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Detecting effective connectivity in neural time series

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Title:
Detecting effective connectivity in neural time series
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BOYKIN,ERIN RAE
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Khargonekar, Pramod
Committee Co-Chair:
Ogle, William
Committee Members:
Principe, Jose C
Talathi, Sachin
Graduation Date:
4/30/2011

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Subjects / Keywords:
Causality ( jstor )
Connectivity ( jstor )
Datasets ( jstor )
Decision trees ( jstor )
Inference ( jstor )
Neurons ( jstor )
Oscillators ( jstor )
Signals ( jstor )
Statistical discrepancies ( jstor )
Time series ( jstor )
EFFECTIVE -- GRANGER -- NEURAL
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Electrical and Computer Engineering thesis, Ph.D.

Notes

Abstract:
Time series inference techniques for analyzing the directionality of interactions in dynamic systems have become popular tools in the neuroscience community for identifying the effective connectivity structure of brain networks. Commonly applied techniques include Granger causality (GC), partial directed coherence (PDC), and phase dynamics modeling (PDM). The goal of my work is to investigate the limitations of several of these time series inference techniques as they are used to analyze effective connectivity in neural time series. Specifically, I investigate the effects of differences in noise variance between time series and find that PDC and PDM are particularly vulnerable to such noise variance differences. I propose and demonstrate the use of a novel modified PDM technique, called generalized PDM (GPDM), which is similar to a previously described modified PDC technique, and is more robust to noise variance differences between time series. Next, I develop a method to assess the ability of various time series inference techniques to accurately determine the true effective connectivity of a given network of coupled, narrow-band neuronal oscillators. My method is based on decision tree classifiers, trained using several time series features that can be computed solely from experimentally recorded data. I show that the classifiers constructed in my study provide a framework for determining whether a particular time series inference technique is likely to produce incorrect results when applied to a given time series recorded from narrow-band oscillators. Finally, I consider the application of time series inference techniques to simulated data from neural mass models (NMMs). While popular time series inference techniques work well for a restrictive class of NMMs, I show that these techniques fail when applied to a more general NMM that captures entire regions of interest (ROI) in the brain. I propose a novel time series inference technique based on GPDM and empirical mode decomposition (EMD), which I call EMD-GPDM, to determine effective connectivity in such NMM-ROI systems and demonstrate its performance. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Khargonekar, Pramod.
Local:
Co-adviser: Ogle, William.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-04-30
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by ERIN RAE BOYKIN.

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Copyright BOYKIN,ERIN RAE. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
4/30/2012
Resource Identifier:
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DETECTINGEFFECTIVECONNECTIVITYINNEURALTIMESERIESByERINR.BOYKINADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ErinR.Boykin 2

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TomyparentsforgivingmetheirloveandDNA 3

