tk. The analog output voltage of the IPFD is given by ∞
Vok = G
∑ r [U(t − t ) − U(t − t )] k
k
k +1
5.4-1
k =2
U(t – tk) is a unit step which occurs at t = tk, by definition, it is 0 for t < tk and 1 for t Š tk. Vok is thus a stepwise series of voltages, each level of which is the instantaneous frequency of the preceding interspike interval. G is a scaling constant, typically, G = 0.01. Qi found that when a vertical stripe was moved from side to side in front of the insect, the object motion from left to right (or anterior to posterior at the right eye) caused the IF of the right NMOT spikes to increase, and at the same time, the IF of the left NMOT decreased. When the stripe centered in front of the fly was given a sinusoidal deflection of known frequency, an interesting set of phase relations emerged between the stimulus position and the averaged left and right NMOT IFs. Figure 5.4-3 illustrates the IF changes of the left and right eye MOT spikes. (Note that MOT spikes follow the NMOT spike in a 1:1 manner without appreciable delay.) Stimulus frequency was about 1 Hz, and the amplitude was about ± 15° at the eyes. The black stripe subtended 3.5° in the horizontal plane, and extended 38° vertically. The IFs of the left and right MOTs were 180° out of phase, and each led the stimulus position by about 90°, suggesting that the nerves were responding to object velocity, rather than position. To verify that object velocity was driving the NMOT frequencies, Qi deflected his centered test stripe ±15° with a triangle wave. The results are shown in Figure 5.4-4. Note that the IF responses to object movement are rounded square waves, which would be expected from taking the bandwidth-limited derivative of a triangular position waveform. Only the IF changes are shown; they represent variations in frequency around an average clock spike frequency of about 80 pps. When both eyes were subjected to a ±15° square-wave lateral displacement of a 0.75° wide stripe centered on the fly, the result was a “impulse function” change in the IF of the right and left MOT spikes, shown in Figure 5.4-5. Note that when the stripe snaps to the left or right, the IF response is different on the
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10HZ
left 15
o
0.4 s
FIGURE 5.4-3 Simultaneous recording of muscle action potentials from the left and right MOT while a vertical stripe was oscillated back and forth (left to right, etc.) in front of the fly. Top trace, IF change of the left MOT spikes; middle trace, IF change of the right MOT spikes; bottom trace, position of the black stripe object. Note the phase difference between the MOT frequency waveforms. (The stripe measured 90 × 7 mm, and was 11.5 cm from the eye. Thus, its width subtended 3.5° to the fly. Stripe oscillations were ±15° an 1 Hz. (From Qi, 1989. With permission.)
5 Hz
left 0
15
0.4 s
FIGURE 5.4-4 Responses of both muscles to triangular motion of the vertical stripe object. Top trace, IF change of the left MOT spikes; middle trace, IF change of the right MOT spikes; bottom trace, position of the black stripe object. Experimental conditions the same as in Figure 5.4-3, except stripe moved in a triangular waveform. Note that the MOT IF waveforms look like the sum of a square wave and a triangle wave, suggesting a proportional plus derivative response. The apparent 180° phase shift between left and right MOT IF suggests that as the stripe is moving right, the right eye is tracking laterally (to the right), and the left eye is tracking medially (also to the right), and vice versa. (From Qi, 1989. With permission.)
left and right MOTs. When the stripe is on the left, the right eye cannot see it, and vice versa. Because of the extra mass of the stripe, the galvanometer pen motor used to move it exhibited a damped overshoot when slewing to the right or left in response to the input square wave. Figure 5.4-6 shows the details of the
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asymmetrical IF response for the left MOT. Because of the expanded timescale, the stripe slew rate is about 300°/s. When the stripe slews to the right, there is a single, impulse-like dip in IF of the left MOT of about 5 Hz peak, or about –6.25% (20 responses are averaged synchronously with the stimulus in each case). When the stripe slews back to the left, there is a distinct doublet response; a broad positive peak of about 5 Hz, followed by a narrow, negative peak of the same amplitude. The more complex waveform may be in response to the slight overshoot of the object over the left eye. Clearly, the animal’s CSEM system responds to high angular velocities, in this case, ±300°/s.
10Hz
right 15o 0.4s
FIGURE 5.4-5 Responses of both muscles to a horizontal step displacement of the vertical stripe object. Upper trace, IF change of the right MOT; middle trace, IF change of the left MOT; bottom trace, stripe position. Note that the right MOT IF spikes positive as the stripe slews to the right, as the right eye’s medial ommatidia attempt to track the stripe. Also note the overshoot in the stripe position; this is an electromechanical artifact that the insect sees. (From Qi, 1989. With permission.)
5 Hz
left 30° 0.1s
FIGURE 5.4-6 Averaged response of a left MOT to a 30° step displacement of a bar object subtending 0.75° (100 mm × 1.5 mm, 11.5 cm from the eye). Left plots, stripe stepped to right; right plots, stripe stepped to left. Note transient increase in the left MOT frequency, as the fly contracts the left MOT in an attempt to track the stripe. The transient decrease in frequency of left MOT spikes is thought to be a response to the overshoot transient in stripe position. The negative slope in the stripe position is about 25 ms in length. (From Qi, 1989. With permission.)
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FIGURE 5.4-7 Schematic of a one-dimensional model for closed-loop control of slip speed, ve, of an object moving in front of anterior (medial) ommatidia. See text for discussion.
Qi (1989a, b) did many tests using the ipsilateral (right) eye with the object (stripe) centered over a point 37° from the anterior axis, over the right eye. The object was a stripe moved ±15° around 37°. To test the hypothesis that the fly’s CSEM system uses feedback to slew the optical axes of the lateral ommatidia of the ipsilateral eye to follow an object moving from front to rear, Qi “opened the loop” of the feedback system by cutting the ipsilateral MOT. He recorded from the ipsilateral NMOT. Because the muscle was cut, the medial ommatidia assumed a fixed position with regard to the head, and obviously could not track an object moving from front to rear, and vice versa. Qi observed two results as a result of cutting the muscle: (1) The peak-to-peak amplitude of the change in frequency decreased about 17% (average of three preparations) for the same ±10° triangular stripe oscillation amplitude over the ipsilateral eye. (2) The time constant of the ipsilateral NMOT square wave rise increased by about 34.5%, signifying that opening the feedback loop caused the system response to slow, i.e., have a lower, dominant natural frequency. Unfortunately, these two observations are incompatible with a simple negative feedback model for the CSEM system. To understand this dichotomy better, refer to Figure 5.4-7, which shows the system schematically in one dimension to facilitate understanding. A one-dimensional array of ommatidia send analog signals from their retinula cells to a “black box” DS processor. There are one or more neuron outputs from the DS processor whose spike frequency is proportional to
fo ∝ v o [1 + cos(2θ)] 2, fo = 0,
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θ < 90° θ > 90°
5.4-2
that is, the product of object speed times the directional factor. θ is the angle between the PD of the DS unit and the velocity, vo of the object. Note that the DSU output, fo, goes to zero for vo directed ±90° to the PD. fo > 0 causes the NMOT frequency to increase, shortening the MOT, and scanning the medial ommatidia in the PD (in this case, toward the rear). In the case where vo is aligned with the PD, θ = 0, fo is maximum, and the optical axes of the medial ommatidia track the object with vr < vo. Thus, the medial array of receptors experiences a reduced apparent object velocity, ve = vo – vr. The DS unit now responds to ve < vo, producing a reduced fo′, etc. This feedback action is summarized for the θ = 0 case in Figure 5.4-8. In the closedloop case, the change in NMOT frequency is given by
FIGURE 5.4-8 Block diagram of a putative model for a closed-loop, MOT control system that attempts to minimize slip speed. See text for discussion.
∆fmn = v o
K ds 1 + G M K ds
5.4-3
When the loop is opened, ∆fmn = voKds
5.4-4
Clearly, for the same object velocity, ∆fmn(OL) > ∆fmn(CL), which is incompatible with Qi’s observation. If the DS system is single-order low-pass, Kds(s) = Kds/(τs + 1), then the closed-loop gain is
∆fmn = v o
K ds (1 + G M K ds )
s τ (1 + G M K ds ) + 1
5.4-5
Note that both the closed-loop system gain and time constant are reduced by 1/(1 + GMKds). The increase of open-loop time constant was observed by Qi, and is compatible with the negative feedback model. Qi neither reported what happened to the steady-state (no object) fmn when the muscle was cut, nor considered the effect considered of not cutting the contralateral MOT at the same time the ipsilateral muscle was cut. The contralateral eye was not stimulated with a moving object, neither was it masked. Hence eye movements caused by changes in frequency to the intact contralateral MOT could enter the system.
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It is clear that the fly’s CSEM system is more complex than the models suggested here. Qi observed that the positive peaks in the IF ∆fmn in response to sinusoidal stripe motion (in the preferred direction toward the rear of the animal) had a nearly 90° phase lead to the object position. However, close inspection of the phase of the negative peaks of ∆fmn caused by the object moving toward the front of the fly (null direction) showed that they led object position by a fixed angle between 70° and 80°, generally constant over 0.5 to 2.0 Hz object oscillation frequency. Qi suggested that this phase difference could mean different mechanisms were involved for perception of front-to-back and back-to-front motion. That is, two separate DS systems were involved. That the fly visual system is far more complex than currently imagined is substantiated by recent behavioral observations on blowfly flight dynamics by van Hateren and Schilstra (1999). The motions of the head and thorax of free-flying blowflies were studied using magnetic sensors. They observed that flying flies performed a series of short, saccade-like turns at a rate of about 10/s in a fixed order, beginning with a roll. The rolled thorax next pitched up (head up), then yawed, resulting in a turn. Finally, the thorax rolled back to a level position. The saccades had amplitudes up to 90°, but 90% were smaller than 50°. Most amazing was the maximum angular velocity, about 2000°/s, and maximum angular acceleration, about 105°/s2. To conserve angular momentum, a fly’s head exhibited counter rolls to the thorax rolls. Yaws of the thorax were accompanied by faster turns of the head, starting later and finishing earlier than the thorax saccades. Between the high angular velocity head and thorax saccades, the thorax and head are well stabilized, as head velocities are generally less than 100°/s for roll, pitch, and yaw. During this “stabilized” phase of flight, the fly’s CSEM system can operate, and visual information is available to stabilize and direct flight. During thorax saccades, the fly’s inertial navigation sensors, the halteres, may be stimulated, providing a different flight stabilization modality (see Section 2.7). The countermovements of head to thorax appear to be for preserving inertial stabilization in flight. In closing, contemplate the evolved purpose of the dipteran CSEM system. From experiments on fixed insects, it appears that one effect of the medial ommatidia tracking a moving object is to provide a longer integration time for low-contrast, moving objects by the retinula cells involved. The object moves at ve < vo with respect to these receptors. Qi (1989a) shows that a better retinula cell signal-to-noise ratio results from slowing the apparent object motion. The overall CSEM system can also provide the animal with yaw stabilization signals, and also advise the flying insect of its relative velocity with respect to objects in its visual space. A fly does not have binocular vision, but the optical axes of its most medial ommatidia of the left and right eyes apparently overlap at a distance of several head diameters in front of the insect. This fact leads to a final question that needs answering: Does the CSEM system permit vergence, i.e., fixation on a moving object coming straight at (or away from) the fly? An interesting experiment would be to move a test object toward and away from the front of the head on the animal’s centerline. If vergence occurs, the IF of both NMOTs should decrease together to track an approaching object, and vice versa.
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5.4.4
FEATURE EXTRACTION CRUSTACEANS
BY
OPTIC LOBE UNITS
OF
The visual information processing characteristics of crustacean CEs has been shown to be similar in many ways to that performed by nonflying insects. Crustacean CEs have, in general, seven retinula cells per ommatidium (Waterman, 1961). Crustaceans (e.g., crabs, lobsters, crayfish) are unique in that their CEs are at the ends of stalks that can be moved by the animal to track moving objects over a limited range, or be moved to a protected position if the animal is threatened. Insect CEs are, of course, fixed to their heads, which can move. As has been seen, certain diptera, however, have internal muscles in their heads that can warp the anterior portions of their CEs medially to track objects binocularly (Qi, 1989a, b). Researchers studying crustacean CE vision have generally recorded from the optic nerve, which, by definition, runs down the eyestalk from the optic ganglia underlying each eye to the anteror-dorsal-lateral portion of the animal’s protocerebrum. To make stable recordings, the eyestalk must be immobilized. The anatomy of certain crustacean optic nerves was described by Nunnemacher (1966). He examined cross sections of the optic nerve, counting fiber numbers and sizes and the number of facets per eye for nine different genera of decapod, specifically, Homarus americanus, Orconectes virilis, Pagurus longicarpus, Upogebia affinus, Emerita talpoida, Cancer borealis, and Uca pugilator. For example, the lobster Homarus had an average of 12,000 facets/eye. Thus, 84,000 retinula cell fibers projected into the lamina ganglionaris. Each Homarus optic nerve contains fibers from (or to) the four optic ganglia. Nunnemacher made histograms of fiber number vs. diameter range; for example, Homarus had 65,000 fibers of 0.15 to 1.0 µm diameter, 3266 of 1 to 1.5 µm diameter, 1625 of 1.5 to 3 µm diameter, 555 of 3 to 6 µm diameter, 48 of 6 to 9 µm diameter, a total of 70,494 in one lobster optic nerve. In making recordings of optic nerve activity, it is obvious that the 603 axons in the 3 to 9 µm diameter range will have the larger recordable spikes. The fiddler crab, Uca pugilator, examined by Nunnemacher was unusual in that it had giant axons in its optic nerve. The Uca CE had about 9100 facets. In its optic nerve, the fibers were distributed 18,384 of 0.15 to 1.0 µm diameter, 665 of 1 to 1.5 µm diameter, 521 of 1.5 to 3 µm diameter, 106 of 3 to 6 µm diameter, 18 of 6 to 9 µm diameter, 4 of 9 to 12 µm diameter, and 4 of 12 to 24 µm diameter, for a total of 19,701 fibers. Some afferent information must be very important to require the high spike conduction velocities belonging to such large fibers. Insects have no nerve tract analogous to the crustacean optic nerve. Afferent signals from the insect OL leave it through a large number of separate tracts that project into the protocerebrum, subesophageal ganglion, etc. To find all the afferent tracts grouped together in the optic nerve is an advantage when working on CE vision in crustaceans. On the other hand, the chitinous exoskeleton protecting the base of the crustacean CE makes single-unit recording from OL units more difficult than with insects. Glantz (1973) recorded from optic nerve units in the hermit crab, Pagurus pollicarus. Basic ON/OFF responses to general illumination and illumination of small numbers of facets with a 100-µm-diameter spot led to the basic classification
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of units by their responses to changes in illumination. It is not surprising that he found tonic on, tonic off, phasic on, phasic off, and phasic on/off units in the optic nerve. Glantz also tested qualitatively for motion and directional sensitivity using white or black disks moved against a contrasting background with various velocities. Of nine phasic on units tested for directional sensitivity, Glantz found only four that exhibited null direction responses. The others showed only slight to moderate differences to different directions of movement. One unit had up as a null direction, and the preferred direction was somewhere between down and forward. A phasic off unit was tested with a black spot object. It adapted rapidly to repeated motions. The response was more prolonged when the spot was given jerky motions. There was no directional sensitivity. (This behavior is reminiscent of the locust DMD units; see Section 5.4.1.) Phasic on/off units responded to moving objects with no directional sensitivity, and less habituation than phasic off units. No information was given by Glantz on object size, contrast, or velocities used. Apparently moving, contrasting edges were not tried. A more-detailed study of crustacean CE vision was done by Waterman, et al. (1964) on the optic nerve of the crab Podophthalmus vigil. The crab and a schematic of its eye, optic ganglia, and optic nerve are shown in Figure 5.4-9. They found four major classes of interneuron in the optic nerve: (1) Afferent visual interneurons of ipsilateral origin. (2) Efferent visual interneurons of contralateral origin. (3) Efferent interneurons carrying mixed mechanoreceptor information including gravity vector signals from the ipsilateral statocyst. (4) “Afferent as well as efferent interneurons carrying mixed ipsilateral or contralateral visual and body mechanoreceptor information.” The optic nerve also carried efferent motor signals to the distal eyestalk muscles, and afferent mechanoreceptor (proprioceptor) signal from the eyestalk distal joints. Waterman et al. (1964) examined in detail the visual properties of 116 afferent visual units from group 1 above. These units had the following summary properties: 1. They had large receptive fields (30° to 180° or more) and are estimated to contain 300 to 104 facets. Thus, the outputs of many retinula cells (hence, dioptric units) are combined to determine optic nerve outputs. 2. There were rapidly adapting, novel movement units that behaved similar to locust and grasshopper DCMD units. These units exhibited area habituation similar to DCMD units, where repeated stimulation by object movement over one area of the eye quickly caused adaptation of the response; moving the object to a different area of the eye restored the response, which would then adapt. No directional sensitivity was noted for this class of unit. Waterman et al. did not test this unit for size preference or limiting resolution; this was a pity, because their narrative suggested that it did respond to high spatial frequencies: “Thus a unit which had reduced or ceased its response to movement of a particular target (such as a fine needle point) (italics added) in a localized part of its field would respond strongly again when the same stimulus was transferred to another, recently unstimulated area within its field.”
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FIGURE 5.4-9 (A) Anterior view of the crab Podophthalmus vigil. Podophthalmus has unusually long eyestalks supporting its two CEs. (B) Schematic section through the right distal eyestalk segment showing the ommatidia, optic ganglia, and optic nerve fibers that run through the eyestalk. Unlike insects, it is relatively easy to record from the efferent and afferent fibers running to the optic ganglia. (From Waterman, T.H. et al., J. Cell Comp. Physiol., 63(2): 135, 1964. With permission from The Wistar Institute, Philadelphia.)
3. Movement units that exhibited directional sensitivity (DS) were found. No systematic catalog of preferred directions was made. However, their narrative suggested that most preferred directions of objects were front to rear, or rear to front in the horizontal plane. Although a variety of test objects were listed under “methods,” the impression is left that most DS units were tested with a vertical black strip or a 45° white spot object. Quantitative information was given in one figure about optimum object speed for two units. It was not clear whether these were DS units, however. 4. “Mixed modality (afferent) interneurons” were also found (see multimodal units in the Romalea OL). There units combined visual and mechanoreceptor sensitivity to touch of the body or legs. Presumably, pure mechanoreceptor efferent information is “mixed” in the optic ganglia with visual signals, then sent back to the brain. 5. ON/OFF units with different levels of response to ON and OFF, and ONsustaining units were also found. Note that any such unit responding to changes of illumination in its receptive field will also fire in response to a moving, contrasting object in its RF having the correct size and polarity. Finally, Waterman et al. examined some Podophthalmus ON responses for specificity to small, moving black spots (in this case, 5.7° diameter). Several units were
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found exhibiting size preference for this small spot and having little response to larger spots. It was not made clear what other unit response category these units fell into, however. In a second, companion paper, Wiersma et al. (1964) examined the optic nerve responses in the crab Podophthalmus to visual stimulation of the contralateral eye. The ipsilateral eye was cut off distal to the recording site, eliminating any responses of ipsilateral origin. Most of the units found gave responses similar to the ipsitateral fibers studied by Waterman et al. (1964). In fact, the authors claimed that the interneurons recorded from normally carried two-way traffic; i.e., signals arising from a visual unit in the right eye was sent to the left eye, and vice versa, on the same fiber; a rather amazing duplexing of visual information. Wiersma et al. found units that appear to behave similarly to Romalea multimodal units; they called these “fast movement fibers.” Their “slow movement fibers” apparently had size preference for small, dark objects. They had true directional sensitivity, but curiously, adapted quickly to repeated target movement over a designated part of the contralateral eye. Response was refreshed for the same movement over another part of the eye. The usual mix of sustaining ON fibers and phasic OFF units were also observed. Much more information could have been forthcoming about feature extraction had these workers been more systematic and quantitative on their tests of object size, directional preference and speed preference. The two most amazing facts presented by these studies is the extent of interoptic lobe information transfer and the high amount of multimodal traffic observed. Interoptic lobe information transfer and multimodal responses have also been observed in insect OLs, and appears to be a design feature of CE visual systems. It is not known whether duplex traffic (i.e., L → R and R → L on the same, decussating interneuron fiber) occurs in insect visual systems. Wiersma and Yamaguchi (1967) recorded from single optic nerve units in the crayfish Procambarus clarki. Several interesting response properties emerged from this study that had not been seen in the Podophthalmus work. First, no DS movement units were reported. Second, jittery-movement (JM) fibers were observed that responded to novel, quick, movements of a contrasting object anywhere over the ipsilateral eye. JM fibers adapted quickly, and moving the object over a new area of the RF restored the response. Again, the JM fiber of the crayfish optic nerve appears to be like the grasshopper DMD units. An important observation made by Wiersma and Yamaguchi was that voluntary or forced movements of the eyestalk inhibited the JM unit responses, and if it was adapted, canceled the adaptation. The other class of ON units of major interest in the crayfish were the space constant (SC) fibers. There are four SC units in a crayfish’s optic nerve: Two behave like tonic ON units (sustaining fibers), one like a JM unit, and the fourth responding to rapidly approaching objects (a looming operator?). Space constancy is a property mostly derived from the animal’s statocysts (gravity vector sensors). The neural outputs from the statocysts, and perhaps signals from other proprioceptors monitoring limb loading, are sent to the CNS, thence to the OLs. In the OLs, this information acts to maintain the receptive fields of the SC units so that they always maintain the same relative position with respect to the Earth’s gravity vector regardless of the
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animal’s roll and pitch. For example, the RF of an SC unit is sector of a hemisphere between 10 and 2 o’clock (Figure 5.4-10) when the animal is resting normally at zero roll and pitch. When the animal is rolled +90° (right-side down), the SC unit RF now occupies a new set of ommatidia so that it still is between 10 and 2 o’clock. The new RF is actually between 1 and 5 o’clock on the eye. Some very interesting neural switching or gating must take place for this to happen. Space constancy is still a neurophysiological enigma.
FIGURE 5.4-10 Diagram illustrating the curious property of space constancy. See text for discussion.
5.4.5
DISCUSSION
In all insect and crustacean CE visual systems studied, feature extraction operations have been observed. Feature extraction is the result of spatiotemporal filtering operations on basic retinula cell outputs by the various ganglion masses in the OLs. Why are feature extraction operations performed in the OLs and not the CNS? The answer is simple. It is physiologically more efficient to perform “hard-wired,” feature extraction operations on basic retinula cell analog outputs before they become attenuated by long-distance, passive propagation on retinula cell axons. Hence, feature extraction operations begin in the lamina ganglionaris, the first neural layer immediately under the receptors, and also occur in the medulla and lobula neuropile masses in the OLs. Feature extraction provides other parts of the insect’s nervous system with infornation needed for survival. A prime example is DS (vector) moving edge units that presumably project into motor centers that control flight attitude (roll, pitch, and yaw), or walking centers that control direction and speed on the ground. Jittery spot units (multimodal units in grasshoppers) may be involved with general arousal of the animal’s nervous system, in effect preparing the animal for flight, or feeding. Crustacean CE systems also perform feature extraction operations. As underwater swimming is a three-dimensional activity like flying, the animal probably uses DS visual movement information for attitude regulation. However, crustaceans also have statocysts to sense static body attitude in the Earth’s gravity field (system redundancy is good for survival). When a crustacean is accelerating, the statolith is
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“pinned” and the statocyst does not work as an up/down sensor, so visual moving edge information can be important. Crustacean CEs are unique because they are on movable eyestalks. Visual control of eyestalk position may make use of vector unit outputs from the OLs. The existence of neural information exchange pathways between the eyes of crustaceans suggests that left–right comparisons of objects are occuring, and operations such as object range estimation by vergence (as well as by looming) could be taking place. It is hoped that someone will investigate vergence in CE systems in the future. As you will be seen in the next chapter, feature extraction also takes place in the retinas of certain vertebrates.
5.5
CHAPTER SUMMARY
This chapter has examined the behavior of CE visual systems of certain insects and crustaceans. The “front-end” of a CE is the photoreceptor array, consisting of the densely packed ommatidia (lenses, rhabdoms, and retinula cells). The ommatidia make up a spatial sampling array in which each ommatidium “looks” at a slightly different portion of the visual object. Light intensities from each portion of the object are weighted by the directional sensitivity functions of the ommatidia. The retinula cell photoreceptors in any ommatidium respond to their absorbed light intensity, which is proportional to the real convolution of the two-dimensional intensity distribution of the object with the ommatidium DSF. The six or so retinula cells in the ommatidium that absorb this light, depolarize. They send this depolarization electrotonically on their axons to the lamina ganglionaris of the OL where visual processing begins. All CE systems perform feature extraction. An OL neuron showing feature extraction can respond as simply as a burst of spikes for ON of general illumination, or as complexly as a multimodal unit that fires for novel jittery motion of a small black object, as well as for mechanical stimulation of the animal’s feet, or a sound. One of the most ubiquitous of the feature extraction responses in all CEs is the DS, movement-sensing neuron. DS neurons can have a slow, random background firing rate in the absence of stimulation. When an object is moved in the preferred direction, it fires faster; when the object is moved in the opposite (null) direction, the firing can fall below the background rate. Most DS units have a broad response for motions away from the preferred direction (perhaps a cosine(θ) directional sensitivity pattern), and an optimum object velocity that will give the highest spike frequency. (If the object is moved faster than the optimum velocity, the DS neuron response falls off.) Object shape, size, and contrast also appear to be selected for. Most DS units respond most strongly to a long black stripe moved in its preferred direction, but the author has found DS units that referred black spots about 5° in diameter. There are also DS units that appear to like white stripes, as well. The many DS units found in flying insects probably are involved with flight stabilization and obstacle avoidance. Mathematical models were developed to account for anomalous resolution in CE visual systems, as well as to describe spatial high-frequency enhancement caused
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by lateral inhibition. A synthetic aperture model was also developed to demonstrate how increased spatial resolution of objects could occur. These models underscore the importance of signal interactions in the CE array that can lead to improved performance over a single photoreceptor.
PROBLEMS 5.1. A “one-dimensional” photoreceptor is modeled by a Gaussian DSF, s(x):
⎡ x2 ⎤ s(x) = exp ⎢− 2⎥ ⎣ 2σ ⎦ x = θm/2 is the acceptance half-angle; the σ parameter is chosen so s(±θm/2) = 0.5. For this to happen, σ = θm/2/ ln( 4) . When the black spot is directly over the receptor (centered at x = 0), the object has a one-dimensional intensity distribution, f(x) = Io[1 – U(x + a) + U(x – a)] (see Figure 5.2-4). a. Derive an expression for the absorbed intensity contrast, Cie ≡ (Iemax – Iemin)/(Iemax + Iemin). Ie is the absorbed intensity in the rhabdom of a retinula cell. In general, Ie is given by
le =
ko 2π
∞
∫ S(u) F(u) e −∞
jux
du
x=0
Clearly, Iemin occurs when the spot is at x = 0, and Iemax occurs when the spot is at ×. b. Plot log(Cie) vs. a/θm/2 for 0 ð a/θm/2 ð 100. (Use semilog paper.) 5.2. A “one-dimensional” photoreceptor is modeled by a hyperbolic DSF, s(x).
s(x) =
(
1
1 + x θm 2
)
2
This receptor views a one-dimensional, square-wave object as shown in Figure P5.2. The square wave has period A and peak intensity, Io. a. Find a general expression for Cie when the object is shifted by A/2 (one-half spatial wavelength. Assume limiting resolution similar to the development in the second example in Section 5.2.2. b. Plot and dimension Cie vs. θm/2/A. c. Now consider the small movement sensitivity of the receptor. The stripes are now shifted by x = A/4 so that the falling edge of the squarewave intensity is at the origin. A small displacement of the square wave to the right by x = +δx will cause a ∆Ie > 0. A small displacement of the object to the left by x = –δx will cause a ∆Ie < 0. Now assume
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FIGURE P5.2
that θm/2 = 1.5° and the minimum useful intensity contrast is Cie = 0.01. Plot the minimum δx vs. A to give Cie = 0.01. Consider A from 0.15° to 15°. (Note: When A is large, only a small δx is required; as A decreases, δx becomes larger until finally δx = A/2 to get Cie = 0.01.) 5.3. A certain photoreceptor subject to general illumination has an output depolarization, vm, which is log-linear over the range, {I1, I2}. See Figure P5.3. In this range, vm can be modeled by vm ≅ A ln(I/B),
A>0
– Assume I –lies in the range, {I1, I2}. Consider a small increase in intensity, ∆I, above I. a. Find an algebraic expression for the contrast in vm, defined as Cvm ≡ ∆vm/ Vm Note that the approximation, ln(1 + x) Ý x, x Ⰶ 1, can be used. b. Find an expression for Cvm/CI in the range {I1, I2}. –Is it possible for Cvm/CI to be > 1? What are the conditions? (CI ≡ ∆I/I). 5.4. An array of three one-dimensional photoreceptors are connected as shown in Figure P5.4: The visual object is an infinite, {0, Io}, spatial square wave with period, A. The receptor center axes are spaced ±b around the x = 0 axis. Each receptor has a Gaussian DSF with half-angle, θm/2. Thus,
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FIGURE P5.3
FIGURE P5.4
[ ( )] s (x) = exp[− (x + b) (2σ )] s (x) = exp[− (x − b) (2σ )], s1 (x) = exp − x 2 2σ 2
2
2
2
2
2
3
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Note : σ = θ m 2
(
)
ln( 4) , so s1 ±θ m 2 = 0.5
Assume
Vrk = A ln(I k B),
3
Vm = K
∑V ,
Vi = F exp(GVm )
rk
k =1
Let A = 0.01°, θm/2 = 1.4°, K = 10, G = 100, and F = 10–3. Derive an expression for the limiting contrast: CVi = Vi/Vi – Vi (as A/θm/2 → 0) as a function of θm/2/A and b. Show how CVi behaves as b is decreased to zero. Assume the square-wave object is moved A/2 so a black stripe is centered on x = 0. Note that at limiting resolution the square wave can be represented by the first harmonic in its Fourier series. A similar development can be used as that described in the Equations 5.2-77 to 5.2-108. 5.5. A one-dimensional, linear, LI system can be modeled by the integral equation:
r ( x ) = e( x ) −
∫
∞
k ( x − σ ) r ( σ ) dσ
−∞
Assuming the instantaneous frequency, r(x) Š 0 everywhere. The spatial inhibition function is given by
k(x) =
k0
1 + (x α )
2
a. Find an expression for K(u), the Fourier transform of k(x). Assume the input e(x) is a spatial sinewave riding on a dc bias so r(x) Š 0. That is, e(x) = A + B sin(ux) over all x. A > B. b. Calculate and plot the one-dimensional, LI system SS frequency response, (R/E) (u), where u is the spatial frequency in rad/mm. Let ko = 2, and α = 1 mm. 5.6. The spatial inhibition function, k(x), of a one-dimensional, linear LI system is shown in Figure P5.6. k(x) is the product of a cosine wave, cos(uox), and a rectangular gating function, g(x) = ko for 冨x冨 ð 3π/(2uo), and g(x) = 0 for 冨x冨 > 3π/(2uo). That is, k(x) = g(x) cos(uox). Note that this k(x) is inhibitory in its center, and excitatory in its periphery. a. Use the Fourier transform theorem, g(t) cos(ωot) ↔ 1/2 {G(ω + ωo) + G(ω – ωo)}, to find K(u). b. Again, let the LI system input be dc plus a spatial sine wave: e(x) = A + B sin(uox). Calculate and plot (R/E) (u). Let ko = 2, uo = 1 r/mm. 5.7. A photoreceptor has a one-dimensional DSF given by
s(x) =
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(
1
1 + x θm 2
)
2
FIGURE P5.6
Of interest is the minimum displacement, δx, of a black spot centered over the eye that will produce a contrast change of 0.01. The intensity contrast change, CI, is defined by
C l (δ x ) = 0.01 =
∆l e l e (δ x = 0 )
a. Find an expression for Ie(δx = 0) in terms of the spot half-width, a, and × the receptor half-intensity angle, θm/2. Recall that Ie = ko ½–× S(u)F(u) du, where u is the spatial frequency in r/°. b. Now shift the spot to the right by δx. Find an expression for Ie(δx). Note that ∆Ie = Ie(δx) – Ie(δx = 0) > 0. c. Find a numerical value for the δx required to make C(δx) = 0.01. Let a = 1.5°, θm/2 = 1.5°; x is in degrees. The answer will require the trialand-error solution of a transcendental equation.
FIGURE P5.7
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5.8. Of interest is the motion sensitivity of a photoreceptor when a onedimensional DSF, s(x) = exp[–x2/(2σ2)], views a one-dimensional, spatial sinewave object given a small displacement, δx. Mathematically this object can be written: f(x, δx) = (Io/2){1 – sin[uo(x – δx)]},
–× ð x ð ×
For algebraic ease, this sinewave can be written as a shifted cosine wave when Fourier transforming. Thus, F(u, δx) = (Io/2)[2π δ(u) + 1/2 δ(u + uo) + 1/2 δ(u – uo]) exp[–ju(λ/4 + δx)]. Note that uo = 2π/λ, where x = λ is the spatial period of the sinewave. The right-hand exponential term shifts the cosine wave by 90° to make a –sinewave, and gives an additional displacement, δx, required to give a contrast change. The contrast is defined here as C(δx) ≡ ∆Ie/Ie(0). Ie(0) is the effective absorbed light intensity with δx = 0. ∆Ie ≡ Ie(0) – Ie(δx). a. Use the Fourier transform approach to find an expression for Ie(0). b. Now find Ie(δx). c. Use the expression for the contrast, C(δx), to calculate the δx required to produce C(δx) = 0.01 as a function of the receptor DSF half-angle, θm/2. Note that σ ≡ θm/2/ ln( 4) , let λ = 5°. Consider 0.25 ð θm/2 ð 2.5°. (Be careful to use proper angle dimensions, degrees or radians, when solving this problem.)
