Dynamical Systems in Neuroscience

1 Dynamical Systems in NeuroscienceComputational NeuroscienceTerrence J. Sejnowski and Tomaso A. Poggio, editorsNeural 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 , Eve Marder, Allen I. Selverston, and Maurice Maulins, 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 withCommentaries, 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 , and David G.

2 Beiser, 1995 Spikes: Exploring the Neural Code, Fred Rieke, David Warland, Rob de Ruyter van Stevenick,and William Bialek, 1997 Neurons, Networks, and Motor Behavior, edited by Paul S. Stein, Sten Grillner, Allen , and Douglas G. Stuart, 1997 Methods in Neuronal Modeling: From Ions to Networks, second edition, edited by ChristofKoch 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 Neural Codes and Distributed Representations: Foundations of Neural Computation, editedby Laurence Abbott and Terrence J. Sejnowski, 1999 Unsupervised Learning: Foundations of Neural Computation, edited by Geoffrey Hinton andTerrence J. Sejnowski, 1999 Fast Oscillations in Cortical Circuits, Roger D.

3 Traub, John Jefferys, and Miles AlWhittington, 1999 Computational Vision: Information Processing in Perception and Visual Behavior, HanspeterA. Mallot, 2000 Graphical Models: Foundations of Neural Computation, edited by Michael I. Jordan andTerrence J. Sejnowski, 2001 Self-Organizing Map Formation: Foundation of Neural 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 Dynamics in Neurobiological Sys-tems, Chris Eliasmith and Charles H. Anderson, 2003 The Computational Neurobiology of Reaching and Pointing, edited by Reza Shadmehr andSteven P. Wise, 2005 Dynamical Systems in Neuroscience : The Geometry of Excitability and Bursting, Eugene , 2007 Dynamical Systems in Neuroscience :The Geometry of Excitability and BurstingEugene M.

4 IzhikevichThe MIT PressCambridge, MassachusettsLondon, Englandc 2007 Massachusetts Institute of TechnologyAll 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 storageand retrieval) without permission in writing from the Press books may be purchased at special quantity discounts for business or salespromotional use. For information, please to Special Sales Department, The MIT Press, 55 Hayward Street, Cambridge,MA 02142 This book was set in LATEX by the author. Printed and bound in the United States of Congress Cataloging-in-Publication DataIzhikevich, Eugene M., 1967 Dynamical Systems in Neuroscience : the geometry of excitability and bursting /Eugene M. cm. (Computational Neuroscience )Includes bibliographical references and 978-0-262-09043-8 (hc.)

5 : alk. paper)1. Neural networks (Neurobiology) 2. Neurons - computer simulation. 3. Dy-namical Systems . 4. Computational Neuroscience . I. Izhikevich, E. M. II Title. 01 13 DC21200604034910 9 8 7 6 5 4 3 2To my beautiful daughters, Liz and Neurons .. What Is a Spike? .. Where Is the Threshold? .. Why Are Neurons Different, and Why Do We Care? .. Building Models .. Dynamical Systems .. Phase Portraits .. Bifurcations .. Hodgkin Classification .. Neurocomputational properties .. Building Models (Revisited) ..20 Review of Important Concepts ..21 Bibliographical Notes ..212 Electrophysiology of Ions .. Nernst Potential .. Ionic Currents and Conductances .. Equivalent Circuit .. Resting Potential and Input Resistance .. Voltage-Clamp and I-V Relation.

6 Conductances .. Voltage-Gated Channels .. Activation of Persistent Currents .. Inactivation of Transient Currents .. Hyperpolarization-Activated Channels .. The Hodgkin-Huxley Model .. Hodgkin-Huxley Equations .. Action Potential .. Propagation of the Action Potentials .. Dendritic Compartments .. Summary of Voltage-Gated Currents ..44 Review of Important Concepts ..49 Bibliographical Notes ..50 Exercises ..503 One-Dimensional Electrophysiological Examples .. I-V Relations and Dynamics .. Leak + InstantaneousINa,p.. Dynamical Systems .. Geometrical Analysis .. Equilibria .. Stability .. Eigenvalues .. Unstable Equilibria .. Attraction Domain .. Threshold and Action Potential .. Bistability and Hysteresis .. Phase Portraits.