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ACKNOWLEDGMENTS Mysincerestthankstomycommitteemembersfortheirtimeandguidance.Inparticular,IthankDr.Ogleforgivingmethefreedomtopursuemyresearchinterestsandforrekindlingmycuriosityinandloveofscience,Dr.Talathi(Sachin)forhisunaggingenthusiasmwhilehelpingmepursuemyacademicgoalsandforencouragingmetoaimhighinmyefforts,Dr.Khargonekarforhisinsightfulfeedbackandforsharinghisincrediblebreadthofknowledge,andDr.Principeforwideningmyresearchperspectivesandforhavingcondenceinmyabilities.Iamhonoredandhumbledtohavetheopportunitytoworkwithagroupofsuchtalentedandesteemedprofessors.IamalsodeeplyindebtedtomyearlyacademicmentorsincludingDr.RobertHolt,Dr.DougJordan,Dr.JieHan,andDr.JoseFortes.Fortheirusefuldiscussionsandconstructivecriticisms,IthankDr.MingzhouDing,Dr.AlexCadotte,andDr.PaulCarney.IhavebeenfortunateenoughtohavethecompanyofanumberofcaringanddelightfullabmatesduringmyacademiccareerincludingXinFu,NanRattanatamrong,SelviKadirvel,GirishVenkatasubramanian,PhilBarish,MattVernon,YingXu,andShanLi.ManythankstomyfellowgraduatestudentcollaboratorsincludingSohanSeth,KateBehrmanandSashaMikheyev.Fortheirunparalleledfriendshipandnever-endingwordsofencouragement,IthankmyFISI's:Gabs,Jenn,Jess,Kelly,Kristine,Monica,andNicole.Iowethedeepestdebtofgratitudetomyparentswhohavesupportedmeemotionally,nancially,andeveryotherwayimaginable.Finally,wordscanhardlydojusticetotheloveandappreciationthatIhaveformycaringandbrillianthusband,Patrick.Hehastrulylledtheunforgivingminutewithsixtyseconds'worthofdistancerun. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1ConnectivityConcepts ............................. 13 1.2TimeSeriesInferenceTechniques ...................... 15 1.3Assumptions .................................. 16 1.4KeyContributions ................................ 17 2GENERALMETHODS ................................ 20 2.1AutoregressiveModeling ............................ 21 2.1.1VectorAutoregressiveModeling .................... 23 2.1.2SpectralPropertiesofVARModels .................. 24 2.2GrangerCausality ............................... 25 2.2.1PairwiseGrangerCausality ...................... 25 2.2.1.1TimedomainpairwiseGrangercausality ......... 25 2.2.1.2SpectraldomainpairwiseGrangercausality ........ 27 2.2.1.3Exampleapplications .................... 30 2.2.2ConditionalGrangerCausality ..................... 32 2.2.2.1TimedomainconditionalGrangercausality ........ 32 2.2.2.2SpectraldomainconditionalGrangercausality ...... 34 2.2.2.3PartitionmatrixGrangercausality ............. 36 2.2.2.4Exampleapplications .................... 38 2.3PartialDirectedCoherence .......................... 41 2.3.1GeneralizedPartialDirectedCoherence ............... 43 2.3.2ExampleApplications ......................... 44 2.4PhaseDynamicsModeling .......................... 44 2.4.1PairwisePhaseDynamicModeling .................. 46 2.4.2MultivariatePhaseDynamicModeling ................ 47 2.4.3ExampleApplications ......................... 48 2.5RecentAdvancesandApplications ...................... 49 2.6SignicanceTesting .............................. 50 5

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3CHARACTERIZATIONANDMITIGATIONOFTHEEFFECTSOFNOISEVARIANCEONTIMESERIESINFERENCETECHNIQUES ........... 63 3.1NoiseVarianceandLinearTimeSeriesInferenceTechniques ....... 63 3.1.1TheImpactofNoiseVarianceonLinearTimeSeriesInferenceTechniques ............................... 63 3.1.2MitigatingtheImpactofNoiseVarianceonLinearTimeSeriesInferenceTechniques .......................... 65 3.2NoiseVarianceandPhaseDynamicsModeling ............... 66 3.2.1TheImpactofNoiseVarianceonPhaseDynamicsModeling .... 66 3.2.2ANewTechniqueforMitigatingtheImpactofNoiseVarianceonPhaseDynamicsModeling ....................... 67 4DETECTINGEFFECTIVECONNECTIVITYINNETWORKSOFCOUPLEDNEURONALOSCILLATORS ............................ 74 4.1Neuron,SynapseandNetworkModels .................... 76 4.1.1NeuronModel .............................. 76 4.1.2SynapseModels ............................ 77 4.1.3NetworkModel ............................. 77 4.2ClassierConstructionMethodology ..................... 78 4.2.1DenitionofTimeSeriesFeatures .................. 78 4.2.2DecisionTreeConstruction ...................... 79 4.2.3ResultingDecisionTrees ........................ 82 4.2.4ExtensiontoLargerNetworks ..................... 86 4.3RecommendationsforDecisionTreeClassierUseinNetworksofCoupledNeuronalOscillators .............................. 87 5DETERMININGEFFECTIVECONNECTIVITYINNERUALMASSMODELNETWORKS ..................................... 96 5.1NeuralMassModels .............................. 100 5.2DeterminingEffectiveConnectivityinNetworksofNeuralMassModelPopulations ................................... 104 5.3DeterminingEffectiveConnectivityinNetworksofNeuralMassModelRegionsofInterest ............................... 106 5.3.1DeterminingEffectiveConnectivityViaGrangerCausality,PartialDirectedCoherenceandPhaseDynamicsModeling ........ 106 5.3.2EMD-GPDM:ANovelTechniqueforDeterminingEffectiveConnectivityinNetworksofNeuralMassModelRegionsofInterest ................................. 107 5.4EmpiricalModeDecomposition ........................ 109 6CONCLUSIONS ................................... 121 6