FIGURE P5.8
5.9. This problem will model the three-ommatidium LI system described by Equation 5.3-2 and Figure 5.3-5. A block diagram for this system is shown in Figure P5.9. Note that LI is modeled here by a feed-forward architecture. The retinula cells of each ommatidium respond to light by depolarizing, but not spiking. A single eccentric cell associated with each ommatidium produces spikes as a result of depolarization excitation being coupled to it from its surrounding retinula cells (see Figure 5.3-1).
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FIGURE P5.9
Summed with the excitation is inhibition coupled from the lateral plexus fibers arising from the retinula cells of neighboring ommatidia. The system can be modeled mathematically. For each retinula cell, use the biochemical, kinetic model considered in Problem 2.1. Here, the depolarization of the first retinula cell is given by vm1 = k6 c1, where a1, b1, and c1 are molecular concentrations in the rhabdom of the first retinula cell. Thus,
a˙ 1 = k 4 c1 − a 1 k 1 log(1 + I1 I o ) b˙ 1 = a 1 k 1 log(1 + I1 I o ) + k 5 c1 − b1 ( k 2 + k 3 c1 ) c˙ 1 = b1 ( k 2 + k 3 c1 ) − c1 ( k 4 + k 5 ) v m1 = k 6 c1 The input to the RPFM spike generator of the first eccentric cell is assumed to be given by Vg1 = vm1 – K13 vm3 – K12 vm2 The first ommatidium RPFM spike generator is modeled by
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u˙ 1 = cVg1 − cu1 − reset 1 w1 = IF u1 > phi THEN 1 ELSE 0 s1 = DELAY (w1 , tau) x1 = w1 − s1 y1 = IF x1 > 0 THEN x1 ELSE 0 reset 1 = y1 phi tau y1 are the output impulses of the first eccentric cell. (Euler integration must be used with delt = tau if using Simnon to run this simulation.) Write a program to simulate a three-ommatidium LI system. Plot the output spikes of the three eccentric cells (y1, y2, and y3), the generator potentials Vg1, Vg2, and Vg3, and the inputs, I1(t), I2(t), and I3(t). Arrange the inputs so that they occur singly, overlap in pairs, and occur together. Parameters for the retinula cell depolarization are a(0) = 1, other ICs = 0, k1 = 4, k2 = 0.3, k3 = 40, k4 = 10, k5 = 0.1, k6 = 100, Io = 1. For the RPFM spike generators of the eccentric cells, let phi = 10 mV, c = 1 rad/ms, and all Kjk = 0.3 (assumes the ommatidia are equidistant). 5.10. This problem will consider the properties of lateral inhibition applied between the mechanosensory hair cells that line the interior of an invertebrate statocyst. (See Section 2.3.5 for a description of statocysts.) As in the case of LI applied to visual systems, this problem will examine a mathematical model in which linearity is assumed, and calculations are carried out in continuous form in one dimension. The input variable is a one-dimensional distribution of static, radial force applied to the hair cells; the force is the result of gravity acting on the mass of the statolith (forces from linear and angular acceleration of the statocyst will be neglected). Although the interior of the statocyst is roughly spherical, a major simplification results if the problem is framed in terms of an infinite, linear dimension, x, –× ð x ð ×, instead of θ, –π ð θ ð π. In the case of the spherical statcyst, assume that an output neuron (without LI) fires at a frequency, ro, proportional to the radial force exerted on it by the statolith. The same property is assumed for the linear continuous case: e.g., ro(x) = K f(x). Here ro(x) is the frequency of the mechanosensor located at x, f(x) is the radial force at x, and K is a positive constant. Each stimulated receptor is assumed to be inhibited by its neighbors’ firings according to the rule:
r(x) = K f (x) −
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∫
∞
k i (x − σ ) r(σ ) dσ
−∞
Where ki(x) is the spatial distribution of inhibition from a sensory cell at x to its neighbors. A schematic of a vertical section through statocyst (not all sensory cells are illustrated) is shown in Figure P5.10A. The onedimensional, discrete case is illustrated in Figure P5.10B.
FIGURE P5.10A
a. Find an expression for the transfer function, R(u)/F(u) = H(u), in the frequency domain when ki(x) = kio exp(–α冨x冨). b. Give an expression for, and plot to scale the spatial impulse response, h(x), of the statocyst continuous model LI system. Let f(x) = δ(x). Ignore negative frequencies. c. Assume that f(x) is modeled by f(x) = fo exp(–γ冨x冨) mg. Calculate the continuous distribution of output frequency vs. x, r(x). Neglect negative frequency values. Note that the FT pair,
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FIGURE P5.10B
2a ← ⎯→ exp[−a x ] u2 + a2 can be used, and an expression of the form,
(
1 u2 + a 2 u2 + b2
)(
)
can be expanded by partial fractions to:
(
A B + 2 2 u +a u + b2 2
) (
)
Let K = 10 pps/mg, fo = 20 mg, γ2 = 10, β2 = 2, α = 1, kio = 0.5.
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6
Large Arrays of Interacting Receptors: The Vertebrate Retina
INTRODUCTION This chapter examines the neuroanatomy and feature extraction properties of various vertebrate retinas. It begins by reviewing the function(s) of each of the interneurons that make up the retina. The retinal photoreceptors, the rods and cones, are different from arthropod retinula cells in that they hyperpolarize in response to absorbed light energy. Rods and cones do not spike; they are essentially analog transducers, like retinula cells. The early work on feature extraction was done on the retinas of frogs. Workers generally recorded from ganglion cell (optic nerve) axons. The classic paper on retinal feature extraction, described in Section 6.2, was published by Lettvin et al., (1959). One of the author’s Ph.D. students (Reddy, 1977) recorded from directionally sensitive units in the frog’s brain. Some of his results are described in Section 6.2.2. Finally, feature extraction operations are described in the retinas of pigeons and rabbits in Section 6.3. Again, many of the ganglion cell types found in frogs appear in these animals. Only when cats and primates are considered is it found that certain retinal feature extraction operations are missing, as they are presumably performed more efficiently in the visual cortex.
6.1
REVIEW OF THE ANATOMY AND PHYSIOLOGY OF THE VERTEBRATE RETINA
The retinas of all vertebrates follow a general anatomical pattern. Located on the rear, inside surface of an eyeball, it has ten anatomical layers. Starting from the vitreous humor, the layers are (1) internal limiting membrane; (2) optic nerve axon layer; (3) ganglion cell layer (about 5 × 105 per eye in the frog); (4) inner plexiform layer; (5) inner nuclear layer; (6) outer plexiform layer; (7) outer nuclear layer; (8) external limiting membrane; (9) sensory cells (about 106 rods and cones in the frog); (10) pigment epithelium layer. The pigment epithelial layer is responsible for eye shine in nocturnal animals. Located on the inner surface of the eyeball, it reflects light back through the photoreceptors, increasing visual efficiency in trapping photons at low light levels. There are five major classes of neurons arranged in the ten retinal layers: Ganglion cells (GCs) and their axons are in layers 1 and 2. The inner plexiform layer contains dendrites and synapses from ganglion cells, amacrine cells (> 20
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types), and bipolar cells (3 types). An inner nuclear layer contains the cell bodies (somata) from amacrine cells, bipolar cells, and horizontal cells. The outer plexiform layer contains dendrites and synapses between the bipolar cells, horizontal cells, and the rods and cones. The rods and cones are the photoreceptor (PR) cells that transduce photon energy into cell membrane hyperpolarization. Figure 6.1-1 illustrates schematically the five major types of neuron in the vertebrate retina and their basic interconnections. The ganglion cells are the spiking output neurons of the retina; their axons form the optic nerve. In the cat retina, there are some 23 subtypes of GCs (Kolb et al., 1999).
Rod
Rod
Cone
}
Distal Bipolar cell
Outer nuclear layer
plexiform } Outer layer Vertical information flow
Bipolar cell
Horizontal cell Lateral information flow Amacrine cell
Amacrine cell
Proximal
Ganglion cell Light
}
Inner nuclear layer
}
Inner plexiform layer
}
Ganglion cell layer
To optic nerve
FIGURE 6.1-1 The five major classes of neurons in the vertebrate retina: photoreceptors (rods and cones), horizontal, bipolar, amacrine, and ganglion cells, arranged in 5 anatomical layers. Information from the photoreceptors flows vertically and horizontally. (From Kandel, E.R. et al., Principles of Neural Science, 3rd ed., Appleton & Lange, Norwalk, CT, 1991. McGraw-Hill Companies, with permission.)
Light entering the eye must pass through the inner layers of the retina before it impinges on the outer segments of the rods and cones where transduction occurs. In the dark, the soma of a rod or cone has a resting potential of about –40 mV. Light absorbed in the outer segment causes a chemical reaction that closes cGMP-gated sodium leakage channels in the outer segment. At the same time, potassium ions leak out of the inner segment of the PR cells, unaffected by the light. Metabolically driven ion pumps in the PR cell membrane continuously pump Na+ out while K+ is brought in (Kandel et al., 1991). In very intense light, all the Na+ leakage channels in the outer segment close, and the membrane potential of the PR cell soma hyperpolarizes; it saturates near –70 mV, the K+ Nernst potential. Intermediate values of light intensity cause hyperpolarization that is closely proportional to the logarithm of the light intensity. It is interesting to compare properties of rod and cones. Cones are associated with bright-light (photopic), high-acuity color vision. Each cone contains one of three different photopigments, each having an absorption spectrum at a different wavelength. Because cones contain less photopigment than rods, they are less © 2001 by CRC Press LLC
sensitive, and saturate at higher light intensities. Rods, on the other hand, are adapted for low-light (scotopic) vision; they contain only one type of photopigment, hence give monochromatic vision. They have high transduction gain, responding even to single photons; they saturate in daylight. They have a slower response time than cones to sudden changes in intensity; their flicker fusion frequency is about 12 Hz compared with about 50 Hz in cones. Rods evidently trade off increased sensitivity for reduced acuity. How the graded hyperpolarizations of PR cells leads to spike outputs of various types from the ganglion cells is now described. Consider first only cones and bipolar cells (BCs) in the outer plexiform layer. Cones make chemical synapses with two types of BC, the off-center BC and the on-center BC. In both cases, in the dark, the cone continuously releases a single neurotransmitter, probably glutamate. When light acts on the outer segment of the cone, the soma hyperpolarizes, reducing the continuous rate of release of glutamate by the cone synapses. The reduced neurotransmitter causes the off-center BC to hyperpolarize, and the on-center BC to depolarize (different ion channels are affected by the glutamate in the two types of BCs). Each type of BC makes an excitatory synaptic connection with a corresponding type of GC (off-center GC and on-center GC). In summary, when a flash of light is directed at a cone connected to an on-center BC and an off-center GC, the former depolarizes and the latter hyperpolarizes. The on-center GC depolarizes and generates an ON burst of spikes. The off-center GC is hyperpolarized by the off-center BC, suppressing its output spikes. It is probable that there is inhibitory cross-coupling between BCs and the corresponding GCs. That is, the on-center BC inhibits or hyperpolarizes further the off-center GC. Similarly, the off-center BC may inhibit the on-center GC (Kandel et al., 1991). When the role of the amacrine cells in the function of the retina is examined, the picture becomes much more complex. There are over 40 morphologically and neurochemically distinguishable types of amacrine cells that use at least eight different neurotransmitters (Kolb et al., 1999). It is not difficult to speculate that amacrine cells function in the feature extraction operations found in various retinas. There are two types of horizontal cells (HCs) found in most mammalian retinas. One is an axonless, A-type, that contacts only cones. The B-type has a soma and dendrites contacting cones, and an axon several hundred micrometers long that has terminal arborizations that contact only rods (Kolb et al., 1999). HCs mediate an antagonistic action between neighboring cones. As has been seen, cones in the surround of the receptive field (RF), in the dark, are depolarized and continuously release glutamate neurotransmitter that acts on the connecting HCs to keep them slightly depolarized. In this state, the HCs release an inhibitory neurotransmitter that maintains cones in the center of the RF in a hyperpolarized state. Illumination of cones in the RF surround hyperpolarizes the surround cones, which in turn further hyperpolarizes the HC. This in turn reduces the rate of release of inhibitory neurotransmitter by the HC and results in the depolarization of the cones in the RF center. Now more glutamate is released by the center cone, leading to hyperpolarization of the on-center BCs, leading to a reduced firing rate of the oncenter GC. Thus, light in the RF surround inhibits the on-center GC output. The net effect is a form of lateral inhibition of the feed-forward type. © 2001 by CRC Press LLC
The neural circuitry on the retina used for low-light vision deserves special mention. The rod photoreceptors, as discussed above, have very high transduction gain, responding at the single photon level. At moderately dim light levels (e.g., at dusk), the rods evidently transmit their light-caused hyperpolarizations to adjacent cones by fast electrical synapses (gap junctions). This means that even though the light level is too low for the cones to be effective transducers, the rods share the same high-resolution signal-processing pathways used by the cones in photopic vision. Thus, there is little lost in spatial resolution; color fades as the light dims, however. Under very low light conditions (e.g., starlight), the gap junction synapses connecting rods to cones close, blocking this pathway; instead, the rods stimulate special rod BCs through ribbon synapses. The rod BC outputs go to type A II amacrine cells, which synapse directly with off-center GCs, and indirectly with oncenter GCs via cone BCs. See Figure 6.1-2 for a schematic description of these pathways.
Rods
Cones Electrical synapse
Ribbon synapse Outer plexiform layer Horizontal cell Rod bipolar
Basal synapse
Invaginating synapse
Cone bipolar (off-center) Cone bipolar (on-center)
AII amacrine
Inner plexiform layer
Ganglion cell
FIGURE 6.1-2 Diagram showing the details of the synaptic connections between retinal interneurons. Rod and cones synapse on BCs and ACs. Rods and cones each synapse on different BCs; however, these pathways converge on the GC layer. Electrical synapses occur between cones and rods. Cones connect to two different classes of BCs with morphologically different synapses; basal (flat) synapses make contact with off-center BCs, ribbon synapses contact on-center BCs. The on-center BCs send their dendrites into invaginations in the cone terminal; there they form the central element of three clustered synapses, the other two of which come from dendrites of HCs. Rods synapse with one type of BC, which receives inputs only at ribbon synapses. These BCs do not synapse directly with GCs; instead, they synapse with type AII amacrine cells (ACs). The AII ACs relay their inputs by synapsing directly onto off-center GCs and onto BCs that connect to on-center GCs. (From Kandel, E.R. et al., Principles of Neural Science, 3rd ed., Appleton & Lange, Norwalk, CT, 1991. With permission from the McGraw-Hill Companies.)
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As neural structures go, the retina at first sight appears simple. However, like many physiological systems, the closer the structure, molecular physiology, and function of the retina is observed, the more complexity is revealed in its design. Many neurophysiologists have studied the retina in a wide variety of vertebrates, here listed alphabetically: cat, frog, goldfish, ground squirrel, pigeon, rabbit, salamander, turtle, to mention a few. Obviously, space prevents a detailed discussion of the results with each animal. Described below are some of the results of the classic, early studies on retinal neurophysiology. It is evident that there are many common response properties among animals’ retinas.
6.2 6.2.1
FEATURE EXTRACTION BY THE FROG’S RETINA EARLY WORK
Back in 1938, Hartline reported on results obtained recording from single frog optic nerve fibers (GC axons) when the eye was stimulated with simple spots of light. In this pioneering work, Hartline coined the term receptive field (RF) to describe the area of the retina over which any change in illumination (ON, OFF, and ON/OFF) would affect the base rate firing of the optic nerve fiber under study. A GC RF was often found to be surrounded with an annular peripheral region in which changes in illumination affected the firing caused by illumination in the center of the RF. Hartline found three types of RF in the frog optic nerve: ON, ON/OFF, and OFF. If a small spot of light appears in the RF of an ON fiber (a pulse of light intensity), the instantaneous spike frequency increases to a peak, then falls off to a lower, steady-state value. At off, the discharge is abolished. The response of an ON/OFF fiber to a pulse of light in its RF is a burst at ON, falling off to zero frequency, then another short burst at OFF. OFF units fire briskly at OFF; then their frequency decreases slowly in the dark. ON silences the OFF unit. In 1953 Barlow added to Hartline’s (1938) observations. He noted that OFF cells have adding RFs; i.e., the response to OFF occurs at both center and periphery of the RF. The effect of turning off the light in the periphery adds to the effect of reducing the light in the center of the RF, with a weight decreasing with radial distance from the center. He found that the ON/OFF cells have differencing RFs, where a discharge caused by ON at the center is diminished by a simultaneous ON in the periphery. The same reduction happens for OFF in the center plus OFF in the periphery. No results were given for ON cells. Thus, the early work of Hartline and Barlow suggested that each optic nerve fiber (and its associated RF) sends to the frog’s brain basic information about where dimming has occurred on the retina (OFF operation); ON/OFF fibers signal where the contrasting boundaries are moving, or where local inequalities of contrast are forming. The ON fibers signal where brightening has occurred. If there is no change in time of a contrasting image projected on the retina, the three types of fibers firing rates will decrease to low, basic levels. In 1959, Lettvin and colleagues published the results of a pioneering neurophysiological study on visual feature extraction, entitled, “What the Frog’s Eye Tells the Frog’s Brain.” Frog GC responses were examined when the eye was stimulated with
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simple, contrasting, two-dimensional objects presented on the inside surface of a 14-in.-diameter, matte gray hemisphere centered over the eye. The thrust of their research was to find natural “features” of simple objects that elicited maximum GC responses, rather than simply to shine lights into the frog’s eye. What Lettvin et al., (1959) demonstrated is that the frog’s retina can send more complex information to the brain than simple ON, ON/OFF, and OFF operations on RFs. Instead of using spots of light as stimuli, Lettvin et al., used simple, contrasting, geometric objects such as a 1°-diameter black spot, and a 12° × 30° black rectangle moved with magnets on the inside of the 14-in.-diameter, gray hemisphere centered over the eye. They were able to identify four consistent, separate operations on images projected on the frog’s retina. These operations were found to be substantially independent of the overall illumination. The operations were named (1) sustained contrast detection, (2) net convexity detection, (3) moving edge detection, and (4) net dimming detection. The details of these operations are described below: 1. Sustained contrast detection (SCD): The axons of these ganglion cells are unmyelinated. They are probably Hartline and Barlow’s ON fibers. The SCD fibers do not respond to ON or OFF of general illumination. The SCD unit RFs are from 2° to 4° in diameter. These units fire when a contrasting, 1° or 3°-diameter black disk is moved into the RF. Apparently the SCD fibers are directionally sensitive; i.e., they have a preferred direction that gives maximum response, and if motion is in the opposite direction (antipreferred direction), firing is suppressed. Curiously, SCD fibers have “memory.” That is, if a spot moves into the RF in the preferred direction, the unit fires. It continues to fire (at a reduced rate) if the spot stops in the RF. If general illumination is turned off, the firing ceases, or in some units is greatly reduced. When the general illumination is again turned on, and the spot remains stationary in the RF, the unit again begins to fire, demonstrating “memory.” It was claimed that SCD fibers responded to white as well as black test objects. They also responded to linear edges (as on the 12 × 30° black rectangle). Unfortunately, no quantitative information was given on directional preferences or optimum object speeds in the paper. 2. Net convexity detection (NCD): NCD fibers are also unmyelinated. They, too, are unresponsive to ON and OFF of general illumination. The RFs are from 3° to 7° in diameter. NCD units respond only to dark contrasting spots moved into the RF. If the spot is stopped in the RF, the unit continues to fire. If general illumination is turned off, the unit is quiet. At ON, the unit continues to be silent until the spot is again moved. There appears to be no memory. There appears to be a broad size optimality for response to spots moved into the RF. Responses were noted down to spot diameters of 0.05°. When the spot is about one half the diameter of the RF, a maximum response is noted. Response begins to fall off for spots less that 1° diameter or greater than half the RF diameter. There is no response to long, moving, dark edges that overlap the RF. Also, a large, black and white checkerboard with a repeat distance of one half the RF diameter gave little, if any, response when moved over the RF. Jerky motions appeared to be more effective in eliciting spikes from moving spots than smooth motions. No quantitative data were given on object speed or
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directional preference; presumably NCD units respond to object motion in any direction. It is tempting to call NCD units the frog’s “fly detectors”; perhaps they send information to the CNS alerting the frog that prey may be near; get ready to strike. 3. Moving edge detection (MED): These are myelinated fibers conducting at about 2 m/s. The RFs of MEDs are about 12° diameter. They are the same as Barlow’s ON/OFF units. They respond to long contrasting edges moving through their RF. Their frequency is proportional to edge speed, and is said to fall off at high speeds. No quantitative information was given on directional sensitivity, if any, or the speed that gave maximum firing frequency. The MED fibers project into the third layer of the tectum. 4. Net dimming detection (NDD): NDD unit fibers are myelinated and conduct at 10 m/s. They are the same as Hartline and Barlow’s OFF fibers. They have large, about 15°, RFs; they respond to OFF or general dimming by a prolonged burst. The amount of light at ON required to interrupt the OFF burst gets less and less the longer the eye sits in the dark. If the general illumination is dimmed, firing occurs. Now if a dark object is moved through the RF, the firing is suppressed. Lettvin et al., thought this effect was due to the relative brightening in the RF as the black object passes through it. To summarize the operations performed by the frog’s retina, we quote Lettvin et al.: Let us compress all of these findings in the following description. Consider that we have four [ganglion cell] fibers, one from each group, which are concentric in their receptive fields. Suppose that an object is moved about in this concentric array: 1) The contrast detector tells, in the smallest area of all, the presence of a sharp boundary, moving or still, with much or little contrast. 2) The convexity detector informs us in a somewhat larger area whether or not the object has a [sharply] curved boundary, if it is darker than the background and moving on it; it remembers the object when it has stopped, providing the boundary lie totally within that area and is sharp; it shows most activity if the enclosed object moves intermittently with respect to a background. The memory of the object is abolished if a shadow obscures the object for a moment. 3) The moving edge detector tells whether or not there is a moving [straight] boundary in a yet larger area within the field. 4) The dimming detector tells us how much dimming occurs in the largest area, weighted by distance from the center [of the RF] and by how fast it happens. All of the operations are independent of general illumination. There are 30 times as many of the first two detectors as of the last two, and the sensitivity to sharpness of edge or increments of movement in the first two are higher than in the last two.
It is interesting to note that the shape of the dendritic tree structure of a frog GC can be associated with the feature extraction operation performed. The GCs of
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edge detectors (also known as boundary or sustained contrast detectors) have constricted dendritic fields (<100 µm wide); convex edge detectors (also known as moving spot detectors) have “E-tree” dendritic fields (~200 µm wide); moving contrast detector GCs (also known as moving edge detectors) have “H-tree” dendritic fields (~300 µm wide); and dimness detector (also known as dimming detector) GCs have broad dendritic fields (> 500 µm wide). Function and form are intimately associated in the frog retina not only for GCs, but also for HCs, amacrine cells, and BCs.
6.2.2
DIRECTIONALLY SENSITIVE NEURONS
IN THE
FROG’S BRAIN
Reddy (1977) examined single visual units found in the frog’s brain exhibiting directional sensitivity (DS) to moving visual objects. Reddy made extracellular recordings from visual units in the tectum, thalamus, and cerebellum using glasscoated Pt/Ir microelectrodes. Only 9% of the motion-sensitive units found in these sites were truly DS. A DS unit by definition fires at a maximum rate for an object having constant speed in a preferred direction (PD). The neuron fires at slower rates for directions of motion different from the PD. Some DS units have a null direction (usually 180° different from the PD) for which the DS unit fires at its slowest rate, often lower than its spontaneous background rate (if it has one). Many animals, vertebrate and invertebrate, have DS visual units. The vertebrate DS unit list includes the frog, goldfish, pigeon, ground squirrel, rabbit, and cat. Arthropod DS units have been found in every flying insect studied (locusts, flies, dragonflies, moths, butterflies) and in such nonflying species as the lubber grasshopper, Romalea microptera, and in crabs. Some workers have hypothesized that DS units are used for eye and head movement control (visual tracking), or flight stabilization, wherever applicable. However, it is well known that frogs lack eye tracking movements; they can only retract or elevate their eyes (Walls, 1967). Frogs also lack head movements. (Frogs do not have necks that allow head movements independent of their bodies.) Because they cannot move their eyes or heads to track moving prey, frogs must use information from DS units to predict where and when prey (e.g., a fly) can be struck at successfully. Several workers who have recorded from frog GCs (optic nerve fibers) have been unable to locate DS units per se (Grüsser-Cornehls et al., 1963; Gaze and Keating, 1970). Reddy was motivated to look for DS cells in the frog’s brain in the belief that the animal’s prey capture behavior and its escape from danger behavior required DS unit information. Reddy moved visual objects for his frogs at constant velocity on a large-bed, XY recorder. Object direction vectors relative to the frog were shifted in 22.5° increments (16 major directions) relative to anterior, left, right, and posterior. Object motion was controlled by a triangle wave generator. Different object sizes, shapes, and contrasts were used. Most tests were done with contrasting spot objects. Monocular stimulation was used; the contralateral eye was masked. A window circuit was used to isolate single units; the total number of spikes a unit fired for a given object direction and speed (vector velocity) was counted electronically for a preset number
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of object motions in a given direction, and also for the reverse motion. This data were used to make polar plots of DS unit vector sensitivity. Most of Reddy’s DS units were found in the deep (100 to 300 µm) tectum layers; a few were recorded in the superficial tectum, and in the thalamus and cerebellum. Figure 6.2-1 illustrates the vector response of a DS unit located in the superficial, contralateral tectum (see caption for details). Note the directional response is significantly sharper than a cosine(2θ) law. Figure 6.2-2 shows the vector response of a DS unit recorded in the contralateral deep tectum. Frog tectal DS units have an optimum object speed for maximum response. Figure 6.2-3 illustrates this phenomenon for a DS unit in the contralateral, superficial tectum. Maximum response occurred for object speed of 1.85°/s.
FIGURE 6.2-1 Response sensitivity for a DS neuron located in the frog’s superficial, contralateral tectum to a 2°-diameter black spot (dashed line) and a 4° × 1° black bar (solid line) moved across its RF in various directions. Scale: 2 cm = 40 spikes. A, anterior (0°); P, posterior (180°); D, dorsal (90°); V, ventral (270°). (Redrawn from Reddy, 1977.)
At 4°/s, the response was less than for 1.1°/s Figure 6.2-4 shows the vector response of a DS unit found in the frog’s cerebellum. Reddy found that there was no correlation between RF location on the retina under test, and the preferred direction of some 34 DS units he measured. In fact, the distribution of preferred
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FIGURE 6.2-2 Response sensitivity for a DS unit located in the frog’s deep, contralateral tectum to motion of a 7°-diameter white spot (solid line) and a 2°-diameter black spot (dashed line) moved across its RF in various directions. Scale: 2 cm = 40 spikes. (Redrawn from Reddy, 1977.)
directions was apparently random, as shown in Figure 6.2-5. Reddy found several important differences between superficial and deep tectal DS units. The deep units had larger RFs (15 to 25°), and adapted quickly to repeated object movements. On the other hand, superficial tectal DS units had RFs from 4° to 12°, and showed no adaptation. Superficial tectal DS units also responded to smaller moving objects (2° to 6° diameter) than did deep units. The width of the response lobes was wider for superficial tectal DS units, and also tended to have larger side lobes. Deep tectal, thalamic, and cerebellar DS units in general had narrower response characteristics.
6.3 6.3.1
FEATURE EXTRACTION BY OTHER VERTEBRATE RETINAS THE PIGEON RETINA
Feature extraction is also performed by the pigeon’s retina. Maturana (1974) reported on pigeon GC responses. Recall that the frog is a sessile animal; it normally sits
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FIGURE 6.2-3 Response sensitivity for a DS unit located in the frog’s superficial, contralateral tectum to a 2°-diameter black spot moving at different velocities through its RF. The total response to eight back-and-forth movements of the spot is plotted for each of the four velocities used. Dotted line, 0.32°/s; small dashed line, 1.1°/s; solid line, 1.85°/s; long dashed line, 4°/s. Note that there is an optimum velocity; DS response falls off for 4°/s. (Redrawn from Reddy, 1977.)
and waits for prey to come to it. Pigeons fly, and any flying animal has additional burdens for its visual system. It must stabilize its flight (roll, pitch, and yaw), and it must avoid fixed and moving obstacles such as trees and other birds. Maturana reported six classes of pigeon optic nerve fiber: 1. Verticality detectors have RFs < 1°. They fire for stationary vertical edges in the RF, or for vertical edges moving through the RF. Rotation of edge 20° to 30° from vertical stops the response. 2. Moving horizontal edge detectors have RFs ~1/2° in diameter. To fire, they require vertical motion of a long, contrasting, horizontal boundary. Presumably they fire for downward motion, as well. Tilting the edge ±20° to 30° from horizontal suppresses the response. In order for a response to occur, the horizontal edge must be long enough to overlap the edges of the RF.
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FIGURE 6.2-4 The sharply directional response of a cerebellar DS unit to 7°-diameter black (dashed lines) or white (solid lines) spots. (Redrawn from Reddy, 1977.)
3. General edge detectors (2 kinds): 3a has a large RF, 2° to 3° diameters; 3b has a small RF, < 1/2° diameter. Common properties: • May respond to ON or OFF of general illumination with one or two spikes. • Respond strongly with six to ten spikes to an edge moving across their RFs. • Respond to a spot of light turned on or off or moved across their RF. • Their RFs are uniformly of the ON/OFF type. Differential properties: • Ganglion cells with large RFs do not “see” the details of a drawing (test object background), but they are able to respond to objects a few minutes in diameter that move within their RF against any background. • Ganglion cells with small RFs can “see” background detail as well as small objects moving independently to the background. 4. Directional moving edge detectors have 0.5° RFs to 1° diameter, and the unit responds to ON/OFF of a small spot of light over the RF. These units respond preferentially to a contrasting edge moved through the RF in a preferred direction. There is little or no response to edge movement in
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FIGURE 6.2-5 The distribution of preferred directions of 32 DS units recorded from the frog’s brain. The numbers on the vectors indicate the number of DS units found with that particular preferred direction. (Test axes were at every 22.5°). Note that in the posterior hemicircle there are 22 out of 32 DS units. (Redrawn from Reddy, 1977.)
the anti-preferred direction, and response falls off as the direction of edge motion departs from the preferred direction, although no quantitative data was given. 5. Convex edge detectors have very small RFs, a few minutes of angle in diameter. They do not respond to moving straight edges, only to moving spots. There does not appear to be directional specificity. 6. Luminosity Detectors LDs fire at a rate proportional to the general illumination level. (Maturana gave no data on RFs for LDs.)
6.3.2
THE RABBIT RETINA
It has become clear from the foregoing sections that the vertebrate retina is more than a simple transducer that maps local light intensity on the retina to impulse frequencies on the optic nerve fibers. Clearly, individual GC fibers signal the presence of certain specific, spatiotemporal features of the retinal image, thus performing a kind of pattern recognition preprocessing, which is sent to the visual portions of
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the CNS. Understanding of these operations has been greatly enhanced by recording from single rabbit GC fibers done by Barlow and Hill (1963), Levick (1967), and Oyster and Barlow (1967). Levick (1967) reported a total of eight different types of GC responses (“trigger features”) found in the fovea (visual streak) of the rabbit’s retina. Five of the types of GC responses found in the peripheral retina were also found in the central region; however, there they had smaller, oval RFs. In addition, three types of GC responses were found only in the fovea (6, 7, and 8 below). The eight GC types were (1 and 2) concentric RF (ON-center and OFF-center subclasses); (3) large-field; (4 and 5) direction-sensitive to image motion (ON/OFF and ON subclasses); (6) orientation-selective (horizontal and vertical subclasses); (7) local edge detectors; (8) uniformity detectors (UDs). Complex tests using both projected beams of light and contrasting objects were used to try to clarify the unique properties of each class of rabbit GC. Descriptions of GC types 7 and 8, unique to the visual streak, follow. The local edge detectors (LEDs) of the rabbit visual streak had RFs from 0.5° to 2° diameter. They responded to ON and OFF of a spot of light anywhere in the RF. About one third of these units exhibited true directional sensitivity of the image, having a preferred direction. The response characteristic of the streak LEDs was little affected by the background illumination level over a range of 0.08 to 300 cd/m2. If a small (< 1° diameter) dark spot was moved into the LED unit’s RF center and stopped, the unit fired a burst. It fired a burst again when the spot was moved out of the RF in any direction. If the spot was moved through the RF at a constant speed (~3°/s) in any direction, there was little response. A small spot of light flashed in the center of the RF gave both ON and OFF bursts. A larger spot of light covering the entire RF flashed on and off gave negligible response. If a black-on-white, squarewave grating was moved over the entire RF at 1°/s, there was negligible response. If the surround was masked off, and the grating was moved at the same speed, there was a continuous brisk response, maximum for grating periods of 0.5° to 1°. The rabbit local edge detector is probably similar to the frog’s net convexity detector GC, according to Levick (1967). The UDs in the rabbit’s visual streak constituted only 4 out of 154 visual streak GCs studied by Levick. However, they had consistent, curious properties. Their RFs were about 4° in diameter. Under background illumination of 7 cd/m2, the unit fired continuously at 10 to 20 spikes/Sec. Firing could be interrupted during (1) a flash of the entire field (to 50 cd/m2); (2) a flash of a spot over the RF; (3) a flash of an annulus surrounding the RF; (4) moving a white disk object into the RF; (5) moving a black disk object into the RF; (6) moving a square-wave grating over the field. No stimulus could increase the rate of firing. In some UDs, firing suppression was transient at the onset of the stimulus, then increased again. Oyster and Barlow (1967) measured the preferred directions of 102 DS GC units from the rabbit’s retina; 79 of the “on-off” subtype and 23 of the “on” subtype were examined. The distribution of preferred directions for both types of DS GCs were tightly clustered about certain axes. The on-off type units had four preferred directions clustered around the four principal axes of the eye; anterior, posterior, superior, and inferior. The means of the vector directions were all rotated clockwise a small
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amount from these principal axes, as shown in Figure 6.3-1A (Oyster and Barlow, 1967). The “on” type DS GCs, had their preferred directions clustered around three major axes, approximately 120° apart (see Figure 6.3-1B). If superior is 0°, anterior is 90°, and inferior is 180°, etc., the preferred directions are approximately at 110°, 205°, and 345°. Oyster and Barlow speculated that from the preferred directions of the discrete grouping of the two classes of DS GC units, they may be the sensors for a dynamic image stabilizing motor system in which they send signals to the CNS, which in turn activates motor neurons controlling specific extraocular muscles that move the eyeball so that the DS GC unit outputs are nulled.
