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Computational Neuroscience,Terrence J Sejnowski and Tomaso Poggio editors. Neural Nets in Electric Fish Walter Heiligenberg 1991. The Computational Brain Patricia S Churchland and Terrence J Sejnowski. Dynamic Biological Networks The Stomatogastric Nervous System edited by. Ronald M Harris Warrick Eve Marder Allen I Selverston and. Maurice Moulins 1992, The Neurobiology of Neural Networks edited by Daniel Gardner 1993. Large Scale Neuronal Theories of the Brain edited by Christof Koch and Joel. L Davis 1994, The Theoretical Foundations of Dendritic Function Selected Papers of Wilfrid. Rall with Commentaries edited by Idan Segev John Rinzel and Gordon. M Shepherd 1995, Models of Information Processing in the Basal Ganglia edited by James C. Houk Joel L Davis and David G Beiser 1995, Spikes Exploring the Neural Code Fred Rieke David Warland Rob de.

Ruyter van Steveninck and William Bialek 1997, Neurons Networks and Motor Behavior edited by Paul S G Stein Sten. Grillner Allen I Selverston and Douglas G Stuart 1997. Methods in Neuronal Modeling From Ions to Networks second edition. edited by Christof Koch and Idan Segev 1998, Fundamentals of Neural Network Modeling Neuropsychology and Cognitive. Neuroscience edited by Randolph W Parks Daniel S Levine and Debra. L Long 1998, Neural Codes and Distributed Representations Foundations of Neural. Computation edited by Laurence Abbott and Terrence J Sejnowski 1998. Unsupervised Learning Foundations of Neural Computation edited by. Geoffrey Hinton and Terrence J Sejnowski 1998, Fast Oscillations in Cortical Circuits Roger D Traub John G R Jeffreys. and Miles A Whittington 1999, Computational Vision Information Processing in Perception and Visual.

Behavior Hanspeter A Mallot 2000, Theoretical Neuroscience Computational and Mathematical Modeling of Neural. Systems Peter Dayan and L F Abbott 2001,Theoretical Neuroscience. Computational and Mathematical Modeling of,Neural Systems. Peter Dayan and L F Abbott,The MIT Press,Cambridge Massachusetts. London England,First MIT Press paperback edition 2005.

c 2001 Massachusetts Institute of Technology, All rights reserved No part of this book may be reproduced in any form by any. electronic or mechanical means including photocopying recording or informa. tion storage and retrieval without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or. sales promotional use For information please email special sales mitpress mit edu. or write to Special Sales Department The MIT Press 55 Hayward Street Cam. bridge MA 02142,Typeset in Palatino by the authors using LATEX 2. Printed and bound in the United States of America, Library of Congress Cataloging in Publication Data. Dayan Peter, Theoretical neuroscience computational and mathematical modeling of neural. systems Peter Dayan and L F Abbott,p cm Computational neuroscience.

Includes bibliographical references,ISBN 0 262 04199 5 hc alk paper 0 262 54185 8 pb. 1 Neural networks Neurobiology Computer simulation 2 Human. information processing Computer simulation 3 Computational neuroscience. I Abbott L F II Title III Series,QP363 3 D39 2001,573 8 01 13 dc21. 2001044005,10 9 8 7 6 5 4,To our families,Preface xiii. I Neural Encoding and Decoding 1, 1 Neural Encoding I Firing Rates and Spike Statistics 3. 1 1 Introduction 3,1 2 Spike Trains and Firing Rates 8.

1 3 What Makes a Neuron Fire 17,1 4 Spike Train Statistics 24. 1 5 The Neural Code 34,1 6 Chapter Summary 39,1 7 Appendices 40. 1 8 Annotated Bibliography 43, 2 Neural Encoding II Reverse Correlation and Visual Receptive. 2 1 Introduction 45,2 2 Estimating Firing Rates 45. 2 3 Introduction to the Early Visual System 51,2 4 Reverse Correlation Methods Simple Cells 60.

2 5 Static Nonlinearities Complex Cells 74,2 6 Receptive Fields in the Retina and LGN 77. 2 7 Constructing V1 Receptive Fields 79,2 8 Chapter Summary 81. 2 9 Appendices 81,2 10 Annotated Bibliography 84,3 Neural Decoding 87. 3 1 Encoding and Decoding 87,3 2 Discrimination 89. 3 3 Population Decoding 97,3 4 Spike Train Decoding 113.