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REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 132 7

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LISTOFTABLES Table page 4-1Testerrorofclassiers ................................ 90 4-2Decisiontreesensitivityandspecicityvalues ................... 90 4-3Likelihoodthattimeseriesinferencetechniquescorrectlyidentifycausality ... 90 5-1Neuralmassmodelbasalparametervaluesconsistentacrossallpopulationtypes ......................................... 113 5-2Neuralmassmodelbasalparametervaluesthatdifferamongpopulationtypes 113 5-3SensitivityandspecicityvaluesofGC,GPDC,andPDMwhenappliedtonetworksofthreecoupledNMMpopulations ................... 113 5-4SensitivityandspecicityvaluesofGC,GPDC,andPDMwhenappliedtonetworksoftwocoupledNMMROIs ........................ 113 5-5SensitivityandspecicityvaluesofEMD-GPDMandEMD-PDMwhenappliedtonetworksoftwocoupledNMMROIs ....................... 114 8

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LISTOFFIGURES Figure page 2-1Twoexampletimeseriesgeneratedfromstochastic,autoregressivemodels .. 52 2-2Spectraldomain,pairwiseGrangercausalityresultwhenappliedtostochastic,autoregressivemodels ................................ 53 2-3Twoexampletimeseriesgeneratedfromstochastic,vanderPoloscillators .. 54 2-4Spectraldomain,pairwiseGrangercausalityresultwhenappliedtostochastic,vanderPoloscillators ................................ 55 2-5Spectraldomain,conditionalGrangercausalityresultwithoutpartitionmatrixmethod ........................................ 56 2-6Spectraldomain,conditionalGrangercausalityresultwithpartitionmatrixmethodappliedtoARmodel ............................ 57 2-7Spectraldomain,conditionalGrangercausalityresultwithpartitionmatrixmethodappliedtovanderPoloscillators ..................... 58 2-8Partialdirectedcoherenceappliedtothreewhitenoiseprocesses ....... 59 2-9Generalizedpartialdirectedcoherenceappliedtothreewhitenoiseprocesses 60 2-10PartialdirectedcoherenceappliedtothreetimeseriesgeneratedfromstochasticvanderPoloscillators .......................... 61 2-11Generalizedpartialdirectedcoherenceappliedtothreetimeseriesgeneratedfromstochastic,vanderPoloscillators ....................... 62 3-1Probabilitythatpartialdirectedcoherenceidentiescausalityinasystem ... 69 3-2Probabilitythatgeneralizedpartialdirectedcoherenceidentiescausalityinasystem ........................................ 70 3-3ProbabilitythatGrangercausalityidentiescausalityinasystem ........ 71 3-4ResultsofthephasedynamicsmodelingtechniqueappliedtotwocoupledvanderPoloscillators ................................ 72 3-5ResultsofthegeneralizedphasedynamicsmodelingtechniqueappliedtotwocoupledvanderPoloscillators ......................... 73 4-1ThenetworkstructureofcoupledMorris-Lecarneuronsconsideredinmystudy 91 4-2Flowchartillustratingtheexperimentaldesignofmystudy ............ 92 4-3DecisiontreesconstructedforGrangercausalityacrossthreedifferentcouplingtypesconsidered ................................... 93 9