FIGURE 6.3-1 (A) Approximate distribution of preferred directions from ON/OFF type DS units from the rabbit’s optic nerve. The four major lobes are spaced about 90° apart. (B) Approximate distribution of preferred directions from ON-type DS units from the rabbit’s optic nerve. The three lobes are spaced about 120° apart. (Based on Oyster and Barlow’s, 1967, data.)
DS GC units have been found in just about all vertebrate retinas studied: amphibians, birds, reptiles, fish, and some mammals, but they are rare in cat and primate retinas (Kolb et al., 1999). As seen in the preceding section, DS units have also evolved in the optic lobes of arthropods’ compound eyes. Since insect compound eyes are fixed on their heads, insect DS units are also probably involved with flight stabilization and prey capture.
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6.4
CHAPTER SUMMARY
One important property of vertebrate retinas is that the feature extractions that they perform are designed to lead to the survival of the host animal. Thus, in birds, there are many DS units, no doubt to permit visual flight stabilization and obstacle avoidance. In rabbits, the DS units are organized so that their preferred directions lie grouped around three and four major axes, depending on the DS unit type. Such grouping of preferred directions may be associated with the control of eye movements. (Rabbits sit still to avoid detection by predators, and so must move their eyes to track moving objects.) Certain frog feature extraction operations appear to be associated with detecting flies (food) moving into the visual field. Frogs sit still when they hunt, and wait until prey approaches before striking. Unlike rabbits, frogs cannot move their eyeballs to track their prey. Thus, the entire retina must participate in tracking a moving spot. The vertebrate retina does not fulfill a function similar to the ganglia in the optic lobes. The retina works in concert with the visual tectum and the optical cortex of the brain. The arthropod optic lobes are far older in an evolutionary sense, and have been specialized for the visually mediated survival of their host animal. The retina can be viewed (no pun intended) as a pre-processing network that supplies information to plastic neural networks that can perform learned, cognitive tasks. That feature extraction occurs in both the arthropod optic lobe and in the vertebrate retina argues for the importance of the preprocessing of visual information. Clearly, it has survival value.
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7
Theoretical Models of Information Processing and Feature Extraction in Visual Sensory Arrays
INTRODUCTION This chapter examines some theoretical mathematical models for visual feature extraction in visual systems. The vertebrate retina and compound eye/optic lobe systems operate as spatiotemporal filters. That is, object contrast, size, and shape (its spatial frequency content) and its motion relative to the sensory array are both factors in determining the responses of an output neuron. First examined is the Boolean logic model for visual spatiotemporal filtering described by Zorkoczy (1966). To respond to spatiotemporal properties of simple objects, Zorkoczy’s filters use unit delay elements, as well as conventional AND and OR gates, etc. Zorkoczy sensor arrays have unit spacing, δ degrees, between receptors. The unit velocity is defined as v = δ/T, °/s (T is the unit delay). An object must be moving at v for the filter to produce an output. Next examined is an analog, directional correlation model for moving object detection described by Reichardt (1964). Reichardt’s directional correlator was proposed to describe optomotor behavior in insects; that is, the tendency of the insect to turn and follow an object of moving stripes. The correlator only requires two adjacent receptors, each of whose outputs is delayed and then multiplied by the direct output of the other receptor. The two multiplier outputs are subtracted, and then low-pass-filtered. A third feature extraction model based more closely on retinal anatomy and neurophysiology devised by Fukushima (1969; 1970) is next considered. Fukushima’s model is basically a continuous, linear, spatial filtering model whose analog outputs are non-negative, continuous variables proportional to instantaneous spike frequency. Fukushima’s models are static models; they can “recognize” (detect is a better word) stationary shapes, such as spots, edges, lines at an angle, etc. The concept of a neural spatial matched filter that can detect specific static object shapes is considered in Section 7.2. The matched filter is usually thought of as an engineering communications tool operating in the time domain. The basic matched filter architecture has also been extended into coherent optical signal processing. Its application in neural systems as a prototype model for a static pattern recognizer is considered.
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Finally, this chapter delves into the biological origins of the very active area of artificial neural networks (ANNs). Early ANNs were inspired by what was then known about the neurophysiology of the retina. Rosenblatt’s original Perceptron, then Widrow’s ADALINE and MADALINE, and finally Fukushima’s complex Neocognitron are reviewed.
7.1
MODELS FOR NEURAL SPATIAL FILTERS AND FEATURE EXTRACTION IN RETINAS
Beginning with Ratliff and Hartline’s (1959) description of lateral inhibition in the compound eye of the horseshoe crab Limulus polyphemus and further work in this area by Ratliff et al. (1963), Ratliff et al. (1966) and Lange et al. (1966), it became well established that lateral inhibition could be interpreted in the spatial frequency domain as a linear spatial filtering operation that enhanced contours and improved resolution of the compound eye of Limulus. Bliss and Macurdy (1961) observed that many human visual contrast phenomena, such as Mach’s bands, can be described mathematically in the spatial frequency domain as linear spatial filtering operations, which presumably take place at the retinal and cortical level. Bliss and Macurdy showed how spatial impulse responses (to a point source of light) were related to the spatial filtering operations, and extended their models into discrete space by use of the z-transform. The following section examines a nonlinear, logic-based approach to visual feature extraction system described by Zorkoczy (1966). Zorkocy models used regularly spaced arrays of binary receptors (ON or OFF) that feed into asynchronous, sequential logic circuits with simple operations such as inversion, unit delay, AND, and OR. The features of simple black and white moving objects were discriminated. In Section 7.1.2, the application of two-dimensional, layered neural models emulating linear spatial filters based on the work of Fukushima (1969; 1970) is considered. Again, the vertebrate retina is the basis for these feature extraction models. Fukushima’s models assume five layers of nodes (excluding the receptor layer). Each layer projects non-negative pulse frequency signals to adjacent layers, which are considered non-negative, continuous analog variables in describing their operations. Signals are conditioned by weightings assigned to their paths. More than one path may converge on a node, and signal paths can backpropagate.
7.1.1
THE LOGIC-BASED, SPATIOTEMPORAL FILTER APPROACH ZORKOCZY
OF
The Boolean logical modules used in Zorkoczy’s (1966) model for retinal feature extraction are clocked with a period T. A photoreceptor array lies in a plane with receptor centers on a rectangular grid spaced ∆x = ∆y = δ units apart. The input to the receptor array is some pattern of black and white that changes in time. The Zorkoczy system produces outputs at the clock frequency that depend on the distribution of the pattern in space and its velocity (speed and direction). The receptor at x = kδ, y = jδ has an output a(k, j) = akj of 0 or 1 at t = nT, depending on the
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distribution of the object in space, and its velocity. An object can also be stationary and change its intensity pattern, evoking an output. First described are dyadic operations on two adjacent photoreceptors, a and b, lying on the y axis of the photoreceptor plane. a and b can each be 0 or 1 at each clock cycle. – 1. The elementary contrast operator: c(a,b) = a · b = 1 @ t = nT IF a(nT) = 1 AND b(nT) = 0. This also can be written: c(a,b) = –a · b = 1 @ t = nT IF a(nT) = 0 AND b(nT) = 1. 2. The ON operator: T1(a) = a · –a* = 1 @ t = nT IF a[(n – 1)T] = 0 AND a(nT) = 1, ELSE 0. a* is the output of receptor a one sample period previously. The dot denotes a logical AND operation. The–ON operator can also be used with two adjacent receptors: T1(a,b) = a · b* = 1 @ t = nT IF b[(n – 1)T] = 0 AND a(nT) = 1, ELSE 0. 3. The OFF operator: T2(a) = a* · –a = 1 @ t = nT IF a[(n – 1)T) – = 1 AND a(nT) = 0, ELSE 0. For adjacent receptors: T2(a,b) = a* · b = 1 @ t = nT IF a[(n – 1)T) = 1 AND b(nT) = 0, ELSE 0. 4. The ON/OFF operator: B(a,b) = T1(a,b) + T2(a,b), or B(a) = T1(a) + T2(a). The plus sign denotes a logical OR operation. 5. Multiple input operations include the OR operations: i=N
R=
∑ r = 1 @ nT IF any one or more r (nT) = 1 j
i
7.1-1
i =1
and i=N
P=
∑p
i
= 1 @ nT IF any one or more p i ( nT ) = 1
7.1-2
i =1
6. A contrast operator C operates on two groups – of receptors, P and R, separated by a boundary, f(x, y). C(P, R) = P · R = 1 @ nT if 1 or more elements of receptor set [P] = 1 AND none of set [R] = 1. See Figure 7.1-1. 7. To sense contrasts in both space and time, Zorcoczy defines the functions: N1(P, R) = Σ T1(pi) · Σ T2 ( ri ) = 1 @ nT IF at least one element of [P] was turned ON at t = nT, AND 0 elements of set [R] were turned ON at nT. N2(P, R) = Σ T2 ( p i ) ⋅ Σ T2(ri) = 1 @ nT IF 0 elements of [P] were turned OFF at nT, AND at least one element of [R] was turned off at nT. Note that N1(P, R) and N2(P, R) will not fire for general ON or OFF over [P] and [R]. N1 and N2 operators apply to stationary patterns changing in time only. 8. Now consider the On-Center/Off-Surround operator, which will give a 1 @ nT IFF excitation over [P] (or a subset of it) changes from 0 → 1 at
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FIGURE 7.1-1 Illustration of two possible receptor geometries for a Zorkoczy contrast operator. See text for description.
t = nT, AND simultaneously illumination over [R] remains the same or dims (1 → 0), OR, IF excitation over [R] dims (1 → 0) AND remains constant or brightens (0 → 1) over [P], ELSE 0. See Figure 7.1-2. The On-Center/Off-Surround operator is written N(P, R) = N1(P, R) + N2(P, R).
FIGURE 7.1-2 Illustration of the receptor geometry for an ON-center/OFF-surround Zorkoczy operation. See text for explanation of the Boolean expression.
Similarly, one can create an Off-center/On-surround operator. Define:
()
7.1-3
()
7.1-4
F(P, R) = F1(P, R) + F2(P, R)
7.1-5
F1 ( P, R ) ≡ Σ T2 ( p i ) ⋅ Σ T2 rj and
F2 ( P, R ) ≡ Σ T1 ( p i ) ⋅ Σ T1 rj so the Off-center/On-surround operator is
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The above operations deal with time-varying but spatially fixed patterns of excitation on receptors. Another important input modality is the motion of a fixed pattern shape of contrasting (0.1) excitation in a specified direction at unit velocity, u. (冨u冨 = δ/T.) 9. The direction of movement, y, of an object may be sensed by an application of the T1(a, b) and T2(a, b) operators, and a sensitivity to increased or decreased excitation at a point passed by the moving edge (contrast boundary) by using the T1(a) and T2(a) operators. Thus,
M 1 (a, b) ≡ T1 (a ) ⋅ T1 ( b) = 1 *
7.1-6
@ t = nT IFF the excitation at a goes 0 → 1 @ nT AND this event is NOT preceded at t = (n – 1)T by brightening (0 → 1) at b. [Note: The M1 = 1 primary requirement is brightening at a at t = nT; any time course and change of illumination at b is permissible to get output except brightening at t = (n – 1)T.] For motion causing a decrease in excitation going from a to b,
M 2 (a, b) = T2 (a ) ⋅ T2 ( b)
*
7.1-7
Note that both M1 and M2 operators are dyadic and one dimensional; they both require unit velocities, u, to work. M1 and M2 operators respond ambiguously to ON (M1 = 1) and OFF (M2 = 1) over a and b together. Example 7.1-1 Now some illustrated examples of Zorkoczy operators are examined. The first example will consider the properties of a relation given by Zorkoczy (1966) applied to a linear, one- dimensional array of receptors with unit spacing, δ = v T mm, where v is the unit velocity and the clock period T = * is the signal delay operator in the array. The M(A) operator is defined by
M(A) =
n −1
∑ ⎡⎢⎣B(a ) ⋅ B(a k
k =1
) ⎤⎥⎦ *
k +1
7.1-8
Figure 7.1-3 illustrates the system. As described above, B(ak) is the ON/OFF operator on the kth receptor; the asterisk implies that the signal from B(ak+1) is delayed by one clock period, the overbar signifies logical inversion, the dot logical AND-ing, and the summation, a mass OR-ing. Examine the output of the big, n-fold OR gate for the following inputs: M(A) = 0 For general ON over all n receptors in the line. M(A) = 0 For general OFF over all m receptors in the line.
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FIGURE 7.1-3 A Zorkoczy system that detects contrasting boundaries moving at unit velocity from right to left. Black or white unit spot objects, and black/white or white/black objects, all with v = –1 have the same response.
M(A) = 0 For any stationary pattern of light and dark over the line of n receptors. M(A) = 0 For a light/dark boundary moving to the right at unit velocity. M(A) = 0 For a dark/light boundary moving to the right at unit velocity. M(A) = 011111 … 1 For a dark/light or a light/dark boundary moving to the left at unit velocity. The first zero in the output sequence is because no output occurs when the contrast change first hits the an receptor. Output starts when it hits the (n – 1)th receptor. M(A) = (011111 … 1) For a dark or light spot of width δ moving with v = –1 (to the left). Example 7.1-2 For a second example, consider the Zorkoczy operation given by the triple product:
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VD (A) =
n −1
∑ [T (a ) ⋅ T (a ) ⋅ T (a )] *
2
k
1
k
2
k +1
7.1-9
k =1
This system is illustrated in Figure 7.1-4. Note that there is a triple AND-ing of the T operator outputs. Again, it is possible to list the outputs for various inputs, stationary and moving: VD(A) = 0 For general ON over all n receptors in the line. VD(A) = 0 For general OFF over all n receptors in the line. VD(A) = 0 For any stationary pattern of light and dark over the line of n receptors. VD(A) = 0 For a light/dark boundary moving to the right at unit velocity. VD(A) = 0 For a dark/light boundary moving to the right at unit velocity. VD(A) = 0 For a light/dark boundary moving to the left at unit velocity. VD(A) = 0 For a dark/light boundary moving to the left at unit velocity. VD(A) = 0 For a light unit spot moving to the right at unit velocity. VD(A) = 0 For a light unit spot moving to the left at unit velocity. VD(A) = 0 For a dark unit spot moving to the left at unit velocity. VD(A) = (1111 … 1) For a dark unit spot moving to the right at unit velocity. Thus, VD(A) detects only a unit dark spot moving to the right with unit velocity. Note that if one interchanges the T1 (ON) operators with the T2 (OFF) operators in Figure 7.1-4, the system detects only unit light spots moving to the right at unit velocity. What the Zorkoczy models suggest is that spatiotemporal feature extraction, and the sensing of object velocity, may involve the systematic basic operations of OFF and ON at the single receptor level, and that these outputs are further conditioned by delays and sent to adjacent receptor processing modules where they are appropriately combined. By considering the shape of the receptor arrays (e.g., lines, areas, etc.), it is possible to extend one-dimensional Zorkoczy models to respond to two-dimensional, contrasting objects, either stationary or moving in a particular direction on the receptor array. The formal, binary Zorkoczy models are not supposed to emulate biological behavior, but instead to suggest continuous, fuzzy, nonlinear, neural configurations that might be modeled as candidates for feature extraction circuits.
7.1.2
ANALOG MODELS
FOR
MOTION DETECTION
IN INSECTS
A whole branch of sensory neuroethology is devoted to the study of how animals respond to moving visual stimuli. Such behavior is called the optomotor response of the animal. The optomotor response is generally a hard-wired, reflex, orientation behavior of the eyes, head, and/or body in response to a visual object such as vertical, black and white stripes given uniform angular velocity around the animal. The optomotor responses of flies, beetles, bugs and locusts have been widely studied, both as a means to determine the resolving power of the eyes and to try to model the neural mechanisms responsible for this reflex. In general, insects try to follow
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FIGURE 7.1-4 Another Zorkoczy system that uniquely senses unit dark spots moving with v = +1 across the receptive field. See text for analysis.
moving objects in such a manner as to keep the image on the same part of their compound eyes, much the same way humans track a moving object with their eyes and head to keep an image on their foveas. One early method used to measure the optomotor behavior of beetles used the Y-globe maze. A beetle was suspended inside a vertically striped drum by gluing its back to a fixed, vertical support. The beetle reflexively grasped a Y-globe maze and began to walk. (See Figure 7.1-5 for a description of this setup.) If the drum were stationary, there is about a 50% probability that the beetle will turn right vs. left at each Y junction. When the striped drum is given a uniform velocity, e.g., clockwise looking down on the beetle, the beetle tends to select more right turns than left, trying to follow the stripes. As a measure of the beetle’s directional preference, workers have defined a “reaction” parameter, R = (W – A)/(W + A), where in an experimental run with the drum turning at fixed velocity, W is the number of times the beetle turns with the direction of the drum, and A is the number of times it turns opposite to the drum. A + W = N, the total number of times the beetle turns at a Y. · · It has been found experimentally when plotting R(θ) vs. log(θ) that R makes a bellshaped curve with a peak whose value depends on stripe period and contrast. Figure 7.1-6 illustrates the optomotor reaction for the milkweed bug, Oncopeltus fasciatus,
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FIGURE 7.1-5 Schematic drawing of a Y globe optomotor testing apparatus. A restrained beetle holds a Y globe while viewing a moving object (in this case, vertical stripes moved at constant velocity toward the beetle’s right). As the beetle “walks,” it rotates the Y globe toward it. When the beetle comes to a Y junction, the beetle generally chooses the right-hand path if the stripes are moving to the right. That is, it tries to follow the stripes.
responding with a Y maze to different rotation velocities of a surrounding drum with black/white stripes with a 20° period (Bliss et al., 1964). Figure 7.1-7 illustrates a typical Chlorophanus beetle’s, Y- maze, optomotor R-response to continuously moving, sinusoidal stripes of fixed spatial wavelength (Reichardt, 1964). Note that the curve has a single broad peak and falls off for high drum velocity. The heads of most insects are so articulated that they can swing from side to side in the horizontal plane (yaw), rotate (roll) around the body axis, and nod up and down (pitch). John Thorson (1966a, b) examined the head optomotor response in locusts by measuring the head roll torque in response to sinusoidal striped drum rotation. The drum rotational axis was aligned with the animal’s long body axis, and the drum was centered on and enclosed the head. The animal’s body was fixed down, but the head was free to turn. A torque-measuring sensor was attached to the front of the locust’s head. In one set of data, the stripe period was 8°. The peak-to-peak amplitude of the striped drum oscillation was a tiny 0.03°. Even at this very low input amplitude, the reflex head roll torque was measured reliably over a drum frequency range of 0.01 to 6 Hz. In most cases, the ratio of peak-to-peak, sinusoidal neck torque to peak-to-peak sinusoidal drum amplitude peaked at ~0.5 Hz. The phase was nonminimum. Thorson (1966b) gave several Bode plots of 20 log (peakto-peak neck torque/reference torque) showing this peak response. The Bode plot
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FIGURE 7.1-6 Record of Y globe turning reaction index, R, for the milkweed bug, Oncopeltus fasciatus, as a function of stripe velocity. The restrained bug and its Y globe were in the center of a rotating, striped drum. Note that the striped drum speed that elicits a maximum turning reaction is between 10° and 20°/s. The stripe period was 18°. R ≡ (C – I)/(C + I), where C is the total number of correct turns the beetle makes at a “Y” (i.e., to the right if the stripes are rotating to the right, as seen by the beetle) and I is the total number of incorrect turns at Ys (to the left). (From Bliss, J.C. et al. Final Report for Contract AF49C638)–1112, Stanford Research Institute, Menlo Park, CA, 1964.)
FIGURE 7.1-7 Representative graph of the average Y globe turning index, R, of the beetle Chlorophanus, as a function of drum speed. (Based on a graph from Reichardt, 1964.) A vertical, sinusoidal intensity pattern with a 4.7° period was used.
sloped up to the peak at about +20 dB/decade at low frequencies, and the highfrequency attenuation slope was between 30 and 40 dB/decade. Response was down about 30 dB from the peak at 5 Hz. Thorson (1966a) examined the stripe period that gave maximum neck torque response at a given oscillation frequency (0.25 Hz) and amplitude (0.03°) for both roll and yaw optomotor responses. For yaw, the peak torque amplitudes occurred for stripes with periods between 7.5° and 9°; peak roll torques required stripes with periods between 9.5° and 12°. The cutoff period was about 4° for roll torque, and 3° for yaw torque. These values agree with the limiting resolution for the locust’s third cervical nerve electrophysiological responses to moving stripes. (The left and right, third cervical nerve innervate neck muscles that move the locust’s head.) © 2001 by CRC Press LLC
Northrop (1975) found that below about a 4° stripe period, there was no significant firing on the third cervical nerve when the stripes were moved in front of one eye (the other was covered). A freely-flying fly’s body has six degrees of freedom: rotational — roll, pitch, and yaw — and linear translational — forward, sideways, and vertical. Dragonflies are adept at lateral movements; houseflies are not. In addition to the six degrees of freedom for body movement, the fly’s head can also move with respect to the body. Such head movements are roll around the body axis, side-to-side movement (yaw), and up and down (pitch). Thus, to describe the motions of a freely flying fly, nine vectors or dimensions as functions of time are required. Clearly, tethered flight under conditions of head immobilization offers great simplification by restricting measured parameters to either yaw torque or thrust and lift forces. In a series of papers beginning in 1956, Werner Reichardt and co-workers extensively investigated the optomotor responses of tethered, flying flies. A fly’s back was glued to a vertical probe attached to a torque sensor that measures visually induced yaw torque. The fly is suspended in the center of a display cylinder having one or more contrasting stripes, and caused to beat its wings as if in flight. When the stripes or visual object are moved at constant velocity, or oscillated back and forth sinusoidally, the fly generates yaw torque trying to follow the stripe(s). Reichardt’s experimental system was similar to Thorson’s described above, except the fly flies in place, its head fixed rigidly to its body. An overview of Reichardt’s work can be found in chapter 17 of Sensory Communication (1964). The underlying model for fly optomotor response in this and all of Reichardt’s many papers on fly optomotor response is his dyadic, directional correlator (DDC) model, first proposed in 1956 to describe optomotor behavior. The pooled outputs of many parallel DDCs are postulated to drive the motoneurons responsible for optomotor turning. A simplified example of the DDC is shown in Figure 7.1-8. Unlike the Zorkoczy models, the Reichardt directional correlator is a purely analog system. The two inputs are signals proportional to the luminous flux on two adjacent “receptors.” The model receptors are aligned along the direction of the object motion that is to be sensed. They have a linear spacing of δmm. The analog signal output of the A receptor is delayed by ε seconds, and is multiplied by the direct analog output of receptor B. The output of the multiplier, y, is subtracted from the output of the right-hand multiplier, y′, and the difference, ∆y, is time-averaged by a low-pass filter to form ∆y . The mathematics of this model system are examined below. In deriving an output/input relation for the DDC, for simplicity, assume onedimensional, linear geometry, rather than the natural angular coordinates. Also, assume that the two receptors determine a line in the direction of object motion; and assume the object is a one-dimensional, moving, sinusoidal distribution of intensity: I(x, t) = Io{1 + sin[(2π/λ)(x – v t)]} = Io{1 + sin[(2πx/λ) – ωt]}
7.1-10
where λ is the sine period in mm, v is the translational velocity of the moving sinewave in mm/s, and the temporal frequency of the sinewave is ω ≡ 2πv/λ rad/s. The object intensity ranges from 0 to 2Io, depending on the spatiotemporal argument of the sin(*)
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FIGURE 7.1-8 Block diagram of a DDC. It is assumed that many DDCs exist in a compound eye, and that their outputs are effectively added together. The two receptors of every DDC are aligned in the preferred direction, and separated by δ°. A detailed algebraic analysis of the operation of a DDC’s is given in the text. The DDC was first proposed by Reichardt in the mid-1950s.
function. Note that the traveling wave can be decomposed by trigonometric identity to functions of time multiplied by functions of distance. Because sin(A – B) ≡ sin A cos B – cos A sin B, one can write: I(x, t) = Io{1 + sin(2πx/λ) cos(ωt) – cos(2πx/λ) sin(ωt)}
7.1-11
The output of receptor A is delayed by ε seconds (pure transport lag); the output of receptor B is displaced in space by δ mm, so one can write the left multiplier output, y, as
{
}
y = k 2 l o2 1 + sin(2 πx λ ) cos[ω(t − ε )] − cos(2 πx λ ) sin[ω(t − ε )]
{1 + sin[(2π λ)(x − δ)] cos(ωt) − cos[(2πx λ)(x − δ)] sin(ωt)} Expanding yields
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7.1-12
{
[
]
[
]
y = k 2 l o2 1 + sin (2 π λ )(x − δ ) cos(ωt ) − cos (2 π λ )(x − δ ) sin(ωt ) + sin(2 πx λ ) cos[ω(t − ε )] − cos(2 πx λ ) sin[ω(t − ε )]
[ ] − sin(2 πx λ ) cos[ω(t − ε )] cos[(2 π λ )(x − δ )] sin(ωt ) − cos(2 πx λ ) sin[ω(t − ε )] sin[(2 π λ )(x − δ )] cos(ωt )
+ sin(2 πx λ ) cos[ω(t − ε )] sin (2 π λ )(x − δ ) cos(ωt )
[
]
7.1-3
}
+ cos(2 πx λ ) sin[ω(t − ε )] cos (2 π λ )(x − δ ) sin(ωt )
Now take the time average of y. The first four sin(*) cos(*) terms contain simple zero-average sinusoid terms averaging to zero, so they can be ignored. Thus,
{
[
]
y = k 2 l o2 1 + sin(2 πx λ ) sin (2 π λ )(x − δ ) cos[ω( t − ε )] cos(ωt )
[
]
[
]
[
]
− sin(2 πx λ ) cos (2 π λ )(x − δ ) cos[ω(t − ε )] sin(ωt )
7.1-14
− cos(2 πx λ ) sin (2 π λ )(x − δ ) sin[ω(t − ε )] cos(ωt )
}
+ cos(2 πx λ ) cos (2 π λ )(x − δ ) sin[ω(t − ε )] sin(ωt )
Now, use the trigonometric identities, cos α cos β ≡ 1/2 cos(α – β) + 1/2 cos(α + β), sin α sin β ≡ 1/2 cos(α – β) – 1/2 cos(α + β), sin α cos β ≡ 1/2 sin(α – β) + 1/2 sin(α + β) to write:
[
]
[
]
y = k 2 l o2 {1 + sin(2 πx λ ) sin (2 π λ )(x − δ ) ( 1 2 ) cos(ωε ) + cos(2ωt + ε )
[ ] − cos(2 πx λ ) sin[(2 π λ )(x − δ )]( )[sin( −ωε ) + sin(2ωt + ε )] + cos(2 πx λ ) cos[(2 π λ )(x − δ )]( )[cos( −ωε ) − cos(2ωt + ε )]} [
]
− sin(2 πx λ ) cos (2 π λ )(x − δ ) ( 1 2 ) sin(ωε ) + sin(2ωt + ε ) 1
2
1
↓
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2
7.1-15
[
]
y = k 2 l o2 {1 + sin(2 πx λ ) sin (2 π λ )(x − δ ) ( 1 2 ) cos(ωε )
[ ] + cos(2 πx λ ) sin[(2 π λ )(x − δ )](
− sin(2 πx λ ) cos (2 π λ )(x − δ ) ( 1 2 ) sin(ωε )
[
1
2
7.1-16
) sin(ωε)
}
]
+ cos(2 πx λ ) cos (2 π λ )(x − δ ) ( 1 2 ) sin(ωε ) Now, examine the output of the right-hand multiplier, y′:
y ′ = k 2 l 2o {1 + sin(2 πx λ ) cos(ωt ) − cos(2 πx λ ) sin(ωt )}
{1 + sin[(2π λ)(x − δ)] cos[ω(t − ε)] − cos[(2π λ)(x − δ)] sin[ω(t − ε)]}
7.1-17
Again, after applying trigonometric identities and taking the time average,
[
]
y = y = k 2 l o2 {1 + sin(2 πx λ ) sin (2 π λ )(x − δ ) ( 1 2 ) cos(ωε )
[ ] − cos(2 πx λ ) sin[(2 π λ )(x − δ )](
+ sin(2 πx λ ) cos (2 π λ )(x − δ ) ( 1 2 ) sin(ωε )
[
1
2
7.1-18
) sin(ωε)
]
}
+ cos(2 πx λ ) cos (2 π λ )(x − δ ) ( 1 2 ) cos(ωε )
The DDC output is taken as ∆y = y ′ − y . Combining the terms, using trigonometric identities, and noting that ω ≡ 2πv/λ, finally yields
∆y = ( 1 2 ) sin(2 πδ λ ) sin(2 πε v λ )
7.1-19
Thus, the DDC output to a moving sinewave object shows the first positive peak for v = λ/4ε mm/s (positive v for object motion from left to right). The first negative peak occurs for object motion from right to left at v = –λ/4ε. ∆y also increases as the ratio δ/λ increases. As shown above, the DDC response to a moving, spatial, sinewave object demonstrates that the DDC works as a model motion detector. How can the basic DDC operation be realized in the optic lobes of insects? The angular separation between ommatidia gives the parameter, δ. The temporal delay, ε, might be associated with the propagation time for an analog signal along a thin dendrite or axon. The multiplier is harder to conceptualize. It might involve a linear summation of two logarithmic signals at a nerve membrane, then an exponential operation in the spike generation process to give the antilog product in terms of spike frequency. (Such a y ≅ ln(A) + ln(B) operation, followed by f Ý K exp(y) spike generation was
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proposed by the author in 1975 for a model of signal-to-noise improvement in insect vision.) The outputs of many such dyadic machines would have to be combined so that the whole compound eye displays directional sensitivity, as in nature. The interested reader wishing to pursue this topic further should consult the papers by Martin Egelhaaf (1985a, b, c) on theoretical neuronal models for motion detection circuits in the fly’s optic lobes, as well as the many papers authored or co-authored by Werner Reichardt.
7.1.3
CONTINUOUS, LAYERED VISUAL FEATURE EXTRACTION FILTERS
The fact that a three-dimensional image (intensity, x, y) can be processed using coherent light by successive layers of Fourier optical transform filters to extract image features was described in the engineering literature by Cutrona et al. (1960) and by vander Lugt (1964). The idea that visual feature extraction can proceed in an analogous manner using layers of discrete neurons was first set forth by Fukushima (1969; 1970). Fukushima’s models were inspired by a long series of papers on the neurophysiology of vision in various vertebrates, including frogs (Lettvin et al. 1959), rabbits (Barlow, 1963; Barlow and Hill, 1964; Levick, 1967), cats (Rodiek and Stone, 1965; Rodiek, 1965; Hubel and Wiesel, 1959; 1962; 1965), and also in the horseshoe crab, Limulus (Ratliff, 1964). Fukushima’s (1969) model for visual feature extraction used six layers of neural signal space, including a two-dimensional receptor layer. It was a static model; i.e., no object motion was assumed, and the object did not change in time. Although the signal nodes in the layers were discrete, Fukushima’s approach was to assume their density sufficiently high to justify the simplification of continuous, analog representation of signals and their transformations. Because the signal at each node is assumed to be a spike frequency, a non-negativity operator, ϕ(u), is assumed, following the diagram in Figure 7.1-9. Figure 7.1-10 illustrates the signal node layers. Operations on visual information are performed by the connection weights connecting the nodes between layers. These operations are shown more clearly in the block diagram of Figure 7.1-11. The system is truly nonlinear because the signal at every node must be non-negative (specifically, the ϕ{*} operator). In Fukushima’s (1970) model, the visual object is assumed to have bright details only, i.e., white lines on a dark field. The signals in the first, u0(x, y), receptor layer are processed by the weighting function, c1(x, y) to yield u1(x, y), where low spatial frequencies are attenuated and object contrast is enhanced. A node at some xo, yo in the u1 plane is connected to nodes in the receptor plane such that a point source of light, Io δ(x, y), moved over u0(x, y) maps the weighting function, c1(x – xo, y – yo). This weighting function is of the ON-center/OFF-surround type, shown for the u1(0, 0) node in Figure 7.1-12. The signal at the nodes in the u1(x, y) layer is given by the two-dimensional, real convolution:
⎧ u1 (x, y) = ϕ ⎨ ⎩
⎫ u 0 (ξ, η) c1 (x − ξ, y − η) dξ dη⎬ = ϕ{u 0 (x, y) * *u1 (x, y)} 7.1-20 S1 ⎭
∫∫
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FIGURE 7.1-9 Treatment of signals at a node in a layer of a Fukushima static visual feature extraction model. k non-negative signals {ui} from nodes in previous layers are multiplied by k weights {ci} and added together to form a signal s. s can be positive or negative, depending on the values of the k, {ui ci} values. Because the signals at the nodes represent “neuron” spike frequency, the actual signal at the output node is half-wave-rectified to form uo = ϕ(s) = ϕ{Σ kk =1 u i c i } .