7 Topological Equivalence .. Local Equivalence and the Hartman-Grobman Theorem .. Bifurcations .. Saddle-Node (Fold) Bifurcation .. Slow Transition .. Bifurcation Diagram .. Bifurcations and I-V Relations .. Quadratic Integrate-and-Fire Neuron ..80 Review of Important Concepts ..82 Bibliographical Notes ..83 Exercises ..834 Two-Dimensional Planar Vector Fields .. Nullclines .. Trajectories .. Limit Cycles .. Relaxation Oscillators .. Equilibria .. Stability .. Local Linear Analysis .. Eigenvalues and Eigenvectors .. Local Equivalence .. Classification of Equilibria .. Example: FitzHugh-Nagumo Model .. Phase Portraits .. Bistability and Attraction Domains .. Stable/Unstable Manifolds .. Homoclinic/Heteroclinic Trajectories.

8 Saddle-Node Bifurcation .. Andronov-Hopf Bifurcation ..116 Review of Important Concepts ..121 Bibliographical Notes ..122 Exercises ..1225 Conductance-Based Models and Their Minimal Models .. Amplifying and Resonant Gating Variables .. ,p+IK-Model .. ,t-model .. ,p+Ih-Model .. +IKir-Model .. +IKir-Model .. Ca2+-Gated Minimal Models .. Reduction of Multidimensional Models .. Hodgkin-Huxley model .. Equivalent Potentials .. Nullclines and I-V Relations .. Reduction to Simple Model ..153 Review of Important Concepts ..156 Bibliographical Notes ..156 Exercises ..1576 Equilibrium (Rest State) .. Saddle-Node (Fold) .. Saddle-Node on Invariant Circle .. Supercritical Andronov-Hopf .. Subcritical Andronov-Hopf .. Limit Cycle (Spiking State) .. Saddle-Node on Invariant Circle.

9 Supercritical Andronov-Hopf .. Fold Limit Cycle .. Homoclinic .. Other Interesting Cases .. Three-Dimensional Phase Space .. Cusp and Pitchfork .. Bogdanov-Takens .. Relaxation Oscillators and Canards .. Bautin .. Saddle-Node Homoclinic Orbit .. Hard and Soft Loss of Stability ..204 Bibliographical Notes ..205 Exercises ..2107 Neuronal Excitability .. Bifurcations .. Hodgkin s Classification .. Classes 1 and 2 .. Class 3 .. Ramps, Steps, and Shocks .. Bistability .. Class 1 and 2 Spiking .. Integrators vs. Resonators .. Fast Subthreshold Oscillations .. Frequency Preference and Resonance .. Frequency Preference in Vivo .. Thresholds and Action Potentials .. Threshold manifolds .. Rheobase .. Postinhibitory Spike.

10 Inhibition-Induced Spiking .. Spike Latency .. Flipping from an Integrator to a Resonator .. Transition Between Integrators and Resonators .. Slow Modulation .. Spike Frequency Modulation .. I-V Relation .. Slow Subthreshold Oscillation .. Rebound Response and Voltage Sag .. AHP and ADP ..260 Review of Important Concepts ..264 Bibliographical Notes ..264 Exercises ..265 CONTENTSxi8 Simple Simplest Models .. Integrate-and-Fire .. Resonate-and-Fire .. Quadratic Integrate-and-Fire .. Simple Model of Choice .. Canonical Models .. Cortex .. Regular Spiking (RS) Neurons .. Intrinsically Bursting (IB) Neurons .. Multi-Compartment Dendritic Tree .. Chattering (CH) Neurons .. Low-Threshold Spiking (LTS) Interneurons .. Fast Spiking (FS) Interneurons.