Computational Neuroscience ‘For the neuroscientist or psychologist who cringes at the sight of mathematical formulae and whose eyes glaze over at terms like differential equations, linear algebra, vectors, matrices, Bayes’ rule, and Boolean logic, this book just might be the therapy needed. Britt Anderson guides the reader into the world of computational methods; writing lucidly and grounding this Computational journey with elegantly constructed exercises. His slender book is an invitation to use tools that will help students and scientists think about neural and psychological mechanisms with rigor and and precision.’ Neuroscience Anjan Chatterjee, Professor of Neurology, University of Pennsylvania ‘Anderson provides a gentle introduction to computational aspects of psychological science, managing to respect Cognitive Modelling the reader’s intelligence while also being completely unintimidating. Using carefully-selected computational demonstrations, he guides students through a wide array of important approaches and tools, with little in the a student’s introduction to methods and procedures way of prerequisites ... I recommend it with enthusiasm.’ and Asohan Amarasingham, Assistant Professor, Department of Biology and Department of Psychology, The Graduate Center, The City University of New York Cognitive Modelling This unique, self-contained textbook provides an introduction to computational modelling for neuroscience and is accessible to readers with little or no background in computing or mathematics. Organized into thematic sections, the book spans from modelling integrate and firing neurons to playing the game Rock, Paper, Scissors in ACT-R. This non-technical guide shows how basic knowledge and modern computers can be combined for interesting simulations, progressing from early exercises utilizing spreadsheets, to simple programs in Python. Key features include: • Interleaved chapters demonstrate simple programming concepts and show their similarity to common spreadsheet procedures. • Mathematical facts and notation needed to understand the modelling methods are presented at their most basic and are interleaved with biographical and historical notes for context. • Numerous worked examples to demonstrate the themes and procedures of cognitive modelling. An excellent text for upper-level undergraduate and postgraduate students taking courses in research methods, Anderson computational neuroscience/computational modelling, and cognitive science/neuroscience. It will be especially valuable to psychology students. Britt Anderson is an Associate Professor in the Department of Psychology at the University of Waterloo, Canada. ISBN-13: 978-1-4462-4930-7 Britt Anderson Cover design by Wendy Scott Cover image © SciencePhotoLibrary anderson_methods_aw.indd 1-3 24/10/2013 10:55 Anderson_Prelims.indd 1 28/10/2013 4:56:58 PM Anderson_Prelims.indd 2 28/10/2013 4:56:58 PM Anderson_Prelims.indd 3 28/10/2013 4:56:58 PM SAGE Publications Ltd © Britt Anderson 2014 1 Oliver’s Yard 55 City Road First published 2014 London EC1Y 1SP Apart from any fair dealing for the purposes of research or SAGE Publications Inc. private study, or criticism or review, as permitted under the 2455 Teller Road Copyright, Designs and Patents Act, 1988, this publication Thousand Oaks, California 91320 may be reproduced, stored or transmitted in any form, or by any means, only with the prior permission in writing of the SAGE Publications India Pvt Ltd publishers, or in the case of reprographic reproduction, B 1/I 1 Mohan Cooperative Industrial Area in accordance with the terms of licences issued by the Mathura Road Copyright Licensing Agency. Enquiries concerning New Delhi 110 044 reproduction outside those terms should be sent to the publishers. SAGE Publications Asia-Pacific Pte Ltd 3 Church Street #10-04 Samsung Hub Singapore 049483 Library of Congress Control Number: 2013940255 Editor: Michael Carmichael Editorial assistant: Keri Dicken British Library Cataloguing in Publication data Production editor: Imogen Roome Copyeditor: Neville Hankins A catalogue record for this book is available from Proofreader: Audrey Scriven the British Library Marketing manager: Alison Borg Cover design: Wendy Scott Typeset by: C&M Digitals (P) Ltd, Chennai, India Printed in India at Replika Pvt Ltd ISBN 978-1-4462-4929-1 ISBN 978-1-4462-4930-7 Anderson_Prelims.indd 4 28/10/2013 4:56:59 PM Contents Preface x 1 Introduction 1 Objectives 1 1.1 Overview 1 1.2 Why Model? 2 1.3 Can We Make A Brain? 2 1.4 Computational Models as Experiments 3 1.5 Do Models Need to Be Biologically Plausible? 5 1.6 Do Computational Models Require Computers? 6 1.7 How Do You Evaluate a Model? 7 1.8 Do Models Need Assumptions? 8 1.9 Additional Questions 8 1.10 Road Map 9 I Modelling Neurons 11 2 What Is a Differential Equation? 