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Dynamical Cognitive Science This page intentionally left blank Dynamical Cognitive Science A Bradford Book The MIT Press Cambridge, Massachusetts London, England Lawrence M. Ward ( 2002 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 information storage and retrieval) without permission in writing from the publisher. This book was set in Times New Roman in 3B2 by Asco Typesetters, Hong Kong, and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Ward, Lawrence M. Dynamical cognitive science / Lawrence M. Ward. p. cm. ``A Bradford book''. Includes bibliographical references and index. ISBN 0-262-23217-0 (hc. : alk. paper) 1. Cognitive science. 2. Change (Psychology). 3. TimeÐPsychological aspects. I. Title. BF311 .W2695 2001 153Ðdc21 2001044336 Contents Preface xiii Chapter 1 Magic, Ritual, and Dynamics 1 1.1 Magic and Ritual 1.2 Dynamics 2.1 The Serial Universe 1 4 Chapter 2 Sequence 9 9 2.2 The Problem of Serial Order in Behavior 11 2.3 13 Markovian Analysis of Behavior 3.1 The Dance of Life 3.2 Music and Rhythm 3.3 Rhythms in the Brain 4.1 Space-Time 4.2 The Arrow of Time 4.3 Measuring Time Chapter 3 Rhythms of Behavior 17 17 19 20 Chapter 4 Time 25 25 28 30 vi Contents Chapter 5 Cognitive Processes and Time 35 5.1 Temporal Unfolding of Cognitive Behavior 35 5.2 40 Timing of Cognitive Behavior 5.3 Memory 41 6.1 Systems 46 6.2 General Systems Theory 6.3 Dynamical Systems Theory 7.1 The Mandala of Science 7.2 Formal Theories 7.3 59 Principle of Complementarity 8.1 61 Theories, Models, and Data 8.2 Statical Models 8.3 Dynamical Models Chapter 6 Systems and General Systems Theory 45 48 50 Chapter 7 Science and Theory 53 53 56 Chapter 8 Dynamical versus Statical Models 61 62 64 8.4 Why We Need Both Statical and Dynamical Models 68 Chapter 9 Dynamical and Structural Models 71 9.1 Structural Models 9.2 Graph Theory 71 72 9.3 Interplay between Dynamical and Structural Models 78 Contents vii Chapter 10 Deterministic versus Stochastic Dynamical Models 81 10.1 Deterministic Models 10.2 Stochastic Models 81 84 10.3 Do We Need Both? 87 Chapter 11 Linear Time Series Analysis 89 11.1 Time Series and Noise 11.2 ARIMA ( p; d; q) 89 90 11.3 ARIMA Model of Time Estimation 94 11.4 Mixed Regression-ARIMA Model of Psychophysical Judgment 96 Chapter 12 Probability Theory and Stochastic Models 99 12.1 Dynamical Cognitive Science and Mathematics 99 12.2 Stochastic Processes: A Random Walk to Ruin 100 12.3 Critical Points in Stochastic Models 103 12.4 Ergodicity and the Markov Property 105 Chapter 13 Stochastic Models in Physics 107 13.1 The Master Equation 13.2 Quantum Physics 110 13.3 Complementarity Redux Chapter 14 Noise 115 14.1 What Is Noise? 107 115 113 viii Contents 14.2 Probabilistic Descri