If u1(x, y) Š 0 over the surface, S1, then u1(x, y) = u0(x, y) ** u1(x, y)
7.1-21
Real convolution of two Fourier-transformable functions can be shown to be equal to the inverse Fourier transform of the product of their Fourier transforms providing u1(x, y) Š 0. u1(x, y) = Ᏺ1[C1(u, v) U0(u, v)]
7.1-22
Where u and v are the spatial frequencies in rad/mm. There are several ways to model the “Mexican hat” weighting function mathematically. One can simply add two Gaussian functions with different peak amplitudes and standard deviations. For example, in one dimension,
⎡ x2 ⎤ ⎡ x2 ⎤ c1 (x) = A exp ⎢ 2 ⎥ – B exp ⎢ 2 ⎥ , ⎣ 2σ a ⎦ ⎣ 2σ b ⎦
σb > σa
7.1-23
A plot of c1(x) is shown in Figure 7.1-12. The Fourier transform of c1(x) above gives the spatial frequency response of the c1 system. u is the spatial frequency in rad/mm.
[
]
[
C1 ( u) = A σ a 2 π exp −( 1 2 )u 2 σ a2 − Bσ b 2 π exp −( 1 2 )u 2 σ 2b
]
7.1-24
A plot of C1(u) is shown in Figure 7.1-13. Two points are worth noting: (1) The c1 filter has zero dc response when A/B = σb/σa; (2) The frequency response of the c1 filter is bandpass, with a peak at
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FIGURE 7.1-10 Organization of a discrete, six-layer Fukushima static feature extractor. The first layer consists of an array of discrete receptors whose outputs, Uo(m∆x, n∆y), are non-negative analog signals proportional to the object intensity, I(x, y). The lines going from nodes in signal plane (k – 1) to plane k are signal conditioning weights, {cki}, k = 1, … 5, i = 1, … N.
u pk =
(
2 ln σ a2 σ 2b
(σ
2 a
− σ 2b
)
)
r mm
7.1-25
The latter result is found by taking the derivative of C1(u2) with respect to u2 and setting it equal to zero, then finding upk. Inspection of Figure 7.1-13 shows that low spatial frequencies in the object are attenuated in favor of frequencies around upk; then high spatial frequencies are again attenuated. This boost of spatial frequencies around upk can be interpreted as the c1 filter boosting the contrast of edges, contours, and boundaries.
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FIGURE 7.1-11 Block diagram of a continuous, five-layer Fukushima static feature extractor. For mathematical convenience in using continuous Fourier transforms, let the spacing between nodes ∆ → 0 in the limit, and also the number of nodes → ×, generating a continuous system in x, y coordinates. Thus, a signal in the kth plane is related to that in the (k – 1)th plane by uk(x, y) = uk-1(x, y) ⊗⊗ ck(x, y). ⊗⊗ denotes the operation of two-dimensional, real convolution. All uk are non-negative by the ϕ{•} operation (half-wave rectification).
FIGURE 7.1-12 An ON-center/OFF-surround, one-dimensional weighting function, c1(x), can be shown to be the sum of a positive and a negative Gaussian function (see Equation 7.123).
Note that other models for ON-center/OFF-surround, c1 weighting functions can be used. For example, in one dimension: c1(x) = A exp(– α 冨x冨) – B exp(– β 冨x冨)
7.1-26
Where A > B, α > β, and A/α ≡ B/β for zero response to zero frequency (background) light on the U0 layer. For the dc stop condition, C1(u) can be shown to be bandpass:
C1 ( u ) = At mid-frequencies,
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2(Aα − Bβ) u 2
(u
2
)(
+ α 2 u 2 + β2
)
7.1-27
FIGURE 7.1-13 Plot of the continuous Fourier transform of c1(x), C1(u). Note that low spatial frequencies are attenuated. See text for discussion.
C1 ( u mid ) ≅
2(Aα − Bβ) α2
7.1-28
Next, consider Fukushima’s model for simple line detection. A weighting function, c2(x, y, α) is defined connecting signal nodes in the bandpass-filtered image layer, u1, with nodes in the simple line detection layer, u2. Here α is the angle a bright line makes with the x-axis in the receptor plane. (To detect lines with 15° angular spacing would require 12 independent c2 filters.) As before, the signals in the u2 layer are given by
⎧ u 2 (x, y, α ) = ϕ ⎨ ⎩
⎫
∫∫ c (x, y, α) u (x − ξ, y − η) dξ dη⎬⎭ S2
2
1
7.1-29
Figure 7.1-14, taken from Fukushima (1970), illustrates a typical c2(x, y, α). Note that this figure is a contour plot, i.e., the contours of c2 are seen from “above.” This weighting function is an elongated ON-center/OFF-surround type, symmetrical around the α-direction axis. This c2 can be made up from the difference of two, two-dimensional, elliptical, Gaussian functions. Remember that there are many parallel, overlapping, receptive fields (RFs) of u2 nodes having a given α, so that if an α-oriented line is presented anywhere over u0(x, y), at least one or more signal nodes in the u2 layer will be active. If the line lies outside a particular u2 RF boundary, or lies in the RF but is perpendicular to the α-axis, there will be no output from the node in the u2 plane that has that particular c2 RF. Maximum output occurs when the line axis is at α° and the line is centered over the particular c2 RF. If the line is centered over the RF but at an angle such as α ± 30°, there will be a slight output from the corresponding u2 unit.
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FIGURE 7.1-14 Contour plot of Fukushima’s c2(x, y, α). This weighting function (WF) selects white lines at an angle of α° with the horizontal (x-axis) on the ON-center/OFFsurround RF.
The next layer in Fukushima’s model feature extractor is the complex-type line detector. This operation occurs as a result of signal connectivities, c3(x, y, α) receiving inputs from nodes in layer u2, and producing outputs at nodes in layer u3. One can again write the nonnegative convolution:
⎧ u 3 (x, y, α ) = ϕ ⎨ ⎩
⎫
∫∫ c (x, y, α) u (x − ξ, y − η) dξ dη⎬⎭ S3
3
2
7.1-30
A node, u3(xo, yo, αo), responds to a bright line of orientation αo, but in contrast to nodes in the u2 layer is not sensitive to the location of the line in the receptor plane. Fukushima states, “These elements [nodes] correspond to complex cells in the [cat’s] visual cortex.” Each u3 node effectively adds the outputs of a large number of u2 nodes with similar α sensitivity. Thus, if a point source input is used, a u3 node is found that responds to light over a large, narrow area of u0, perpendicular to α. Thus, a u3 node receives a large number of u2 outputs responding to a line of orientation α over a large area of u0. The c3(x, y, α) weighting function for a u3 node is shown in Figure 7.1-15A and B.
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A
B FIGURE 7.1-15 (A) Contour plot of Fukushima’s c3(x, y, α), the complex line detector of layer 3. u3 elements respond to a bright line of orientation α anywhere over the receptor layer. See text for discussion. (B) The same c3 WF as in A, portrayed as a three-dimensional surface.
7.1.4
DISCUSSION
This section has examined some speculative models advanced by Zorkoczy (1966), Reichardt (1964), and Fukushima (1969) that have attempted to describe certain aspects of feature extraction observed in vertebrate and insect visual systems. Zorkoczy’s models are particularly interesting because he introduces the basis for spatiotemporal filters that respond to an object’s shape as well as its velocity. The use of a delay between signals from adjacent receptors appears as a recurrent theme in many other visual receptor array models exhibiting DS. Indeed, Reichardt’s basic
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neural correlator model (see Figure 7.1-8), used to describe insect optomotor responses not only uses delays, but also signal multiplication. (A review of DS models can be found in Kien, 1975.) Fukushima’s pattern recognition models work on static, nonmoving objects. They are based on the neural equivalents of Fourier optic image processors. Continuous equivalents of discrete weighting functions are convolved with signals in a sequence of planes to extract an object’s features such as straight lines at some angle to a reference axis (such as vertical). While these models may appear simple, they laid the groundwork for and were the basis of subsequent innovations in machine vision and visual signal processing by artificial neural networks (ANNs).
7.2
MODELS FOR NEURAL MATCHED FILTERS IN VISION
A central question in cognitive psychology is how do animals recognize visual objects that they have learned? That is, how do they recall the name and the learned properties of an image on the retina? The image might be a shape, a letter of the alphabet, a face, an automobile, etc. Image properties and features such as relative size, shape, color, and contrast all must figure in the identification/recognition process. Clearly, information stored in the CNS is compared with the incoming information to arrive at a match or probable match. This section examines a simple, highly-speculative model for object recognition, i.e., the neural matched filter. Before examining the application of a matched filter model to the recognition of elementary visual objects, it is important to note that application of the matched filter concept, as used in communications, requires the assumption of a linear system. However, an animal’s CNS is in general quite nonlinear, although piecewise linearity might be argued under certain input operating conditions. The matched filter was originally developed as a tool for communications in the time domain (Schwartz, 1959). It permits the design of the statistically optimum filter to allow detection of a singular event (such as a pulse or small group of pulses) combined with broadband, Gaussian noise. Curiously, the exact implementation of matched filters in the time domain is not possible because of causality. However, the exact implementation of realizable matched filters in the space domain is possible. They can also be extended to two dimensions and be discretized (Papoulis, 1968). The big question is, does pattern recognition in animal visual systems rely on some form of matched filter operation? The design of a spatial matched filter (SMF) model is now examined with a view to evaluating it as a candidate visual signal-processing strategy.
7.2.1
THE CONTINUOUS, ONE-DIMENSIONAL SPATIAL MATCHED FILTER
An image of intensity s(x) is projected on the x-axis (assume it is a one-dimensional, continuous “retina”). Additive Gaussian white noise, n(x), also impinges on the
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x-axis. In general, f(x) = s(x) + n(x). f(x) is processed by a linear, SMF, hm(x), giving an output, g(x). Thus, because real convolution in the space domain is equivalent to multiplication in the frequency domain,
g(x o ) =
∫
∞
f ( v) h m (x o − v) dv =
−∞
1 2π
∫
∞
F( ju) H m ( ju) exp( jux o ) du
7.2-1
−∞
The noise is assumed to have a two-sided power density spectrum, φnn(f) = η/2 mean squared watts/cycle/mm. Thus the total noise output power of the SMF, No, can be written:
No =
∫
∞
−∞
( η 2) H m ( ju)
2
df
mean squared watts
7.2-2
Note that u ≡ 2πf (rad/mm ≡ 2π cycles/mm). The SMF output due to the signal is given by the inverse Fourier transform:
g s (x) =
1 2π
∞
∫ S( ju) H −∞
m
( ju) exp( jux) du
7.2-3
The magnitude squared of gs(x) at a particular position, xo, is
g s (x o ) = 2
∫
1 4π 2
∞
S( ju) H m ( ju) exp( jux) du
2
7.2-4
−∞
xo is chosen so 冨gs(x)冨 is a maximum. That is,
ρ=
g s (x o )
2
is maximum at x = x o
No
7.2-5
That is, the local squared peak signal-to-noise ratio at x = xo is maximum. In other words, the SMF acts to give a maximum output when a particular s(x) for that Hm(ju) is present. The SMF system does not try to recover and reproduce the image, s(x), but only detects its presence by examining ρ(xo) for a maximum. The input signal, s(x), must have finite “energy.” By Parseval’s theorem,
E=
∫
∞
s 2 (x) dx =
−∞
∫
∞
−∞
S( ju) du = 2
∞
∫ S * ( ju) S( ju) du
7.2-6
−∞
To find the desired SMF weighting function or transfer function, examine the conditions whereby the ratio below is maximized:
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g s (x o ) NoE
2
2
∞ 1 S( ju) H m ( ju) exp( jux) du 2 4 π −∞ = ( η 2) ∞ H ju 2 df ∞ S ju 2 du ( ) ( ) 4 π 2 −∞ m −∞
∫
∫
7.2-7
∫
To maximize the ratio, use Schwarz’s inequality:
∫
∞
2
X( ju) Y( ju) du ≤
−∞
∫
∞
−∞
X( ju) du 2
∫
∞
−∞
Y( ju) du 2
7.2-8
One way of interpreting Schwarz’s inequality for complex integrals is to use the familiar vector inequality: 冨a + b冨 ð 冨a冨 + 冨b冨. Equality is satisfied if a = Kb, or if a and b are collinear. In the case of complex functions, it can be shown that equality is satisfied if Y(ju) = K X*(ju)
7.2-9
(See, for example, Schwartz, 1959, for a detailed proof of Schwarz’s inequality.) To apply Equation 7.2-9 to the SMF problem, we note that X(ju) = S(ju) exp(juxo) and Y(ju) = Hm(ju). Thus, the ratio, (η/2) 冨g(xo)冨2/(E No) must be ð 1, and
∫
∞
2
S( ju) H m ( ju) exp( jux) du ≤
−∞
∫
∞
−∞
H m ( ju) df 2
∫
∞
−∞
S( ju) du 2
7.2-10
To find the form of the SMF, note that the ratio, Equation 7.2-7, is maximized when the equality holds, which is when Hm(ju) = K [S(ju) exp(juxo)]* = K S*(ju) exp(–juxo)
7.2.11
The asterisk denotes taking the complex conjugate of the vector (e.g., if S = p + jq, then S* = p – jq). Now if s(x) is an even function, i.e., s(x) = s(–x), then clearly S(ju) = S(u) is real and even, and S*(ju) = S(u). Thus, Hm(ju) = K S(u) exp(–juxo). It follows that for the even s(x), hm(x) = F–1{Hm(ju)}, and hm(x) = K s(x – xo). If s(x) is not even, then in general, hm(x) = K s[–(x – xo)]. That is, s(x) is reversed in x and shifted some xo (see Figure 7.2-1). Example 7.2-1 As a first example of an SMF calculation, consider a one-dimensional, continuous SMF for an image that is a light spot of radius r, centered at x = 0. Thus, s(x) = Io[U(x + r) – U(x – r)] is a pulse of height Io, which has the well-known Fourier transform, S(u) = (Io2r) sin(ru)/(ru), which is real. g(x) is given by real convolution.
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FIGURE 7.2-1 Plots of the one-dimensional, input signal intensity, s(x), reversed and translated in the form for real convolution. s(x) is made asymmetrical for illustrative purposes.
g(x) =
∞
∫ s(v)h −∞
m
(x − v) dv
7.2-12
The SMF is also a pulse of radius r centered at x = 0, and height KIo. In the real convolution process, shown in Figure 7.2-2, g(x) emerges as a triangle of base 4r and a peak of height KI2o 2r at x = 0. Interestingly, if the input object has width 2w, where w > r, the height of gs(x) will still be KI2o 2r, maximum at x = 0. The triangular shape of this g(x) is 2(r + w) wide, however. The ms noise output of the SMF is
N o = ( η 2)
1 2π
∫
∞
−∞
H m ( ju) du = ( η 2) 2
= ( η 2)(Kl o ) 2 r 2
The peak MS SNR is at x = xo = 0:
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K2 2π
∫
∞
−∞
l o 2 r[sin( ru) ( ru)] du 2
7.2-13
FIGURE 7.2-2 Illustration of steps in the real convolution process. In this case, it is the matched filter weighting function, hm(v), that is reversed and translated. v is the variable used for convolution; x is the amount of translation. The result of the real convolution process is g(x).
ρ=
g s (0) NoE
2
=
[
(Kl 2r) 2 o
( η 2)(Kl o )
2
2
][
2 o
2r l 2r
]
=
1
( η 2)
7.2-14
Example 7.2-2 As a second example, consider the same, one-dimensional image, s(x), but a nonmatched spatial filter of the form, h(x) = K exp(–a冨x冨). Thus, H(u) = 2a/(a2 + u2). The image energy is the same: E = I2o 2r. The MS noise is
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No =
1 2π
∫
∞
−∞
( η 2 ) H( u )
2
( η 2)
du =
2η
∞
4a 2 K 2
∫ (a −∞
2
+ u2
)
2
du = K 2 ( η 2) a 7.2-15
The peak convolved filter output is
g s (0) =
∫
∞
s( ν) h(0 − ν) dν = l o
−∞
r
∫ K exp(−a ν ) dν = 2l K(1 − e ) a − ar
o
−r
7.2-16
Thus, ρ is given by
ρ=
(
4l 2o K 2 1 − e − ar
[K
2
][
( η 2)
)
2
a l o2 2 r
]
=
(
2 1 − e − ar
)
2
7.2-17
( η 2) ar
To illustrate that this ρ is less than that for the SMF shown in Equation 7.2-14, let a = r = 1. Thus,
ρ=
(
2 1 − e −1
)
2
= 0.799 ( η 2)
( η 2) (1)
7.2.18
which is less than the maximum response with the SMF.
7.2.2
THE CONTINUOUS, TWO-DIMENSIONAL, SPATIAL MATCHED FILTER
Consider a two-dimensional image, s(x, y), bounded in energy, as in the onedimensional case:
E=
∞
∫∫
∞
−∞ −∞
s(x, y) dx dy = 2
1 4π 2
∞
∫∫
∞
−∞ −∞
S( u, v) du dv 2
7.2-19
Also the two-dimensional ms noise is
No =
1 4π 2
∞
∫∫
∞
Φ nn ( u, v) H m ( u, v) du dv 2
7.2-20
−∞ −∞
The maximum of function g(x,y) occurs at x = xo, y = yo. It can be written
g(x o , y o ) =
∫∫ s(ξ, γ ) h R
1 = 4π 2
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∞
m
(x o − ξ, y o − γ ) dξ dγ 7.2-21
∞
∫ ∫ S(u, v)H −∞ −∞
m
(u, v) exp[+ ju x o + jv y o ]du dv
As in the one-dimensional case, the ratio, ρ, will be maximum for the matched filter, Hm(u, v). That is,
ρ=
g(x o , y o )
2
7.2-22
NoE
is max for Hm(u, v) ≡ K S*(u, v) exp[–j(uxo + vyo)]. Note that if s(x, y) is an even function in x and y, then S(u, v) = S*(u, v), and Hm(u, v) are real and even (i.e., they have no phase).
7.2.3
DISCUSSION
To be used effectively in visual pattern recognition, SMFs must be used in large neural arrays, as shown in Figure 7.2-3. Some elemental image, sk(x, y) impinges on the sensory array. It is processed in parallel by N, different SMFs, the outputs of each are normalized by dividing by Ek, the image’s “energy.”
Ek =
∞
∫∫
∞
−∞ −∞
s k (x, y) dx dy = 2
1 4π 2
∞
∫∫
∞
−∞ −∞
S k ( u, v) du dv 2
7.2-23
FIGURE 7.2-3 A SMF array for object classification. An object, sk(x, y) is presented to a retina whose output feeds an array of different SMFs. The SMF outputs are fed to a peak detecting classifier. If the kth SMF with the largest output is selected, the kth feature is assumed to be present and is acted upon by the animal.
The normalized output of the kth SMF given the jth image, sj(x, y) can be written:
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gˆ kj (x o , y o )
2
⎡ 1 ⎢ 4π 2 =⎣
∞
⎤ S j ( u, v) H mk ( u, v) exp + ju x o + jv y o ) du dv⎥ −∞ ⎦ 2 1 ∞ ∞ S ( u, v) du dv 4 π 2 −∞ −∞ k
∫∫ −∞
∞
[
]
2
∫∫
7.2-24
ⲏˆgkj(xo, yo)ⲏ2 will always be less than *gkk(xo, yo)2. It is easy to show that
g kk (x o , y o )
2
⎡ K ⎢ 4π 2 =⎣ 1 4π 2
∞
∞
∫∫ ∫∫ −∞ ∞
2
2 ⎤ S k ( u, v) du dv⎥ −∞ ⎦ = K2E ∞ 2 S k ( u, v) du dv
7.2-25
−∞ −∞
for xo = yo = 0 (even s(x,y)). It is left for the reader to conjecture how one could implement such a massively parallel SMF array for elementary pattern recognition using neurons (or model neurons). There is no evidence that the SMF is an operational entity in any living pattern recognition system. Neural networks are used in the CNS, but little is known how the cognition process really works.
7.3
MODELS FOR PARALLEL PROCESSING: ARTIFICIAL NEURAL NETWORKS
Artificial neural networks (ANNs) are layered, information-processing systems whose designs have distant origin based on real neural structures such as the vertebrate retina. The architecture of ANNs is very similar to the feature extractor systems of Fukushima (1969; 1970) (see Section 7.1.3). That is, there are two or more planes containing signal nodes. Connecting the nodes between the planes are signal paths with numerical weights. In Fukushima’s systems, the weights are fixed; in ANNs, the weights can be changed by a training rule to optimize the performance of the ANN. The inspiration for rule-based optimization of weights can be traced to a seminal text by Hebb (1949), The Organization of Behavior: A Neuropsychological Theory. Hebb hypothesized that when neuron A persistently and frequently causes neuron B to fire, the excitatory synaptic connections between A and B are strengthened. Mathematically, Hebb’s hypothesis can be approximated by the discrete equation: wj, k+1 = wj, k + c xj, k yk
7.3-1
where xj,k is the jth input at t = kT to the node whose output is y at t = kT, wj,k is the weight of the path connecting the jth input to the output node at t = kT, wj,k+1 is the new weight, and c is a positive learning rate constant. A more general learning rule can be written (Khanna, 1990):
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wj,k+1 = wj,k + c rj,k
7.3-2
where rj,k is a general reinforcement rule. The Hebbian rule is a special case where rj,k = xj,k yk.
7.3-3
Hecht-Nielsen (1990) defines an ANN as follows: A neural network is a parallel, distributed information processing structure consisting of processing elements (which can possess a local memory and can carry out localized information processing operations) interconnected via unidirectional signal channels called connections. Each processing element has a single output connection that branches (“fans out”) into as many collateral connections as desired: each carries the same signal — the processing element output signal. The processing element output signal can be of any mathematical type desired. The information processing that goes on within each processing element can be defined arbitrarily with the restriction that it must be completely local; that is, it must depend only on the current values of the input signals arriving at the processing element via impinging connections and on values stored in the processing element’s local memory.
Modern, powerful ANNs have many technical uses in areas such as: Pattern recognition, with applications including, but not limited to, recognition of visual objects including faces, fingerprints, retinas, iris patterns, recognition of printed and written words, recognition of speech, classification of two-dimensional curves such as spectrophotometer and chromatogram outputs, ECG and EEG records, detection of cancers and aneurysms in X rays, CT scans, and MRI images, detection of targets in radar and sonar scans, and oil, gas, and water exploration, etc. Control applications, including, but not limited to, autopilots, weapons guidance, control of robot manipulators, adaptive control of systems with timevariable plant parameters, guidance of autonomous vehicles and robots, etc. Financial applications, including, but not limited to, credit application evaluations, market analysis, bond rating, stock trading advisory systems, real estate appraisal, mortgage application screening, manufacturing expense reduction, machine maintenance analysis, etc.
7.3.1
ROSENBLATT’S PERCEPTRON
One of the first ANNs, the Perceptron, was developed by Rosenblatt (1958; 1962). His Mark I perceptron was a pattern recognition system that could learn to recognize simple visual patterns presented to its “compound eye” receptor plane, which contained 400 CdS photosensors in a square, 20 × 20 array. This perceptron had available a total of 512 connections with adjustable weights. The weights were varied by motor-driven potentiometers. The connections could be set by a patch panel, and were usually connected “randomly.” The training law sent signals to the servomotors that adjusted the potentiometers. An ingenious but clumsy system by today’s standards.
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In its simplest form, the Rosenblatt perceptron was a one-layer system (for some reason, the input nodes are not counted as a layer); 400 receptors in the input layer, and one output “neuron” is in the output layer. This perceptron was a simple binary classifier; the output of the neuron was taken as zero if the sum of weighted inputs plus the bias input was < 0; it was “1” if the bias plus weighted inputs was Š 0. Mathematically:
y′ = 0
y′ = 1
IF
IF
⎡ ⎢w 0 + ⎢⎣ ⎡ ⎢w 0 + ⎢⎣
400
∑ k =1
400
∑ k =1
⎤ wkxk ⎥ < 0 ⎥⎦
7.3-4
⎤ wkxk ⎥ ≥ 0 ⎥⎦
Note that w0 = 1 is the bias. The complete set of weights, w0, w1, w2, … w400, is called the weight vector, w. Similarly, the sensor outputs, x1, x2, x3, … x400, constitute the input vector, x. Thus, the net input to the output “neuron” is y = wT x = w0 + w1 x1 + w2 x2 + … + w400 x400. A simple pattern is “shown” to the sensor array, generating x. The perceptron operates as an iterative machine. It adjusts w repetitively so that after a finite number of iterations if the pattern belongs to class 0, y′ = 0, and if it belongs to class 1, y′ = 1. The basic perceptron training law (TL) determines the next, (k + 1)th, set of values for the weights. C is the correct class number ( 0 or 1) of the object presented, x. y is the perceptron output (0 or 1). α is the positive constant that adjusts the learning rate. The TL can be written: wk+1 = wk + (C – y)xT α
7.3-5
In this TL, if the perceptron makes an error (C – y) in its output, this error indicates a need to reorient the w line in x0, x1 space so that the perceptron will be less likely to make an error on this particular x vector again. Note that the output error (C – y) = 0 if the perceptron output is correct and w will not be changed. Otherwise, (C – y) can be ±1, and w will be modified to improve performance. It should be noted that perceptrons can have more than one layer, and more than one output element. For example, x is the input vector, connected by weights w1 to a first layer of K “neurons,” y1. y1 is connected to an output layer of M “neurons,” yo by weights w2. The y1 layer is called a hidden layer. Lippmann (1987) argues that no more than three layers (excluding the receptors) are required in perceptronlike, feed-forward ANNs because a three-layer net can generate arbitrarily complex decision regions when the ANN is trained for binary discrimination. (Recall that the simple Mark I perceptron, a one-layer ANN, is only capable of a straight line decision boundary in x1, x0 space that will separate (classify) x1 and x0 members.) The multilayer (Š 2) perceptrons are made to converge more swiftly on trained weight vectors (w1, w2, w3) by use of more sophisticated learning algorithms, a topic that beyond the scope of this chapter.
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7.3.2
WIDROW’S ADALINE
AND
MADALINE
The ADALINE (for ADAptive LINear Element) was developed by Bernard Widrow and colleagues (1985; 1962; 1960) for use as a pattern classifier ANN, similar to the perceptron. Multilayer assemblies of ADALINE units were called MADALINES (Many ADALINES). Except for its improved w training algorithm, an ADALINE was essentially a single-layer perceptron. In the early 1960s, Widrow’s Ph.D. student, W. C. Ridgway, III (1962), found a means to obtain non-linear separability in classifying inputs, x. A two layer MADALINE is shown in Figure 7.3-1. Two independent ADALINES are connected to the array of input elements; their twostate outputs are ANDed by a single element in layer two, which produces the desired 0, 1 output of the ANN when both ADALINES are trained, and the receptors are presented with patterns that are either of class x0 or x1. Figure 7.3-2 illustrates the classification boundary lines from the ADALINES, and their intersection. Note that an acute angle intersection is required to separate os from ·s in this case.
FIGURE 7.3-1 A two-layer, two ADALINE, MADALINE feature extractor by Widrow (1985). In the training process for this ANN, the MADALINE “learns” to discriminate two types of inputs. Feedback is used to iteratively adjust the input weights, {wjk} , so e1 and e2 are minimized. Widrow used the LMS algorithm to adjust the weights.
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FIGURE 7.3-2
The ADALINE’s decision boundaries used to separate the •s from the os.
Three-layer ANNs with multiple output neurons can be made from ADALINES. The w3 weights of the output layer can be optimized or trained by use of the Widrow–Hoff LMS algorithm, described below. A back-propagation algorithm is generally required to train the w2 and w1 weights of the hidden layers (Widrow and Winter, 1988). The LMS training algorithm for a single MADALINE adjusts w to minimize the mean squared error (MSE) of the classifier. The MSE can be written in matrix notation (Hecht-Nielsen, 1990):
[
ε 2 = E (d k − y ′k )
[(
2
]
= E dk − wTxk
7.3-6A
)] 2
7.3-6B
[(
= E d 2k − 2d k w T x k + w T x k x Tk w
[ ]
[
]
[
)]
7.3-6C
]
= E d 2k − 2w T E d k x k + w T E x k x Tk w
7.3-6D
= p − 2w T q + w T Rw
7.3-6E
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where dk is the desired response to the kth pattern input (0 or 1), and yk′ is the ADALINE output for the kth pattern input, xk. Clearly, the dependence of ε 2 on w is quadratic. The R matrix is square (n + 1) × (n + 1), and can be shown to be symmetric and positive semidefinite. This quadratic form determines a paraboloidal surface for ε 2 as a function of each w value. There will be some w = w* that will make ε 2 (w) a minimum. To find the w* that minimizes ε 2 (w), we can start at any w, and run downhill to ε 2 (w*), where the slope will be zero. Since ε 2 (w) is a closed-form expression, vector calculus can be used to find its gradient and set it equal to zero, and solve for w*. Thus,
(
)
∇ε 2 ( w ) = ∇ p − 2 w T q + w T Rw = 0
7.3-7
w* = R–1 q
7.3-8
which leads to
where R = E[xk xkT], and q = E[dk xk]. Clearly, the vector calculus required to evaluate w* is tedious. Fortunately, Widrow and Hoff approached the problem of finding the optimum w from a pragmatic, heuristic engineering viewpoint. Again, using the vector gradient, they derived a simpler iterative learning law: wk+1 = wk + α δk xk
7.3-9
This is the Widrow–Hoff delta training law in which α is a positive constant smaller than 2 divided by the largest eigenvalue of the square matrix, R, δk ≡ (dk – yk′), and k is the input index used in training. As a further heuristic, R is not calculated, and the effective α value is estimated by trial and error. Usually, 0.01 ð α ð 10; α0 = 0.1 is generally used as a starting value. If α is too large, values for w will not converge; if α is too small, convergence will take too long. Two refinements of the Widrow–Hoff TL can also be used; the batching version and the momentum version. The interested reader should see Hecht-Nielsen (1990) for a complete description of these variant TLs.
7.3.3
FUKUSHIMA’S NEOCOGNITRON
There has been an explosion of effort and innovation in the field of ANNs since the original work of Rosenblatt and Widrow. It is not the purpose here to describe the evolution of modern ANNs or the details of their operation. Instead, this section will examine the general properties of one of the more biologically inspired ANN systems, the neocognitron. Fukushima, whose model of a static pattern recognition system based on the vertebrate retina was introduced in Section 7.1.3, went on to develop an amazingly complex ANN that he called the neocognitron (Fukushima, 1980, 1984, 1988a,b). The neocognitron is an hierarchical ANN; it has strong roots from the retina in its
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design. Hecht-Nielsen (1990) calls it “the largest and most complicated neural network yet developed.” The purpose of the neocognitron was to recognize members of a set of lowresolution, binary, alphanumeric character images such as the numerals 0 to 9. One neocognitron has an amazing eight layers of nodes (Fukushima, 1988a). In the layers, there are 156 “slabs” or processing subunits. For example, there is a 19 × 19 element receptor array followed by the first (US1) layer, which has 12, 19 × 19 element slabs in it. The second (UC1) layer has 8, 21 × 21 element slabs in it; the third (US2) layer has 38, 21 × 21 element slabs in it; the fourth (UC2) layer has 19, 13 × 13 element slabs in it; the fifth (US3) layer has 35, 13 × 13 element slabs in it; the sixth (UC3) layer has 23, 7 × 7 element slabs in it; the seventh layer (US4) has 11, 3 × 3 element slabs in it, and finally, the eighth layer has 10 1-element slabs in it. (Element is synonymous with neuron or node.) There is a total of 34,980 nodes in the eight layers, and over 14 million connections with weights; Figure 7.3-3 illustrates the basic architecture of the eight-layer neocognitron and its slab subunits. Clearly, space does not permit description of the operation and training of this neocognitron in detail here. (The interested reader should read Hecht-Nielsen’s, 1990, section 6.3 for a clear summary description of how the neocognitron works.)
FIGURE 7.3-3 The eight layers with subunits of Fukushima’s vastly complex ANN, the Neocognitron. The heritage of the neocognitron can be traced to Fukushima’s early multilayer, static feature extractors, which were based on retinal neurophysiology.
