Progress in Motor Control

Dagmar SternadEditor

Progress in Motor Control

A Multidisciplinary Perspective

1 3

Editor

Dagmar SternadPennsylvania State UniversityUniversity Park, PA, USAdxs48@psu.edu

ISBN: 978-0-387-77063-5 e-ISBN: 978-0-387-77064-2DOI 10.1007/978-0-387-77064-2

Library of Congress Control Number: 2007940148

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Preface

It has becomewidely acknowledged, and almost trivial to state, that the study of thecontrol and coordination of biological movement – motor control – is inherentlymultidisciplinary. From the investigation of overt functional behavior to the intri-cacies of neuronal activations, the issues are numerous and invite many differentlevels of analysis, methods, and perspectives. Clearly, the biological movementsystem is simultaneously a dynamical, neurophysiological, electrophysiological,and intentional system, in short, a complex system in the technical senseof theword.

While multidisciplinarity in motor control research is a necessity, it alsopresents a stumbling block to developing a coherent body of knowledge thatrepresents the science of the control and coordination of movement. Researchthrusts are developing from different academic backgrounds that are not easilyunderstood by peers with entirely different disciplinary training. Not only forthe student of motor control, but also for the advanced researcher, it can bedaunting to make connections, for example, between cognitive issues like plan-ning or attention and functional properties of the peripheral nervous system,between motor cortical activation and the biomechanics of the multi-joint limbsystem. Yet, all of these approaches aim to shed light on the same phenomenon –the astonishing ability of biological systems to move, perceive, grow, adapt,use tools, and do infinitely more things. For the science of motor control toprogress more integration of disciplines is therefore necessary.

The present book is an attempt to facilitate connections across differentstrands of research and thereby contribute towards developing a more coherentbody of knowledge. Organized into seven core topics, 38 contributions wereselected from leading researchers to represent the study of movement in all itsbreadth and facets. In each of these topic sections four to six differentapproaches are juxtaposed to entice readers to go beyond their immediatefocus and become acquainted with different lines of thought. An introductionat the beginning of each section provides a commentary and guideline bydrawing connections between the individual contributions. The chapters areexplicitly written in accessible form and provide some review followed by amore focused treatment of the authors’ own research. The diversity aims to givethe student of motor control not only understanding of the extent of the fieldbut also, hopefully, some orientation for their own research.

v

This book is aimed at graduate students to provide both an introduction toand an overview of motor control. To date, there is no textbook that presents asystematic progression through motor control in all its breadth. On the onehand, this is a sign of a growing but not yet mature research; on the other hand,the sheer expanse of our area of research would make writing such a text adaunting undertaking for any single person. Evidently, there are a number ofexcellent single-authored textbooks that focus on, for example, the neurophy-siology of movements, or on computational approaches, or on more clinicallyapplied questions. While such books are eminently valuable, they necessarilypresent a selected subsection of the area, simply because they are written by asingle author. A book that provides an informed and unbiased overview overthe entire field would require a psychologist/physiologist/nonlinear dynamicist/kinesiologist/engineer to faithfully represent the advances made in all thesedomains. For this reason, the present volume invited a number of leadingresearchers to present their perspectives on seven selected core topics.

This book originally developed from the conference ‘‘Progress in MotorControl V – A Multidisciplinary Perspective’’ which took place at thePennsylvania State University on August 19–22, 2005. The conference featuredan impressive array of speakers and the oral and poster presentations spanned anunprecedented range of disciplinary approaches tomotor control. The 350 atten-dees came from 23 countries to State College, PA to make this an exciting andspecial event. An amazing realization was that while all researchers were focusedon motor control, due to their disciplinary affiliation, many typically attendeddifferent conferences, published in different journals and were part of differentresearch communities. After the four conference days, such boundaries wereovercome and discussions were spontaneous and mutually informative.

This book first began as a conference volume collecting this state-of-the-artresearch in its breadth and depth. Very soon, I realized that it would be beneficialtomake this collection of papers less of a conference volume, but rathermore of atextbookwith an introductory flavor for both students and researchers.With thisgoal in mind, I subsequently also invited other contributors to complement theoriginal conference presentations. The last section on the equilibrium-pointhypothesis has a somewhat special status: the papers were presented as part ofa symposium dedicated to the ‘‘40-year-anniversary’’ of the equilibrium-pointhypothesis and simultaneously honored Anatol Feldman who spearheaded thisresearch. The contributions presented an excellent discussion of this particularline of theorizing and are therefore included in this book.

One clear message from the conference was that the study of the control andcoordination of biological behavior, in shortmotor control, is a thriving field ofresearch with a fast growing body of knowledge. At the present stage ofdevelopment the field of research may gain from more integration and a text-book covering and connecting the many strands of this research. This volume isone attempt in this direction.

Dagmar SternadState College, PA October 2007

vi Preface

Contents

Part I The Nature of Motor Control

Nature of Motor Control: Not Strictly ‘‘Motor’’, Not Quite ‘‘Control’’ . . . 3Michael T. Turvey

Beyond Control: The Dynamics of Brain-Body-Environment Interaction

in Motor Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Randall D. Beer

Towards Testable Neuromechanical Control Architectures for Running . . 25Shai Revzen, Daniel E. Koditschek, and Robert J. Full

Control from an Allometric Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bruce J. West

Synergies: Atoms of Brain and Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 83J.A. Scott Kelso

Nature of Motor Control: Perspectives and Issues . . . . . . . . . . . . . . . . . . . 93Michael T. Turvey and Sergio Fonseca

Part II What is Encoded in the Brain?