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4-4Compositedecisiontreesforalltimeseriesinferencetechniques ........ 94 4-5Thetrajectoriesoftwoinstantaneousphasetimeseriesfromtwodatasets ... 95 5-1Asystemdiagramofasinglepopulationneuralmassmodel ........... 114 5-2Powerspectraldensitiesofneuralmassmodelpopulations ........... 115 5-3Twoneuralmassmodelpopulationnetworks ................... 116 5-4Twocoupledneuralmassmodelregionsofinterest ................ 117 5-5Powerspectraldensityofasingleneuralmassmodelregionofinterest .... 117 5-6Anexampleneuralmassmodelpopulationnetwork ............... 118 5-7Anexampleneuralmassmodelregionofinterestsignalandtwoconstituentintrinsicmodefunctions ............................... 119 5-8Theempiricalmodedecomposition-generalizedpartialdirectedcoherencetechnique ....................................... 120 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDETECTINGEFFECTIVECONNECTIVITYINNEURALTIMESERIESByErinR.BoykinMay2011Chair:PramodP.KhargonekarMajor:ElectricalandComputerEngineeringTimeseriesinferencetechniquesforanalyzingthedirectionalityofinteractionsindynamicsystemshavebecomepopulartoolsintheneurosciencecommunityforidentifyingtheeffectiveconnectivitystructureofbrainnetworks.CommonlyappliedtechniquesincludeGrangercausality(GC),partialdirectedcoherence(PDC),andphasedynamicsmodeling(PDM).Thegoalofmyworkistoinvestigatethelimitationsofseveralofthesetimeseriesinferencetechniquesastheyareusedtoanalyzeeffectiveconnectivityinneuraltimeseries.Specically,IinvestigatetheeffectsofdifferencesinnoisevariancebetweentimeseriesandndthatPDCandPDMareparticularlyvulnerabletosuchnoisevariancedifferences.IproposeanddemonstratetheuseofanovelmodiedPDMtechnique,calledgeneralizedPDM(GPDM),whichissimilartoapreviouslydescribedmodiedPDCtechnique,andismorerobusttonoisevariancedifferencesbetweentimeseries.Next,Idevelopamethodtoassesstheabilityofvarioustimeseriesinferencetechniquestoaccuratelydeterminethetrueeffectiveconnectivityofagivennetworkofcoupled,narrow-bandneuronaloscillators.Mymethodisbasedondecisiontreeclassiers,trainedusingseveraltimeseriesfeaturesthatcanbecomputedsolelyfromexperimentallyrecordeddata.Ishowthattheclassiersconstructedinmystudyprovideaframeworkfordeterminingwhetheraparticulartimeseriesinferencetechniqueis 11

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likelytoproduceincorrectresultswhenappliedtoagiventimeseriesrecordedfromnarrow-bandoscillators.Finally,Iconsidertheapplicationoftimeseriesinferencetechniquestosimulateddatafromneuralmassmodels(NMMs).WhilepopulartimeseriesinferencetechniquesworkwellforarestrictiveclassofNMMs,IshowthatthesetechniquesfailwhenappliedtoamoregeneralNMMthatcapturesentireregionsofinterest(ROI)inthebrain.IproposeanoveltimeseriesinferencetechniquebasedonGPDMandempiricalmodedecomposition(EMD),whichIcallEMD-GPDM,todetermineeffectiveconnectivityinsuchNMM-ROIsystemsanddemonstrateitsperformance. 12

