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Dynamical Systems in Neuroscience.pdf by IzhikevichDynamical Systems in Neuroscience.pdf by Izhikevich
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Dynamical Systems
in Neuroscience
Eugene M. Izhikevich
The Geometry of Excitability and Bursting
neuroscience/computational neuroscience
Dynamical Systems in Neuroscience
The Geometry of Excitability and Bursting
Eugene M. Izhikevich
In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must
call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical sys-
tems theory for researchers and graduate students in neuroscience. It also provides an overview of neu-
roscience for mathematicians who want to learn the basic facts of electrophysiology.
Dynamical Systems in Neuroscience presents a systematic study of the relationship of electro-
physiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information
processing in the brain depends not only on the electrophysiological properties of neurons but also on
their dynamical properties.
The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-
type models and going on to describe bursting systems. Each chapter proceeds from the simple to the
complex, and provides sample problems at the end. The book explains all necessary mathematical con-
cepts using geometrical intuition; it includes many figures and few equations, making it especially suit-
able for non-mathematicians. Each concept is presented in terms of both neuroscience and mathemat-
ics, providing a link between the two disciplines.
Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is
not a standard part of the graduate neuroscience curriculum—or taught by math or physics departments
in a way that is suitable for students of biology. This book offers neuroscience students and researchers
a comprehensive account of concepts and methods increasingly used in computational neuroscience.
An additional chapter on synchronization, with more advanced material, can be found at the author’s
website, www.izhikevich.com.
Eugene M. Izhikevich is Senior Fellow in Theoretical Neurobiology at the Neurosciences Institute, San
Diego, coauthor of
Weakly Conducted Neural Networks, and editor-in-chief of Scholarpedia, the free
peer-reviewed encyclopedia.
Computational Neuroscience series
“This book will be a great contribution to the subject of mathematical neurophysiology.”
—Richard Fitzhugh, former researcher, Laboratory of Biophysics, National Institutes of Health
“Eugene Izhikevich has written an excellent introduction to the application of nonlinear dynamics to the
spiking patterns of neurons. There are dozens of clear illustrations and hundreds of exercises ranging
from the very easy to Ph.D.-level questions. The book will be suitable for mathematicians and physicists
who want to jump into this exciting field as well as for neuroscientists who desire a deeper underst
and
-
ing of the utility of nonlinear dynamics applied to biology
.”
—Bard Ermentrout, Department of Mathematics, University of Pittsburgh
“
A stimulating, entert
aining, and scenic tour of neuronal modeling from a nonlinear dynamics viewpoint.
”
—John Rinzel, Center for Neural Science and Courant Institute, New Y
ork University
The MIT Press
Mass
ac
husetts Institute of T
echnology
Cambridge, Massachusetts 02142
http://mitpress.mit.edu
0-262-09043-0
978-0-262-09043-8
Dynamical Systems in Neuroscience Izhikevich
49924Izhikevich 6/1/06 6:01 AM Page 1
Dynamical Systems in Neuroscience
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, 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 Stevenick,
and William Bialek, 1997
Neurons, 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
Neural 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 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 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
Cambridge, 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,
MA 02142
This book was set in L
A
T
E
X 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.
ISBN 978-0-262-09043-8 (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.I94 2007
573.8’01’13—DC21 2006040349
10987654321
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