Past, Present, and Emerging Principles in the Neural Encoding

of Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Timothy J. Ebner, Claudia M. Hendrix, and Siavash Pasalar

From Intention to Action: Motor Cortex and the Control of Reaching

Movements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139John F. Kalaska

vii

Control of Muscle Synergies by Cortical Ensembles . . . . . . . . . . . . . . . . . 179Michelle M. Morrow, Eric A. Pohlmeyer, and Lee E. Miller

Behavioral and Neurophysiological Aspects of Target Interception . . . . . . 201Hugo Merchant, Wilbert Zarco, Luis Prado, and Oswaldo Perez

Learning from Learning: What Can Visuomotor Adaptations

Tell us About the Neuronal Representation of Movement?. . . . . . . . . . . . . 221Rony Paz and Eilon Vaadia

The Problem of Parametric Neural Coding in the Motor System . . . . . . . 243Jacob Reimer and Nicholas G. Hatsopoulos

Part III Perception and Action

Introduction to Section on Perception and Action . . . . . . . . . . . . . . . . . . . 263Brett R. Fajen

Mutuality in the Perception of Affordances and the Control of Movement 273Claudia Carello and Jeffrey B. Wagman

Object Avoidance During Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293David A. McVea and Keir G. Pearson

The Roles of Vision and Proprioception in the Planning of Reaching

Movements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Fabrice R. Sarlegna and Robert L. Sainburg

Using Predictive Motor Control Processes in a Cognitive Task:

Behavioral and Neuroanatomical Perspectives . . . . . . . . . . . . . . . . . . . . . . 337James Stanley and R. Christopher Miall

The Human Mirror Neuron System and Embodied Representations . . . . . 355Lisa Aziz-Zadeh and Richard B. Ivry

Disorders of the Perceptual-Motor System . . . . . . . . . . . . . . . . . . . . . . . . 377Steven A. Jax and H. Branch Coslett

Part IV Motor Learning

Some Contemporary Issues in Motor Learning . . . . . . . . . . . . . . . . . . . . . 395Karl M. Newell and Rajiv Ranganathan

Motor Learning and Consolidation: The Case of Visuomotor Rotation . . . 405John W. Krakauer

viii Contents

Cortical Processing during Dynamic Motor Adaptation . . . . . . . . . . . . . . 423Simon A. Overduin, Andrew G. Richardson, and Emilio Bizzi

Motor Learning: Changes in the Structure of Variability

in a Redundant Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Hermann Muller and Dagmar Sternad

Time Scales, Difficulty/Skill Duality, and the Dynamics

of Motor Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Karl M. Newell, Yeou-Teh Liu, and Gottfried Mayer-Kress

Part V Bridging of Models for Complex Movements in 3D

Bridging of Models for Complex Movements in 3D . . . . . . . . . . . . . . . . . . 479Stan Gielen

The Posture-Based Motion Planning Framework: New Findings Related

to Object Manipulation, Moving Around Obstacles, Moving in Three

Spatial Dimensions, and Haptic Tracking . . . . . . . . . . . . . . . . . . . . . . . . . 485David A. Rosenbaum, Rajal G. Cohen, Amanda M. Dawson,Steven A. Jax, Ruud G. Meulenbroek, Robrecht van der Wel,and Jonathan Vaughan

Grasping Occam’s Razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Jeroen B.J. Smeets, Eli Brenner, and Juul Martin

Review of Models for the Generation of Multi-Joint Movements in 3-D . . 523Stan Gielen

Part VI The Hand as a Complex System

Why the Hand? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553Francisco J. Valero-Cuevas

Selective Activation of Human Finger Muscles after Stroke

or Amputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559Marc H. Schieber, C.E. Lang, K.T. Reilly, P. McNulty, and A. Sirigu

Neural Control of Hand Muscles During Prehension . . . . . . . . . . . . . . . . . 577Jamie A. Johnston, Sara A. Winges, and Marco Santello

Multi-Finger Prehension: Control of a Redundant Mechanical System . . . 597Mark L. Latash and Vladimir M. Zatsiorsky

Contents ix

A Mathematical Approach to the Mechanical Capabilities of Limbs

and Fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619Francisco J. Valero-Cuevas

Part VII Forty Years of Equilibrium-Point Hypothesis

Origin and Advances of the Equilibrium-Point Hypothesis . . . . . . . . . . . . 637Anatol G. Feldman

The Biomechanics of Force Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 645Denis Rancourt and Neville Hogan

The Implications of Force Feedback for the l Model. . . . . . . . . . . . . . . . . 663Richard Nichols and Kyla T. Ross

Control and Calibration of Multi-Segment Reaching Movements . . . . . . . 681James R. Lackner and Paul DiZio

The Equilibrium-Point Hypothesis – Past, Present and Future . . . . . . . . . 699Anatol G. Feldman and Mindy F. Levin

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

x Contents

Contributors

Lisa Aziz-ZadehBrain and Creativity Institute, Department of Occupational Sciences,University of Southern California, Los Angeles, CA, USA,lisa.azizzadeh@usc.edu

Randall D. BeerCognitive Science Program, Department of Computer Scienceand Informatics, Indiana University, Bloomington, IN 47406, USA,rdbeer@indiana.edu

Emilio BizziDepartment of Brain and Cognitive Sciences and McGovern Institute forBrain Research, Massachusetts Institute of Technology, 43Vassar Street,Cambridge, MA 02139, USA, ebizzi@mit.edu

Eli BrennerResearch Institute MOVE, Faculty of Human Movement Sciences, VUUniversity Amsterdam, van der Boechorststraat 9, NL-1081 BT Amsterdam,The Netherlands, e.brenner@fbw.vu.nl

Claudia CarelloCenter for the Ecological Study of Perception and Action, University ofConnecticut, CESPA U-1020, 406 Babbidge Road, University of Connecticut,Storrs, CT 06269-1020, USA, Claudia, claudia.carello@uconn.edu

Rajal G. CohenDepartment of Psychology, Pennsylvania State University, University Park,PA 16802, USA, rajal@psu.edu

Branch CoslettMoss Rehabilitation Research Institute, University of Pennsylvania MedicalSchool, Department of Neurology, Philadelphia, PA, USA,hbc@mail.med.upenn.edu

xi

Amanda M. DawsonMoss Rehabilitation Research Institute, University of Pennsylvania MedicalSchool, Department of Physical Medicine and Rehabilitation, 213 KormanBuilding, 1200 West Tabor Road, Philadelphia, PA, USA,dawson@einstein.edu

Paul DiZioAshton Graybiel Spatial Orientation Laboratory, Brandeis University,MS033, Waltham, MA 02454-9110, USA, dizio@brandeis.edu

Timothy J. EbnerDepartment of Neuroscience, University of Minnesota, Lions ResearchBuilding, Room 421, 2001 Sixth Street S.E., Minneapolis, MN 55455, USA,ebner001@umn.edu

Brett R. FajenDepartment of Cognitive Science, Rensselaer Polytechnic Institute, CarnegieBuilding 308, 110 8th Street, Troy, NY 12180-3590, USA, fajenb@rpi.edu

