Nonlinear Dynamics in Human Behavior PDF
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RaoulHuysandViktorK.Jirsa(Eds.) NonlinearDynamicsinHumanBehavior StudiesinComputationalIntelligence,Volume328 Editor-in-Chief Prof.JanuszKacprzyk SystemsResearchInstitute PolishAcademyofSciences ul.Newelska6 01-447Warsaw Poland E-mail:[email protected] Furthervolumesofthisseriescanbefoundonour homepage:springer.com Vol.317.RogerLee(Ed.) ComputerandInformationScience2010 Vol.306.TruHoangCao ISBN978-3-642-15404-1 ConceptualGraphsandFuzzyLogic,2010 Vol.318.OscarCastillo,JanuszKacprzyk, ISBN978-3-642-14086-0 andWitoldPedrycz(Eds.) Vol.307.AnupamShukla,RituTiwari,andRahulKala SoftComputingforIntelligentControl TowardsHybridandAdaptiveComputing,2010 andMobileRobotics,2010 ISBN978-3-642-14343-4 ISBN978-3-642-15533-8 Vol.319.TakayukiIto,MinjieZhang,ValentinRobu, Vol.308.RogerNkambou,JacquelineBourdeau,and ShaheenFatima,TokuroMatsuo, RiichiroMizoguchi(Eds.) andHirofumiYamaki(Eds.) 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Nonlinear Dynamics in Human Behavior 123 RaoulHuys TheoreticalNeuroscienceGroup CNRS&UniversitédelaMéditerranée,UMR6233“MovementScienceInstitute" FacultédesSciencesduSport 163av.DeLuminy 13288,Marseillecedex09 France E-mail:[email protected] ViktorK.Jirsa TheoreticalNeuroscienceGroup CNRS&UniversitédelaMéditerranée,UMR6233“MovementScienceInstitute" FacultédesSciencesduSport 163av.DeLuminy 13288,Marseillecedex09 France and CenterforComplexSystems&BrainSciences FloridaAtlanticUniversity 777GladesRoad BocaRatonFL33431 USA ISBN 978-3-642-16261-9 e-ISBN 978-3-642-16262-6 DOI 10.1007/978-3-642-16262-6 Studiesin Computational Intelligence ISSN1860-949X Library of Congress Control Number:2010937350 (cid:2)c 2010 Springer-VerlagBerlin Heidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpart of the material is concerned, specifically therights of translation, reprinting,reuse ofillustrations, recitation,broadcasting, reproductiononmicrofilmorinanyother way, and storage in data banks. Duplication of this publication or parts thereof is permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution undertheGerman Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset&CoverDesign:ScientificPublishing ServicesPvt. Ltd., Chennai, India. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Preface In July 2007 the international summer school “Nonlinear Dynamics in Movement and Cognitive Sciences” was held in Marseille, France. The aim of the summer school was to offer students and researchers a “crash course” in the application of nonlinear dynamic system theory to cognitive and behavioural neurosciences. The participants typically had little or no knowledge of nonlinear dynamics and came from a wide range of disciplines including neurosciences, psychology, engi- neering, mathematics, social sciences and music. The objective was to develop sufficient working knowledge in nonlinear dynamic systems to be able to recog- nize characteristic key phenomena in experimental time series including phase transitions, multistability, critical fluctuations and slowing down, etc. A second emphasis was placed on the systematic development of functional architectures, which capture the phenomenological dynamics of cognitive and behavioural phe- nomena. Explicit examples were presented and elaborated in detail, as well as “hands on” explored in laboratory sessions in the afternoon. This compendium can be viewed as an extended offshoot from that summer school and breathes the same spirit: it introduces the basic concepts and tools adhering to deterministic dynamical systems as well as its stochastic counterpart, and contains correspond- ing applications in the context of motor behaviour as well as visual and auditory perception in a variety of typically (but not solely) human endeavours. The chap- ters of this volume are written by leading experts in their appropriate fields, re- flecting ta similar multi-disciplinary range as the one of the students. This book owes its existence to their contributions, for which we wish to express our gratitude. We are further indebted to the Technical Editor Dr. Thomas Ditzinger for his advice, guidance, and patience throughout the editorial process, and the Series Editor Janusz Kacprzyk for inviting and encouraging us to produce this volume. Contents Dynamical Systems in One and Two Dimensions: A Geometrical Approach .................................... 