Dynamical Systems in Neuroscience.pdf by Izhikevich

【实例简介】
Dynamical Systems in Neuroscience.pdf by IzhikevichDynamical Systems in Neuroscience.pdf by Izhikevich
Computational Neuroscience Terrence J Sejnowski and Tomaso A. Poggio, editors Neural Nets in Electric Fish, Walter Heiligenberg, 1991 The Computational Brain, Patricia S Churchland and Terrence J Sejnowski, 1992 Dynamic Biological Networks: The Stomatogastric Nervous System, edited by Ronald M Harris-Warrick. Eve marder. Allen i selverston, and Maurice Maulins. 19 92 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 Stevenick and William Bialek, 1997 eurons, Networks, and Motor Behavior, edited by Paul s. 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. Levin, and Debra L Long, 1998 eural Codes and Distributed Representations: Foundations of Neural Computation, edited by Laurence abbott and Terrence J. Sejnowski, 1999 Unsupervised Learning: Foundations of Neural Computation, edited by Geoffrey Hinton and Terrence J. Sejnowski, 1999 Fast Oscillations in Cortical Circuits, Roger D. Traub, John G.R. Jefferys, and Miles Al Whittington. 1999 Computational vision: Information Processing in Perception and Visual behavior, Hanspeter A. Mallot. 2000 Graphical Models: Foundations of Neural Computation, edited by Michael I. Jordan and Terrence J. Sejnowski, 2001 Self-Organizing Map Formation: Foundation of Nearal Computation, edited by Klaus Ober mayer and Terrence J. Sejnowski, 2001 Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems Peter Dayan and L. F. Abbott, 2001 Neural Engineering: Computation, Representation, and Dymamics in Neurobiological Sys tems. Chris Eliasmith and Charles H. Anderson. 2003 The Computational Neurobiology of Reaching and Pointing, edited by Reza Shadmehr and Steven P. Wise. 2005 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Eugene M Izhikevich. 2007 Dynamical Systems in Neuroscience The Geometry of Excitability and Bursting Eugene m. Izhikevich The mit Press ambridge, Massachusetts London, england C 2007 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any elec- tronic or mechanical means(including photocopying, recording, or information 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, Cambridge MA02142 This book was set in iAteX by the author. Printed and bound in the United States of America Library of Congress Cataloging-in-Publication Data Izhikevich, Eugene M, 1967- Dynamical systems in neuroscience: the geometry of excitability and bursting/ Eugene m. Izhikevich p. cm.-( Computational neuroscience Includes bibliographical references and index ISBN978-0-26209043-8(hc.:alk. paper) 1. Neural networks(Neurobiology)2. Neurons- computer simulation. 3. Dy- namical systems. 4. Computational neuroscience. I Izhikevich, E. M. II Title. III Series QP363.3.1942007 573.8:0113DC21 2006040349 10987654321 To my beautiful daughters, Liz and Kate Contents Preface 1 Introduction 1.1 Neurons 1.1.1 What Is a spike? 1.1.2 Where Is the Threshold? 1. 1.3 Why are Neurons Different, and why Do We care? 1.1.4 Building models 1. 2 Dynamical Systems 11236688 1.2.1 Phase portraits 1.2.2 Bifurcations 1.2.3 Hodgkin Classification 1.2.4 Neurocomputational properties 16 1.2.5 Building Models(revisited) Review of Important Concepts 21 Biblic 21 2 Electrophysiology of Neurons 25 2.1 Ions 25 2.1.1 Nernst Potential 2.1.2 Ionic Currents and conductances 27 2. 1.3 Equivalent circuit 1.4 Resting Potential t resist 2. 1.5 Voltage-Clamp and I-V relation 2.2 Conductances 32 2.2.1 Voltage-Gated Channels 33 2.2.2 Activation of Persistent Currents 2.2.3 Inactivation of Transient Currents 2.2.4 Hyperpolarization-Activated Channels 2.3 The Hodgkin-Huxley Model 37 2.3.1 Hodgkin-Huxley Equations 37 2.3.2 Action Potential 41 2.3.3 Propagation of the Action Potentials 42 CONTENTS 3.4 Dendritic co 43 2.3.5 Summary of Voltage-Gated Currents Review of Important Concepts Bibliographical Notes 50 Exercises 50 3 One-Dimensional Systems 53 3.1 Electrophysiological Examples 53 3.1.1 I-V Relations and dynamics 54 3.1.2 Leak+ Instantaneous INa, p 55 3.2 Dynamical systems 3.2.1 Geometrical Analysis 59 3.2.2 equilibria 3.2.3 Stability 3.2.4 Eigenvalues 61 3.2.5 Unstable equilibria 61 3.2.6 Attraction domain 62 3.2.7 Threshold and Action Potential 3.2.8 Bistability and hysteresis 3.3 Phase portraits 3.3.1 Topological Equivalence 3.3.2 Local Equivalence and the Hartman-Grobman Theorem 3.3.3 Bifurcations 3.3.4 Saddle-Node(Fold) Bifurcation 3.3.5 Slow Transition 3.3.6 Bifurcation diagram 3.3.7 Bifurcations and i-v relations 3.3.8 Quadratic Integrate-and-Fire Neuron Review of important Concepts Bibliographical Notes E Xercises 83 4 Two-Dimensional Systems 89 4.1 Planar vector fields 4.1.1 Nullclines 92 4.1.2 Trajectories 94 4. 1. 3 Limit Cycles 4.1.4 Relaxation Oscillators 4.2 equilibria 4.2.1 Stability 4.2.2 Local Linear Analysis 101 4.2.3 Eigenvalues and eigenvectors 4.2. 4 Local equivalence 103 CONTENTS 4.2.5 Classification of equilibria 103 4.2.6 Example: FitzHugh-Nagumo Model 4.3 Phase portraits 108 4.3.1 Bistability and attraction Domains .108 4. 3. 2 Stable/Unstable Manifolds 109 4.3.3 Homoclinic/Heteroclinic Trajectories 4.3.4 Saddle-Node Bifurcation 113 4.3.5 Andronov-Hopf Bifurcation Review of Important Concepts 121 Bibliographical notes 122 E Xercises 122 5 Conductance- Based models and their reductions 127 5.1 Minimal models 5. 1.1 Amplifying and resonant gating Variables 5.1.2 INa p+lk-Model 132 5.1.3 INat-model 133 5.1.4 INap+lh-Model 5.1.5 Ih+IKir-Model 138 5.1.6 IK+IKir-Model 140 51.7ⅠA- Model 142 5.1.8 Ca2+-Gated Minimal Models 147 5.2 Reduction of multidimensional models 147 5.2.1 Hodgkin-Huxley model 147 5.2.2 Equivalent Potentials 151 5.2.3 Nullclines and i-V relations 151 5.2.4 Reduction to Simple model 153 Review of Important Concepts 156 Bibliographical notes 156 Exercises 157 6 Bifurcations 159 6.1 Equilibrium(rest State 159 6.1.1 Saddle-Node(Fold) 162 6.1.2 Saddle-Node on invariant Circle 164 6.1.3 Supercritical Andronov-Hopf 168 6.1.4 Subcritical Andronov-Hopf 174 6.2 Limit Cycle(Spiking State 6.2.1 Saddle-Node on invariant circle 180 6.2.2 Supercritical Andronov-Hopf .181 6.2.3 Fold Limit Cvcle 181 6.2.4 Homoclinic 6.3 Other Interesting cases 【实例截图】
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