Anatol G. FeldmanDepartment of Physiology, University of Montreal, School of Physical andOccupational, Therapy McGill University; Center for InterdisciplinaryResearch in Rehabilitation, Montreal Rehabilitation Institute and JewishRehabilitation Hospital, Canada, feldman@med.umontreal.ca

Sergio FonsecaFederal University of Minas Gerais, Brazil, and, Center for the EcologicalStudy of Perception and Action, University of Connecticut, Storrs, CT, USA,sfonseca@pib.com.br

Robert J. FullDepartment of Integrative Biology, University of California at Berkeley, CA,USA, rjfull@socrates.berkeley.edu

Stan GielenDepartment of Biophysics, Radboud University Nijmegen, Geert Grooteplein25, NL 6525EZ Nijmegen, The Netherlands, S.Gielen@science.ru.nl

Nicholas G. HatsopoulosDepartment of Organismal Biology and Anatomy, University of Chicago,Chicago, IL 60637, USA, nicho@uchicago.edu

xii Contributors

Claudia M. HendrixDepartment of Neuroscience, University of Minnesota, Minneapolis MN55455, USA, hendrix@me.umn.edu

Neville HoganDepartment of Mechanical Engineering, Department of Brain andCognitive Sciences, Massachusetts Institute of Technology, Cambridge,MA, USA, neville@mit.edu

Richard B. IvryDepartment of Psychology, 3210 Tolman Hall, University of California,Berkeley, CA 94720-1650, USA, ivry@berkeley.edu

Steven A. JaxMoss Rehabilitation Research Institute, University of PennsylvaniaMedical School, Department of Physical Medicine & Rehabilitation,213Korman Building, 1200West Tabor Road, Philadelphia, PA, USA,jaxs@einstein.edu

Jamie A. JohnstonFaculty of Kinesiology, University of Calgary, 2500 University Dr. NW,Calgary, AB, T2N 1N4, johnston@kin.ucalgary.ca

John F. KalaskaDepartement de Physiologie, Faculte de Medecine, Universite de Montreal,Montreal, Quebec H3C 3 J7, Canada, john.francis.kalaska@umontreal.ca

J.A. Scott KelsoCenter for Complex Systems and Brain Sciences, Florida Atlantic University,Boca Raton, FL 33435, USA, kelso@ccs.fau.edu

Daniel E. KoditschekGRASP Laboratory, Department of Electrical and Systems Engineering,University of Pennsylvania, Philadelphia PA, USA, kod@seas.upenn.edu

John W. KrakauerThe Neurological Institute, Columbia University Medical Center, 710West168th Street, NY 10032, USA, jwk18@columbia.edu

James R. LacknerAshton Graybiel Spatial Orientation Laboratory, Brandeis University,MS033, Waltham, MA 02454-9110, USA, lackner@brandeis.edu

Contributors xiii

Catherine E. LangDepartment ofNeurology,WashingtonUniversity in St. Louis, St. Louis,MO,USA, langc@wustl.edu

Mark L. LatashDepartment of Kinesiology, Rec Hall 267, The Pennsylvania State University,University Park, PA 16802, USA, mll11@psu.edu

Mindy F. LevinDepartment of Physiology, University of Montreal and Schoolof Physiotherapy, McGill University; Center for Interdisciplinary Researchin Rehabilitation,Montreal Rehabilitation Institute and Jewish RehabilitationHospital, Canada, mindy.levin@mcgill.ca

Yeou-Teh LiuGraduate Institute of Exercise and Sport Science, National Taiwan NormalUniversity, Taipei, Taiwan, yeouteh@ntnu.edu.tw

Juul MartinResearch Institute MOVE, Faculty of Human Movement Sciences, VUUniversity Amsterdam, van der Boechorststraat 9, NL-1081 BT Amsterdam,The Netherlands, Juul.Martin@snr.nl

Gottfried Mayer-KressDepartment of Kinesiology, The Pennsylvania State University, UniversityPark, PA 16802, USA, gxm21@psu.edu

Penelope McNultyPrince of Wales Medical Research Institute, University of New South Wales,Sydney, Australia, p.mcnulty@unsw.edu.au

David A. McVeaDepartment of Physiology and Centre for Neuroscience, 715Medical SciencesBuilding, University of Alberta, Edmonton AB, T6G 2H7, Canada,dmcvea@ualberta.ca

Hugo MerchantInstituto de Neurobiologıa, UNAM, Campus Juriquilla, Queretaro Qro.76230Mexico, USA, merch006@umn.edu

Ruud G. MeulenbroekNijmegen Institute for Cognition and Information, Radboud UniversityNijmegen, Department of Cognitive Psychology, P.O. Box 9104, 6500HENijmegen, The Netherlands, r.meulenbroek@nici.ru.nl

xiv Contributors

Christopher MiallSchool of Psychology, University of Birmingham, Birmingham B15 2TT, UK,r.c.miall@bham.ac.uk

Lee E. MillerDepartment of Physiology, Northwestern University, Chicago 1L60611,lm@northwestern.edu

Michelle M. MorrowCenter for the Neural Basis of Cognition, University of Pittsburgh, 4074 BST3,3501 Fifth Avenue, Pittsburgh PA 15261, mim37@pitt.edu

Hermann MullerInstitute for Movement Science, Justus-Liebig-University of Giessen,Germany, hermann.mueller@sport.uni-giessen.de

Karl M. NewellDepartment of Kinesiology, Pennsylvania State University, 267 Rec Hall,University Park, PA 16802, USA, kmn1@psu.edu

Richard NicholsDepartment of Biomedical Engineering, Georgia Institute of Technology,Atlanta, GA 30332, trn@gatech.edu

Simon A. OverduinDepartment of Brain and Cognitive Sciences and McGovern Institute forBrain Research, Massachusetts Institute of Technology, 43Vassar Street,Cambridge, MA 02139, USA, overduin@mit.edu

Siavash PasalarDepartment of Neuroscience, University of Minnesota, Minneapolis MN55455, USA, pasalar@me.umn.edu

Rony PazDepartment of Neurobiology, Weizmann Institute of Science, Room 203,Leonesco Building, Rehovot, Israel 76100, rony.paz@weizmann.ac.il

Keir G. PearsonDepartment of Physiology and Centre for Neuroscience, 715Medical SciencesBuilding, University of Alberta, Edmonton AB, T6G 2H7, Canada,keir.pearson@ualberta.ca

