Estimating degrees of freedom in motor systems

arXiv:q-bio/0610058v1[q-bio.QM]30Oct2006EstimatingdegreesoffreedominmotorsystemsRobertH.Clewley,JohnM.Guckenheimer,andFranciscoJ.Valero-CuevasFebruary6,2008AbstractStudiesofthedegreesoffreedomor“synergies”inmusculoskeletalsystemsrelycriticallyonalgorithmstoestimatethe“dimension”ofkinematicorneuraldata.Linearalgorithmssuchasprincipalcomponentanalysis(PCA)areusedalmostexclusivelyforthispurpose.However,biologicalsystemstendtopossessnonlinearitiesandoperateatmultiplespatialandtemporalscalessothatthesetofreachablesystemstatestypicallydoesnotlieclosetoasinglelinearsubspace.WecomparetheperformanceofPCAtotwoalternativenonlinearalgorithms(Isomapandournovelpointwisedimensionestimation(PD-E))usingsyntheticandmotioncapturedatafromaroboticarmwithknownkinematicdimensions,aswellasmotioncapturedatafromhumanhands.WefindthatconsiderationofthespectralpropertiesofthesingularvaluedecompositioninPCAcanleadtomoreaccuratedimensionestimatesthanthedominantpracticeofusingafixedvariancecapturethreshold.WeinvestigatemethodsforidentifyingasingleintegerdimensionusingPCAandIsomap.Incontrast,PD-Eprovidesarangeofestimatesoffractaldimension.Thishelpstoidentifyheterogeneousgeometricstructureofdatasetssuchasunionsofmanifoldsofdifferingdimensions,towhichIsomapislesssensitive.Contrarytocommonopinionregardingfractaldimensionmethods,PD-Eyieldedreasonableresultswithreasonableamountsofdata.WeconcludethatitisnecessaryandfeasibletocomplementPCAwithothermethodsthattakeintoconsiderationthenonlinearpropertiesofbiologicalsystemsforamorerobustestimationoftheirdegreesoffreedom.1IntroductionTheabilitytousesensordatatoobjectivelyquantifythenumberofactiveorcontrolledskeletaldegreesoffreedom(DOFs)duringnaturalbehavioriscentraltothestudyofneuralcontrolofmusculoskeletalredundancy.AlongstandingprobleminthestudyofneuromuscularsystemsiswhetherandhowthenervoussystemusesthenumerousDOFsprovidedbytheneuro-musculo-skeletalsystem.Forexample,severalstudieshavesoughttodeterminewhetherthenervoussystemcouplesthemechanicalDOFsofthehandtosimplifythecontrolofhandshapingforgrasporsignlanguage[1,2].Otherimportantproblemsaretheestimationofdimensionoftheneuralcontroller1fromelectromyographicsignals[3]orextracellularneuralrecordingsfromthebrain[4].Theoriesofmotorlearningalsoaddressproblemsofdimensionestimationbyproposingthattheacquisitionofcomplextasksprogressesbyinitially“freezing”someskeletaldegreesoffreedomandgraduallyreleasingthemasthenervoussystemsisabletoincorporatethemintothemotortask[5].1.1AlgorithmicmethodstoestimatethedimensionofdataThispaperdiscussesalgorithmicmethodsthatmeasurethedimensioninstatespaceoccupiedbyobserveddynamicalbehaviours.Ourgoalistocompareandcontrasttheperformanceoftoday’smathematicallyrelated,butdistinct,definitionsofdimensionandvariedapproachestoestimatingthedimensionofadynamicalsystemfromsam-pleddata.Wefocusoncomparingtheperformanceoftwoestablishedalgorithms—principalcomponentanalysis(PCA)[6]andIsomap[7]—withanewalgorithmthatestimatespointwisedimension(PD-E).PCA,linearregressionandmulti-dimensionalscaling(MDS)[8]arelinearmeth-odsthattestwhetheradatasetliesclosetoalinearsubspace,inwhichcasethecoordinatesfromthissubspacecanbeusedtoparameterizethedata.However,thesemethodsdonotdeterminewhetherthedatamaylieonalowerdimensionalsetwithinthesubspace.Indeed,asinglearmrotatingrelativetothebodyinaplaneproducesamotioncapturedatasetthatliesalongacircle,Thecircleisaonedimensionalsetbecauseitcanbeparameterizedbyasinglecoordinate(e.g.,anangle),butthecircledoesnotlieclosetoaonedimensionallinearsubspace.Linearmethodssuggestthattwocoordinatesaremostappropriateinthisexample,wherethesecoordinatesdescribetheplaneinwhichthecircularmotiontakesplace.Thisisanexamplewherelinearmethodsdonotsufficeindeterminingthedimensionevenofasimplegeometricobjectunderlyingthemotionofasimplekinematicsystem.Isomap[7],locallinearembedding(LLE)[9],andLaplacianorHessianeigen-maps[10,11]aremethodsthathavebeendevelopedwithinthesettingofmachinelearninganddimensionreductiontofindcoordinatesystemsfornonlinearmanifolds.TheyincludeproceduresfordiscoveringthedimensionofdatasetsthatlieonsmoothRiemannianmanifolds.Isomapseeksasetofglobalcoordinatesforthismanifoldviasingularvaluedecompositionofamatrixofinterpointdistancesofthedata.Inourapplicationtobiomechanicstheunderlyingstructureofthebiomechani-calsystemgoverning,say,locomotionormanipulation,maynotberepresentableasmotioninasmoothmanifold.Thestructuremightinsteaddecomposeastheunionofsubmanifoldshavingdifferentdimensionfordifferentphasesinthegaitcycle(e.g.,swingversusdoublesupport)orgraspacquisitionversusmanipulation.Weneedtech-niquesthatcanidentifytheappropriatedecompositionandestimatethedimensionofthedifferentphasesofthetask.ThispapershowsthatPD-Eaidsintheprocessofexploringthiskindofcomplexgeometricstructureindatasets.Pointwisedimensionisaquantityassignedtoprobabilitydensitiesormeasuresthataredefinedonmetricspaces.LikeIsomap,algorithmsforcomputingpointwisedimensionarebaseduponanalysisofthedistancesbetweenpairsofdatapoints.However,thewayinwhichthisinformationisusedisqui