Fukushima (1988b) also described a six-layer neocognitron, one of whose “neurons,” or cells, is shown in Figure 7.3-4. N excitatory inputs from the previous layer are summed to form the parameter, e. One or more inhibitory inputs are summed to form h. e and h are combined according to the function, y(e, h):
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FIGURE 7.3-4 Block diagram of a basic “neuron” used in Fukushima’s neocognitron. e, summed excitatory inputs (e Š 0); h, summed inhibitory inputs (h ð 0). Note that for 冨h冨 > e, y < 0. Since y represents an instantaneous frequency of a neuron, it must be non-negative, hence the actual output, y′, is the rectified y.
y(e, h ) =
e−h 1+ h
7.3-10
As in the earlier (1969) Fukushima feature extractor models, y is half-wave-rectified to form the neuron output, y′. That is, y′ = y,
yŠ0
y′ = 0,
y<0
7.3-11
Also, N
e=
∑a u k
k
7.3-12
k
7.3-13
k =1
M
h=
∑b v k
k =1
where a and b are weight vectors, the same as w in the previous notations, u is the excitatory input vector, and v is the inhibitory input vector. There are a total of 41,399 such neurons in this version of the neocognitron. Figure 7.3-5 illustrates in summary form the architecture of this neocognitron. Not only is the system hierarchical and multilayered, but it also uses complex feedback connections. Its design was motivated by known behavior of single units in the vertebrate visual cortex, and in the author’s opinion, a good deal of ANN art.
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FIGURE 7.3-5 Block diagram of the neocognitron. This ANN uses over 41,000 neurons of the type shown in Figure 7.3-4. This is a very complex ANN. See text for description.
For the reader to appreciate Fukushima’s design rationale, his 1988b paper section, “Physiology,” is quoted: In the visual area of the cerebrum, neurons respond selectively to local features of a visual pattern, such as lines and edges in particular orientations. In the area higher than the visual cortex, cells exist that respond selectively to certain figures like circles, triangles, squares or even human faces [SMFs ?]. Accordingly, the visual system seems to have a hierarchical structure, in which simple features are first extracted from a stimulus pattern, then integrated into more complicated ones. In this hierarchy, a cell in a higher stage generally receives signals from a wider area of the retina and is more insensitive to the position of the stimulus. Within the hierarchical structure of the visual system are forward (afferent or bottomup) and backward (efferent or top-down) signal flows. For example, anatomical observations show that the major visual areas of the cerebrum interconnect in a precise topographical and reciprocal fashion. Such neural networks in the brain are not always complete at birth. They develop gradually, neurons extending branches and making connections with many other neurons, adapting flexibly to circumstances after birth. This kind of evidence suggests a network structure for the model.”
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Unlike the binary neurons found in perceptrons and MADALINEs, the neocognitron uses analog-type neurons of the form shown in Figure 7.3-4. The neuron’s non-negative analog output, y′, corresponds to the instantaneous firing frequency of the neuron. Figure 7.3-5 shows that some of the interconnecting weights are fixed, while others are variable and are subject to training or optimization. The system operates in the following manner: A stimulus pattern is presented to the lowest stage of the forward paths, the input layer, which consists of a two-dimensional array of receptor cells. The highest stage of the forward path is the recognition layer. After the process of learning ends, the final result of the pattern recognition shows in the response of the cells of the highest stage. In other words, cells of the recognition layer work as gnostic cells (or grandmother cells); usually one cell is activated, corresponding to the category of the specific stimulus pattern. Pattern recognition by the network occurs on the basis of similarity of shape among patterns, [it is] unaffected by deformation, changes in size, and shifts in the position of the input patterns.
Clearly, of all the ANN architectures to date, Fukushima’s neocognitron is the most biological. Like the vertebrate brain, its complexity defies analysis. Its development followed neurophysiologically inspired heuristics and led to a working system.
7.3.4
DISCUSSION
The purpose of this section has been to introduce the reader to three examples of the ANN, and to show how the neurophysiological work on visual receptor arrays inspired their designs. By modern standards, Rosenblatt’s (1958; 1962) perceptrons were the simplest of ANNs. Widrow’s (1985) ADALINE and MADALINE ANNs were capable of more sophisticated pattern recognition tasks, able to generate complex decision boundaries. Fukushima (1988a) developed a series of very complex ANNs he called neocognitrons. As in the case of the earlier static feature extractors (see Section 7.1.3), Fukushima used the layered approach based on vertebrate retinal organization. While the neocognitrons performed amazing feature extraction operations, their complexity challenges most potential users, and other types of ANNs are generally used today for commercial applications.
7.4
CHAPTER SUMMARY
A wide variety of mathematical and structural models that have been proposed over the past 35 years or so to describe some of the feature extraction operations found in arthropod compound eyes and vertebrate retinas have been examined. The detection or preferred response to an object’s shape may in fact occur as the result of signal processing by the neural equivalent of a spatial filter. Fukushima illustrated how these operations could occur as visual signals propagate from one layer to the next of his static, visual pattern feature extractor. Spatiotemporal filtering to select shape and velocity properties of a moving visual object was advanced by Zorkoczy, who used Boolean (logic) elements in
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arrays underlying the receptors, and also signal delays. It is the delay elements that enable his filters to respond to moving objects. The specialized property of insect optomotor behavior (walking or in flight) has given rise to several models for the visual-to-motor reponse system responsible. A number of workers (Reichardt, Thorson, Bliss, Chapple, Northrop) have examined the turning tendency of walking insects, the torque on the neck, and neck muscle motor nerve activity as an insect tries to turn its head to follow the moving stripes. These optomotor outputs vary in intensity with stripe velocity, period, and contrast. Reichardt proposed an analog signal-processing model that used the outputs of two receptors (or pairs of receptors over the whole compound eye). This chapter examined Reichardt’s simple DDC correlation model mathematically and derived an expression for its output response. Requirements for and properties of speculative neural matched filters were examined. A matched filter in optics gives a maximum output (at one point) when presented with the object to be detected in noise. A neural matched filter would presumably to the same. The template filter for the object to be detected would have to be stored in the optic lobes, perhaps as some Fukushima-type filter based on synaptic connectivities between layers of neuropile. ANNs, like any radically new technology, had a slow start in their development and applications. Three early ANNs whose designs were motivated by visual system feature extraction operations were examined. The purpose of ANNs is not to model BNNs, but to perform various pattern recognition operations with diverse applications. The salient feature of an ANN is that it can be trained, i.e., it “learns” to perform its discrimination tasks. The concept of a trainable BNN goes back to the seminal work of Hebb (1949). Hebb, a physiological psychologist, postulated that a BNN could learn some discrimination task by a purposeful readjustment of its synaptic connections. All ANNs have training algorithms that adjust the weights (transmission gains) between layers of signal nodes; such adjustments are analogous to Hebbian synaptic changes. Modern ANNs are not very “neural” in their architecture of operation; their roots were neural, however.
PROBLEMS 7.1. A Fukushima-type, one-dimensional, linear, spatial filter has the spatial impulse response:
c BP (x) =
A sin(x π λ 1 ) A sin(x π λ 2 ) − λ1 (x π λ1 ) λ 2 (x π λ 2 )
where A = 1, λ1 = 0.2 mm, λ2 = 0.3 mm. a. Plot and dimension CBP(x). b. Plot and dimension the spatial frequency response, CBP(u). u is the spatial frequency in rad/mm. c. Calculate the Q of CBP(u). (Q = center frequency/bandwidth.)
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7.2. A Fukushima-type, one-dimensional linear spatial filter has the spatial impulse response: cF(x) = (A/2) exp[–α冨x冨] – (B/2) exp[–β冨x冨] Assume α > β, and (A/α) = (B/β). a. Plot and dimension cF(x). b. Plot and dimension, CF(u). c. Can this filter pass dc (constant, overall illumination)? 7.3. A one-dimensional Zorkoczy system is shown in Figure P7.3. The receptors emit pulses at a constant rate when illuminated. All receptors are separated by mm. The unit delay, *, is equal to the receptor clock period, T, and T = δ/u. u is the “unit object speed.” The one-dimensional, black and white object can be stationary, or have a unit velocity of ±u (in the ±x direction). T2(ak) is the unit OFF operator. a. Write the Boolean expression for the system output. Give the system output for: b. ON of general illumination. c. OFF of general illumination. d. A light edge moving to the right at +u. e. A light edge moving to the left with –u. f. A dark edge moving to the right with +u. g. A dark edge moving to the left with –u. h. A light unit spot (width = δ) moving to the right with +u. i. A light unit spot (width = δ) moving to the left with –u. j. A dark unit spot moving to the right with velocity +u. k. A dark unit spot moving to the left with velocity –u. 7.4. a. Design a one-dimensional Boolean Zorkoczy system that will respond selectively only to a unit white spot moving with –u (to the left at unit speed). Receptors produce a clocked output when illuminated. The objects are black and white. The unit spots can cover only one receptor at a time. Use T1, T2, AND, OR, and unit delay operators. Find the Boolean expression for the output, Qo. b. Verify the design by finding the responses to tests b through k in the preceding problem. c. Sketch a one-dimensional retinal neuron equivalent that will respond selectively to a white spot moving to the left. That is, replace the AND, OR, and delay elements of the Zorkoczy system with neuronal elements that will approximate the same behavior. 7.5. This problem will simulate the dynamics of Reichardt’s DDC described in Section 7.1.2. The organization of the DDC is shown in Figure 7.1-8 in the text. The input signal is to be an intensity sinewave pattern moving in the x direction with velocity v mm/ms. At the left (A) receptor, it is ia(x, t) = Io{1 + sin[k(x – vt)]}
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FIGURE P7.3
At the right (B) receptor, the intensity is ib(x, t) = Io{1 + sin[k(x – δ – vt)]} where Io is the intensity when sin[*] = 0, v is the pattern velocity (v > 0 moving left to right), δ is the spacing between the two receptors in mm, λ is the spatial period of the input sinewave in mm, and k ≡ 2π/λ. The output of the left-hand multiplier is the product of the delayed left receptor and the nondelayed right (B) receptor; that is,
{
[
]}{1 + sin[k(x − δ − vt)]}
y(t ) = K R l o2 1 + sin k(x − v(t − ε ))
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In Simnon notation, y = ib*DELAY(ia, ε), and yprime = ia*DELAY(ib, · ε). The difference, ∆y = y′ – y, is low-pass-filtered by the ODE, yo = –a*yo + a*∆y. The DDC output is the analog signal, yo. Simulate the DDC with Simnon using the following parameters: a = 0.2 r/ms, ε = 0.333 ms, δ = 0.1 mm, x = 0.1 mm, Io = 1, λ = 0.5 mm. Use Euler integration with delT = 0.001 ms. t is in ms. [Note that Simnon will not take Greek letters or subscripts, so users will have to use their own notation.] a. Plot ia, ib, ∆y, and yo in the steady state. Let v = 0.5 mm/ms. Vertical scale: 2, 3. Horizontal scale: 15, 20 ms. b. Now find the DDC static transfer function. Plot the steady-state yo vs. v. Use v values between –1 and +1 mm/ms. Clearly, when v = 0, y o = 0. Note that yo(v) is an odd function so it can sense the direction of object motion. 7.6. Repeat Problem 7.5 using moving black and white stripes (a spatial square-wave pattern). In Simnon notation, ia = Io*(1 +SQW(k*(x – v*t))) and ib = Io*(1 + SQW(k*(x – dx – v*t))). dx is δ used above. Use the constants: a = 0.2 r/ms, λ = 1 mm, dx = 0.5 mm, dT = ε = 0.333 ms. Let v range from –0.25 to +0.25 mm/ms. 7.7. The input to a one-dimensional is Gaussian white noise with power spectrum, φ(f) = η/2 msu/(cycle/mm) plus a signal s(x) = x 2c − x 2 that is non-negative for 冨x冨 ð xc and zero for 冨x冨 Š xc (a semicircle centered at the origin). Thus, the input to the SMF is v(x) = s(x) + n(x). a. Find an expression for hm(x), the SMF weighting function. b. Give an expression for the maximum output of the SMF due to s(x) at the origin, rs(0). c. Find an expression for the mean squared noise output from the matched filter, No. (Hint: Use Parseval’s theorem.) d. Give the ms signal-to-noise ratio, ρ, at x = xo = 0.
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8
Characterization of NeuroSensory Systems
REVIEW OF CHARACTERIZATION AND IDENTIFICATION MEANS FOR LINEAR SYSTEMS Because neural system identification and systems-level modeling are very broad subjects in which many workers have developed a variety of approaches, one cannot hope to review them all in this chapter. However, those classical approaches that have had some impact on neural systems will be examined. Many schemes have been devised by systems and control engineers for the characterization of single-input/single-output (SISO) linear systems, including those exhibiting nonminimum phase transfer functions, or having transport lags. Linear systems can be described in several ways. The most fundamental method is by a set of first-order, linear, time-invariant, ordinary differential equations (ODEs) to which the parameters (gains, natural frequencies, or A matrix) are known. Such state equations are generally the result of the analysis and modeling of the physics and biochemistry of the system. When a linear system is investigated experimentally, its unique response to an impulse input yields its weighting function or impulse response. The impulse response can also be obtained by solution of the state ODEs for zero initial conditions and an impulse input. If a steady-state sinusoidal input is applied to the linear system, the output will, in general, be sinusoidal with the same frequency but having a different phase and amplitude from the input. The sinusoidal frequency response function of the system is defined as the vector:
H( jω ) =
Yo ∠θ(ω ) Xi
8.1
Where Yo is the peak amplitude of the output sinusoid at frequency ω rad/s, Xo is the peak amplitude of the input sinusoid, and θ(ω) is the phase angle by which the output lags (or leads) the phase of the input. Often system frequency responses are presented as Bode plots for convenience. A Bode plot consists of two parts, a magnitude and a phase plot: 20 log10[Yo/Xo] vs. ω (log scale), and θ(ω) vs. ω (log scale). Note that the Fourier transform of the linear system impulse response function, h(t), is the frequency response function, H(jω). It is generally difficult to go from h(t) or H(jω) to the state ODEs. One must estimate the natural frequencies (poles and zeros) of the system graphically, which is not an accurate process; hence, the actual order of the system may be underestimated.
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In some situations, it is desirable to measure the parameters of a linear system “online” without being able to use either an impulse input or a purely sinusoidal input with variable frequency. Under these conditions, one records simultaneously the system input, x(t), and its output, y(t), over a long period of time. Over that period of time, the system must remain stationary (i.e., the system parameters must remain fixed in time). x(t) can be a random signal, including broadband Gaussian noise. The system cross-correlation function, ϕxy(τ) is defined here as
1 T→∞ 2 T
ϕ xy ( τ) ≡ lim
∫
T
x(t ) y(t + τ) dt
8.2
−T
(In practice, a cross-correlogram, ϕxy(τ), is computed because of the finite length of data.) The convolution integral can be used to bring the system weighting function into the expression:
1 T→∞ 2 T
ϕ xy ( τ) ≡ lim
∫
T
x( t )
−T
∫
∞
h( ν) x(t + τ − ν) dν dt
8.3
−∞
Inverting the order of integration (a legitimate bit of mathematical legerdemain) yields:
ϕ xy ( τ) ≡
∫
∞
∫
1 T→∞ 2 T
h( ν) dν lim
−∞
T
x(t ) x(t + τ − ν) dt
8.4
−T
The second integral is the autocorrelation function of the input signal, defined as
1 T→∞ 2 T
ϕ xx ( τ − ν) ≡ lim
∫
T
x(t ) x(t + τ − ν) dt
8.5
−T
Thus the cross-correlation function can finally be expressed as
ϕ xy ( τ) =
∫
∞
h( ν) ϕ xx ( τ − ν) dν
8.6
−∞
This expression is simply the real convolution of the linear system weighting function with the autocorrelation function of the system input. If the Fourier transform of both sides of Equation 8.6 is taken, the well-known result is obtained: Φxy(jω) = H(jω) Φxx(ω)
8.7
Note that Φxy(jω) and H(jω) are complex, while Φxx(ω) is a real, even function. Equation 8.7 can be used to find the frequency response function of the system:
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H( jω ) =
Φ xy ( jω ) Φ xx (ω )
8.8
In practice, the relation above yields an estimate of H(jω) because of the finite data length of x and y. A measure of how well H(jω) is estimated is found in the coherence function, γ2(ω), for the system (Bendat and Piersol, 1966). The coherence function is real, and is defined as
γ 2 (ω ) ≡
Φ xy ( jω )
2
Φ xx (ω )Φ yy (ω )
8.9
Φ yy (ω) is the autocorrelogram of the system output, Φ xx (ω) is the autocorrelogram of the input, and, of course, Φ xy (jω) is the system cross-correlogram, all calculated with finite length data. Note that 0 ð γ2(ω) ð 1. A coherence function approaching unity means that the system has constant parameters, is linear, and the inputs are stationary and clearly defined random signals of great length. When γ2(ω) is less than unity it can mean (1) the system is nonlinear and/or time-variable; (2) extraneous noise is present in the measurement of x(t) and y(t); (3) the output y(t) not only depends on x(t), but also on other (hidden) inputs. In general, γ2(ω) will drop off at high frequencies because of windowing and aliasing effects on the sampled x(t) and y(t). Often linear systems are “identified” or characterized using an input of broadband Gaussian noise. Such noise can be assumed to be white if its frequency spectrum is flat and extends well beyond the frequencies where 20 log冨H(jω)冨 is down by 40 dB. If the white assumption can be justified, then one can let Φxx(ω) ≅ K meansquared units/rad/s. The inverse Fourier transform of Φxx(ω) is the autocorrelation function, ϕxx(τ). In the case of white noise, ϕxx(τ) = Kδ(τ), that is, a delta function of area K. This means that ideally, H(jω) ≅ Φxy(jω)/K. Suppose a system is described by having two inputs, x(t) and u(t), each of which is acted on by linear systems (LS) having frequency response functions, Hx(jω) and Hu(jω), respectively. The LS outputs, yx(t) and yu(t) are added to make y(t), the overall system output (See Figure 8.1). By superposition, the autopower spectrum of y(t) can be written Φyy(ω) = Φxx(ω)冨Hx(jω)冨2 + Φuu(ω)冨Hu(jω)冨2
8.10
The output cross-power spectra are given by the simultaneous vector equations: Φxy(jω) = Hx(jω) Φxx(ω) + Hu(jω) Φxu(jω)
8.11A
Φuy(jω) = Hx(jω) Φux(ω) + Hu(jω) Φuu(jω)
8.11B
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FIGURE 8.0-1
Block diagram of a linear system with a common (summed) output.
One can solve for the transfer functions, Hx(jω) and Hu(jω) using Cramer’s rule. Thus,
H x ( jω ) =
Φ xy ( jω )Φ uu (ω ) – Φ uy ( jω )Φ xu ( jω )
[Φ
xx
(ω )Φ uu (ω ) − Φ ux ( jω )Φ xu ( jω )]
8.12
If x and u are uncorrelated, then the messy vector quotient for Hx(jω) → Φxy(jω)/Φxx(ω), as above. Rather than generate analog, broadband Gaussian noise to characterize linear systems by the cross-correlation method, use has been made of pseudo-random, binary noise (PRBN) generated by a clocked, digital circuit using sequential and combinational logic (O’Leary and Honrubia, 1975; Graupe, 1976; Chapter 4). Figure 8.2 illustrates a 32-bit PRBN generator designed and built by the author and his graduate students. This system uses a 50-kHz clock with a period of ∆t = 20 µs. The PR digital sequence at the output repeats itself after N = (232 – 1) = 4.29496730 E9 clock cycles, or after N∆t = 85.8993459 E3 seconds, or 23.861 hours. The TTL output of the PRBN generator is a TTL square wave with random high and low intervals, clocked at 50 kHz. If HI = 3.6 V is assumed, and LO = 0.2 V, then subtraction of 1.9 Vdc from the output wave will produce a random square wave with zero mean and peak height, a = 1.7 V. In the figure, the zero-mean TTL PRBN wave is passed through a digitally controlled, biquad, analog LPF (Northrop, 1990). The digital control input allows the user to set the break frequency of the biquad while holding the filter gain and damping factor constant. O’Leary and Honrubia (1975) show that the autocorrelation function for a PRBN random square wave having zero mean and peak amplitude a is given by
τ⎞ ⎛ ϕ nn ( τ) = ⎜1 − ⎟ a 2 (1 + 1 N ) − a 2 N ⎝ ∆t ⎠ (for 0 ≤ τ ≤ ∆t ) (for all τ )
[
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]
8.13
FIGURE 8.0-2 Schematic of a 32-bit PRBN generator built by the author. The random square wave, n(t), has zero mean. n(t) is low-pass-filtered to generate a continuous, bandwidthlimited Gaussian noise, Vn(t).
ϕnn(τ) is periodic in T; i.e., ϕnn(τ) = ϕnn(τ + T). However, in this system, T Ⰷ ∆t or N Ⰷ 1, so the periodicity in ϕnn(τ) will be neglected and the continuous Fourier transform (CFT) of Equation 8.13, Φnn(ω) calculated. This is easily shown to be
[
Φ nn (ω ) = a 2 ( N + 1) N
]
sin 2 (ω ∆t 2)
(ω ∆t 2)
2
(
)
− 2 πa 2 N δ(ω )
8.14
Considering the size of N = (232 – 1) = 4.29496730 E9, Φnn(ω) can be finally approximated by
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Φ nn (ω ) ≅ a 2
sin 2 (ω ∆t 2)
(ω ∆t 2)2
mean - squared V/Hz
8.15
Figures 8.3A and B illustrate the autocorrelation function of the PRBN offset output, n(t), and its approximate autopower spectrum (PDS).
FIGURE 8.0-3 (A) Autocorrelation function of the PRBN square wave, n(t). (B) Approximate autopower spectrum of n(t). (The small negative level in the autocorrelation function was neglected.) Vn(t) is used to make a random noise input for neuro-sensory systems (e.g., light, sound) being studied by a noise technique.
At low frequencies, the peak amplitude of the PDS is a2. At a frequency of ω10, the magnitude of Φnn(ω) is down 10% from peak. That is, Φnn(ω10) = 0.9a2. Solving algebraically by trial and error, when (ω∆t/2) = 0.56 rad, Φnn(ω10) = 0.9a2, and ω10 = 5.60 E4 r/s and f10 = 8.91 E3 Hz. Thus, the PDS of the raw dc-offset TTL output of this PRBN generator is substantially “white” (flat) out to about 9 kHz, given the 50-kHz clock and 32-state shift registers. Recall that if the raw n(t) is conditioned by being passed through a linear lowpass filter, G(jω), the autopower spectrum at the filter output, z, is given by Φzz(ω) = Φnn(ω) 冨G(jω)冨2
8.16
If the conditioned PRBN, z(t), is used to identify H(jω), then by the development above,
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H(jω) = Φzy(jω)/[Φnn(ω) 冨G(jω)冨2]
8.17
The major requirements of the PRBN method are that the flat bandwidth of Φzz(ω) must exceed that of H(jω) by a significant amount, and that the PRBN repeat time, N∆t, must be much longer than the settling time of the impulse response, h(t). Often when a physiological system is characterized, the interest is in obtaining an accurate black-box model of the system dynamics to use in a model-reference control system, such as the Smith delay compensator (Northrop, 1999). The ability to be able to subdivide the model into subsystems that can be related to the physiological and biochemical components of the system is sacrificed. To design a controller to regulate some variable in the physiological system, the model only needs to be accurate mathematically (constants, natural frequencies, order, etc.). Widrow and Stearns (1985) have described a simple black-box approach in which a discrete finite impulse response (FIR) filter of order N is “tuned” to match the input/output characteristics of an unknown plant (system). The plant is discretized, i.e., its input is discrete (sampled) broadband noise, [xk], and its output [dk] is also sampled at the same rate. The same sampled noise is the input to the adaptive model FIR filter, whose output [yk] is subtracted from the plant’s output to generate an error, [ek]. The sequence [ek] is used to calculate the MSE, e 2k . The least-mean-squared (LME) error method is used to find an optimum set of the j = L + 1 coefficients, [wjk], used in the adaptive linear combiner form of the model system, shown in Figure 8.0-4. That is, the optimum vector, W* = [w*0 w*1 w*2 … wL],* where the w*j are the optimum coefficients produced by the LMS process. The LMS process is too complicated to describe here; the interested reader should consult Widrow and Stearns (1985). These authors give examples of using the LMS modeling approach to characterize physiological systems (but not neural systems).
8.1
PARSIMONIOUS MODELS FOR NEURAL CONNECTIVITY BASED ON TIME SERIES ANALYSIS OF SPIKE SEQUENCES
A basic problem in neurophysiology has been to establish minimal models for neural connectivity, including excitatory and inhibitory connections, when recording from two or more functionally interconnected interneurons. It has been observed by the author (and many other workers) that certain interneurons in neuropile associated with sensory transduction fire in the absence of a stimulus. The stimulus might make them fire faster, or slower, but in every case alters the statistics of the zero-stimulus firing point process. For example, a neuron in the optic lobe of a grasshopper might fire fairly randomly in the absence of a stimulus, but given an appropriate visual stimulus, its firing would become faster and more regular (periodic). If the stimulus were moved in the opposite direction, the zero-stimulus firing would still remain random, but would fire more slowly, or not at all (Northrop, 1970). Some insight into how this behavior could arise can be inferred from repeating the experiment
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FIGURE 8.0-4 A discrete, self-tuning system that can mimic (model) a biological system. The input is sampled, broadband noise [xk]. The L + 1 model weights {wjk} are adjusted iteratively by an LMS algorithm (Widrow and Stearns, 1985) until the model closely matches the behavior of the neuro-sensory system.
and recording from two or three closely situated neurons that can be shown to interact functionally. One of the tools that sidesteps the need to calculate various individual and joint statistical functions for each neuron is the use of the joint peri-stimulus time (JPST) histogram, introduced by Gerstein and Perkel (1969). The JPST diagram is a visual tool, and offers qualitative (and some quantitative) evidence for canonical neural interconnections. It is normally viewed as a two-dimensional dot density plot when characterizing two neurons, but can be extended to three-dimensional, dot volumes for three neurons. The JPST diagram is described in the next section.
8.1.1
THE JPST DIAGRAM
The poststimulus (firing) time (PST) diagram is a well-known neurophysiological tool that is used to illustrate adaptation and habituation in a single responding neuron. Every time a stimulus is given, the time base is triggered on a storage oscilloscope; when the neuron fires, the spike is discriminated and the z-axis (beam intensity) is brightened to make a dot to mark the event on the CRT face. On the next stimulus, the trace is stepped up and another sweep is started, etc. This process builds a raster on the CRT screen full of dots, each one representing the time history of firing of the neuron following a stimulus. (Stimulation can be sensory, i.e., a flash of light,
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FIGURE 8.1-1
Schematic of a PST diagram.
or can be electrical, applied directly to an “input” neuron. Stimulation is generally periodic.) A typical PST diagram is shown schematically in Figure 8.1-1. Consider another scenario where a stimulus of some type is given to a nervous system and electrical recordings are made from two nearby interneurons, A and B. The responses of A and B can be described statistically by the JPST PST defined as: ξ(t, u) dt du = Pr{a neuron A spike occurs in the interval, (t, t + dt) and a B spike occurs in (u, u + du), given a stimulus at t = u = 0}
8.1-1
where the local times t and u are measured from each stimulus event. The sets of point tallies made in vertical and horizontal columns do not in general correspond to the PST diagrams for the individual neurons. This may be shown by writing the joint density as the product of a conditional density and a marginal density: ξ(t, u) = ξA冨B(t, u) ξB(u) = ξB冨A(t, u) ξA(t)
8.1-2
where the marginal densities ξA(t) and ξB(u) are the ordinary PST densities for neurons A and B. The conditional density is defined as ξA冨B (t, u) dt = Pr{an A spike occurs in (t, t + dt) 冨 a B spike at u and a stimulus at 0}
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8.1-3
A JPST diagram is used to estimate ξ(t, u). It is constructed from N repetitions of a stimulus. At the jth stimulus, spikes from the two neurons are discriminated and converted to a joint point process consisting of their occurrence times. The jth, two-dimensional, scatter diagram is made up as shown in Figure 8.1-2, and its dots are superimposed with the dots from the previous (j – 1) JPST scatter diagrams. Assume neuron A fires on the average 15 spikes in response to a typical stimulus, and B fires 20 in the time frame considered. Thus, there will be on the average 15 × 20 = 300 dots on each scatter diagram. If the stimulus is given N = 25 times, then the final JPST diagram will have about 7.5 × 103 dots. To prevent the loss of information caused by coincident dots, one can overlay a rectangular grid over the JPST surface and have the computer count the number of dots in each differential cell area, ∆t ∆u. These numbers can be converted to a height above each cell, generating a smooth, three-dimensional surface with contour lines using the proper software, such as found in Matlab.™ In viewing a two-dimensional JPST histogram, point densities taken on lines parallel to the 45o principal diagonal (the t = u line) correspond to the crosscorrelation histogram between neurons A and B. Because of obvious geometrical relations, the timescale of a cross-correlation histogram is in the ratio of 1: 2 to that of the JPST scatter diagram. Gerstein and Perkel (1972) in a paper on statistical techniques for display and analysis of spike trains, presented a number of interesting examples of how twodimensional, JPST scatter diagrams can be used to deduce parsimonious models for the interaction of three and four neuron groups. The deduced models of neural interaction are qualitative and not unique. They do suggest neural anatomical structures to be searched for, however. Some examples taken from Gerstein and Perkel (1972) are shown in Figure 8.1-3A to H. On these figures, the gray background of the JPST diagrams represents a nearly uniform, random dot density. The darker bars represent wide regions of high-dot-density. Single, bold diagonal lines represent sharp, high-dot-density diagonal plots, and multiple diagonal lines represent broad, high-dot-density diagonal bands. The noise sources are assumed to be present to account for steady-state, random firing of the recorded neurons. Synapses with arrows are excitatory; inhibitory inputs have a small circle synaptic ending. The neural circuits are examples of connections that are capable of producing the associated JPST diagrams, according to Gerstein and Perkel (1972). In circuit A, there is no functional connection between the stimulus, S, and the recorded neurons, so the JPST diagram has a uniform, random-dot-density; the firing of neurons A and B is uncorrelated. In circuit B, superimposed on the random firing, the stimulus briefly increases the firing rate of A, hence the dark band along the t axis. Both A and B are driven by the stimulus in C, producing two dense dot bands, one along the t axis and the other along the u axis. The uncorrelated background firing is still present. In circuit D, the stimulus again has no effect on interneurons A and B; however, A drives B in a nearly 1:1 manner, producing a high degree of correlation and a sharp, bold line along the 45° diagonal of the JPST plot. The correlation between A and B in plot E is less sharp because they are each driven by an independent noise source and also by a third, randomly firing neuron, C. The broad, more diffuse
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FIGURE 8.1-2 Diagram of how a JPST diagram is generated from the simultaneous spike recording of two neurons with a common, periodic, transient input.
diagonal is represented by three parallel lines in the figure. In circuit F, both A and B are driven by the stimulus, and also A drives B. Thus, broad bands parallel to the t and u axes from S are seen; as well as diffuse density on the diagonal from the A → B connection. The responses in G are similar to those in D and E. The sharp diagonal band in the lower JPST diagram is the result of taking data from neurons B and C (presumably one would see that same JPST response from A and C, from symmetry). The diffuse diagonal band in the upper JPST figure is from a vs. B and the presence of three independent noise sources. Finally, in the circuit of H, the stimulus excites both A and B. C, if not inhibited by S, drives B from an excitatory input from A. The JPST diagram shows a greater latency in the response of A than B, and the interrupted, sharp diagonal is the result of the stimulus inhibiting the C interneuron whereby A drives B.
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FIGURE 8.1-3 (A to H) Simple neural configurations capable of producing the associated JPST diagrams. Light gray, sparse random points; dark gray, denser random dots; dark lines, dense lines of dots. Note in H that an inhibitory synapse to interneuron C causes a break in the diagonal line. See text for descriptions.
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FIGURE 8.1-3
(Continued)
Other interesting neural circuit scenarios can be devised and described with the JPST technique. More challenging to interpret are the three-neuron, three-dimensional JPST dot volume displays described by Kristan and Gerstein (1970). (The axes of a three-dimensional JPST cube are shown in Figure 8.1-4.) These authors generated three-dimensional JPST volume dot displays in stereo pairs from three interneurons in the pleural ganglion of the sea-slug, Aplysia. The stereoscopic pairs permit viewing the dot density in a three-dimensional volume, rather than as a projection on a two-dimensional page. Interpretation of three-dimensional JPST
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FIGURE 8.1-3
(Continued)
point volume displays takes some experience. Kristan and Gerstein (1970) offer some advice: Bands of increased point density parallel to the three coordinate axes [t, u, v] represent the direct effects of the stimulus on the firing of the observed neurons. The lack of symmetry in the scatter diagram shows that the three neurons responded with different temporal patterns. Diagonal bands of increased point density, which are most visible near the ac and ab planes, correspond to the time structure of the probability for nearly simultaneous firing of the corresponding neuron pairs. Finally, a band of increased point density along the principal diagonal of the [display] cube represents the enhanced probability for nearly simultaneous firing of all three neurons.
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FIGURE 8.1-4 A three-dimensional JPST cube display. Correlations in the firing of three neurons can be studied. See text for description.
The three-dimensional JPST cube shown schematically in Figure 8.1-4 would normally be filled with a random volume of dots because of independent random factors causing all three neurons to fire in the absence of stimulus. The bold line in the AC plane is because A drives C in a nearly 1:1 manner. The bands in the AB plane are because the stimulus drives neuron A in an uncertain manner. Finally, the dense line running along the cube diagonal from the origin is because the stimulus also drives neurons A, B, and C in a nearly 1:1:1 manner.
8.1.2
DISCUSSION
The simple PST diagram is seen to be a useful semiquantitative means of displaying the responses of a single neuron to a repeated stimulus, and allows the researcher to see at once if the neuron shows nonstationary behavior. Many sensory neurons exhibit adaptation and/or habituation to repeated stimuli. The two-dimensional JPST diagram applied to two neurons allows conjectures to be made on the structure of models of the basic neural circuits involved in their responses. The JPST technique can be used with a mathematical neural model to verify putative neural circuit structures. By displaying dot densities as heights, a two-dimensional JPST diagram can be converted to a three-dimensional surface, which is easier to interpret.