Oswaldo PerezInstituto de Neurobiologıa, UNAM, Campus Juriquilla, Queretaro Qro.76230Mexico, USA, dawalreti@gmail.com

Contributors xv

Eric A. PohlmeyerDepartment of Physiology, Northwestern University, Chicago, IL 60611,USA, e-pohlmeyer@northwestern.edu

Luis PradoInstituto de Neurobiologıa, UNAM, Campus Juriquilla, Queretaro Qro.76230Mexico, USA, pradol@inb.unam.mx

Denis RancourtDepartement de Genie Mecanique, Universite de Sherbrooke, Quebec,Canada, denis.rancourt@usherbrooke.ca

Rajiv RanganathanDepartment of Kinesiology, Pennsylvania State University, 275 Rec Building,University Park, PA 16802, USA, rxr259@psu.edu

Jacob ReimerDepartment of Organismal Biology and Anatomy, University of Chicago,Chicago, IL 60637, USA, jreimer@uchicago.edu

Shai RevzenIntegrative BiologyDepartment, University of California, Berkeley, CA,USA,shrevz@berkeley.edu

Andrew G. RichardsonDivision of Health Sciences and Technology, Massachusetts Institute ofTechnology and Harvard Medical School, 45 Carlton Street, Cambridge, MA02142, USA, drewrich@mit.edu

David A. RosenbaumDepartment of Psychology, Pennsylvania State University, University Park,PA 16802, USA, dar12@psu.edu

Kyla T. RossDepartment of Biomedical Engineering, Georgia Institute of Technology,313 Ferst Drive, Atlanta, GA 30332, kyla.ross@gatech.edu

Robert L. SainburgDepartment of Kinesiology, Pennsylvania State University, 29 RecreationBuilding, University Park, PA, 16802, USA, rls45@psu.edu

Marco SantelloDepartment of Kinesiology, Arizona State University, Tempe, AZ 85287,USA, marco.santello@asu.edu

xvi Contributors

Fabrice R. SarlegnaLaboratoire Mouvement and Perception, CNRS and University of theMediterranean, 163, avenue de Luminy-CP 910, 13 288 Marseille, France,fabrice.sarlegna@univmed.fr

Marc H. SchieberDepartment of Neurology, University of Rochester, 601 Elmwood Ave, Box673, Rochester, NY 14642, USA, mhs@cvs.rochester.edu

Karen T. ReillyCentre for Cognitive Neuroscience, CNRS, Lyon, France, kreilly@isc.cnrs.fr

Angela SiriguCentre for Cognitive Neuroscience, CNRS, Lyon, France, sirigu@isc.cnrs.fr

Jeroen B.J. SmeetsResearch Institute MOVE, Faculty of Human Movement Sciences, VUUniversity Amsterdam, van der Boechorststraat 9, NL-1081 BT Amsterdam,The Netherlands, J.Smeets@fbw.vu.nl

James StanleySchool of Psychology, University of Birmingham, Birmingham B15 2TT, UK,j.stanley@bham.ac.uk

Dagmar SternadDepartments of Kinesiology and Integrative Biosciences, 266 Rec Hall,University Park, PA, Pennsylvania State University, USA, dxs48@psu.edu

Michael T. TurveyCenter for the Ecological Study of Perception and Action, Universityof Connecticut, Storrs, and Haskins Laboratories, New Haven, CT, USA,michael.turvey@uconn.edu

Eilon VaadiaDepartment of Physiology, Hadassah Medical School, The InterdisciplinaryCenter for Neural Computation (ICNC), Faculty of Medicine, The HebrewUniversity, POB 12271, Jerusalem, 91120, Israel, eilon@md.huji.ac.il

Francisco J. Valero-CuevasDepartment of Biomedical Engineering, The University of SouthernCalifornia, 3710McClintock Ave, Los Angeles, CA 90089-2905, USA,valero@usc.edu

Contributors xvii

Robrecht van der WelDepartment of Psychology, Pennsylvania State University, University Park,PA 16802, USA, rxv906@psu.edu

Jonathan VaughanDepartment of Psychology, Hamilton College, Clinton, NY 13323, USA,jvaughan@hamilton.edu

Jeffrey B. WagmanDepartment of Psychology, Illinois State University, Campus Box 4620Normal, Illinois 61790-4620 USA, JeffreyWagman@ilstu.edu

Bruce J. WestMathematical & Information Science Directorate, U.S. ArmyResearch Office,Research Triangle Park, NC, USA, Bruce.J.West@us.army.mil

Sara A. WingesDepartment of Neuroscience, University of Minnesota, 6–145 Jackson Hall,321 Church Street SE, Minneapolis, MN 55455, USA, swinges@umn.edu

Wilbert ZarcoInstituto de Neurobiologıa, UNAM, Campus Juriquilla, Queretaro Qro.76230Mexico, USA, zarco@inb.unam.mx

Vladimir M. ZatsiorskyDepartment of Kinesiology, Rec Hall 267, Rec Hall 267, Pennsylvania StateUniversity, University Park, PA 16802, USA, vxz1@psu.edu

xviii Contributors

Part I

The Nature of Motor Control

Nature of Motor Control: Not Strictly ‘‘Motor’’,

Not Quite ‘‘Control’’

Michael T. Turvey

The five chapters directed at the Nature of Motor Control share much in

common: they are concept oriented, each expressing respect for the level of

abstraction needed to meet the scientific challenges of animal movement. That

level of abstraction, as the reader will ideally discern, is at some remove from the

ordinary meanings of the terms motor and control.For Beer and for Revzen, Koditscheck and Full, control resides in nervous

system, body, and environment, viewed as dynamical systems in continuous

interaction. Beer enters the complexity of the formulation expressed through

Fig. 1 by means of genetic algorithms that permit the study of the evolution of

model variants of Fig. 1. Thesemodel variants bring into view potential tradeoffs

among nervous system, body, and environment. For Beer, dynamical systems

theory is a toolbox for revealing the principles behind an evolved system’s

behavioral achievements. With similar sentiments, Revzen et al. approach

Fig. 1’s complexity through a kinematic strategy based in the observable changes

in phase and frequency of component movements induced by select perturba-

tions. Identifying the nature of the perturbations is a primary challenge for

the strategy. For Revzen et al. they must follow from ecological (animal-

environment system) based considerations rather than engineering convenience.