13 Objectives 13 2.1 Overview 13 2.2 Modelling Individual Neurons 14 2.3 Differential Equations: The Mathematical Method of the Hodgkin and Huxley Model 14 2.4 The D in DE 16 2.5 Summary 21 3 Numerical Application of a Differential Equation 23 Objectives 23 3.1 Overview 23 3.2 First, We Assume a Spherical Chicken 23 3.3 Numerical Simulations 26 3.4 Numerical Integration and Differential Equations: Euler’s Method 28 3.5 Solving the Spring Differential Equation 30 3.6 Summary 31 Anderson_Prelims.indd 5 28/10/2013 4:56:59 PM vi Contents 4 Intermezzo: Computing with Loops 33 4.1 There Is More Than One Type of Computer Programming Language 33 4.2 Imperative Programming 33 4.3 The Harmonic Exercise in Python with Loops 37 5 Integrate and Fire 39 Objectives 39 5.1 Overview 39 5.2 Action Potentials 40 5.3 Computational Models of the Action Potential in Action 40 5.4 The Integrate and Fire Neuron Equation 41 5.5 Coding the Integrate and Fire Model of the Neuron 45 5.6 Summary 48 6 Intermezzo: Computing with if Statements 49 6.1 Code for a Simple Version of the Integrate and Fire Model 50 7 Hodgkin and Huxley: The Men and Their Model 53 Objectives 53 7.1 Overview 53 7.2 Who Were Hodgkin and Huxley? 53 7.3 The Hodgkin and Huxley Model 55 7.4 Simulating the Hodgkin and Huxley Model with a Spreadsheet 57 7.5 A Final Word about Differential Equations 60 8 Intermezzo: Computing with Functions 61 8.1 Conclusion 64 II Neural Networks 65 9 Neural Network Mathematics: Vectors and Matrices 67 Objectives 67 9.1 Overview 67 9.2 Linear Algebra 68 9.3 Elementary Vector and Matrix Operations 69 9.4 Think Geometric 70 9.5 Functions for Matrices and Vectors 74 9.6 Test Your Knowledge 75 9.7 Summary 75 10 Intermezzo: Interactive Computing 77 10.1 Compiling 77 10.2 Interpreting 78 10.3 Interpreted Languages for Linear Algebra 78 10.4 An Introduction to Interacting at the Command Line 79 Anderson_Prelims.indd 6 28/10/2013 4:56:59 PM Contents vii 11 An Introduction to Neural Networks 81 Objectives 81 11.1 Overview 81 11.2 What Are Neural Networks? 81 11.3 Some Neural Network History 82 11.4 Global Structure from Local Interactions 84 11.5 Cellular Automata 84 11.6 The Perceptron 86 11.7 Another Learning Rule: The Delta Rule 90 11.8 Why Aren’t Perceptrons the Only Type of Neural Networks? 93 11.9 Summary 94 12 Intermezzo: Interactive Exploration of the Delta Rule with Octave 97 12.1 Writing Functions 97 12.2 Octave Functions for the Delta Rule Exercise 98 12.3 Using Octave to Solve the Delta Rule Exercise 99 13 Auto-associative Memory and the Hopfield Net 103 Objectives 103 13.1 Overview 103 13.2 The Many Paths to Enlightenment 103 13.3 Hopfield Networks: Similarities and Differences to the Perceptron 104 13.4 Basic Structure of a Hopfield Network 106 13.5 A Hopfield Network in Action 107 13.6 Asynchronous Updating 109 13.7 Hopfield Insights 110 13.8 Summary 113 III Probability and Psychological Models 115 14 What Are the Odds? 117 Objectives 117 14.1 Overview 117 14.2 Paradox and Probability: Sets and Urns 118 14.3 Probability as Counting 118 14.4 Probability as a Measure of Set Size 119 14.5 Some Basic Relations of Probability 120 14.6 Humans Are Not Rational 123 14.7 Conclusion 124 15 Decisions as Random Walks 125 Objectives 125 15.1 Overview 125 15.2 Random Walks 126 15.3 Measuring Reaction Time 128 15.4 Testing Models of Reaction Time 129 15.5 Conclusion 135 Anderson_Prelims.indd 7 28/10/2013 4:56:59 PM viii Contents 16 Intermezzo: Programming Psychophysical Experiments with Python 137 16.1 Writing Python Programs for Research 137 16.2 The PsychoPy Library 138 16.3 Conclusion 143 IV Cognitive Modelling as Logic and Rules 145 17 Boolean Logic 147 Objectives 147 17.1 Overview 147 17.2 Origins of Mathematical Logic 148 17.3 There IS More Than One Type of Logic 150 17.4 Summary 155 18 Intermezzo: Computing with Functional Languages 157 18.1 Functional Programming 157 18.2 Advantages of Functional Programming 158 18.3 Functional Programming Languages 159 18.4 Summary 160 19 Production Rules and Cognition 161 Objectives 161 19.1 Overview 161 19.2 Productions 161 19.3 History of Production Systems 163 19.4 Production Systems 164 19.5 Summary 167 20 Intermezzo: Functional Coding of a Simple Production System 169 20.1 A Haskell Implementation 169 20.2 Summary 172 21 ACT-R: A Cognitive Architecture 173 Objectives 173 21.1 Overview 173 21.2 A Full-Fledged Cognitive Architecture: ACT-R 173 21.3 Summary 190 22 Agent-Based Modelling 193 Objectives 193 22.1 Overview 193 22.2 History of Agent-Based Models 194 22.3 Components of an Agent-Based Model 195 22.4 Building an Agent-Based Model 195 22.5 An Introduction to NetLogo 195 Anderson_Prelims.indd 8 28/10/2013 4:56:59 PM Contents ix 22.6 Making a NetLogo Model 197 22.7 Running the Eye Movement Model 206 22.8 Conclusion 206 23 Concluding Remarks 209 Notes 211 References 223 Index 225 Anderson_Prelims.indd 9 28/10/2013 4:56:59 PM Preface My own efforts to develop a little computational competency have followed a fairly stand- ard path.