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The use of three-dimensional JPST dot volume displays derived from simultaneous recordings from three interneurons gives a common stimulus and is also potentially useful in deriving basic neural circuits. Its features are necessarily more complicated to interpret, however. Both the two- and three-neuron JPST displays are subject to error from nonstationary behavior of the neurons. The simple PST diagram easily detects nonstationary responses. However, because the addition of JPST data recorded for each stimulus is required to build up a display, nonstationarity means that JPST features may change during the stimulation process, creasing uncertain or fuzzy features. JPST features from stationary responses generally become more pronounced, certain, and reliable as the number of stimuli increases.
8.2
TRIGGERED CORRELATION APPLIED TO THE AUDITORY SYSTEM
Triggered correlation (TC), first described by de Boer (1967) and deBoer and Kuyper (1968), is a conditional expectation statistic, which under certain conditions allows an estimate to be made of an equivalent linear system weighting function associated with frequency discrimination in auditory systems. Broadband Gaussian noise is generated and used as a stimulus to the animal’s “ear.” The sound pressure level at the ear is x(t). A single eighth nerve or cochlear nucleus neuron responding to the auditory noise stimulus is isolated and recorded from electrophysiologically. A parsimonious TC system model is assumed, shown in Figure 8.2-1. The linear portion of the system is assumed to precede the spike generation process. Below is shown that the TC process provides a biased statistical estimate of the impulse response, h(t), of the equivalent linear filter. The impulse response of the filter is the primary, time-domain descriptor of the linear filter. It allows estimation of the set of ODEs that describe the filter, and calculation of the filter frequency response function by Fourier transform.
FIGURE 8.2-1 A parsimonious model for a frequency-selective auditory neuron. A linear. narrow bandpass filter processes the auditory information. Its output is the input to a spike generator.
The TC algorithm allows estimation of the cross-correlation function, Rxy(τ), between the linear filter input, x(t), and its unobservable output, y(t). It is well known that in the limiting case where x is Gaussian white noise with a two-sided power density spectrum described by Φxx(f) = η/2 msV/Hz (or Φxx(ω) = η/(4π) msV/r/s),
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the cross-correlation function Rxy(τ) can be shown to be related to the impulse response by (Lee, 1960): Rxy(τ) = η h(τ)/(4π)
8.2-1
The spike generation process is assumed to occur when the filter output, y(t), crosses a positive threshold, b, with positive slope. That is, w(t) = δ(t – tk) when · ) > 0. The spike generation process is illustrated in Figure 8.2-2. y(tk) = b and y(t k As will be seen below, a conditional expectation, x+(τ) = E{x(t – τ) 冨 y = b, y· > 0} is estimated each time an output nerve spike occurs, and x+(τ) is averaged NT times for NT spikes. x(t – τ) are past values of x relative to tk that led to w(t) = δ(t – tk).
FIGURE 8.2-2 Block diagram of the spike generator model assumed to operate in the TC model architecture. The AND gate output m goes high (to 1) when the input derivative is > 0, and the input exceeds a positive threshold, b. The event m → 1 generates a positive output spike, w.
The sample mean of x+(τ) is µ. The underlying assumptions for the probability calculations that follow is that x, y, and y· = z are joint Gaussian variables, characterized by a mean matrix η in which ηx ≡ E{x(t)}, ηy = E{y(t)}, ηz = 0, and a 3 × 3 covariance matrix, ρ:
ρxx (0) ρ = ρ yx ( τ) ρzx ( τ)
ρxy ( τ) ρ yy (0) 0
ρxz ( τ) 0
8.2-2
ρzz (0)
The zeros in ρ are based on the reasonable assumption that ρyz(τ) and ρzy(τ) are negligibly small.
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8.2.1
DEVELOPMENT OF AN EXPRESSION EXPECTATION, X+(τ)
FOR THE
CONDITIONAL
In general, one can write for the elements of ρ,
{[
] }≡R
{[
][
ρ uu (0) = E u( t ) − ηu
2
uu
(0) – η2u ≡ Var{u} ≡ σ 2u
ρ uv ( τ) ≡ E u(t − τ) − ηu v(t ) − ηv
]} ≡ R
uv
( τ) – ηu ηv
8.2-3
8.2-4
Wu (1970), using probability theory, has shown that, when ρyz(0) = 0, E{x(t – τ) 冨 y(t), z(t)} = E{x(t – τ) 冨 y(t)} + E{x(t – τ) 冨 z(t)} ηx
8.2-5
Hence,
x + = E{x(t − τ) y = b, z > 0} = E{x(t − τ) y = b} + E{x(t − τ) z > 0} − ηx
8.2-6
Now the two terms on the right-hand side of Equation 8.2-6 will be evaluated. The conditional Gaussian density, f(x 冨 y), given by
f (x y) =
1 2 πσ 2x 1 − r 2
(
)
[
]
(
)
⎧ x − η 2 σ 2 + y − η 2 ρ2 − 2 r(x − η ) y − η σ σ ⎫ x] x y y x y x y⎪ ⎪[ exp ⎨ ⎬ 2 2 1− r ⎪⎩ ⎪⎭
(
)
8.2-7
Where
r = ρxy ( τ) σ x σ y =
{[R
[R
xx
xy
( τ ) − ηx ηy ]
(0) − η ][R yy (0) − η 2 x
2 y
]}
8.2-8
Hence,
E{x(t − τ) y = b} =
∫
∞
x( t − τ) f (x( t − τ) y = b) dx
−∞
Integrating the integral in Equation 8.2-9 by completing the square yields
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8.2-9
)[
(
]
(
E{x(t − τ) y = b} = ηx + rxy b − ηy σ x σ y = ηx + b + ηy
)[ R [
R xy ( τ) − ηx ηy yy
(0) − η
2 y
]
]
8.2-10 Now to evaluate E{x(t – τ) 冨 z > 0}, the integral;
E{x(t − τ) z > 0} =
∫
∞
x(t − τ) f (x( t − τ) z > b) dx
8.2-11
−∞
must be solved. To solve the integral, f(x(t – τ) 冨 z > 0) must be found from f(x, y) and f(z): ∞
f ( x( t − τ ) z > 0 ) = f ( x( t − τ ) 0 < z ≤ ∞ ) =
∫ f (x, z) dz ∫ f (z) dz
8.2-12
∞
0
where
f (x, z) =
1 2 πσ x σ y
(1 − r ) 2
⎧⎪ (x − η )2 σ 2 + (z − η )2 σ 2 − 2 r(x − η ) (z − η ) σ σ ⎫⎪ x x z z xr x z x z exp ⎨ ⎬ 2 2 1 − rxz ⎪⎭ ⎪⎩
(
8.2-13
)
and
f (x) =
1 σ z 2π
{
exp − (z − ηz )
2
(2σ )} 2 z
8.2-14
Also, note that y· ≡ z, ηz ≡ 0, and rxz ≡ ρxz(τ)/(σxσz). Integrating yields
f ( x( t − τ ) z > 0 ) =
1 σx
⎧ ⎡ r (x − η ) 2 ⎪ x exp − (x − ηx ) 2σ 2x ⎨1 + erf ⎢ xz 2 2π ⎢ 2σ x 1 − rxz2 ⎪⎩ ⎣
{
}
(
)
⎤ ⎫⎪ ⎥ ⎬ 8.2-15 ⎥⎪ ⎦⎭
Now, performing the required integration of f(x(t – τ) 冨 z > 0), the conditional expectation can be written:
E{x(t − τ) z > 0} = rx σ x
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(2 π) + ηx = (2 π)R xz (τ) σ z + ηx
8.2-16
Substituting Equations 8.2-16 and 8.2-10 into Equation 8.2-6, the conditional expected value of x + (t – τ) is:
x + = E{x(t − τ) y = b, z > 0} = ηx
)[
(
]
+ b − ηy R xy ( τ) − ηx ηy σ 2y + R xz ( τ)
(2 π )
8.2-17
σz
Because the input x(t) is the sound pressure level, it has zero mean, thus ηx → 0, and the average output of the linear filter H(s), ηy, will also be zero. Thus Equation 8.2-17 can be simplified:
x + = E{x(t − τ) y = b, z > 0} = b R xy ( τ) σ 2y + R xz ( τ)
(2 π )
σz
8.2-18
The statistic, x + , is a biased estimator of the cross-correlation function of the wideband Gaussian noise input, x(t), and the output, y(t), of H(s). Thus x+ can be used to estimate h(t). It is also of interest to examine the error statistics associated with the computation of the TC function. Wu (1970) has shown:
{
} ≡ E{x } − x
VAR[x + ] ≡ E [x + − x + ]
2
2 +
2 +
↓
8.2-19
[
VAR[x + ] = σ 2x − R xy ( τ) − ηx ηy
]
2
σ 2y − R 2xz ( τ) 2 π σ 2z
Wu also examined the effect of TC performed on uncorrelated spikes added to the spikes elicited by the spike generator operating on the filter output, y. Assume that a total of NT spikes are used to compute the sample mean of the TC function, and of these spikes, NU are uncorrelated with the input x, and NC are due to x. Thus, N T = NC + NU .
µ=
1 NT
⎧⎪ ⎨ ⎪⎩
∑
⎫
Nu
Nc
x +i +
i =1
∑ x ⎪⎬⎪⎭ j
8.2-20
j=1
Hence,
E{µ} =
{
}
N N N 1 N c E[X + ] + N u E[x] = c x + + u E[x] = c x + NT NT NT NT
Wu (1970) has also shown:
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8.2-21
VAR{µ} =
1 NT
⎡ Nc ⎤ N VAR{x + } + u σ 2x ⎥ ⎢ NT ⎦ ⎣ NT
8.2-22
Equation 8.2-22 is a measure of the noisiness of the x+ statistic, µ, given uncorrelated pulses in the neural output. As remarked in the introduction, the purpose of the TC algorithm is to estimate the linear filter weighting function. If the noise spectrum is broadband, one can solve Equation 8.2-17 for Rxy(τ) and write:
ˆ ( τ) = η η + R xy x y
σ 2y
(b − η ) [
µ − ηx − R xz (t )
(2 π )
]
σ z ≅ h( τ ) P ( 4 π )
y
8.2-23
P is the msV/Hz of a flat, one-sided PDS of the broadband noise input, x. Thus, the sample mean, µ, is a biased estimator of h(τ).
8.2.2
OPTIMUM CONDITIONS TC ALGORITHM
FOR
APPLICATION
OF THE
In the auditory system case, ηx → 0, ηy → 0, which is desirable. It is also desirable for σz → ×. σz → × for a linear system with one more pole than zero (N – M = 1). (Wu shows other special conditions that will make σz → ×. In some cases σz → 0, which makes TC impossible.) Thus, for broadband noise:
µ ≅ h( τ )
R ( τ) P b + 2 π xz σz 4 π σ 2y
8.2-24
Wu examined the theoretical case where H(ω) is an ideal bandpass filter (IBPF), shown in Figure 8.2-3. The Q of the IBPF is simply Q = ωo/2B. The PDS of the input noise is rectangular, and has an amplitude of Io msV/r/s for 冨ω冨 ð ωi. Thus, the autocorrelation function of the noise is
FIGURE 8.2-3 Block diagram of a TC system in which bandwidth-limited Gaussian noise is the input to an ideal (rectangular) bandpass filter with a Q = ωo/(2B). The BPF output, y(t) is the input to the model spike generator of Figure 8.2-2. See text for derivation and discussion.
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R xx ( τ) =
1 2π
∫
∞
lo 2π
Φ xx (ω )e jωτ dω =
−∞
∫
ωi
e jωτ dω
− ωi
8.2-25
l ω sin(ω i τ) l o ω i = o i = sinc(ω i τ π) π π (ω i τ ) and
l oω i π
R xx (0) = σ 2x =
8.2-26
The PDS at the output of the linear filter is
for (ω o − B) ≤ ω ≤ (ω o + B)
Φ yy (ω ) = Φ xx (ω ) H( jω ) = G 2 l o =0
8.2-27
otherwise
The autocorrelation function of the narrowband noise at the filter output is found from
R yy ( τ) =
1 2π
∫
∞
G 2 l o e jωτ dω =
−∞
G2lo 2π
−ωo +B
∫(
e jωτ dω +
− ω o + B)
G2lo 2π
∫
ωo +B
ωo −B
e jωτ dω
8.2-28
↓
R yy ( τ) =
2G 2 l o B sinc(Bτ π) cos(ω o τ) π
8.2-29
and
σ 2y = R yy (0) =
2G 2 l o B π
8.2-30
Wu shows VAR{z} = σz2 = Rzz(0) to be
σ 2z = R zz (0) =
1 2π
∫
∞
ω 2 Φ yy (ω ) dω =
−∞
2 G 2 l o B3 12Q 2 + 1 3π
[
]
8.2-31
Thus, the higher the filter Q, the better, because it is desired that σz → ×, so that the Rxz(τ) bias term in Equation 8.2-24 for µ(τ) will → 0. Finally, the theoretical cross-correlation function, Rxy(τ) for the IBPF is examined. Wu shows this to be
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R xy ( τ) =
∫
1 2π
∞
Φ xx (ω ) H( jω ) e jωτ dω =
−∞
2Gl o B sinc(Bτ π) cos(ω o τ) π
8.2-32
Compare this expression for Rxy(τ) with the inverse Fourier transform of the IBPF, h(t):
h( t ) =
1 2π
∫
∞
H( jω )e jωτ dω =
−∞
G 2π
B−ω 0
∫(
e jωτ dω +
− ω o + B)
G 2π
∫
ωo +B
ωo −B
e jωτ dω
8.2-33
↓
h( t ) =
2GB sinc(B t π) cos(ω o t ) π
8.2-34
Another important statistic is the expected rate of spike firing at the system output, Nb. Wu shows that Nb is given by
Nb =
1 ⎛ σz ⎞ exp − b 2 2σ 2y 2 π ⎜⎝ σ y ⎟⎠
[
( )]
8.2-35
Substituting for σy and σz found in the case of the IBPF above yields
(
[
)
( )]
N b = (B 2 π) 12Q 2 + 1 6 exp − b 2 2σ 2y
8.2-36
Wu also derived an expression for the ms SNR of the cross-correlogram found by the TC process for the IBPF. This is
(SNR)C
{
}
⎛ [sin(Bτ) Bτ] cos(ω τ) 2 ⎞ R XY ( τ) 2 BT o ⎜ ⎟ = = π ⎜ 1 + [sin(2 Bτ) 2 Bτ] cos(2ω o τ) ⎟ VAR{R XY ( τ)} ⎝ ⎠
8.2-37
But for τ → 0, (SNR)c = BT/π. Another SNR, (SNR)t, was found to be
[
]
(SNR)t ≡ N T x 2+ VAR{x + } = N b T b 2 ρxy (τ) σ 2y (1 − ρ2xy (τ))
8.2-38
where T is the total observation time over which spikes are processed, and ρxy(τ) is defined as the standardized cross-correlation function, given by
(
) (2B ω ) [sin(Bτ) (Bτ)] cos(ω τ)
ρxy ( τ) = R xy ( τ) σ x σ y =
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i
o
8.2-39
for the IBPF. Wu goes on to define a “noise figure,” F, for TC applied to the IBPF.
F≡
(SNR)c (SNR)t
8.2-40
At τ = 0, the noise figure reduces to
F=
2G 2 l o (ω i − 2 B) πb
2
(12Q
2
)
+1 6
[
]
8.2-41
[
]
8.2-42
exp b 2 π 4G 2 l o B
And for ωi Ⰷ 2B, F becomes
F=
2G 2 l o ω i πb
2
(12Q
2
)
+1 6
exp b 2 π 4G 2 l o B
As a performance index, F is desired to approach unity. Thus, the TC process gives the best results when the IBPF Q is large. Interestingly, too large a noise bandwidth, ωi, works against one in making F large. However, ωi must be large enough so that the estimate of Rxy(τ) is a valid representation of h(τ), the linear system weighting function. Common sense dictates that, in any case, ωi must be > (ωo + B).
8.2.3
ELECTRONIC MODEL STUDIES
OF
TC
Figure 8.2-4 illustrates the system configuration Wu (1970) used to compute the TC function of a known bandpass filter. A broadband Gaussian noise generator (QuanTech model 420) was used as the source. Its output was conditioned by a bandpass filter with adjustable –3 dB frequency and a –36 dB/octave rolloff (SKL model 302). The filter output was amplified by a broadband amplifier whose output was x(t). x(t) was the input to a linear bandpass filter set up to emulate H(s) in the TC model. The BPF output voltage was y(t). y(t) was the input to a comparator-pulse generator. When y(t) exceeded the comparator threshold voltage, b, the comparator output went high. This event was sensed by a one-shot multivibrator (74123) that produced a 50-µm TTL output pulse only for positive-going crossings of b. Both x(t) and the output pulses were digitized and recorded on magnetic tape using a PI model 6200 analog instrumentation tape recorder. This recorder is unique in that it can play back in reverse at the same speed that the data were recorded in the forward direction. To compute individual x+(τ), whenever a recorded pulse occurred, the reverse x(t) was averaged by a signal averager. This process is given physical significance in Figure 8.2-5 where the kth pulse has just occurred. By reverse playback, one can obtain x(tk – τ) for 0 ð τ ð TA . x(tk – τ) is then plotted in positive time as x+k(τ). TA is the maximum analysis time over which each x+k(τ) is averaged. The average of NT records of x+k(τ) gives us the desired x + ( τ) .
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FIGURE 8.2-4
Instrumentation system used by Wu (1970) to compute the TC func-
tion, x + ( τ) . It was used on electronic bandpass filters as well as on the grasshopper’s tympanal organ.
Figure 8.2-6C illustrates a cross-correlogram “gold standard,” computed using the broadband noise input, x(t), and the bandpass filter output, y(t). A PAR model 100 signal correlator was used to compute Rxy(τ). Figure 8.2-6A shows a noisy x + ( τ) , computed by averaging NT = 2500 elicited spikes. The bandpass filter center frequency was at 1 kHz, and the noise spectrum cutoff was at 100 kHz. When
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FIGURE 8.2-5 Diagram showing how reverse playback of the analog tape allows averaging of the signal, x(–t), that led to the kth spike.
the noise bandwidth, fi, was reduced to 3.E3 Hz, the SNR of x + ( τ) is slightly improved (see Figure 8.2-6B). Finally, with a 3-kHz fi, and 2500 spikes, the SNR is better still. Clearly, x + ( τ) appears to estimate Rxy(τ) with acceptable accuracy. (Data from Wu, 1970.)
8.2.4
NEUROPHYSIOLOGICAL STUDIES USING TC
OF
AUDITORY SYSTEMS
de Boer (1967; de Boer and Kuyper, 1968) was the first worker to apply TC to the study of audiofrequency selectivity associated with single units in the cat cochlear nucleus, presumably auditory nerve fibers. Using TC to find x + ( τ) , de Boer found that many cat auditory neurons appeared to be preceded by a filter whose estimated h(τ) denoted a high degree of frequency selectivity, or Q. The results of his TC studies say nothing about the detailed neural mechanism responsible for the high Q, but they do indicate what to look for (e.g., lateral inhibition at the level of the
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FIGURE 8.2-6 Three triggered correlograms using a SKL-302 variable electronic filter set to 1-kH bandpass (1 kHz low-pass in series with a 1 kHz high-pass). (a) The conditional expectation, x + ( τ) , with the filter given 100-kHz bandwidth Gaussian noise. (The noise associated with 2500 spikes was averaged.) (b) The conditional expectation, x + ( τ) , with the filter given 3-kHz bandwidth Gaussian noise. (The noise associated with 2500 spikes was averaged.) (c) The true cross-correlogram, R xy ( τ) , of the analog noise output of the filter with 140-kHz bandwidth limited Gaussian noise. A PAR Model 100 signal correlator was used. This was the gold standard for the TC conditional expectations. R xy ( τ) has the shape of the impulse response of the filter. (From Wu, J.M., Ph.D. dissertation, University of Connecticut, Storrs, 1970.)
cochlea). An example of a cat auditory neuron x + ( − τ) is shown in Figure 8.2-7. (Note that time τ increases from right to left, τ = 0 is at the right, because of the way the TC conditional averaging was done.) The relatively long settling time of x + ( − τ) suggests that the filter is of the bandpass type with a high Q. To demonstrate this behavior, note that the transfer function of a simple, high-Q, quadratic bandpass filter can be written:
H(s) =
s
[(s + a)
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2
+b
2
]
=
s s = 2 2 2 s + s2 a + a + b s + s ω n Q + ω 2n 2
8.2-43
FIGURE 8.2-7
The reversed, conditional expectation, R xy ( τ) , calculated from a single
fiber in a cat auditory nerve. Noise bandwidth was 200 to 2000 Hz; the noise input for 4403 spikes was processed. The peak sinusoidal frequency response for this unit was 1.5 kHz. (From DeBoer, E. and Kuyper, P., IEEE Trans. Bio. Med. Eng., 15(3): 169, 1968. © 1968 IEEE. With permission from IEEE.)
Taking the inverse LaPlace transform of H(s) yields its weighting function or impulse response, h(t):
a 2 + b 2 − at e sin( bt + ϕ), b
h( t ) =
where ϕ = tan −1 ( b −a )
8.2-44
In terms of the more familiar system parameters, Q and ωn, h(t) can be written:
h( t ) =
(
[
]
exp −(ω n 2Q)t sin ⎡ω n ⎢⎣ 4Q − 1 2Q 2
where: a = ωn/2Q, b = ωn
)
(1 − 1 4Q ) , ω 2
n
(1 − 1 4Q ) t + ϕ⎤⎥⎦ 2
8.2-45
is the undamped natural frequency of
the filter, ζ is the damping factor of the quadratic pole pair, and Q = 1/2ζ. Note that as the Q increases, the damping envelope on the sinusoid is sustained. That is, there are more cycles visible in h(t), and it has a longer settling time. Wu (1970) applied TC to both electronic models and the auditory system of the grasshopper, Romalea microptera. Romalea has paired, external, tympanal organs on the sides of the first abdominal segment. The tympanic membrane is oval, measuring about 3.6 mm long by 2 mm wide. A tympanic ganglion with 60 to 70 sensillae is attached to the inner surface of the tympanic membrane. The auditory nerves run from the tympanic ganglia to the third thoracic ganglion. Wu obtained
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single-unit auditory responses with microelectrodes inserted into a tympanic ganglion. Both steady-state sinusoidal acoustic excitation was used, as well as broadband Gaussian noise. Wu found that typical units responded to the sinusoidal stimuli with relatively broad bandpass characteristics. A typical unit had its peak (0 dB) response at 3 kHz. From the width of its frequency response curve, this unit had a Q = 1.15, or an equivalent quadratic damping factor of ζ = 0.43. These numbers suggest that the equivalent h(t) would have one major peak followed by one small undershoot. Wu found that under broadband acoustic noise stimulation, for every unit studied, no significant x + ( τ) of any shape emerged from the computational noise. Using the same recording apparatus and calculation techniques, and an analog electronic model of the bandpass filter, Wu did find a well-defined x + ( τ) of the expected shape for the low-Q condition, however. One explanation for the lack of results was that the auditory sensillae did not fire in a 1:1 manner with the applied sinusoidal frequency. This behavior suggested that coherent, mechanical stimulation by a number of cycles at a given frequency were required for a sensillum to fire. Under broadband noise stimulation, the tympanal membrane may have jumped randomly from one mechanical resonance mode to another, never providing enough sustained cycles of mechanical stimulation of the sensory neuron to preserve coherent causality.
8.2.5
SUMMARY
Triggered correlation has been shown to be a useful tool to characterize the frequency selectivity of single units in the vertebrate auditory system. Such characterization is simplified mathematically because the means of the noise stimulus, x(t), the equiv· are all zero. Once the x τ esalent bandpass filter output, y(t), and its derivative, y, +
( )
timate of h(t) is obtained, it can be Fourier-transformed to find the equivalent frequency response of the bandpass filter, H(jω). If a sensor is low-pass or has a broad bandpass characteristic in its response to stimuli, TC can also be applied. Most physiologically relevant physical input parameters are non-negative, and thus the means of x and y would need to be considered. Theoretically, TC can be applied to the definition of the linear spatiotemporal transfer function of ganglion cells (GCs) in the retina. In this case, the stimulus can be a two-dimensional checkerboard overlapping the GS RF. The elements are made to shift rapidly and randomly in intensity, using either white light or monochrome illumination. Every time a GC spike occurs, a two-dimensional average of stimulus previous patterns is made, giving an l + (x, y, τ) . There are probably four reasons TC has not enjoyed wider application in sensory neurophysiology: 1. The theory is difficult to understand. 2. The TC algorithm is difficult to implement.
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3. TC says nothing about the underlying physiology giving rise to frequency selectivity, etc. It is a “black box” approach. 4. TC is best applied to “tuned” neuro-sensory systems, such as the vertebrate auditory system and the electroreceptor system of mormyrid electric fish (see Section 2.5.2). Results when the linear filter is not sharply tuned are hard to interpret.
8.3
THE WHITE NOISE METHOD OF CHARACTERIZING NONLINEAR SYSTEMS
Many of the extensions of linear system theory to the characterization of frankly nonlinear systems presuppose a particular configuration or partitioning of separable linear and nonlinear portions of the system. While such techniques may have some utility in the study of neural systems, it is clear that a more “black box” method that makes no supposition about system configuration could also be useful in the characterization of neuro-sensory systems. The white noise method of characterizing nonlinear systems, as introduced by Wiener (1958), offers in theory this utility as well as concise, quantitative descriptions of the system dynamics and nonlinearity. The Wiener white noise method allows one to build a model that emulates the behavior of the nonlinear system under study (NSUS) that is optimum in a minimum MSE sense, but sheds little light on the biology of the system; it provides an optimum black box model. The Wiener white noise approach is related to the Volterra (1959) expansion of the input/output characteristics of the NSUS in terms of a power series with functionals as terms. (A functional is a term whose argument is a function, and whose value is a number. A definite integral is a functional; a real convolution is a functional.) Wiener showed that a nonlinear dynamic system excited by Gaussian white noise could be described in terms of an infinite series of orthogonal functionals in an elegant derivation. The output of the NSUS can be written:
y( t ) =
∞
∑ G [h , x(t)] k
8.3-1
k
k =0
where {G} is a complete set of orthogonal functionals with respect to the input Gaussian white noise, x(t), and {h} is the set of kernels that characterizes the impulse responses of the NSUS. The first four Wiener functionals are
[
]
G 0 h 0 , x(t ) = E[y(t )] = y G1[ h1 , x(t )] =
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8.3-2
∞
∫ h (τ) x(t − τ) dτ = y (t) 0
1
1
8.3-3
where h1(t) is the linear weighting function of the NSUS. If the NSUS is in fact linear, then all hk = 0 for k Š 2.
G 2 [ h 2 , x(t )] =
[
]
G 3 h 3 , x( t ) =
∞
∫∫
∞
0
0
∞
∞
h 2 ( τ1 , τ 2 ) x(t − τ1 ) x(t − τ 2 ) dτ1dτ 2 − P
∞
∫ h (τ, τ) dτ
∞
∫ ∫ ∫ h (τ , τ , τ ) x(t − τ ) x(t − τ ) x(t − τ ) dτ dτ dτ 0
−3P
0
0
∞
∞
3
1
2
3
1
2
∫ ∫ h (τ , τ , τ ) x(t − τ ) dτ dτ 0
8.3-4
2
0
0
3
1
2
3
1
1
3
1
2
3
8.3-5
2
where P = Φxx(f) is the power density spectrum of the input Gaussian white noise in mean-squared units/Hz. Beyond order k = 3, the physical significance of the kernels and functionals strains the imagination; they can be considered hypersurfaces. Marmarelis (1972) noted that each kernel hk is a symmetrical function of its arguments, and that the nonlinear kernels (k Š 2) give a quantitative description of the nonlinear “cross-talk” between different portions of the input as it affects the present system response. In a second-order nonlinear system, the kernel h2(t, t – to) describes the deviation from linear superposition between an impulse at t = 0 and another at time to. For example, let the input to a “pure” second-order NL system be x(t) = δ(t) + δ(t – to). Assume h0 = 0 and hk = 0 for k Š 3. Hence, the Wiener formulation becomes
y( t ) = G1[ h1 , x( t )] + G 2 [ h 2 , x(t )] = +
∞
∞
∫ h ( τ ) [δ ( t − τ ) + δ ( t − t 0
1
∞
∫ ∫ h ( τ , τ ) [δ ( t − τ ) + δ ( t − t 0
0
2
1
2
1
o
o
− τ1 ) dt
][
]
− τ 1 ) δ ( t − τ 2 ) + δ ( t − t o − τ 2 ) dτ 1 d τ 2 8.3-6
Integrating Equation 8.3-6 yields y(t) = h1(t) + h1(t – to) + h2(t, t) + h2(t, t – to) + h2(t – to, t) + h2(t – to, t – to)
8.3-7
Subtracting the output of the second-order, nonlinear system caused by each input acting alone, from Equation 8.3-7 yields ∆y = h2(t, t – to) + h2(t – to, t)
8.3-8
Since h2(t1, t2) is symmetrical around the line t1 = t2, Equation 8.3-8 can be written: ∆y = 2h2(t – to, t) = 2h2(t, t – to)
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8.3-9
Equation 8.3-9 gives a measure of the cross-talk between the two inputs. It can be shown that if the system consists of a no-memory nonlinearity, f(x), followed by a linear dynamics, then h2(τ1, τ2) ≡ 0 for τ1 ¦ τ2, and the system obeys “time superposition”; i.e., in this case the responses of the system to the sum of two or more impulses is equal to the sum of the responses of the system to each impulse separately. The values of h2(τ1, τ2) for τ1 = τ2 are a continuous series of impulses of varying areas. Thus, the magnitudes of the kernels gives an indication of the nonlinear cross-talk between different (in past time) portions of the input (Marmarelis and Marmarelis, 1978). Marmarelis (1972) clearly showed the awesome computational complexity required to find estimates of a finite set of Wiener kernels, {hˆ k}, required to characterize the nonlinear system. Wiener’s approach, although possessing great mathematical sophistication, is so very unwieldy in an experimental situation that it cannot satisfactorily cope even with a low-order linear system.
8.3.1
THE LEE–SCHETZEN APPROACH
TO
WHITE NOISE ANALYSIS
Lee and Schetzen (1965) showed how the terms of the Wiener functional series may be estimated through cross-correlation (or cross-power spectrum) techniques to arrive at a useful, general, quantitative model for a nonlinear dynamic system. Marmarelis and Naka (1973a) extended the Lee–Schetzen method of white noise analysis to nonlinear systems with two inputs and one output, and have applied this approach to describing the dynamics of signal processing in the catfish retina. They also analyzed the statistical errors associated with finding the kernel estimates, restricting their analysis to second-order nonlinear systems having all {hk} = 0 for k Š 3. Lee and Schetzen (1965) showed that the Wiener kernels are, in general, given by hk(τ1, τ2, … τk) = 1(k! Pk) E[y(t) x(t – τ1) x(t – τ2) … x(t – τk)]
8.3-10
which is valid for τm ¦ τn. This expectation is a k-dimensional cross-correlation. A more general form for hk is valid for τm = τn:
(
h k ( τ1 , τ 2 , … τ k ) = 1 k! P k
)
⎧⎪⎡ E ⎨ ⎢ y( t ) − ⎪⎩⎢⎣
k −1
∑ n=0
⎫⎪ 8.3-11 ⎤ G n [ h n , x( t )]⎥ x(t − τ1 )x(t − τ 2 ) … x(t − τ k )⎬ ⎥⎦ ⎪⎭
To clarify the meaning of Equation 8.3-11, the first three kernels are written out:
[
]
h 0 = G 0 h 0 , x( t ) = E[y( t )] = y
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8.3-12
{[
}
]]
[
[
]
h1 ( τ1 ) = (1 P ) E y( t ) − G 0 h 0 , x(t ) x(t − τ1 ) = (1 P ) ϕ xy ( τ1 ) − y x 8.3-13
(
) {[
(
)[
[
]
}
]
h 2 ( τ1 , τ 2 ) = 1 2 P 2 E y( t ) − G 0 h 0 , x( t ) − G1[ h1 , x( t )] x(t − τ1 ) x(t − τ 2 )
]
= 1 2 P 2 ϕ xxy ( τ1 , τ 2 ) − yϕ xx ( τ1 − τ 2 ) − ϕ xxy1 ( τ1 , τ 2 )
8.3-14 where G1[h1, x(t)] = y1(t) is the linear output term found from the real convolution:
y1 ( t ) =
∞
∫ h (τ) x(t − τ) dτ 0
1
8.3-15
The terms, –y ϕxx(τ1 – τ2) and ϕxxy1(τ1, τ2) are zero for τ1 ¦ τ2. The functions, h2(τ1, τ2) and ϕxxy(τ1, τ2) can be visualized as three-dimensional surfaces over a τ1, τ2 plane. Also, the h2(τ1, τ2) kernel is symmetrical around the line τ1 = τ2. According to Marmarelis and Marmarelis (1978): “The value of h2(τ1, τ2) gives a quantitative measure of the nonlinear deviation from superposition due to interaction between portions of the stimulus signal that are τ1 and τ2 sec in the past, in affecting the system’s response in the present. Or if τ1 = τ2, it denotes the amplitudedependent nonlinearities.” Because of the computational complexity of finding k Š 3 cross-correlation functions and kernels, and problems in visualizing Š four-dimensional functionals, they are seldom used in white noise analysis of nonlinear systems. This means that if the system is sharply nonlinear, its description only in terms of h0, h1, and h2 will be inaccurate. The number of terms required to accurately describe a nonlinear system can be estimated by examining the response of the system to a steady-state sinusoidal input. The number and relative magnitudes of the harmonics in the system output can predict how many G terms will be required, because theoretically, Gn can produce at most an nth order harmonic in y(t). Maramarelis and Naka (1974) pointed out that computation of the system kernels by using the fast Foutier transform (FFT) to find the cross-power spectrums, and then inverting these functions by FFT, provides a considerable economy in time and computer effort. The catfish retina studies reported by Marmarelis and Naka (1973a,b; 1974) use no higher than h2 terms in the Wiener functional series models of retinal behavior. They justify the truncation of the series by arguing that the biosystem is weakly nonlinear and does not generate a significant amount of thirdand higher-order harmonics when driven sinusoidally. They also observed very relevantly: The fact that we cannot easily [in terms of computational effort] estimate crosscorrelations of higher (than third) degree limits the applicability of the method to systems whose nonlinearities allow an acceptable representation in terms of the first few terms of the series. Thus systems with “sharp” nonlinearities (thresholds, sharp
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limiters, etc.) cannot be described accurately. However, if the series is truncated after the nth order term, the resulting approximation is the best nth order characterization in the MSE sense. This derives directly from the fact that the terms of the series are orthogonal.