The strategy is designed to constrain experimental evaluations of four neurome-

chanical architectures that are candidate instantiations of the dynamics inherent

in Fig. 1 for the canonical control problem of n-legged running. As is the case for

Beer, the toolbox for Revzen et al. is dynamical systems theory.The control perspective detailed in West’s chapter is aimed at a principle

that would tie together temporally overlapping faster and slower processes

within the course of movement. A major lesson from the study of locomotion

and posture is that the variability (of stride interval, of center of pressure) is

fractal—meaning that it possesses long-time correlations with fluctuations at

any one time scale exhibiting statistically similar behavior to the fluctuations

M.T. Turvey (*)Center for the Ecological Study of Perception and Action, University of Connecticut,Storrs, and Haskins Laboratories, New Haven, CT, USAe-mail: michael.turvey@uconn.edu

D. Sternad (ed.), Progress in Motor Control,DOI 10.1007/978-0-387-77064-2_1, � Springer ScienceþBusiness Media, LLC 2009

3

at any other time scale. West draws upon the tradition of allometry in biologi-

cal and zoological inquiry: the identification of power laws (alias, scaling laws)

that tie together smaller and larger animals in respect to an anatomical or

physiological attribute. Figure 2 schematizes the relation between allometry in

NB

E

neural-bodydynamics

neural-environmentaldynamics

body-environmentaldynamics

NB

E

NB

E

Fig. 1 A schematic ofembedded, embodied motorcontrol identifying thecouplings of nervous system,body, and environment thatare the focus of concern inthe theory and researchdiscussed in Beer’s chapterand the chapter by Revzen,Koditscheck and Full

100

104

108

respiratory cycle

cardiac cycle

fast muscle contraction

circulation of blood volume

metabolism of fat

maximum life span

1 g

BODY MASS

1 kg 1000 kg

TIME(min)

a b

large scale

small scale

renormalization group theory

10–4

Fig. 2 (a) A sample of physiological events occurring at several magnitudes of time scaleaccording to body mass. Available data suggest that the different events might share acommon power law exponent (Lindstedt & Calder, 1981). The horizontal arrow depictshow allometric laws have been traditionally employed to tie together smaller and largeranimals. The vertical arrow depicts how allometric scaling might be conceived for controlof these many events (in particular those central to movement) in a single animal: the fittingtogether of faster and slower time scales. (b) In locomotion over uneven terrain, there arefluctuations at many time scales. Larger and smaller time scales of head fluctuations and footsurface fluctuations would need to be tied together. The tool of renormalization group theoryis promoted in West’s chapter as potentially key to achieving this concinnity (parts fittingparts, and fitting the whole system)

4 M.T. Turvey

traditional guise and allometric control as envisaged byWest. This scale invariant

image of control relies on a wide variety of mathematics and concepts from

statistical physics for its expression. Much of the mathematics and concepts in

question is relatively new to the field of motor control, and to control theory in

general.Kelso’s focus is the concept of synergy. Figure 3a is one way of commu-

nicating the central idea. The interactions of the parts entail a unitary collec-

tive state; the unitary collective state entails the interactions of the parts. Such

coupling relations between muscles

emergent synergy

entail entails

a

b

Fig. 3 (a) A schematic of the view of synergy as an emergent, self-organizing functional form aspresented in Kelso’s chapter. (b) A hyperset graph expressing a minimal two-muscles synergy.Construction of the graph follows two common set-theoretic conventions. First, functionsare represented as ordered pairs. That is, if P is a function from Q to R, P is represented asP=<Q, R>. Second, to preserve their ordering, each ordered pair is represented as two sets,one set containing the ordered pair’s first member and one set containing the ordered pair’sfirst and secondmembers. Thus<Q, R> is represented as { { Q }, { Q, R } }. Notable featuresof the graph are (i) the loops from synergy to coupling: these signify that the synergy’s efficientcause is internal to the synergy, and (ii) the absence of loops involving the muscles asindividual: this identifies the muscles as material causes, resources for the synergetic organiza-tion. Further details of hyperset representations can be found in Chemero and Turvey (2008)

Nature of Motor Control 5

an organization is a hallmark feature of self-organizing systems. It is ageneral organizational form that includes the synergies of biological move-ment as one of its many manifestations. Conventionally the idea of a move-ment synergy can be defined in an uncomplicated way by standard (well-founded) set theory as a set of muscles and the set of interactions that couplethem: synergy = {(muscles), (coupling interactions)}. This formulation viastandard set theory however does not do justice to Kelso’s view. It does notcapture Fig. 3a. What is needed is non-standard (non well-founded) settheory (Aczel, 1988). Figure 3b (courtesy A. Chemero) introduces the morechallenging but apposite idea of a movement synergy as a hyperset of muscles,coupling, and synergy. In the standard case synergy is defined predicatively (non-circularly), in the non-standard case the definition is impredicative. In a (circular)impredicative definition, the thing defined participates in its own definition. It isthis impredicative definition, permissible within non-standard set theory andexpressed inminimal form inFig. 3b, which underwrites the issues and argumentsraised by Kelso. Recent developments in computer science have shown thecomputability of hypersets (e.g., Lisitsa & Sazonov, 1999), opening the door ona potentially richer synergy-based foundation for modeling and understandingmotor control.

The chapters of Beer, Revzen et al., West, and Kelso, exemplify researchand thinking on the nature of motor control that is not standard textbookfare. Undergraduates and graduates will tend to encounter dissections of themotor control problem into functionally distinct neural components whoseroles are described fairly strictly in the language of computation. The presentvolume presents several excellent chapters that are representative of theorthodox perspective. The chapter by Turvey and Fonseca in this sectionon the ‘‘The Nature of Motor Control’’ is aimed at providing an overview onthe major perspectives, both orthodox and heterodox, that drive contempor-ary inquiry into biological movement. Their chapter takes advantage of thecontrasts among the perspectives to identify issues, both current and poten-tial, whose resolution can be expected to have important consequences forfuture developments. Ideally, the five chapters of this section should give thereader much to think about as she or he takes the intellectual journey offeredby the collection of papers that compose the present volume on ‘‘Progress inMotor Control’’.