1 Armin Fuchs Benefits and Pitfalls in Analyzing Noise in Dynamical Systems – On Stochastic Differential Equations and System Identification ................................................ 35 Andreas Daffertshofer The Dynamical Organization of Limb Movements............ 69 Raoul Huys Perspectives on the Dynamic Nature of Coupling in Human Coordination ................................................ 91 Sarah Calvin, Viktor K. Jirsa Do We Need Internal Models for Movement Control? ....... 115 Fr´ed´eric Danion Nonlinear Dynamics in Speech Perception ................... 135 Betty Tuller, No¨el Nguyen, Leonardo Lancia, Gautam K. Vallabha A Neural Basis for Perceptual Dynamics .................... 151 Howard S. Hock, Gregor Sch¨oner Optical Illusions: Examples for Nonlinear Dynamics in Perception................................................... 179 Thomas Ditzinger A Dynamical Systems Approach to Musical Tonality ........ 193 Edward W. Large Author Index................................................ 213 Dynamical Systems in One and Two Dimensions: A Geometrical Approach ArminFuchs Abstract.Thischapterisintendedasanintroductionortutorialtononlineardynami- calsystemsinoneandtwodimensionswithanemphasisonkeepingthemathematics aselementaryaspossible.Byitsnaturesuchanapproachdoesnothavethemath- ematicalrigorthatcan be foundin mosttextbooksdealingwith this topic.On the otherhanditmayallowreaderswithalessextensivebackgroundinmathtodevelop anintuitiveunderstandingoftherichvarietyofphenomenathatcanbedescribedand modeledbynonlineardynamicalsystems.Eventhoughthischapterdoesnotdealex- plicitlywithapplications–exceptforthemodelingofhumanlimbmovementswith nonlinearoscillatorsinthelastsection–itneverthelessprovidesthebasicconcepts andmodelingstrategiesallapplicationsarebuildupon.Thechapterisdividedinto twomajorpartsthatdealwithone-andtwo-dimensionalsystems,respectively.Main emphasisisputonthedynamicalfeaturesthatcanbeobtainedfromgraphsinphase spaceandplotsofthepotentiallandscape,ratherthanequationsandtheirsolutions. Afterdiscussinglinearsystemsinbothsections,weapplytheknowledgegainedto theirnonlinearcounterpartsandintroducetheconceptsofstabilityandmultistabil- ity,bifurcationtypesandhysteresis,hetero-andhomoclinicorbitsaswellaslimit cycles,andelaborateontheroleofnonlineartermsinoscillators. 1 One-Dimensional DynamicalSystems Theone-dimensionaldynamicalsystemswearedealingwithherearesystemsthat canbewrittenintheform dx(t) =x˙(t)= f[x(t), λ ] (1) dt { } In(1)x(t)isafunction,which,asindicatedbyitsargument,dependsonthevariable t representing time. The left and middle part of (1) are two ways of expressing ArminFuchs CenterforComplexSystems&BrainSciences,DepartmentofPhysics, FloridaAtlanticUniversity e-mail:[email protected] R.HuysandV.K.Jirsa(Eds.):NonlinearDynamicsinHumanBehavior,SCI328,pp.1–33. springerlink.com (cid:2)c Springer-VerlagBerlinHeidelberg2010 2 A.Fuchs howthefunctionx(t)changeswhenitsvariablet isvaried,inmathematicalterms calledthederivativeofx(t)withrespecttot.Thenotationinthemiddlepart,with adotontopofthevariable,x˙(t), isusedin physicsasa shortformofaderivative with respect to time. The right-handside of (1), f[x(t), λ ], can be any function { } ofx(t)butwewillrestrictourselvestocaseswhere f isalow-orderpolynomialor trigonometricfunctionofx(t).Finally, λ representsasetofparametersthatallow { } forcontrollingthesystem’sdynamicalproperties.Sofarwehaveexplicitlyspelled outthefunctionwithitsargument,fromnowonweshalldropthelatterinorderto simplifythenotation.However,wealwayshavetokeepinmindthatx=x(t)isnot simplyavariablebutafunctionoftime. In commonterminology(1) is an ordinaryautonomousdifferentialequation