Thus, it is apparent that because of difficulties in interpretation of high-order kernels, and computational difficulties, it is not practical to use kernels of order three and higher in implementing the Lee–Schetzen approach to white noise analysis. These are important, practical restrictions.
8.3.2
PRACTICAL ASPECTS OF IMPLEMENTING WHITE NOISE ANALYSIS
THE
LEE–SCHETZEN
One of the first requirements in implementing white noise analysis of a nonlinear system is that the system must be stationary; i.e., its parameters (gains, rate constants, natural frequencies, etc.) must not change in time, at least over the time required to sample the data (x(t) and y(t)) required to compute the kernels h0, h1, h2, etc. White noise is an engineering idealization (similar to an ideal voltage source, or an ideal op amp), made for computational expedience. In practice, noise is called “white” when its power density spectrum is flat and overlaps the low- and highfrequency limits of the NSUS bandpass by at least two octaves. Because kernel estimations are carried out digitally, attention must be given to problems caused by aliasing, quantization noise, and to data window functions. To avoid aliasing, the sampling rate for the noise x(t) must be well above the frequency where the amplitude of the autopower spectrum, Φxx(f), of the noise is down to 10% of its peak value. The noise autopower spectrum must also be sharply attenuated above its cutoff frequency to avoid aliasing. Maramarelis and Naka (1974) have shown that aside from the bandwidth requirements imposed on Φxx(f) by the NSUS and by alias-free sampling, an excessive input noise spectral bandwidth will contribute to large statistical errors in computing the estimate of the linear kernel, h1 (and presumably also to the higher-order kernels as well). They show that, for a strictly linear system where kernels of order Š 2 are zero, when the input noise has a two-sided, rectangular spectrum with cutoff frequency, ωo r/s, then the variance of the Lee–Schetzen estimate of h1(τ) is the variance of ϕxy(τ), and is given by
(
VAR[ h1 ( τ)] = ω o 4 π 3
)∫
ωo
−ωo
(
H 1 ( jω ) dω + 1 4 π 2 2
)∫
ωo
−ωo
2
H 1 ( jω ) exp( jωτ) dω 8.3-16
where H1(ω) = F{h1(t)}, and Φxx(ω) = 1 for 冨ω冨 ð ωo, and Φxx(ω) = 0 for 冨ω冨 > ωo. If h1(t) = Aδ(t), (h1 is an ideal amplifier with gain A), then it can be shown:
(
) (
)[
VAR[ h1 ( τ)] = ω 2o 4 π 4 + 1 4 π 4 τ 2 1 − cos (2ω o τ) 2
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]
2
8.3-17
If h1(t) = exp(–αt), then they show VAR[h1(τ)] = (ωo/2π3 α) tan–1(ωo/α) + [(1/π) tan–1(ωo/α)]2
8.3-18
Note that both variances are increasing functions of ωo, the input noise spectrum cutoff frequency. Thus, it appears that it is important not to have the input spectrum exceed the system bandwidth by too much for two important reasons; the first is potential problems with aliasing, and the second is excessive variances in the kernel estimates. See Marmarelis and Naka (1974) for details on the theoretical effects of uncorrelated and measurement noise on the variances of the estimates of the kernels. A summary of the steps required to calculate the NSUS kernels, h0, h1(t), and h2(τ1, τ2) in the time domain by the Lee–Schetzen method is shown in Figure 8.3-1. Note that the kernels can be estimated in the frequency domain following the procedure shown in Figure 8.3-2. It is necessary to use the average spectra of x(t) and yo(t) to find estimates of H1(jω) and H2(jω1, jω2) and their inverse transforms, h1(t) and h2(τ1, τ2). This means that x(t) and yo(t) are broken into N successive sampling epochs, each having 4096 samples, for example. N might be 32.
8.3.3
APPLICATIONS OF THE WHITE NOISE METHOD NEUROBIOLOGICAL SYSTEMS
TO
Many of the nonlinear neural systems that can be studied by the white noise method present special problems. The systems described above have assumed that both the white noise input and the NSUS output, y(t), are continuous analog signals. Many neural systems differ in that they can have nerve spike inputs and/or nerve spike outputs. Nerve spike trains are best characterized as point processes; that is, by a train of unit impulses whose occurrence times are the peak times of the recorded spikes, as shown in Equation 8.3-19:
y p (t ) =
∞
∑ δ( t − t )
8.3-19
k
k =1
To apply the white noise method effectively, it is expedient to convert yP(t) to a continuous function. One way to do this is to compute the elements of instantaneous frequency, rk, for yP(t). rk is defined at each spike occurrence time, tk as rk ≡ 1/(tk – tk1),
k = 2,3,4 …
8.3-20
rk is also a point process, but it can be converted simply to a stepwise-continuous analog signal by integrating each rkδ(t – tk) and holding the resulting step over the interval, {tk, tk+1}. The output of the process is the stepwise waveform, q(t), given by
q( t ) =
∞
∑ r [U(t − t ) − U(t − t )] k
k =2
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k
k +1
8.3-21
FIGURE 8.3-1 Block diagram describing how the zeroth, first, and second-order kernels are calculated for a SISO nonlinear system, using the white noise method developed by Marmarelis (1972). Note that the second-order kernel, h2(τ1, τ2), is presented as a contour plot.
where U(t – tk) is a unit step that occurs at t = tk, and U(t – tk+1 ) is a unit step at tk+1. Figure 8.3-3 illustrates the generation of the instantaneous frequency (IF) signal, q(t). This function is physically realizable because rk is not defined until the kth pulse in the sequence occurs, triggering a step at tk of height rk and lasting until the next pulse at tk+1. An application of the IF description of a spike train used in the white noise method was given by Poliakov (Poliakov et al., 1997; Poliakov, 1999). Poliakov injected a dc plus broadband noise current into a motoneuron soma with a microelectrode and recorded its spike activity on its axon. A mild, second-order nonlinearity was assumed in which the input noise first acted on the linear kernel, h1(t). The output of h1(t), u, was assumed to be the input to a nonlinearity of the form: y(t) = h0 + u(t) + au2(t). y(t) is the output of the NSUS, in this case, the IF of the motoneuron’s spike train. Figure 8.3-4 illustrates the typical form of h1(t), h2(τ1, τ1)
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FIGURE 8.3-2 Block diagram showing how the Fourier transforms of the first- and secondorder kernels can be calculated.
and h2(τ1, τ2) found for the motoneurons he studied. Note that the second-order kernel is symmetrical around the line τ1 = τ2. Another example of the application of the white noise method to a neuro-sensory system was described by Marmarelis and Naka (1973a). In this case the system was the single-input/single output catfish retina photoreceptors to a spiking ganglion cell (GC). The input was noise-intensity-modulated light in the form of the general RF, a spot at the center of the RF, or an annulus around the center of the RF. The first system studied by Marmarelis and Naka (1973a) using white noise analysis was the analog input/analog output light (whole RF) to horizontal cell (HC) pathway. (For a description of the anatomy of the vertebrate retina, see Section 6.1.) Because of the spatiotemporal filtering properties of the retina, and the sigmoidal (log intensity)
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FIGURE 8.3-3 A spike train, showing how the spikes’ instantaneous frequency elements can be converted to a stepwise, analog output, ry(t), by use of a holding process. The height of each step of ry(t) is proportional to the IF of the preceding pulse interval.
response characteristic of photoreceptors, the calculated kernels for light input/HC response are different for different mean light intensities and stimulus shapes (spot, annulus, whole RF). Figure 8.3-5 illustrates the gross cellular features of the retina. Figure 8.3-6 show the linear kernels for a horizontal cell membrane potential when its whole RF was stimulated by “white” noise at two different mean intensities. (Note that the RF of the HC recorded from contains many photoreceptors and other HCs.) Figure 8.3-7 shows two second-order kernels calculated for the same two mean intensities. The higher mean intensity gives more pronounced peaks and valleys in the h2(τ1, τ2) plots. Finally, Figure 8.3-8 shows h1(t) plots, each calculated for three increasing mean intensities (C highest) for a spot stimulus (A) and for an annulus stimulus (B). Note that the annulus impulse response becomes more underdamped with increasing mean intensity. When Marmarelis and Naka (1973a, b) used white noise analysis to characterize the light to GC responses of the catfish retina, they faced the problem of converting the point process describing the GC spikes to a continuous signal proportional to frequency. In this case, they did not use IF, but instead used the following procedure. Five to ten runs were made using identical noise records. A pooled, PST histogram was made from the resulting GC spikes for each run. (The pooled PST diagram had the dimensions of spikes/s.) The pooled PST diagram was smoothed with an “appropriate smoothing window,” to form y(t) used in kernel calculations. Figure 8.3-9 shows three h1(t) kernels, all from the same GC system calculated for uniform RF stimulation, and spot and annulus stimuli. Note the strong biphasic response for the annulus and uniform RF stimulation, denoting temporal differentiation of the input. The first and second-order kernel contour plots for the GC system with entire RF stimulation are shown in Figure 8.3-10. Note the sharp peaks and valleys in h2(τ1, τ2). To interpret these features, note that a positive impulse of light intensity (a short flash) given 100 ms in the past followed by a second flash at 80 ms in the past will
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FIGURE 8.3-4 Representative forms for the kernels of motoneurons studied by Poliakov (Poliakov et al., 1997; Poliakov, 1999). Poliakov’s study was unique in that he applied the white noise method to a nonsensory neural system.
give a strong positive response. On the other hand, a sharp suppression of the response will occur when the first impulse is given at 150 ms in the past followed by a second flash at 100 ms.
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FIGURE 8.3-5 Block diagram illustrating the basic features of applying the white noise method to the retina. The input noise (visual object) can have both spatial and temporal properties that vary randomly.
Marmarelis and Marmarelis (1978) considered the canonical structure of a second-order, nonlinear dynamical system having two independent white noise inputs, and one output. A block diagram showing the system architecture is shown in Figure 8.3-10A. The Wiener system approximating the NSUS is shown in Figure 8.3-10B. The MSE between the actual system output, y(t), and the Wiener system output, yˆ (t), theoretically decreases as high-order kernels (k Š 3) are added to the system. However, the computational complexity and inability to understand and interpret kernels of order 3 and higher has limited their use in any practical application. The two-input, one-output approach was used by Marmarelis and Naka (1973b) to explore interactions between spot (center) and annulus (surround) stimulation of the RF of a catfish GC. Their results will not be described here; the interested reader should examine the original papers.
© 2001 by CRC Press LLC
First Order Kernel, h1 (τ)
28
Response
A
32
B
24
A B
20
Log
16
I
12 8 4 0 -4 -8
-12
64
128
192
256
320
Time (msec), τ FIGURE 8.3-6 First-order kernel computed for a system consisting of noise-modulated light on the entire RF (input); the output was the transmembrane potential of an HC in the catfish retina that had that RF. Two different, mean input intensities were used. (From Marmarelis, P.Z. and K.I. Naka, J. Neurophysiol., 36(4): 619, 1973. American Physiological Society. With permission from The American Physiological Society.)
h2 (τ1 τ2) , HORIZONTAL CELL τ1 , sec 0
.032
τ1 , sec
.064
.096
.128
.16
0
0
.032
.064
.128
.16
0
0 0
.032
0
−8
+8
−9
−6
6
14
-23 -21 τ2 , sec
.064
0 7
.032 0
τ2 , sec
.096
0
.064
0
-14
0
0
-7 7 0
.096
−9
14
.096 0
+8
0
.128
0 0
.128 0 0
.16
0 .16
Low mean intensity
High mean intensity
FIGURE 8.3-7 Two second-order kernel contour plots for the system described in Figure 8.3-6. (From Marmarelis, P.Z. and K.I. Naka, J. Neurophysiol., 36(4): 619, 1973. With permission.)
© 2001 by CRC Press LLC
40
Horizontal cell
A
C Spot
First Order Kernels h1
20
B A
0 B C
80
Annulus B
40 A 0
-20 0
0.1
0.2
0.3
Time τ, sec FIGURE 8.3-8 (A) Linear [h1(τ)] kernels for the system: Spot of noise-modulated light on the center of the RF (input); the output was the transmembrane potential of a horizontal cell in the catfish retina that had that RF. (B) Linear [h1(τ)] kernels for the system: Annulus of noise-modulated light around the periphery of the RF (input); the output was the transmembrane potential of a horizontal cell in the catfish retina that had that RF. In both A and B, mean intensities are about I/2 for curve C, –0.8 log units in B, and –1.6 log units in A. Note that as the average light intensity increases, the h1 response becomes underdamped and develops an undershoot. (From Marmarelis, P.Z. and K.I. Naka, J. Neurophysiol., 36(4): 619, 1973. With permission.)
8.3.4
DISCUSSION
The utility of the white noise method in characterizing neuro-sensory systems is limited by several factors. Because of the difficulty in calculating and interpreting kernels of order 3 and higher, applications of white noise analysis have generally been limited to first- and second-order system descriptions. This limits the accurate application of the method to systems that are weakly nonlinear. Also, it is difficult to go from h1 and h2 to meaningful conclusions about the anatomy and physiology of the NSUS. Because there are so few other effective systems characterization methods available to study nonlinear systems (e.g., inverse describing functions, triggered correlation), white noise analysis is expected to be used in the future to characterize and model other weakly nonlinear neuro-sensory systems.
8.4
CHAPTER SUMMARY
A major challenge in studying any neuro-sensory system is to obtain a general mathematical model of the system input/output behavior. (The rationale for finding a model of a complex nonlinear system should now be abundantly clear to the reader.) In simplest terms, the input stimulus to the system can be controlled, and spikes or graded potentials recorded from one or more interneurons that are causally driven
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B 25
20
FIRST ORDER KERNELS hI
15 10
UNIFORM LIGHT SPOT OF LIGHT ANNULUS OF LIGHT
60 MSEC
-5
-10 -15
TIME τ
0.54 SEC
CURRENT INJECTION S - POTENTIAL
15 10
5 0
20
FIRST ORDER KERNELS hI
A 25
60 MSEC
5 0 -5
TIME τ
0.54 SEC
-10 -15
-20
-20
-25
-25
FIGURE 8.3-9 (A) First-order kernels for the system: Noise-modulated light → GC (spike frequency output). Three input conditions were used: (1) entire RF stimulated; (2) center of RF illuminated with spot; (3) annulus of light illuminated peripheral RF. (B) —♦— is the h1 kernel for the system: Noise-modulated light to the entire RF → HC (positive peak is from hyperpolarization). —x— is the h1 kernel for the system: Noise current injected intracellularly to a HC → GC spike frequency output. Note the difference in h1 peak polarities. (From Marmarelis, P.Z. and K.I. Naka, J. Neurophysiol., 36(4): 619, 1973. With permission.)
by the input receptor(s). These input/output signals can be used to create mathematical models describing the neuro-sensory system. Very often one can gather clues about why a neuro-sensory system behaves the way it does from neuroanatomical studies, and from the way the neuro-sensory system behavior changes when it is given certain drugs, such as TTX or TEA. This chapter has described three methods of characterizing neuro-sensory systems. In Section 8.1 discussed that a qualitative statistical tool, the JPST diagram, that can be used to make putative, parsimonious models for neural interactions. The JPST does not yield a mathematical description of the input/output behavior of a neuro-sensory system; rather, it leads to a structural model whose properties can be examined with neural modeling software such as GENESIS or XNBC validate the model. (If it quacks like a duck, it may be a duck.) The triggered correlation algorithm (TCA), introduced in Section 8.2, was shown to be applicable in neuro-sensory systems with narrow-band or tuned behavior operating on the stimulus. Thus, it has application in auditory systems that exhibit frequency selectivity. (Hypothetically, the TCA could also be used in visual systems exhibiting spatial frequency selectivity.) The TCA was shown not to give useful results if the bandpass Q is low. The TCA yields an equivalent linear weighting function for the frequency selectivity of the neuro-sensory system. Since the TCA
© 2001 by CRC Press LLC
FIGURE 8.3-10 (A) Canonical structure of a nonlinear system having two independent, uncorrelated noise inputs. The F12, 2I/SO system allows mixing of the two inputs. (B) Wiener system approximating the 2I/SO nonlinear system. Only the first two kernels and the secondorder cross kernel are used. See text for comments.
is used on what is basically a nonlinear system, different input stimulus noise conditions can give different results. The Lee–Schetzen–Marmarelis white noise method described in Section 8.3 is more general and powerful than the TCA. The white noise method gives a very abstract description of a neuro-sensory system that is characterized by kernels or weighting functions. The first-order kernel is basically the weighting function of the neuro-sensory system if it is purely linear, the same result as x+(τ) from the TCA. If the neuro-sensory system is nonlinear (and it always is), the white noise method also gives higher-order kernels or system weighting functions corresponding to the nonlinear behavior. A major problem seen with the white noise method is that kernels
© 2001 by CRC Press LLC
of order 3 or higher, which are necessary to describe a very nonlinear system, have more than three dimensions, and cannot easily be visualized or interpreted. The second-order kernel however, h2(τ1, τ2), is a three-dimensional function that is usually visualized as a two-dimensional contour plot. The white noise method has been applied successfully to retinal systems and to spinal motoneurons. The white noise method yields a black-box description of the neuro-sensory system being studied. The usefulness of this type of model is enhanced if it can be correlated with neuron structure and interconnections.
© 2001 by CRC Press LLC
9
Software for Simulation of Neural Systems
INTRODUCTION Why is it important to simulate the electrical and molecular behavior of individual neurons and small neural networks? There are several answers to this question: (1) To prove that how they work is truly understood. (2) To predict neural behavior not yet seen in nature by altering ionic conductances in membranes with models of drug action. (3) To verify connectivity structures in small biological neural networks that exhibit unique firing behavior (e.g., two-phase bursting). (4) To model CNS functions (on a greatly reduced scale) such as the motor control of eye movements, the detection of objects by electric fish, or a basic learning behavior. Biological neural networks (BNNs), such as in the retina or cochlear nucleus, are too complex and nonlinear to permit their neurophysiological behavior to be predicted from anatomy alone. Neuroanatomists have the tools to identify the neurotransmitters in synapses, and enough neuropharmacology is known to identify whether a given synapse is excitatory or inhibitory, fast or slow, and what ions are gated, etc. Such details can be inserted in the detailed, conductance-based dynamic models to approach verisimilitude. In the past 10 years, as personal computers and desktop workstations have become ever more powerful, there has been a proliferation of specialized software applications designed to simulate the electrical behavior of individual neurons and the information-processing properties of assemblies of biological neurons. As will be seen, some of these programs deal best with the molecular and ionic events in and around the cell membranes of individual neurons and their dendrites, while other programs appear to be better used to investigate the properties of BNNs. There is a trend to sacrifice molecular and ionic details of neuron function as the number of neurons in the BNN increases. For example, instead of relying on the Hodgin–Huxley (HH) model or one of its variations for spike generation, the spike generator locus (SGL) can be modeled by the RPFM (leaky integrator) spike generator (see Section 4.3.2). A more primitive SGL can use the IPFM (integrate-fire-reset) voltage-tofrequency converter (see Section 4.3.1). Most of the neural modeling (NM) applications described below are freeware, downloadable from Web sites. Also, most of them have been designed to run on workstations having UNIX operating systems (DECOSF, Ultrix, AIX, SunOS, HPux, etc.) using Xwindows, or on systems using LINUX. Some NM applications have been modified to run on personal computers using Microsoft Windows™ or Windows NT™.
© 2001 by CRC Press LLC
There are many Web sites from which the interested reader can download descriptions of NM software and, in most cases, the software itself. The reader should be warned that information on the World-Wide Web is ephemeral; Web sites can move and also can close down. Information on the Web does not have the same permanence as books and journals on library shelves, and CDs. A summary listing of Computational Neuroscience Software can be found at wysiwyg://2/http://home.earthlink.net/~perlewitz/sftwr.html. Under the heading, Compartmental Modeling, this site lists and has hot links to the following NM programs: CONICAL, EONS, GENESIS, NEURON, NeuronC, NODUS, NSL, SNNAP, the Surf-Hippo Neuron Simulation System, and XPP. (Note that “compartmental,” used in an NM context means a closed membrane volume over which the same electrical potential exists, which is quite different from “compartment” as used in pharmacokinetics.) Realistic Network Modeling includes BIOSIM, SONN, and XNBC. The GENESIS and NSL programs are also supported by detailed textbooks: Bower and Beeman (1998) wrote The Book of GENESIS, and Weitzenfeld et al. (1999) wrote The Neural Simulation Language (NSL). As will be seen below, some programs such as NSL and NEURON can easily model networks, and probably should have been listed under both categories above. Figure 9.1 illustrates the evolution of the compartmental model of a typical neuron having dendrites, a soma, and axon. One chemical synapse is shown at the tip of the top dendrite. The first step in the compartmental modeling approach is to subdivide the features of the neuron into cylindrical “compartments,” each of which has an area of membrane, Ak = dkπLk cm2; dk is the diameter of the kth cylinder, and Lk is its length. The modeler must choose Lk small enough to give an accurate lumped-parameter model, and large enough to keep a reasonable number of ODEs and auxiliary equations. Each compartment is characterized by a total transmembrane capacitance, cmk, in farads, and one or more specific ionic conductance parameters, gjk; some may be fixed, or functions of the membrane voltage of that compartment, Vmk. Other conductances can depend on calcium ion concentration, or the local concentration of neurotransmitter. Dendrite tips are modeled by cones of area Adk = dkπLk/2. Most simulation programs using the compartmental modeling architecture allow the user to specify input currents to individual compartments, and also to place certain compartments under voltage clamp conditions. Note that every compartment is joined to its neighbors by longitudinal spreading resistances based on local axoplasm resistivity and the length of the compartment(s) involved. One method to simulate the action of chemical synapses is to treat the arrival of the presynaptic spike as a delta function, U–1(t), which acts as the input to a pair of concatenated, first-order ODEs, the output of which is the postsynaptic ion conductivity function in time that generates the epsp or ipsp at the site of the synapse. For the so-called alpha function governing Na+ conductance, the defining ODEs are
x˙ 1 = −ax1 + U −1 ( t )
a ≡ 1 / τ1
( )
x˙ 2 = −ax 2 + a e1 x1
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9.1 9.2
FIGURE 9.0-1 (A) A bipolar neuron with axon, soma, and dendrites. Thin lines denote compartment boundaries. One synapse is shown. (B) The neuron of A is modeled with linked compartments made up from cylinders of nerve membrane. The area of the kth cylinder is Ak = π DkLk cm2. (D, diameter; L, length.) (C) Each compartment is characterized by (1) a transmembrane capacitance, (2) a transmembrane conductance that can be fixed (in the case of passive membrane on dendrites and soma), or voltage dependent (gK and gNa in the axon), or chemically dependent, as in the compartments receiving synapses. Connecting adjoining compartments are the two ri/2 of the adjoining compartments (ri is the axonal longitudinal resistance.)
( )
G(t ) = G max x 2 (t ) = G max a e1 t exp( −at )
9.3
For added flexibility, one sometimes uses a two-time-constant model for postsynaptic conductivity increase.
x˙ 1 = −ax1 + U −1 ( t )
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a ≡ 1 τ1
9.4
x˙ 2 = − bx 2 + K x1
ba ≡ 1 τ 2
G(t ) = G max x 2 (t ) = G max [K ( b − a )][exp( −at ) − exp( − bt )] ,
9.5
b>a
9.6
The HH spike generation formalism can be applied to the axon compartment sections, and thus generate an action potential traveling wave as the voltage-dependent conductance activity spreads from one axon compartment to the next. Considerable neurophysiological detail can result from the compartmental approach. Single neurons and small BNNs can also be modeled by the general-purpose, nonlinear ODE solver Simnon™, at any level of detail desired. Using the modular locus approach, the programmer develops sets of ODEs describing SGLs, epsps, ipsps, delays, dendritic attenuations, etc., then a connection matrix giving the modeled BNN structure. The following sections describe some of the currently available neural modeling programs, their applications, and where they can be found.
9.1
XNBC V8
XNBC stands for Xwindow Neuro_Bio_Clusters. Xwindows is the UNIX windowing system, Neuro_Bio is for biological neurons, and Clusters is for the way the neurons are grouped. (A cluster is a group of model neurons sharing the same membrane properties.) This powerful and flexible program was first developed in France about 12 years ago by computational neurobiologists and software engineers to run on Unix workstations. Its current version is 8.25 (July, 1999). The user manual, written by Jean-François Vibert, the XNBC project leader, is available on line at http://www.b3e.jussieu.fr:80/logiciels/xnbc8_manual/index.html. XNBC v8 has a user-friendly, interactive interface from which the following tools can be accessed: Two neuron (SGL) editors (a phenomenological model editor and a conductance-based (HH) model editor) Two network editors (simple and full-featured) A drug editor tool (drugs affect ion channels) A simulator tool A visualization tool A time series analysis tool (to perform point-process statistics on the spike trains of the model). A cluster activity analysis tool. An expert system (under development). The user can visualize in time neuron spikes, transmembrane potentials, epsps, ipsps, and ionic conductances and currents. To quote from the History and Implementation portion of the online user manual.
© 2001 by CRC Press LLC
XNBC is written in portable ANSI C, and was compiled on [computers running under] Ultrix, Digital Unix, IBM AIX, Sun Solaris, HP Ux, Linux and DEC VMS and open VMS [operating systems]. XNBC runs on Xwindow workstations and needs the Motif library. When possible, the GNU C compiler (gcc) should be preferred. XNBC produces generally simple ASCII data files that can easily be converted to any format required by common graphic programs or spreadsheets. It produces native color PostScript files (that can be directly used to prepare figures). XNBC is a public domain software package available freely for academic research purpose on Internet (ftp://ftp.b3e.jussieu.fr/pub/XNBC) and informations [sic] about new versions at URL http://www.b3e.jussieu.fr/logiciels/xnbc.html.
The XNBC v8 simulation program is user-friendly and interactive. For example, the user interface for the conductance-based model graphic editor (G_neuron) allows setting 12 different transmembrane currents and two synaptic currents. Simplified HH kinetics are used. The user can select one of three integration routines for the simulation (Euler, fourth order Runge-Kutta, and exponential). Simulation parameters are adjusted with a mouse by moving dial cursors, or typing in numerical values, while a real-time display shows the neuron membrane potential (including spikes) and the ionic currents as the parameters are adjusted. A graphic utility also allows one to plot any variable vs. any other variable. Voltage clamp and current clamp experiments can be simulated to adjust better the conductance parameters. (Note that the current clamp is in effect an ideal current source or sink to the neural element, while the voltage clamp requires a negative feedback system that adjusts the current to maintain a preset membrane potential.) The program also allows the user to simulate the effects of the drugs TEA (tetraethylammonium) and TTX (tetrodotoxin) on conductances, as well as introduce the effects of long-term NMDA neuromodulation from glutamate-releasing synapses. Whether conductance-based neurons are modeled or the more succinct leaky integrator (RPFM) spike generator is used, the user can add broadband Gaussian noise to the membrane potential (generator potential) specified by standard deviation and mean to emulate synaptic noise (see Section 1.3.4). (Noise bandwidth is evidently not an adjustable parameter.) In summary, it appears that XNBC v8 is a very versatile, user-friendly, and flexible neural modeling program, whether single neurons or small assemblies are being considered. The program graphic user interfaces (GUIs) make it easy to use.
9.2
NEURAL NETWORK SIMULATION LANGUAGE, OR NSL
NSL is a product of computational neurobiologists at the University of Southern California Brain Program (USCBP). The latest version of this program is NSL3.0j (23 April 1999). It is designed to run on UNIX OS computers; however, NSL3.0m has been adapted to run on personal computers running Windows 95, 98, or NT 4.0 (22 September, 1999). The files occupy some 81.25 MB, and the m version runs under either the DOS window environment or the Cygnus/cygwin window environment. NSL is written in NSL C++. NSL is public-domain software for educational institutions; however, NSL is licensed and copyrighted. To read about the protocol
© 2001 by CRC Press LLC
for downloading NSL, see http://kona.usc.edu/~nsl/unprotected/status99.4.14.html. To obtain the PC/Windows version, see: http://kona.usc.edu/~nsl/unprotected/ns/NSL_3_0_m/SetupForPCs.html. Because NSL was developed to model CNS behavior in specific situations, its models typically deal with hundreds if not thousands of neural elements, and proportionally more synapses. A typical neural SGL uses the leaky integrator model, in which the generator potential, mp is described by the simple ODE:
˙ = − mp + s τ mp
9.2-1
where τ is the time constant of the low-pass filter, and s is its net input voltage. s can be given as N
s=
∑w r
k k
9.2-2
k =1
where rk is the (instantaneous) firing frequency of the kth input neuron, and wk is the input weight for the kth synapse. Note that wk can be positive or negative (inhibitory inputs). There are N inputs to the single neuron. Unlike the RPFM SGL model, the output frequency of the leaky integrator neuron is not determined by a pulse generator and reset mechanism. Instead, the generator potential is passed through a no-memory nonlinearity, r = f(mp), to determine the instantaneous output frequency. Figure 9.2-1 illustrates four possible nonlinearities, or threshold functions, that can be used in NSL. Note that mp and r are analog variables. In the NSL book (Weitzenfeld et al., 1999) some 11 CNS neural models applying NSL are described. All use very large numbers of neurons. In many ways, NSL is a bridge program between BNNs and ANNs. NSL can use more-detailed biological features of neurons than a typical ANN program; yet it also allows ANN structures to be simulated, albeit for neurophysiological purposes. The 11 example models in the NSL book include: Grossberg’s adaptive resonance theory, Dev and House’s depth perception, modeling the retina, receptive fields, the associative search network – landmark learning and hill climbing, a model of primate visual-motor conditional learning, the modular design of the oculomotor system in monkeys, the CrowleyArbib saccade model, a cerebellar model of sensorimotor adaptation, learning to detour, and face recognition by dynamic link matching.
9.3
NEURON
The Neuron program was initially written by Michael J. Hines at Yale University about 10 years ago. Originally for use on UNIX computers, it now has versions that run on MSWindows 3.1, 95, 98, and NT 4.0, as well as the MacOS. Neuron, too, is public-domain software for use in universities. Like XNBC described above, Neuron has evolved to have a user-friendly, interactive GUI. It uses an objectoriented interpreter to define the anatomical and biophysical properties of the model,
© 2001 by CRC Press LLC
FIGURE 9.2-1 Four nonlinearities relating generator potential (mp) to (analog) instantaneous frequency (r) in the NSL modeling language.
set up the GUI, control the simulations and plot results. Neuron effects the simulation on a molecular and ionic level, using a modified HH approach. It permits both voltage- and current-clamping simulations on active nerve membranes. Axons, dendrites, and soma are modeled as cylindrical sections; they can be sized and connected, and given specified membrane properties (e.g., HH or passive). Synapses are not modeled per se, but the action potential at the end of an axon can be used to activate specific conductance changes on the soma or dendrites of the postsynaptic neuron through point processes. To achieve synaptic connection between a presynaptic axon and a postsynaptic dendrite or soma, the user must write a subroutine in the Neuron MODL modeling language. This process must be performed for each synapse. The delay, size, and time course of the conductance changes producing the epsp or ipsp can be specified in the KINETIC section of the MODL subroutine. When using multiple neurons, their locations are specified in three-dimensional space. The user has a choice of two integration routines to solve the conductance ODEs: backward Euler and a “variant of Crank–Nicholson (C-N).” The reader interested in learning more about Neuron and what it can do is urged to visit the following web sites: http://www.neuron.yale.edu/neuron/about/what.html (“What is NEURON?”); http://cs.unc.edu/~martin/tutorial.html (Kevin E. Martin’s “NEURON Programming Tutorial”); http://www.cnl.salk.edu/~spillman/neuron.html (Kimberly Spillman’s “Beginning with Neuron”); http:/neuron.duke.edu/environ/techhtml/nc1.htm (N.T. Carnevale and M.L. Hines’ digital preprint of “The NEURON Simulation Environment”). Neuron software is located at the anonymous ftp site: http://neuron.duke.edu/userman/5/version3.html.