References

Aczel, P. (1988). Non-well-founded Sets. Stanford: CSLI.Chemero, A., & Turvey, M. T. (2008). Autonomy and hypersets. Biosystems, 91, 320–330.Lindstedt, S. L., & Calder, W. A. (1981). Body size, physiological time, and longevity of

homeothermic animals. Quarterly Review of Biology, 56, 1–16.Lisitsa, A. P., & Sazonov, V. Yu. (1999). Linear ordering on graphs, anti-founded sets and

polynomial time computability. Theoretical Computer Science, 224, 173–213.

6 M.T. Turvey

Beyond Control: The Dynamics

of Brain-Body-Environment Interaction

in Motor Systems

Randall D. Beer

Abstract Discussions of motor behavior have traditionally focused on how anervous system controls a body. However, it has become increasingly clear thata broader perspective, in which motor behavior is seen as arising from theinteraction between neural and biomechanical dynamics, is needed. This chap-ter reviews a line of work aimed at exploring this perspective in a simple modelof walking. Specifically, I describe the evolution of neural pattern generators fora hexapod body, present a neuromechanical analysis of the dynamics of theevolved agents, characterize how the neural and biomechanical constraintsstructure the fitness space for this task, and examine the impact of networkarchitecture.

Introduction

Discussions of motor behavior typically center on the nervous system. On thisview, an animal’s environment is just a stage on which its behavior plays out, itsbody is merely a collection of muscles to activate and masses to move, and itssensors are simply sources of error signals to be compensated. Indeed, when wespeak of the neural control of behavior, our very language betrays our bias, as ifan animal’s body were a mere puppet whose only task was to respond faithfullyto the commands of its nervous system.

However, it is becoming increasingly clear that a broader perspective isneeded. There is no question that nervous systems vastly increase the range ofbehavior in which an animal can stably engage. But an animal’s nervous system,its body, and its environment each posses their own complex intrinsic dynamics,which are in continuous interaction. A bird, for example, flies not only because ofthe patterns of muscle activation produced by its nervous system, but alsobecause of the shape and composition of its feathers and the hydrodynamical

R.D. Beer (*)Cognitive Science Program, Department of Computer Science, Department ofInformatics, Indiana University, Bloomington, IN 47406e-mail: rdbeer@indiana.edu

D. Sternad (ed.), Progress in Motor Control,DOI 10.1007/978-0-387-77064-2_2, � Springer ScienceþBusiness Media, LLC 2009

7

properties of the air through which it moves. Furthermore, evolution selects onlyfor the behavioral efficacy of this entire package, and it seems likely that it wouldtake full advantage of any available freedom in distributing behavioral mechan-isms. This suggests that behavior is best viewed as a property of a complete brain-body-environment system (Fig. 1A; Beer, 1995a; Chiel & Beer, 1997), and cannotproperly be assigned to any individual component of this coupled system. Evi-dence for this view has come not only from neuromechanical studies of motorbehavior, but also work in robotics, perception, developmental psychology,cognitive science, and philosophy of mind (Gibson, 1979; Brooks, 1991; Thelen &Smith, 1994; Clark, 1997).

This integrated perspective on behavior raises both experimental andtheoretical challenges. It is difficult enough to study any one component ofa brain-body-environment system in isolation, let alone the simultaneousinteraction of all three. Not only must one be able to measure and manipulateneural activity in a behaving animal, but also the relevant properties of itsbody and environment. While important progress along these lines is begin-ning to be made in several systems (Winters & Crago, 2000), significantchallenges remain. Perhaps even more fundamentally, even if the requiredexperimental tools were available, we currently lack the theoretical frame-work necessary for an integrated understanding of the tangled, counter-intuitive designs that evolution produces.

Given current experimental and theoretical limitations, another possiblestrategy involves the systematic study of idealized models of complete brain-body-environment systems. Like Galileo’s frictionless planes, such frictionlessbrains (and bodies, and environments) can help us to build intuition and,ultimately, the conceptual framework and mathematical and computationaltools necessary for understanding the mechanisms of behavior (Beer, 1990).

Body

Environment

NervousSystem

Neural Dynamics

Biomechanics

Fitness Evaluation

Fitness

Behavior

Motor Pattern

Neural and Synaptic Parameters

A BFig. 1 A new perspective onmotor systems. (A) Ananimal’s nervous system, itsbody and its environment areeach viewed as dynamicalsystems, which are incontinuous interaction.(B) The relationship betweenneural parameters andbehavioral performance isvery indirect, passingthrough several layers oftransformations. Becauseeach of these transformationsmay be degenerate, a givenlevel of fitness may beobtained by a wide range ofdifferent neural parameters

8 R.D. Beer

A specific approach that we and others have pursued for many years is the useof evolutionary algorithms to evolve model nervous systems embedded inmodel bodies situated in model environments (Beer & Gallagher, 1992; Cliff,Harvey & Husbands, 1993; Nolfi & Floreano, 2000). Evolutionary algorithmsallow an exploration of possible brain-body-environment tradeoffs becausemodel agents are selected only for their overall behavioral efficacy, with aminimum of a priori theoretical bias. The mathematical tools of dynamicalsystems theory are then used to analyze the operation of the evolved systems(Beer, 1995a). This approach has been successfully applied to a wide range ofbehavior, including chemotaxis, walking, learning, categorical perception andselective attention (Beer, 1997).

As a specific illustration of this approach, this chapter will review a longline of work on the evolution and analysis of walking. The Evolution ofWalking in a Legged Model Agent describes the neural and body modelsemployed in this work and reviews the results of the evolutionary experimentsthat have been run to date. In Neuromechanical Analysis of Evolved Walkers,I describe how particular evolved walkers have been analyzed and some of theinsights that have been gained. The Structure of Fitness Space attempts tosituate these analyses of particular walkers within a broader picture of thespace of all possible solutions to this task. In The Impact of Network Archi-tecture, I explore the impact of neural architecture on the performanceand evolution of walking. The chapter ends with a discussion of the implica-tions of this work for the brain-body-environment perspective in motorcontrol and some directions for future work. The long term goal of thiswork is nothing less than a complete understanding of the relationshipbetween neural architecture, neural parameters, neural activity, peripheralbiomechanics, sensory feedback, behavior, performance and evolution in asimple model agent.