3 5 Chapter Summary 118,3 6 Appendices 119,3 7 Annotated Bibliography 122. 4 Information Theory 123,4 1 Entropy and Mutual Information 123. 4 2 Information and Entropy Maximization 130,4 3 Entropy and Information for Spike Trains 145. 4 4 Chapter Summary 149,4 5 Appendix 150,4 6 Annotated Bibliography 150. II Neurons and Neural Circuits 151,5 Model Neurons I Neuroelectronics 153.

5 1 Introduction 153,5 2 Electrical Properties of Neurons 153. 5 3 Single Compartment Models 161,5 4 Integrate and Fire Models 162. 5 5 Voltage Dependent Conductances 166,5 6 The Hodgkin Huxley Model 173. 5 7 Modeling Channels 175,5 8 Synaptic Conductances 178. 5 9 Synapses on Integrate and Fire Neurons 188,5 10 Chapter Summary 191.

5 11 Appendices 191,5 12 Annotated Bibliography 193. 6 Model Neurons II Conductances and Morphology 195. 6 1 Levels of Neuron Modeling 195,6 2 Conductance Based Models 195. 6 3 The Cable Equation 203,6 4 Multi compartment Models 217. 6 5 Chapter Summary 224,6 6 Appendices 224,6 7 Annotated Bibliography 228. 7 Network Models 229,7 1 Introduction 229,7 2 Firing Rate Models 231.

7 3 Feedforward Networks 241,7 4 Recurrent Networks 244. 7 5 Excitatory Inhibitory Networks 265,7 6 Stochastic Networks 273. 7 7 Chapter Summary 276,7 8 Appendix 276,7 9 Annotated Bibliography 277. III Adaptation and Learning 279,8 Plasticity and Learning 281. 8 1 Introduction 281,8 2 Synaptic Plasticity Rules 284.

8 3 Unsupervised Learning 293,8 4 Supervised Learning 313. 8 5 Chapter Summary 326,8 6 Appendix 327,8 7 Annotated Bibliography 328. 9 Classical Conditioning and Reinforcement Learning 331. 9 1 Introduction 331,9 2 Classical Conditioning 332. 9 3 Static Action Choice 340,9 4 Sequential Action Choice 346. 9 5 Chapter Summary 354,9 6 Appendix 355,9 7 Annotated Bibliography 357.

10 Representational Learning 359,10 1 Introduction 359. 10 2 Density Estimation 368,10 3 Causal Models for Density Estimation 373. 10 4 Discussion 389,10 5 Chapter Summary 394,10 6 Appendix 395. 10 7 Annotated Bibliography 396,Mathematical Appendix 399. A 1 Linear Algebra 399,A 2 Finding Extrema and Lagrange Multipliers 408.

A 3 Differential Equations 410,A 4 Electrical Circuits 413. A 5 Probability Theory 415,A 6 Annotated Bibliography 418. References 419,Exercises http mitpress mit edu dayan abbott. Series Foreword, Computational neuroscience is an approach to understanding the infor. mation content of neural signals by modeling the nervous system at many. different structural scales including the biophysical the circuit and the. systems levels Computer simulations of neurons and neural networks are. complementary to traditional techniques in neuroscience This book series. welcomes contributions that link theoretical studies with experimental ap. proaches to understanding information processing in the nervous system. Areas and topics of particular interest include biophysical mechanisms for. computation in neurons computer simulations of neural circuits models. of learning representation of sensory information in neural networks sys. tems models of sensory motor integration and computational analysis of. problems in biological sensing motor control and perception. Terrence J Sejnowski,Tomaso Poggio, Theoretical analysis and computational modeling are important tools for.