© 2001 by CRC Press LLC
9.4
GENESIS
GENESIS stands for GEneral NEural SImulation System. This simulation program originated at CalTech about 12 years ago, and has been steadily evolving. GENESIS is a ubiquitous simulator, enabling the user to model at many different levels, from subcellular components to single neurons to small networks. GENESIS models can be programmed in script, and by a flexible, interactive XODUS GUI in an objectoriented manner. (XODUS stands for X-based Output and Display Utility for Simulators.) GENESIS is widely used in the computational neurobiology community, and there is a clearly written text available describing how to use it, with examples (Bower and Beeman, 1998). Most GENESIS applications appear to be at the microlevel; i.e., detailed neural models of single neurons of small assemblies of neurons such as found in models for central pattern generators (CPGs). Such detailed models include soma, dendrites, axon, synapses, and many different ion channels. GENESIS allows user selection of five integration routines: forward Euler, backward Euler, exponential Euler (the GENESIS default), Adams–Bashford, and Crank– Nicholson. When GENESIS is used to simulate very large arrays of neurons, the parallel processing form of GENESIS, PGENESIS can be used. GENESIS is used to define neuron properties in the large network, then PGENESIS is used to run the simulation efficiently. Information about PGENESIS can be found at the Pittsburgh Superconducting Center web site: www.psc.edu/general/software/packages/pgenesis/project_ docs/pgenesis-home.html. GENESIS runs exclusively on UNIX OS machines and their variants. Recently, it was adapted to run under LINUX on 486 and Pentium PCs. The interested reader wishing to investigate GENESIS further should visit the URLs: http://www.bbb. caltech.edu/GENESIS, http://www.bbb.caltech.edu/hbp/database.html and http://www. caltech.edu/hbp/GOOD.
9.5
OTHER NEURAL SIMULATION PROGRAMS
It appears that almost every major computational neurobiology group at various universities at one time or another has written neural modeling software. Most of these programs run on UNIX-type systems, with some exceptions. Below are listed some of the “minor” or less well known programs that the author found on the Web. Some may be more flexible and user-friendly than the programs described above.
9.5.1
EONS
EONS stands for Elementary Objects of Neural Systems. This program originated at The University of Southern California (Los Angeles) Brain Program around 1996. It appears oriented toward simulating medium-large neural networks (e.g., 800 neurons) using fairly detailed neuron and synaptic models (including a variety of specified ion channels involved in generating a psp). There are two major components to the EONS program: an EONS library and a user interface. The library contains neuron, synapse, synaptic cleft, postsynaptic spine, receptor channels, voltage-gated ion channels, and
© 2001 by CRC Press LLC
neural network. EONS has been used successfully to validate a hypothesis that receptor channel aggregation on the postsynaptic membrane is the cellular mechanism underlying the expression of long-term potential. Although not specifically stated in the “Summary Description,” EONS probably is freeware and runs on UNIX systems. To learn more, visit the URL: http://www-hbp.usc.edu/Projects/eons.htm.
9.5.2
SNNAP
SNNAP is the acronym for Simulator for Neural Networks and Action Potentials. It was developed around 1994 at the University of Texas Health Science Center in Houston to do detailed, realistic modeling of single neurons and small neural networks. SNNAP runs under the UNIX environment; it was developed using ANSI C and Xlib. A DOS/Windows version exists that can simulate up to 20 neurons. Both versions have GUIs. UNIX SNNAP allows the user to simulate the injection of external currents into multiple cells, to simulate ion channel blocking by drugs by removing specific conductances, to modulate membrane currents with modulatory transmitters, and to simulate the voltage clamping of cells. Some 16 sample simulations can be downloaded, including burst generation, CPGs, etc. This appears to be an easy-to-use simulation program, and may be suited for use in an introductory neurobiology class. For details, see the URL: http://snnap.med.uth.tmc.edu/ /overview.htm. Questions and inquiries about obtaining SNNAP can be e-mailed to:
[email protected].
9.5.3
SONN
SONN stands for Simulator of Neural Networks. SONN was developed at the Hebrew University, Jerusalem, Israel, and, like most neural modeling software, runs under the UNIX operating system and its variants. A new PC/Windows version was released in August 1999. SONN v1.0 can be downloaded free; however, the UNIX SONN manual costs U.S.$40. SONN is oriented toward the fairly detailed simulation of single neurons and small neural networks. It allows specification of the following functional units: soma, axon; presynapse, postsynapse. Interested readers should contact http://icnc.huji. ac.il/Research/neuro.html, and http://www.Is.ac.il/~litvak/Sonn/sonn.html. Email requests for the manual to:
[email protected] or
[email protected].
9.5.4
NODUS 3.2
Nodus v3.2 is a state-of-the-art neural modeling program originating in Belgium that has been written to run on Apple Macintosh™ computers (specifically, a Power Mac running System 7, or a Mac II series SE30, Centris 650/660, any Quadra with Š 4 MB RAM and OS 6 or 7). Three different versions of Nodus v3.2 are available to run on various versions of Mac (3.2, 3.2P, and 3.2Q). See the URL: http://bbfwww.uia.ac.be/SOFT/NODUS_system.html for details. To quote from the Nodus Web information blurb:
© 2001 by CRC Press LLC
Nodus combines a powerful simulator with sophisticated model database management. Models are defined in separate files: conductance definition files, neuron definition files and network definition files. All files specifying one model are linked together in a hierarchical structure and [are] automatically loaded when the top file is opened. Several conductance and neuron files can be open at the same time. A simulation database is build [sic] from user specified definition files and can be saved in a separate file, together with specific settings for graphic or text output, experiments, etc.
Two integration methods are available, a fifth-order Runge-Kutta/Fehlberg and a fast-forward Euler integrator, both with variable time steps. The value of any simulation database parameter can be manipulated by the user during a simulation. Networks can be “hard-wired” with up to 200 neurons and a maximum of 60 synapses with delays and/or 20 electric connections for each neuron. Currents of various waveforms can be injected in any compartment of the model. Two neurons can be simultaneously voltage-clamped. Selected ionic currents can be blocked to emulate drug action (e.g., TTX and TEA). Up to 13 ionic conductances can be simulated. Synaptic neurotransmitter release can be constant, voltage-dependent, or concentration pool dependent. Postsynaptic conductance changes can follow the standard alpha model, with the synaptic input an impulse function, g(t) = gmax(t/τ1) exp(1 – t/τ1)], or follow a time course determined by the impulse response of two concatenated, first-order ODEs, giving g(t) = [gmax /(τ1 – τ2)] [exp(–t/τ1) –exp(–t/τ2)]. Voltage-dependent conductances can follow the standard HH model format, or any sort of dynamics described by a user equation. Conductances can also be made calcium-dependent. See the URL: http://bbf-www.uia.ac.be/SOFT/NODUS_index.shtml to download Nodus.
9.6
NEURAL MODELING WITH GENERAL, NONLINEAR SYSTEM SIMULATION SOFTWARE
The sections have discussed a wide variety of specialized programs developed to model the dynamics of neurons and BNNs. Historically, the desktop computers at the beginning of the 1990s that were powerful and fast enough to support the simulation of large, complex BNNs were the systems running UNIX and its variations. Thus, it was only logical that most of the neural modeling programs developed in the early 1990s were written to run on computers with UNIX-type operating systems (OSs) with Xwindows, rather than less powerful DOS machines. Today (02/00), personal computers are priced so that many graduate and undergraduate students have their own, powerful Pentium PCs. A typical PC now has a Pentium III processor running at 600 MHz, 64 to 128 MB RAM, a huge (> 6 GB) hard drive, a CD drive, modem, etc. and costs less than $2000 with monitor. Such computers usually run the Windows 95, 98, 2000 or NT4 OS, although some users use Linux. Some of the academic custodians of the neural modeling software packages described above have seen the handwriting on the wall and now offer, or are in the process of developing, Windows/DOS versions of their simulation programs. This trend is encouraging.
© 2001 by CRC Press LLC
The student or researcher with a PC who wishes to obtain an introduction to neural modeling, and who wants to avoid the large investment in time and effort to master one of the large simulation programs described above, does have an alternative, i.e., the use of a general, nonlinear system simulation language such as Simnon™ or the Matlab Simulink™, both of which run on PCs. The author has done the examples in this text using Simnon. Simnon and Simulink are described below.
9.6.1
SIMNON
Simnon was developed at the Department of Automatic Control at the Lund Institute of Technology, Sweden in the late 1980s. The program in its early versions (V1.0 to V3.2) was written to run under DOS on PCs. A simple, algebraic input script was used. The author found in 1988 that Simnon v2.0 was particularly well suited for simulation of compartmental pharmacokinetic systems, chemical kinetic systems, and physiological regulators and control systems because its input modality is in the form of algebraic ODEs, which arise naturally from the analysis and modeling of these three classes of systems. Simnon also allows the simultaneous simulation of a discrete controller, if desired. The latest Windows version, Simnon 3.0/PCW, is well suited to solve sets of stiff ODEs. The user has the choice of one of four integration algorithms: RungeKutta/Fehlberg 2nd/3rd, Runge-Kutta/Fehlberg 4th/5th, Euler, and Dormand-Prince 4th/5th. Simnon 3.0/PCW can handle up to 10,000 states (ODEs), 100 subsystems, 50 pure time delays, 50 function tables, 32 plot variables, and up to 100 stored variables. Calculations are performed with double precision. Simnon has a userfriendly, interactive, GUI, and it has quality graphic outputs on monitors and to laser and bubblejet printers. Solutions of sets of nonlinear ODEs by Simnon can be displayed in the time domain, or parametrically as phase-plane plots. Simnon data files can also be exported to Matlab. Matlab can be used to do time- and frequencydomain operations not found in Simnon. The author has also run the DOS Simnon V3.2 on a Pentium PC with the Windows NT4® operating system. Simnon V3.2 has color graphics that can be printed out as such with a suitable color printer. Simnon V3.2 cost about $750, the student version, $95. The new windows version, Simnon 3.0/PCW with a user-friendly GUI is available from SSPA Maritime Consulting, P.O. Box 24001, S-400 22 Göteborg, Sweden, e-mail
[email protected]. Simnon can be ordered online from: http://www.sspa.se /simnon/simnon.htm. Its price is ECU 99 from SSPA (about U.S. $103, 11/99). Truly a bargain. The user manual is on the CD.
9.6.2 SIMULINK® Simulink runs with Matlab; both are products of The MathWorks, Natick, MA. (Simulink is now at v3.0 (11/99); different versions of Matlab and Simulink run on PCs with Windows 95, 98, or NT4.0, or on Macs or UNIX platforms ). Simulink is an icon-driven, dynamic simulation package that allows the user to represent a nonlinear dynamic process with a block diagram. As the block diagram is built, the
© 2001 by CRC Press LLC
user has to specify numerical values for the parameters in the blocks, and, of course, the interconnections between them. Before the simulation is run, the user specifies the integration routine to be used, the stepsize, and start and stop times. A diverse selection of integration routines includes R-K 23, R-K 45, Euler, Gear, Adams, and Linsim (plain vanilla for purely linear state systems). Gear is recommended for stiff nonlinear systems. Because Simulink runs in the Matlab “shell,” it can make use of all of the many features of Matlab and its various toolboxes. Although extremely versatile for the simulation of linear analog and/or discrete systems, Simulink does not shine in the simulation of systems of nonlinear ODEs, such as found in chemical kinetics or neural modeling. Its block diagram format becomes unwieldy, and it is clear that a program such as Simnon that accepts the algebraically written ODEs directly is easier to set up for running a simulation. The reader interested in learning more about Simulink can visit the URL: http://www.mathworks.com/products/simulink.
9.7
CONCLUSION
As has been seen, there are a bewildering number of neural modeling programs available for the asking at various university URLs on the Internet. In selecting one, the potential user must weigh and decide: 1. On what platform (type of computer and OS) the program will be run. 2. At what level will the modeling will take place? That is, single neuron, small groups of neurons (e.g., CPGs), large groups of neurons (e.g., the retina), or very large neural networks (e.g., brain and its components), or at all levels. 3. What degree of flexibility is required in the fine structure of the model? That is, how many different types of ion channels are required, and can their parameters be easily edited? 4. Can the user trade off detail for number in simulating networks? For example, can one abandon the HH formalism for spike generation and use RPFM (leaky integrator) spike generators? Are the action of chemical synapses describable by the two-state, alpha model for subsynaptic membrane conductance change, or can one directly generate epsps (or ipsps) from presynaptic spikes, and condition and sum these transients to form the generator potential directly at the SGL? 5. What level of support is available? That is, is there a text or an online tutorial available? Are there example simulations and neuron parameter libraries available online? The major neural modeling programs such as GENESIS and NSL are complex and detailed; they have a huge effort overhead for the user to learn to use them effectively. They have evolved over time through the efforts of many neurobiologists, computational neuroscientists, and computer scientists. Both GENESIS and NSL have large cadres of users and, better yet, are supported by textbooks with examples, etc. The Shepherd Lab at Yale offers an online, cerebral neuron database (NeuronDB)
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with simulation parameters listed for approximatley 20 types of neurons, including spinal type Ia interneurons and motoneurons. It lists voltage-gated conductances, neurotransmitter receptors and substances, canonical forms of neurons (dendrites, soma, axon hillock, axon, and axon terminal), as well as tools for the integration of these properties in a given type of neuron, and comparison of properties across different neuronal types. The URL for the NeuronDB is http://spine.med.yale.edu /neurondb/. To run many simulation programs, (free) Linux can be easily installed on a Pentium PC. (Actually, it is easier to install Linux if one first removes Windows; the PC is then dedicated to neural modeling and other applications compatible with Linux.) Because of the intellectual and time overheads associated with mastering the large simulation programs, the entry-level neural modeler should consider first using Simnon to model single neurons and small neural circuits. There is very little overhead in using Simnon, and the model structure is in terms of algebraic, firstorder ODEs. Neural models with Simnon are well suited to the phenomenological locus approach, where individual details such as individual ionic conductance functions and connected compartments are avoided in favor of directly simulating epsps and ipsps, modeling their propagation to an RPFM SGL through low-pass, dendrite transfer functions, and then modeling the spike propagation on the axon as it travels from SGL to synapse with a pure transport lag. Simnon can easily handle the ODEs describing parametric conductance changes, however. As has been seen, any level of detail (and realism) is possible in neural modeling, given the wide choices of programs available. With all of this preoccupation with modeling, one should not forget from where the numbers come that are used in the models, and the model architecture itself. That is, actual “wet,” experimental neurophysiology. This is where it began, and is where it should end.
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Appendix 1 A program written for Simnon to model Arenivaga CNS positional unit behavior is listed below. The program’s file name is ARENcns3.T. (The *.T is not used in the program text below.) Comment and space lines begin with quotes, “. The actual *.txt program is case insensitive, and should not have boldface characters; they were used here for emphasis. If Microsoft Word is used to prepare the program, it should be saved as a “text only” file named ARENcns3.T, not ARENcns3.txt.(Simnon does not recognize the *.txt extension.) 10 point, Courier New type should be used. Continuous system ARENcns3 “ Simnon program. V. 2/27/99 “ Use EULER integration with delT = tau. “ Neural model to simulate enhancement of positional orientation “ by protocerebral PPI units, as recorded by Willey, 1981. “ There are 47 states. STATE r1 r2 r3 r4 STATE v1 v2 v3 v4 v5 v6 v7 STATE p1 q1 p2 q2 p3 q3 p4 q4 p5 q5 p6 q6 p7 q7 p8 q8 p9 q9 STATE p10 q10 p11 q11 p12 q12 p13 q13 p14 q14 p15 q15 STATE p16 q16 p17 q17 p18 q18 “ DER dr1 dr2 dr3 dr4 DER dv1 dv2 dv3 dv4 dv5 dv6 dv7 DER dp1 dq1 dp2 dq2 dp3 dq3 dp4 dq4 dp5 dq5 dp6 dq6 dp7 dq7 dp8 dq8 dp9 dq9 DER dp10 dq10 dp11 dq11 dp12 dq12 dp13 dq13 dp14 dq14 dp15 dq15 DER dp16 dq16 dp17 dq17 dp18 dq18 “ TIME t “ t is in ms. “ “ ANALOG DRIVES FOR 4 VNC PIs: (Theta can range 0 to 360 deg.) “ frcpi =(maxF1/2)*(1 + cos(2*(theta – 45)/R)) “Choose theta in deg. rcpi is + rcpi1 = IF THETA > 135 THEN 1 ELSE 0 “ Suppresses rcpi for theta rcpi2 = IF THETA < 315 THEN 1 ELSE 0 “ between 135 - 315°. rcpi3 = rcpi1*rcpi2 rcpi = IF rcpi3 > 0 THEN 0 ELSE frcpi “ fripi = (maxF2/2)*(1 + cos(2*(theta - 135)/R)) “ cos arg must be in rads. ripi1 = IF theta > 225 THEN 1 ELSE 0 ripi2 = IF theta < 45 THEN 1 ELSE 0 ripi3 = ripi1 + ripi2 ripi = IF ripi3 > 0 THEN 0 ELSE fripi “ flipi = (maxF2/2)*(1 + cos(2*(theta - 225)/R)) lipi1 = IF theta > 315 THEN 1 ELSE 0
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lipi2 = IF theta < 135 THEN 1 ELSE 0 lipi3 = lipi1 + lipi2 lipi = IF lipi3 > 0 THEN 0 ELSE flipi “ flcpi = (maxF1/2)*(1 + cos(2*(theta - 315)/R)) lcpi1 = IF theta > 45 THEN 1 ELSE 0 lcpi2 = IF theta < 225 THEN 1 ELSE 0 lcpi3 = lcpi1*lcpi2 lcpi = IF lcpi3 > 0 THEN 0 ELSE flcpi “ “ IPFM VFCs TO GENERATE INPUT SPIKES: Driven by analog f(theta)s above. “ dr1 = rcpi - zi1 wi1 = IF r1 > phi2 THEN 1 ELSE 0 si1 = DELAY(wi1, tau) xi1 = wi1 - si1 b1 = IF xi1 > 0 THEN xi1 ELSE 0 zi1 = b1*phi2/tau uRCPI = zi1*Do “ Pulse train emulating VNC positional interneuron, RCPI. “ dr2 = ripi - zi2 wi2 = IF r2 > phi2 THEN 1 ELSE 0 si2 = DELAY(wi2, tau) xi2 = wi2 - si2 b2 = IF xi2 > 0 THEN xi2 ELSE 0 zi2 = b2*phi2/tau uRIPI = zi2*Do “ Pulse train emulating VNC positional interneuron, RIPI. “ dr3 = lipi - zi3 wi3 = IF r3 > phi2 THEN 1 ELSE 0 si3 = DELAY(wi3, tau) xi3 = wi3 - si3 b3 = IF xi3 > 0 THEN xi3 ELSE 0 zi3 = b3*phi2/tau uLIPI = zi3*Do “ Pulse train emulating VNC positional interneuron, LIPI. “ dr4 = lcpi - zi4 wi4 = IF r4 > phi2 THEN 1 ELSE 0 si4 = DELAY(wi4, tau) xi4 = wi4 - si4 b4 = IF xi4 > 0 THEN xi4 ELSE 0 zi4 = b4*phi2/tau uLCPI = zi4*Do “ Pulse train emulating VNC positional interneuron, LCPI. “ “ THE 3 OUTPUT RPFM NEURONS “ dv1 = -co*v1 + co*e1 - z1 “ RPFM Output Neuron # 1 @45°. z1 resets v1 to 0. z1 = y1*phi/tau “ e1 is sum of e- & i- psps to N1. w1 = IF v1 > phi THEN 1 ELSE 0 s1 = DELAY(w1, tau)
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x1 = w1 - s1 y1 = IF x1 > 0 THEN x1 ELSE 0 “ + pulse generator when v1 > phi. u1 = y1*Doo/tau e1 = ge1*q1 - gi1*q2 “ Sum of epsps - ipsps for N1. “ dv2 = -co*v2 + co*e2 – z2 “ Output Neuron 2 @ 90°. z2 = y2*phi/tau w2 = IF v2 > phi THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 – s2 y2 = IF x2 > 0 THEN x2 ELSE 0 u2 = y2*Doo/tau e2 = ge2*(q7 + q8) – gi2*(q6 + q9) “ dv3 = -co*v3 + co*e3 – z3 “ Output Neuron 3 @ 135°. z3 = y3*phi/tau w3 = IF v3 > phi THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 - s3 y3 = IF x3 > 0 THEN x3 ELSE 0 u3 = y3*Doo/tau e3 = ge3*q14 - gi3*q13 “ “ 4 INHIBITORY INTERNEURONS: “ dv4 = –ci*v4 + ci*e4 -z4 z4 = y4*phi/tau w4 = IF v4 > phi THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 – s4 y4 = IF x4 > 0 THEN x4 ELSE 0 u4 = y4*Doi/tau e4 = ge4*(q3 + q17 + q18) “ dv5 = –ci*v5 + ci*e5 – z5 z5 = y5*phi/tau w5 = IF v5 > phi THEN 1 ELSE 0 s5 = DELAY(w5, tau) x5 = w5 – s5 y5 = IF x5 > 0 THEN x5 ELSE 0 u5 = y5*Doi/tau e5 = ge5*(q4 + q5) “ dv6 = -ci*v6 + ci*e6 - z6 z6 = y6*phi/tau w6 = IF v6 > phi THEN 1 ELSE 0 s6 = DELAY(w6, tau) x6 = w6 - s6 y6 = IF x6 > 0 THEN x6 ELSE 0 u6 = y6*Doi/tau e6 = ge6*(q10 + q11) “ dv7 = –ci*v7 + ci*e7 – z7 z7 = y7*phi/tau w7 = IF v7 > phi THEN 1 ELSE 0
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s7 = DELAY(w7, tau) x7 = w7 – s7 y7 = IF x7 > 0 THEN x7 ELSE 0 u7 = y7*Doi/tau e7 = ge7*(q12 + q15 + q16) “ “ SYNAPTIC BALLISIC FILTER ODEs. (There are 14 synapses) “ c1 = uRCPI dp1 = –ae*p1 + c1 “ c1 is spike input to BF. dq1 = –ae*q1 + p1 “ q1 is BF analog output, > 0. “ c2 = u4 dp2 = –ai*p2 + c2 dq2 = –ai*q2 + p2 “ c3 = u2 dp3 = –ae*p3 + c3 dq3 = –ae*q3 + p3 “ c4 = u1 dp4 = –ae*p4 + c4 dq4 = –ae*q4 + p4 “ c5 = uLCPI dp5 = –ae*p5 + c5 dq5 = –ae*q5 + p5 “ c6 = u5 dp6 = -ai*p6 + c6 dq6 = -ai*q6 + p6 “ c7 = uRIPI dp7 = -ae*p7 + c7 dq7 = -ae*q7 + p7 “ c8 = uRCPI dp8 = -ae*p8 + c8 dq8 = -ae*q8 + p8 “ c9 = u6 dp9 = -ai*p9 + c9 dq9 = -ai*q9 + p9 “ c10 = uLIPI dp10 = -ae*p10 + c10 dq10 = -ae*q10 + p10 “ c11 = u3 dp11 = -ae*p11 + c11 dq11 = -ae*q11 + p11 “ c12 = u2 dp12 = -ae*p12 + c12 dq12 = -ae*q12 + p12
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“ c13 = u7 dp13 = -ai*p13 + c13 dq13 = -ai*q13 + p13 “ c14 = uRIPI dp14 = -ae*p14 + c14 dq14 = -ae*p14 + p14 “ c15 = uRCPI dp15 = -ae*p15 + c15 dq15 = -ae*q15 + p15 “ c16 = uLIPI dp16 = -ae*p16 + c16 dq16 = -ae*q16 + p16 “ c17 = uRIPI dp17 = -ae*p17 + c17 dq17 = -ae*q17 + p17 “ c18 = uLCPI dp18 = -ae*p18 + c18 dq18 = -ae*q18 + p18 “ “ OFFSET SCALED SPIKE OUTPUTS FOR PLOTTING: “ bo1 = b1/5 + .1 “ VNC INPUTS set by theta. bo2 = b2/5 + .4 bo3 = b3/5 + .7 bo4 = b4/5 + 1.0 “ yo1 = y1/4 + 1.3 “ Positional unit outputs: #1 @ 45°, #2 @ 90°, #3 @ 135°. yo2 = y2/4 + 1.6 yo3 = y3/4 + 1.9 “ yo4 = y4/4 + 2.2 “ Inhib interneuron outputs yo5 = y5/4 + 2.5 yo6 = y6/4 + 2.8 yo7 = y7/4 + 3.1 angle = theta/100 “ theta = t “ Plot over 0 to 225 sec.= degrees. “ “ CONSTANTS: pi:3.14159 ge1:1 “N1 epsp input weight. gi1:3 “N1 ipsp input weight. ge2:1 gi2:3 ge3:1 gi3:3 ge4:1 ge5:1 ge6:.3 “N6 epsp input weight.
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ge7:1 ae:.5 ai:.5 co:1 ci:1 phi:2.5 phi2:3 tau:0.01 “(sec) Do:1. Doo:.5 Doi:2 maxF1:1 maxF2:1 R:57.296 “ END
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Appendix 2 The following is a Simnon program, SZGRIsys.T, used to simulate the candidate, neural CPG generator model described in Sec. 4.5.2. Following the program are listed the parameters used in the simulation in file, Ri3.t Continuous system SZGRIsys“ V. 3/14/99 “ 4 Neuron RI model after Szentagothai in B&H, Fig. 5.16. 14 states. “ STATE v1 v2 v3 v4 p1 p2 p3 p4 p5 p6 p7 p8 va vb DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dp5 dp6 dp7 dp8 dva dvb TIME t “ “ IPFM VFCa TO GENERATE INPUT SPIKES: dva = Ea - za wa = IF va > phia THEN 1 ELSE 0 sa = DELAY(wa, tau) xa = wa - sa ya = IF xa > 0 THEN xa ELSE 0 za = y*phia/tau ua = z*Doa “ “ IPFM VFCb TO GENERATE INPUT SPIKES dvb = Eb - zb wb = IF vb > phib THEN 1 ELSE 0 sb = DELAY(wb, tau) xb = wb - sb yb = IF xb > 0 THEN xb ELSE 0 zb = y*phib/tau ub = z*Dob “ THE RPFM RI SYSTEM: dv1 = -c1*v1 + c1*E1 w1 = IF v1 > phi1 THEN s1 = DELAY(w1, tau) x1 = w1 - s1 y1 = IF x1 > 0 THEN x1 z1 = y1*phi1/tau “ dv2 = -c2*v2 + c2*E2 w2 = IF v2 > phi2 THEN s2 = DELAY(w2, tau) x2 = w2 - s2 y2 = IF x2 > 0 THEN x2 z2 = y2*phi2/tau “ dv3 = -c3*v3 + c3*E3 w3 = IF v3 > phi3 THEN s3 = DELAY(w3, tau) x3 = w3 - s3 y3 = IF x3 > 0 THEN x3
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z1 “ Output Neuron 1. 1 ELSE 0
ELSE 0
z2 “ Output Neuron 2. 1 ELSE 0
ELSE 0
z3 “ Inhibitory interneuron 3. 1 ELSE 0
ELSE 0
z3 = y3*phi3/tau u3 = y3*Do3/tau “ dv4 = -c4*v4 + c4*E4 - z4 “ Inhibitory interneuron 4. w4 = IF v4 > phi4 THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 - s4 y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau “ “ EXCITATORY SYNAPSE BFs: “ dp1 = -a1*p1 + ua “ Synapse 1 ballistic filter (ipsp) “ dp2 = -a2*p2 + ub “ Synapse 2 ballistic filter. q2 is output. “ dp3 = -a3*p3 + ua “ dp4 = -a4*p4 + ub “ “ INHIBITORY SYNAPSE BFs: “ dp5 = -a5*p5 + u4 “ dp6 = -a6*p6 + u3 “ dp7 = -a7*p7 + u3 “ dp8 = -a8*p8 + u4 “ “ Inputs to RPFM neurons E1 = p1 - p5 “ epsp - ipsp input to output neuron 1. E2 = p2 - p6 “ epsp - ipsp input to output neuron 2. E3 = p3 - p8 “ epsp - ipsp input to inhibitory interneuron 3 E4 = p4 - p7 “ epsp - ipsp input to inhibitory interneuron 4. “ “ Offset outputs for plotting. oa = ya/5 + 1 ob = yb/5 + 1.3 o1 = y1/5 + 1.6 o2 = y2/5 + 1.9 o3 = y3/5 + 2.2 o4 = y4/5 + 2.5 “ “ ANALOG INPUTS TO IPFM VFCs: Ea = A*(1 - cos(wo*t)) “ Nonzero input to IPFM VFCa. Eb = B*(1 + cos(wo*t)) “ Nonzero input to IPFM VFCb. wo = 6.28*fo “ “ PARAMETERS A:2. B:2 fo:. 05
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tau:0.001 a1:1 a2:1 a3:1 a4:1 a5:1 a6:1 a7:1 a8:1 c1:1. c2:1 c3:1 c4:1 phia:1 phib:1 phi1:1 phi2:1 phi3:1 phi4:1 Doa:1. Dob:1 Do3:1 Do4:1 “ END
Actual parameters used: [SZGRIsys] “ Data file RI3.t For burst sharpening by RI. 3/16/99. v1:0. v2:0. v3:0. v4:0. p1:0. p2:0. p3:0. p4:0. p5:0. p6:0. p7:0. p8:0. va:0. vb:0. phia:1. tau:1.E-3 Doa:1. phib:1. Dob:1. c1:1. phi1:1. c2:1. phi2:1. c3:1. phi3:1.
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Do3:1. c4:1. phi4:1. Do4:1. a1:1. a2:1. a3:1. a4:1. a5:1. a6:1. a7:0.3 a8:0.3 A:0.8 B:0.8 fo:0.05
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Appendix 3 This appendix lists a Simnon program to simulate the author’s proposed burst generator model in Section 4.5.3. The file, RINH1.T contains the parameters and ICs used in generating Figure 4.5-9. Continuous system BURSTNM1" V. 3/16/99 " 4 Neuron RI model by RBN. " STATE v1 v2 v3 v4 p1 p2 p3 p4 p5 p6 va DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dp5 dp6 dva TIME t " " IPFM VFCa TO GENERATE INPUT SPIKES: dva = Ea - za wa = IF va > phia THEN 1 ELSE 0 sa = DELAY(wa, tau) xa = wa - sa ya = IF xa > 0 THEN xa ELSE 0 za = ya*phia/tau ua = ya*Doa/tau " " THE RPFM RI PAIR: " dv1 = -c1*v1 + c1*E1 - z1 " Output Neuron 1. RPFM model. w1 = IF v1 > phi1 THEN 1 ELSE 0 s1 = DELAY(w1, tau) x1 = w1 - s1 y1 = IF x1 > 0 THEN x1 ELSE 0 z1 = y1*phi1/tau u1 = y1*Do1/tau " dv2 = -c2*v2 + c2*E2 - z2 " Output Neuron 2. RPFM model w2 = IF v2 > phi2 THEN 1 ELSE 0 s2 = DELAY(w2, tau) x2 = w2 - s2 y2 = IF x2 > 0 THEN x2 ELSE 0 z2 = y2*phi2/tau u2 = y2*Do2/tau " " THE INHIBITORY INTERNEURONS: " dv3 = -c3*v3 + c3*e3 - z3 " Ihibitory interneuron 3. w3 = IF v3 > phi3 THEN 1 ELSE 0 s3 = DELAY(w3, tau) x3 = w3 - s3 y3 = IF x3 > 0 THEN x3 ELSE 0
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z3 = y3*phi3/tau u3 = y3*Do3/tau u3d = DELAY(u3, D) " dv4 = -c4*v4 + c4*e4 - z4 " Inhibitory interneuron 4. w4 = IF v4 > phi4 THEN 1 ELSE 0 s4 = DELAY(w4, tau) x4 = w4 - s4 y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau " " 1 TC SYNAPTIC BFs: dp1 = -a1*p1 + ua " Synapse 1 ballistic filter (ipsp) " dp2 = -a2*p2 + ua " Synapse 2 ballistic filter. q2 is output. " dp3 = -a3*p3 + u3d " Delayed FB around N1 " dp4 = -a4*p4 + u4 " dp5 = -a5*p5 + u1 " dp6 = -a6*p6 + u1 " " Inputs to RPFM neurons E1 = p1 - p3 " epsp - ipsp input to output neuron 1. E2 = p2 - p4 " epsp - ipsp input to output neuron 2. E3 = p6 " epsp input to inhibitory interneuron 3 E4 = p5 " epsp input to inhibitory interneuron 4. " " Offset outputs for plotting. oa = ya/5 + 1 o1 = y1/5 + 1.3 o2 = y2/5 + 1.6 o3 = y3/5 + 1.9 o4 = y4/5 + 2.2 " " " PARAMETERS (Adjusted to get 180o phase between N1 and N1 bursts) A:2. fo:.025 tau:0.001 D:3" msec. a1:1 a2:1 a3:.5 a4:5 a5:5 a6:.5 c1:1. c2:1 c3:1 c4:1 phia:1
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phi1:0.5 phi2:0.7 phi3:2.1 phi4:0.2 Doa:1. Do1:2 Do2:1 Do3:.5 Do4:3 " END
The parameter file, RINH1.t: [BURSTNM1] v1:0. v2:0. v3:0. v4:0. p1:0. p2:0. p3:0. p4:0. p5:0. p6:0. va:0. phia:1. tau:1.E-3 Doa:1. c1:1. phi1:0.5 Do1:2. c2:1. phi2:0.7 Do2:1. c3:1. phi3:2.1 Do3:0.5 D:3. c4:1. phi4:0.2 Do4:3. a1:1. a2:1. a3:0.5 a4:5. a5:5. a6:0.5 A:2. fo:0.025 a7:1. a8:1.
" 3/16/99. Pars for BURSTNM1.t
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