The Evolution of Walking in a Legged Model Agent

We examined the evolution of pattern generators for walking in a simplesix-legged body (Fig. 2A; Beer, 1990; Beer & Gallagher, 1992). Each leg wascomposed of a joint actuated by two opposing swing ‘‘muscles’’ and a binaryfoot (Fig. 2B). When the foot was ‘‘down’’, any torque produced by the musclesapplied a translational force to the body under Newtonian mechanics. Whenthe foot was ‘‘up’’, any torque produced by the muscles served to swing the legrelative to the body. Each leg was only able to generate force over a limitedrange of motion (modeling how mechanical advantage changes with limb geo-metry) with snaps back to these limits after foot release when a stancing legstretches outside this range (modeling the passive restoring forces of muscle)and hard kinematic limits (modeling skeletal constraints). The body could onlymove when it was statically stable (i.e., the center of mass was contained within

The Dynamics of Brain-Body-Environment Interaction in Motor Systems 9

the polygon of support formed by the stancing legs). Each leg also possessed anangle sensor whose output was proportional to the angular deviation of the legfrom perpendicularity to the long axis of the body. Complete details of the bodymodel can be found in (Beer, Chiel & Gallagher, 1999).

The model body was coupled to a continuous-time recurrent neural network(Beer, 1995b):

�i _yi ¼ �yi þXN

j¼1wji�ðyj þ �jÞ i ¼ 1; :::;N

where yi is the mean membrane potential of the ith neuron, _yi denotes the timerate of change of this potential, �i is the neuron’s membrane time constant, wij isthe strength of the synaptic connection from the jth to the ith neuron, �i is a biasterm, and �ðxÞ ¼ 1=ð1þ e�xÞ represents the neuron’s mean firing rate. A self-connection wii is interpreted as a simple nonlinear active conductance ratherthan as a literal synapse. While this simple model is computationallyand analytically tractable, it is known to be a universal approximator of smoothdynamics (Kimura&Nakano, 1998). Thus, the use of continuous-time recurrentneural networks (CTRNNs) implies no essential restriction whatsoever on therange of dynamical behavior that can be generated. Three of these neurons ineach leg circuit are always motor neurons that control the two opposing musclesof the leg (labeled BS for Backward Swing and FS for Forward Swing)

FT

BS FS

AS

� �� �� �� ��

FT

BS

FS

INT1

INT2

time

CStance Swing Stance

A BFig. 2 Neuromechanics ofwalking. (A) Schematicbody model. (B) Operationof an individual leg(FT=Foot, BS=BackwardSwing, FS = ForwardSwing, AS=Angle Sensor).(C) Activity of the top ten5-neuron CPG motorpatterns over one step, withthe optimal pattern shown ingray

10 R.D. Beer

and the foot (labeled FT), while any additional neurons are interneurons(labeled INTn) with no preassigned function. Individual leg circuits were fully-interconnected, whereas homologous neurons in each adjacent leg circuit werebidirectionally interconnected. Several symmetries were also imposed on theneural circuits, so that all leg controllers shared the same parameters andcorresponding connections were mirrored both across and along the body.

CTRNNs capable of generating walking in these legged agents were pro-duced using evolutionary algorithms. An evolutionary algorithm is a searchtechnique whose operation is loosely based on natural evolution (Goldberg,1989; Mitchell, 1996). A population of individuals is maintained. In the initialpopulation, the parameters describing each individual are set randomly. Eachindividual is then evaluated on its performance on some task of interest. Thefitness on this task is then used to select individuals to serve as parents for theproduction of a new population. The parameters are then mutated and/orcrossed over between parents to produce children. Once a new population hasbeen created, all individuals are once again evaluated on the task and the cyclerepeats.

Although our earliest work on the evolution of walking utilized a traditionalbinary genetic algorithm, we switched to a real-valued evolutionary algorithmin subsequent work (Back, 1996). In this case, each individual is encoded as avector of real numbers representing the time constants, biases and connectionweights. Elitist selection was used to preserve the best individual each genera-tion, whereas the remaining children were generated by mutation of selectedparents. Individuals were selected for mutation using a linear rank-basedmethod. A selected parent was mutated by adding to it a random displacementvector with uniformly distributed direction and normally distributedmagnitudewith 0mean. Connection weights and biases were constrained to lie in the range–16, while time constants were constrained to the range [0.5, 10].

The walking performance measure optimized by the evolutionary algorithmwas average forward velocity of the body. This average velocity was computedin two ways. During evolution, truncated fitness was evaluated by integratingthe model for a fixed length of time using the forward Euler integration methodand then computing the average velocity (total forward distance covereddivided by the time interval). During analysis, asymptotic fitness was evaluatedby integrating the model for a fixed length of time to skip transients and thencomputing its average velocity for one stepping period (with a fitness of 0assigned to nonoscillatory circuits). Although asymptotic fitness more accu-rately describes the long-term performance of a circuit, truncated fitness ismuch less expensive to compute during evolutionary searches.

In our first set of experiments (Beer & Gallagher, 1992), 30-neuron CTRNNcircuits for walking were evolved under three different conditions: (1) sensoryfeedback from the leg angle sensor was always available during evolution, (2)sensory feedback was never available, and (3) sensory feedback was intermit-tently available. In all cases, the best walkers utilized a tripod gait, in which thefront and back legs on each side of the body step in phase with the middle leg on

The Dynamics of Brain-Body-Environment Interaction in Motor Systems 11

the opposite side. The evolution of successful locomotion typically passedthrough four distinct stages. In the first stage, walkers produced limited for-wardmovement by pushing with all six legs simultaneously until they fell. In thesecond stage, walkers evolved the ability to swing their legs in a rhythmic butuncoordinated fashion, taking multiple steps but still falling quite often. In thethird stage, walkers utilizing statically stable gaits appeared, allowing them tomake steady forward progress but with suboptimal coordination. In the fourthand final stage, the efficiency of locomotion slowly improved as the walkingpattern was fine-tuned.