characterizing what nervous systems do determining how they function. and understanding why they operate in particular ways Neuroscience. encompasses approaches ranging from molecular and cellular studies to. human psychophysics and psychology Theoretical neuroscience encour. ages crosstalk among these subdisciplines by constructing compact repre. sentations of what has been learned building bridges between different. levels of description and identifying unifying concepts and principles In. this book we present the basic methods used for these purposes and dis. cuss examples in which theoretical approaches have yielded insight into. nervous system function, The questions what how and why are addressed by descriptive mecha. nistic and interpretive models each of which we discuss in the following. chapters Descriptive models summarize large amounts of experimental descriptive models. data compactly yet accurately thereby characterizing what neurons and. neural circuits do These models may be based loosely on biophysical. anatomical and physiological findings but their primary purpose is to. describe phenomena not to explain them Mechanistic models on the mechanistic models. other hand address the question of how nervous systems operate on the. basis of known anatomy physiology and circuitry Such models often. form a bridge between descriptive models couched at different levels In. terpretive models use computational and information theoretic principles interpretive models. to explore the behavioral and cognitive significance of various aspects of. nervous system function addressing the question of why nervous systems. operate as they do, It is often difficult to identify the appropriate level of modeling for a partic. ular problem A frequent mistake is to assume that a more detailed model. is necessarily superior Because models act as bridges between levels of. understanding they must be detailed enough to make contact with the. lower level yet simple enough to provide clear results at the higher level. Organization and Approach, This book is organized into three parts on the basis of general themes. Part I Neural Encoding and Decoding chapters 1 4 is devoted to the. coding of information by action potentials and the representation of in. xiv Preface, formation by populations of neurons with selective responses Model. ing of neurons and neural circuits on the basis of cellular and synaptic. biophysics is presented in part II Neurons and Neural Circuits chapters. 5 7 The role of plasticity in development and learning is discussed in. part III Adaptation and Learning chapters 8 10 With the exception of. chapters 5 and 6 which jointly cover neuronal modeling the chapters are. largely independent and can be selected and ordered in a variety of ways. for a one or two semester course at either the undergraduate or the grad. uate level, Although we provide some background material readers without previ.

background ous exposure to neuroscience should refer to a neuroscience textbook such. as Kandel Schwartz Jessell 2000 Nicholls Martin Wallace 1992. Bear Connors Paradiso 1996 Shepherd 1997 Zigmond et al 1998. or Purves et al 2000, Theoretical neuroscience is based on the belief that methods of mathemat. ics physics and computer science can provide important insights into ner. vous system function Unfortunately mathematics can sometimes seem. more of an obstacle than an aid to understanding We have not hesitated. to employ the level of analysis needed to be precise and rigorous At times. this may stretch the tolerance of some of our readers We encourage such. readers to consult the Mathematical Appendix which provides a brief re. view of most of the mathematical methods used in the text but also to. persevere and attempt to understand the implications and consequences. of a difficult derivation even if its steps are unclear. Theoretical neuroscience like any skill can be mastered only with prac. tice Exercises are provided for this purpose on the web site for this. exercises book http mitpress mit edu dayan abbott We urge the reader to. do them In addition it will be highly instructive for the reader to con. struct the models discussed in the text and explore their properties beyond. what we have been able to do in the available space. Referencing, In order to maintain the flow of the text we have kept citations within. the chapters to a minimum Each chapter ends with an annotated bibliog. raphy containing suggestions for further reading which are denoted by. a bold font information about works cited within the chapter and ref. erences to related studies We concentrate on introducing the basic tools. of computational neuroscience and discussing applications that we think. best help the reader to understand and appreciate them This means that. a number of systems where computational approaches have been applied. with significant success are not discussed References given in the anno. tated bibliographies lead the reader toward such applications Many peo. ple have contributed significantly to the areas we cover The books and. review articles in the annotated bibliographies provide more comprehen. sive references to work that we have failed to cite. Preface xv,Acknowledgments, We are extremely grateful to a large number of students at Brandeis the. Gatsby Computational Neuroscience Unit and MIT and colleagues at. many institutions who have painstakingly read commented on and crit. icized numerous versions of all the chapters We particularly thank Bard. Ermentrout Mark Goldman John Hertz Mark Kvale Zhaoping Li Eve. Marder and Read Montague for providing extensive discussion and ad. vice on the entire book A number of people read significant portions. Theoretical neuroscience is based on the belief that methods of mathemat ics physics and computer science can provideimportant insights into ner vous system function Unfortunately mathematics can sometimes seem more of an obstacle than an aid to understanding We have not hesitated to employ the level of analysis needed to be preciseand

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