of first order. It is a differential equation because it represents a relation between a function(herex)anditsderivatives(herex˙).Itiscalledordinarybecauseitcontains derivativesonlywithrespecttoonevariable(heret)incontrasttopartialdifferential equations that have derivatives to more than one variable – spatial coordinates in additiontotime,forinstance–whicharemuchmoredifficulttodealwithandnot ofourconcernhere.Equation(1)isautonomousbecauseonitsright-handsidethe variablet doesnotappearexplicitly.Systems thathave an explicitdependenceon time are called non-autonomous or driven. Finally, the equation is of first order becauseitonlycontainsafirstderivativewithrespecttot;weshalldiscusssecond ordersystemsinsect.2. Itshouldbepointedoutthat(1)isbynomeansthemostgeneralone-dimensional dynamical system one can think of. As already mentioned, it does not explicitly dependontime,whichcanalsobeinterpretedasdecoupledfromanyenvironment, henceautonomous.Equallyimportant,thechangex˙atagiventimet onlydepends onthestateofthesystematthesametimex(t),notatastateinitspastx(t τ)or − itsfuturex(t+τ).Whereasthelatterisquitepeculiarbecausesuchsystemswould violate causality, one of the most basic principles in physics, the former simply means that system has a memory of its past. We shall notdeal with such systems here;inallourcasesthechangeinasystemwillonlydependonitscurrentstate,a propertycalledmarkovian. Afunctionx(t)whichsatisfies(1)iscalledasolutionofthedifferentialequation. Asweshallseebelowthereisneverasinglesolutionbutalwaysinfinitelymanyand all of them together built up the general solution. For most nonlinear differential equationsitisnotpossibletowritedownthegeneralsolutioninaclosedanalytical form,whichisthebadnews.Thegoodnews,however,isthatthereareeasyways to figure out the dynamical properties and to obtain a good understanding of the possiblesolutionswithoutdoingsophisticatedmathorsolvinganyequations. 1.1 Linear Systems Theonlylinearone-dimensionalsystemthatisrelevantistheequationofcontinuous growth x˙=λx (2) DynamicalSystemsinOneandTwoDimensions 3 wherethechangeinthesystemx˙isproportionaltostatex.Forexample,themore membersof a givenspeciesexist, the moreoffspringstheyproduceand the faster thepopulationgrowsgivenanenvironmentwithunlimitedresources.Ifwewantto know the time dependenceof this growthexplicitly,we have to find the solutions of(2),whichcanbedonemathematicallybutinthiscaseitiseveneasiertomake an educated guess and then verify its correctness. To solve (2) we have to find a functionx(t)thatisessentiallythesameasitsderivativex˙timesaconstantλ.The familyoffunctionswiththispropertyaretheexponentialsandifwetry x(t)=eλt wefind x˙(t)=λeλt hence x˙=λx (3) andthereforex(t)isasolution.Infactifwemultiplytheexponentialbyaconstant citalsosatisfies(2) x(t)=ceλt wefind x˙(t)=cλeλt andstill x˙=λx (4) Butnowtheseareinfinitelymanyfunctions–wehavefoundthegeneralsolutionof (2)–andweleaveittothemathematicianstoprovethatthesearetheonlyfunctions thatfulfill(2)andthatwehavefoundallofthem,i.e.uniquenessandcompleteness of the solutions. It turns out that the general solution of a dynamical system of nth orderhasn openconstantsand as we are dealingwith one-dimensionsystems here we have one open constant: the c in the above solution. The constant c can be determined if we know the state of the system at a given time t, for instance x(t=0)=x 0 x(t=0)=x =ce0 c=x (5) 0 0 → wherex iscalledtheinitialcondition.Figure1showsplotsofthesolutionsof(2) 0 fordifferentinitialconditionsandparametervaluesλ<0,λ=0andλ>0. Wenowturntothequestionwhetheritispossibletogetanideaofthedynamical properties of (2) or (1) without calculating solutions, which, as mentioned above, isnotpossibleingeneralanyway.Westartwith(2)asweknowthesolutioninthis x(t) x(t) x(t) λ<0 λ>0 λ=0 t t t Fig.1Solutionsx(t)fortheequationofcontinuousgrowth(2)fordifferentinitialconditions x0(solid,dashed,dottedanddash-dotted)andparametervaluesλ<0,λ=0andλ>0on theleft,inthemiddleandontheright,respectively.
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