Interdisciplinary Applied Mathematics Volume 25 Editors S.S.Antman J.E. Marsden L. Sirovich S. Wiggins Geophysics and Planetary Sciences Mathematical Biology L. Glass, J.D. Murray Mechanics and Materials R.V.Kohn Systems and Control S.S. Sastry, P.S. Krishnaprasad Problems in engineering, computational science, and the physical and biologica! sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines bas led to the establishment ofthe series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, frrstly, by encouraging the ways that that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and, secondly, by encouraging other scientific disciplines to engage in a dialog with mathernaticians outlining their problems to both access new methods and suggest innovative developments within mathernatics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology. Interdisciplinary Applied Mathematics Volumes published are listed at the end of the book. Springer Science+Business Media, LLC Anne Beuter Leon Glass Michael C. Mackey Michele S. Titcombe Editors Nonlinear Dynamics in Physiology and Medicine With 162 Illustrations ~Springer Anne Beuter Leon Glass Institut de Biologie Department of Physiology Universite de Montpellier 1 McGill University 4, Boulevard Henri IV Montreal, Quebec H3G 1Y6 Montpellier Cedex 1, 34060 Canada France [email protected] [email protected] Michael C. Mackey Micbe1e S. Titcombe Department of Physiology Department of Physio1ogy McGill University McGill University Montreal, Quebec H3G 1Y6 Montreal, Quebec H3G 1Y6 Canada Canada [email protected] [email protected] Editors s.s. Antman J.E. Marsden Department of Mathematics Control and Dynamical Systems and Mai! Code 107-81 Institute for Physical Science and Technology California Institute ofTechno1ogy University ofMaryland Pasadena, CA 91125 College Park, MD 20742 USA USA [email protected] [email protected] L. Sirovich S. Wiggins Division of Applied Mathematics School ofMathematics Brown University University of Bristol Providence, RI 02912 Bristol BSS 1TW USA UK [email protected] [email protected] Mathematics Subject Classification (2000): 92Cxx, 37-01 Library of Congress Cata1oging-in-Publication Data Nonlinear dynarnics in physiology and medicine 1 editors, Anne Beuter ... [et al.]. p. cm-(Interdisciplinary applied mathematics ; v.25) lncludes bibliographical references and index. 1. Physiology-Mathematical models. 2. Nonlinear systerns. 3. Dynarnics. l. Beuter, Anne. Il. Series. QP33.6.M36N663 2003 612".001"5118-dc21 2003044936 ISBN 978-1-4419-1821-5 ISBN 978-0-387-21640-9 (eBook) DOI 10.1007/978-0-387-21640-9 © 2003 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2003 Softcover reprint of the hardcover lst edition 2003 AU rights reserved. This work may not be translated or copied in whole or in part without the written pennission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any forrn of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade narnes, trademarks, service marks, and similar terrns, even if they are not identi­ fied as such, is not tobe taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 SPIN 10911 792 Typesetting: Pages created by the editors using a Springer LaTeX macro package. www.springer-ny.com We dedicate this book to the memory of Arthur Winfree, a man before his time in life and death. Contributors J acques Belair Caroline Haurie Departement de Mathematiques Department ofPhysiology et de Statistique McGill University Universite de Montreal AnneBeuter Andre Longtin Laboratoire de Physiologie Department ofPhysics Institut de Biologie University of Ottawa Universite de Montpellier 1 Mare Courtemanche Michael C. Mackey Centre de Recherche Department ofPhysiology Institut de Cardiologie de Montreal McGill University Roderick Edwards John Milton Department ofMathematics Department ofNeurology and Statistics The University of University of Victoria Chicago Hospitals Leon Glass Michele S. Titcombe Department ofPhysiology Department ofPhysiology McGill University McGill University Michael R. Guevara Alain Vinet Department ofPhysiology Departement de Physiologie McGill University Universite de Montreal Preface Je tiens impossible de connaître les parties sans connaître le tout, non plus que de connaître le tout sans connaître particulierement les parties -Pascal The eterna[ mystery of the world is its comprehensibility -Einstein This book deals with the application of mathematical tools to the study of physiological systems. It is directed toward an audience of physiologists, physicians, physicists, kinesiologists, psychologists, engineers, mathemati­ cians, and others interested in finding out more about the complexities and subtleties of rhythmic physiological processes from a theoretical per­ spective. We have attempted to give a broad view of the underlying notions behind the dynamics of physiological rhythms, sometimes from a theoretical perspective and sometimes from the perspective of the experimentalist. This book can be used in a variety of ways, ranging from a more tra­ ditional approach such as a textbook in a biomathematics course (at either the advanced undergraduate or graduate level) to a research re­ source in which someone interested in a particular problem might look at the corresponding discussion here to guide their own thinking. We hope that researchers at all levels will find inspiration from the way we have dealt with particular research problems to tackle completely new areas of investigation, or even approach these in totally new ways. Mathematically, we expect that readers will have a solid background in differential and integral calculus as well as a first course in ordinary differ­ ential equations. From the physiological side, they should have a minimal training in general physiology at the organismal level. We have endeav­ ored in every area to make the book as self-contained as possible given this mathematical and physiological preparation. Furthermore, many of the later chapters stand on their own and can be read independently of the others. When necessary we have added references to other chapters or to appendices. This book is divided roughly into two sections. Following an introduc­ tory Chapter 1, the first section (sort of theory) mainly introduces concepts from nonlinear dynamics using an almost exclusively biologica! setting for viii Preface motivation. This is contained in Chapters 2 through 5. Chapter 2 intro­ duces concepts from nonlinear dynamics using the properties of difference and diffcrential equations as a basis. Chapter 3 extends and draws on this material in considering bifurcations from fixed points and limit cycles, illus­ trating a number of bifurcation patterns with examples drawn from data. This is continued in Chapter 4, which focuses on excitable cell electrophys­ iology. The first section concludes with a consideration in Chapter 5 of the properties of biologica! oscillators when perturbed by single stimuli or periodic inputs. The second part ( applications, more or less) consists of fi ve in-depth examples of how the authors have used the concepts of the first part of this book in their research investigations of biologi cal and physiological systems. Thus, Chapter 6 examines the infiuence of noise in nonlinear dynamical systems using examples drawn from neurobiology. Chapter 7 looks at the properties of spatially extended nonlinear systems as illustrated by the properties of excitable cardiac tissue. In Chapter 8 attention is directed to the dynamics of cellular replication, illustrating how periodic diseases can illuminate underlying dynamics. Chapter 9 returns to the neurobiology arena in considering the properties of a simple neural feedback system, the pupil light reflex. The book continues with this neurobiological emphasis to conclude in Chapter 10 with an examination of the dynamics of tremor. We have not considered data analysis techniques as a separate subject, but rather have included an Appendix C, explaining those techniques we have found valuable (e.g., Fourier analysis, Lomb periodogram analysis). No doubt, other investigators will have their favorites that are not mentioned. Since a combination of both analytical and computational techniques are essential to maximize understanding of complex physiological dynamics, we have included both analytic and computer exercises throughout the book using either the freeware XPP or the generally available commercial package Matlab. Introductions to both XPP and Matlab are to be found in Appendices A and B, respectively. The source code and data files (and in some cases, help files) for the computer exercises are available on the