Nonlinear Dynamics in Human Behavior

216 Pages·2013·5.89 MB·English

by  Huys, Raoul, Jirsa, Viktor K

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.)
AdvancesinIntelligentTutoringSystems,2010
InnovationsinAgent-BasedComplex
ISBN978-3-642-14362-5
AutomatedNegotiations,2010
Vol.309.IsabelleBichindaritz,LakhmiC.Jain,SachinVaidya, ISBN978-3-642-15611-3
andAshleshaJain(Eds.)
Vol.320.xxx
ComputationalIntelligenceinHealthcare4,2010
ISBN978-3-642-14463-9 Vol.321.DimitriPlemenosandGeorgiosMiaoulis(Eds.)
IntelligentComputerGraphics2010
Vol.310.DiptiSrinivasanandLakhmiC.Jain(Eds.) ISBN978-3-642-15689-2
InnovationsinMulti-AgentSystemsand
Applications–1,2010 Vol.322.BrunoBaruqueandEmilioCorchado(Eds.)
ISBN978-3-642-14434-9 FusionMethodsforUnsupervisedLearningEnsembles,2010
ISBN978-3-642-16204-6
Vol.311.JuanD.Vela´squezandLakhmiC.Jain(Eds.)
AdvancedTechniquesinWebIntelligence,2010 Vol.323.YingxuWang,DuZhang,andWitoldKinsner(Eds.)
ISBN978-3-642-14460-8 AdvancesinCognitiveInformatics,2010
ISBN978-3-642-16082-0
Vol.312.PatriciaMelin,JanuszKacprzyk,and
Vol.324.AlessandroSoro,VargiuEloisa,GiulianoArmano,
WitoldPedrycz(Eds.)
andGavinoPaddeu(Eds.)
SoftComputingforRecognitionbasedonBiometrics,2010
InformationRetrievalandMininginDistributed
ISBN978-3-642-15110-1
Environments,2010
Vol.313.ImreJ.Rudas,Ja´nosFodor,and ISBN978-3-642-16088-2
JanuszKacprzyk(Eds.) Vol.325.QuanBaiandNaokiFukuta(Eds.)
ComputationalIntelligenceinEngineering,2010 AdvancesinPracticalMulti-AgentSystems,2010
ISBN978-3-642-15219-1 ISBN978-3-642-16097-4
Vol.314.LorenzoMagnani,WalterCarnielli,and Vol.326.SherylBrahnamandLakhmiC.Jain(Eds.)
ClaudioPizzi(Eds.) AdvancedComputationalIntelligenceParadigmsin
Model-BasedReasoninginScienceandTechnology,2010 Healthcare5,2010
ISBN978-3-642-15222-1 ISBN978-3-642-16094-3
Vol.315.MohammadEssaaidi,MicheleMalgeri,and Vol.327.SlawomirWiakand
CostinBadica(Eds.) EwaNapieralska-Juszczak(Eds.)
IntelligentDistributedComputingIV,2010 ComputationalMethodsfortheInnovativeDesignof
ISBN978-3-642-15210-8 ElectricalDevices,2010
ISBN978-3-642-16224-4
Vol.316.PhilippWolfrum
InformationRouting,CorrespondenceFinding,andObject Vol.328.RaoulHuysandViktorK.Jirsa(Eds.)
RecognitionintheBrain,2010 NonlinearDynamicsinHumanBehavior,2010
ISBN978-3-642-15253-5 ISBN978-3-642-16261-9
Raoul Huys andViktor K.Jirsa (Eds.)
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.