Although the normal behavior of the best walkers in all experiments wasindistinguishable, lesions of the leg angle sensor revealed that the organizationof the evolved pattern generators differed significantly depending on the con-ditions under which they were evolved. Walkers evolved with reliable sensoryfeedback utilized reflexive pattern generators (RPGs). RPGs exhibit a chainreflex organization, depending on sensory feedback to trigger the transitionfrom stance to swing. In the absence of sensory feedback, RPGs becometrapped in a permanent stance phase and are thus not robust to sensory failure.However, RPGs are capable of adjusting their motor pattern to environmentalcontingencies such as a perturbed leg.Walkers evolved in the absence of sensoryfeedback utilized central pattern generators (CPGs). Unlike RPGs, CPGsare capable of intrinsically generating the rhythmic motor pattern necessaryfor walking. However, pure CPGs cannot make use of sensory feedback to fine-tune their motor pattern. Finally, walkers evolved with intermittent sensoryfeedback utilized mixed pattern generators (MPGs). MPGs represent the bestof both worlds. Like RPGs, they can use sensory feedback when it is availableto improve their operation, but like CPGs they can function in its absence ifnecessary.

For simplicity, our subsequent work has focused on the generation of step-ping movements in a single leg controlled by from 3 to 5 neurons. This simpli-fication retains the problem of rhythmically coordinating multiple effectors toachieve efficient walking, but removes the issue of coordinating multiple legs soas to maintain postural stability. Over the past several years, we have evolvedmillions of single-leg pattern generators under a variety of different conditionsand systematically studied the characteristics of the resulting walkers in avariety of ways. We have found that single-leg walkers still exhibit distinctstages of evolution. The population always first plateaus at nonrhythmicsingle-steppers with a truncated fitness of �0.125. It then exhibits a series ofincreasingly fit rhythmic but suboptimal walkers. Finally, it asymptotes to fine-tuned walkers with a truncated fitness of up to about 0.6 (Fig. 3). Evolvedsingle-leg pattern generators also still exhibit reflexive, central and mixedorganizations depending on the conditions under which they were evolved.We have also found pattern generators in which the rhythmic walking patterncould be initiated or terminated by a transient stimulus. In addition, we havefound evidence of dynamical reorganization in response to sensory lesions inmixed pattern generators, as well as adaptation to leg growth.

12 R.D. Beer

Thus, although one might expect that this restriction to a single leg wouldleave the walking task too trivial to be interesting, we have in fact foundquite the opposite to be true. Not only does single-leg coordination engage awide variety of fundamental questions in motor control, but the very sim-plicity of the task actually makes it possible to begin to answer some of them.In the remainder of this chapter, I review various analyses of these evolvedwalkers.

Neuromechanical Analysis of Evolved Walkers

These evolved model walkers present us with a remarkable opportunity.Because we have complete access to and control over all neural parametersand activity, we can analyze the operation of individual pattern generators ingreat detail. Because we have similar access to the model body, we can alsostudy the interplay between central and peripheral properties in the generationof a walking pattern. Because we have such a large population of evolvedwalkers, we can ask questions about general principles and individual varia-bility. Because we have access to their complete evolutionary history, we canalso examine the process by which successful walkers evolve, and the impact ofvarious modeling assumptions on their evolution.

Let us begin with an examination of the motor patterns produced by the bestpattern generators (Beer, Chiel & Gallagher, 1999). For example, the neuralactivities over one step of the top ten 5-neuron CPGs are shown in Fig. 2C. Thewalking performance of these ten CPGs differ by only a few percent. Note thatthe overall patterns of motor outputs are rather similar across these ten CPGs,while the interneuron activities are quite different. Since the interneuron activ-ities are far less constrained by the body than the motor neurons, this is to beexpected. However, there are variations even in the motor outputs. Specifically,

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

Generation

BestFitness

Fig. 3 A typical plot of fitness of the best individual of the population vs. the generationnumber during an evolutionary search. Note the initial plateau around a fitness of 0.125 andthe final fitness around 0.6

The Dynamics of Brain-Body-Environment Interaction in Motor Systems 13

while both FT transitions and the swing-to-stance transitions of BS and FSare fairly tightly clustered, the stance-to-swing transitions in BS and FS showconsiderably larger variability.

What is the behavioral significance of this motor pattern variability? Froma purely neuronal point of view, one might argue that the larger variability inthe stance-to-swing transition is obviously the important one, accounting for theobserved differences in walking performance. However, since it is the body, notthe nervous system, that actually walks, inferences from neuronal activity tobehavior must take into account the properties of the body. Thus, we undertooka detailed analysis of themechanical properties of ourmodel body (Beer, Chiel &Gallagher, 1999). Because of the simplicity of the bodymodel, the optimalmotorpattern could be formulated as a pair of minimum-time optimal control pro-blems that could be solved analytically. We calculated that the maximumtruncated and asymptotic fitness is 0.627, which is just above the fitness of thebest pattern generators that we evolved. We also calculated that the maximumtruncated fitness for walkers taking a single step was 0.125, which matches thefitness plateau observed in our evolutionary searches before the discovery ofoscillatory pattern generators (Fig. 3).

In order to assess the significance of the motor pattern variability weobserved, we compared the motor patterns of the top ten CPGs to the optimalmotor pattern that we calculated (gray pattern in Fig. 2C). Interestingly, wefound that the optimal motor pattern is degenerate; it consists not of a singletrajectory, but rather an infinite family of trajectories. This degeneracy arises intwo ways. First, because the foot can only be up or down in this model, the onlyproperty of the output of FT that matters is whether it is above or belowthe threshold for raising and lowering the foot. Second, because a supportingleg can only generate force over a limited range due to changing mechanicaladvantage, the force output of the forward and backward swing musclesbecomes irrelevant near the end of stance. These degeneracies are shown asgray rectangles in Fig. 2C. Any motor trajectory that remains within the greyregions will have identical walking performance.

Note that the stance-to-swing variability in BS and FS falls almost entirelywithin degenerate regions, where it makes no difference to performance andtherefore has not been selected away during evolution. Thus, the inference thatthis larger neuronal variability is significant is in fact wrong. It is actually themuch smaller variability in the swing-to-stance transition in BS and FS thatis significant, because it falls within a very tightly constrained portion of theoptimal motor pattern. This clearly demonstrates that it is absolutely essentialto consider the properties of the body when making inferences from neuralactivity to behavior, even in the case of pure CPGs.

Now let us turn to the question of how these evolved pattern generatorsactually work (Beer, 1995). From a neuromechanical perspective, the operationof the CPGs is the most straightforward. CPGs exhibit stable limit cycles whichare tuned to the body in order to match the optimal motor pattern as closely aspossible. Despite the fact that individual CTRNN neurons cannot oscillate,

14 R.D. Beer