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CHAPTER1INTRODUCTIONThroughoutitshistory,thestudyofneurosciencehasfocusedextensivelyontheissueofneuralconnectivity.ThisthemecanbeobservedintheearlyworksofGustavFritschandEduardHitzigwhodemonstratedinthelate1800'sthatelectricalstimulationofdifferentareasofthecerebrumcausedinvoluntarycontractionsofspecicmusclesindogs( FritschandHitzig 1870 ).ItwasalsoastrongmotivatingfactorintheworkofWilderPeneld,aneurosurgeonwhostimulatedthebrainsofhisepilepticpatientswithelectricalprobesinordertoobservethepatients'responses.Peneld'sstudiesenabledhimtocreatemapsillustratingtheconnectionsbetweensensoryandmotorcorticesofthebrainandthevariouslimbsandorgansofthebody,whicharestillusedtoday( PeneldandJasper 1954 ).Morerecently,theHumanConnectomeProject,sponsoredbytheNationalInstitutesofHealth,hasspurredeffortstoconstructacomprehensivemapofneuralconnectionsinthehumanbrain,termedthehumanconnectome( Spornsetal. 2005 ).Traditionalmethodsofbrainresearch,suchasaxonaltracing,histologicaldissection,anddegenerationarenotidealforuseintheseefforts,however,duetotheirinvasivenature( Morecraftetal. 2009 ).Instead,researchersarefocusedprimarilyonusingnon-invasiveimagingandbrainrecordingtechnologiessuchasfunctionalmagneticresonanceimaging(fMRI),electroencephalograms(EEG),magnetoencephalograms(MEG),andpositronemissiontomography(PET)( Kandeletal. 2000 ).Withthesetechnologiescontributingtotherapidgrowthinrecordeddatafromthebrain,neuroscientistshaverecognizedtheneedforstatisticaltoolsthatcanuncoverbrainconnectivityfromtimeseriesofneuralactivity. 1.1ConnectivityConceptsInthepastdecade,manysuchstatisticaltools,referredtohereastimeseriesinferencetechniques,havebeendevelopedoradaptedforthispurpose.Timeseries 13

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inferencetechniquescanbeappliedtotimeseriesrecordingsofneuralactivityatmultiplescales,frompatchclamprecordingswhichrecordtheringofasingleneuron,toEEGrecordingswhichcapturetheelectriceldsgeneratedbypopulationsofneurons,tofMRIswhichmeasuresthehemodynamicresponseofthebraintolarge-scaleneuralactivity( Kandeletal. 2000 ).Ingeneral,timeseriesinferencetechniquesdonotrevealtheanatomicalconnectivityofneurons,cellpopulations,orbrainregions,termedstructuralconnectivity( Sporns 2010 ; Spornsetal. 2005 ).Rather,theyreectwhatisknownasthefunctionaloreffectiveconnectivityofthebrain.Functionalconnectivityisdenedasastatisticaldependencebetweenneuronalunits.Timeseriesinferencetechniquesthatmeasurecorrelation/covariance,spectralcoherenceorphaselockingbetweenpairsoftimeseriesarecommontoolsfordeterminingfunctionalconnectivity( Malsburg 1994 ).Thesemeasuresarehighlytimedependentandaremodulatedbyfactorssuchassensorystimulationandanorganism'sinternalstate.Also,becausetheyonlyconsiderstatisticalindependenceorthelacktherefore,theyarepurelyundirectedmeasuresanddonotmakeanyexplicitreferencetothephysical,anatomicalstructureofthebrainortothecausalrelationshipsamongneuronalunits( Friston 1994 ).Instead,suchdirected,causalrelationshipsarecapturedbyeffectiveconnectivity.Effectiveconnectivity,denedasthecausalinuenceoneneuronalunitsexertsoveranother,canbedeterminedthroughtimeseriesinferencetechniques( BaccalaandSameshima 2001 ; Granger 1969 ; Malsburg 1994 ; Schreiber 2000 ),statisticalmodeling( Fristonetal. 2003 ; Kline 2010 )orphysicalstimulationandablationexperiments( Massiminietal. 2005 ; McCulloch 1945 ).Likefunctionalconnectivity,effectiveconnectivityisdynamic,uctuatingwithchangesinexternalstimuliorinternalstate( Friston 1994 ; Sporns 2010 ).Acompleteandthoroughunderstandingofthebrainrequiresbothstructuralandfunctional/effectiveconnectivityinformation.Evenifthehumanconnectome,thecompletewiringdiagramofthebrain,isconstructed,therewillstillbeaneedfor 14

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temporalinformationregardingpathwaysthatarecriticalinlearninganddiseasesstatesaswellastheplasticityofthesepathwaysovertime.Sincethesequestionscannotbeaddressedwithstructuralinformationalone,thereisanevidentneedforfunctional/effectiveconnectivitydatatohelpcompleteourunderstandingofhowthebrainfunctionstemporally( Sporns 2010 ). 1.2TimeSeriesInferenceTechniquesWiththisneedinmind,timeseriesinferencetechniquesformeasuringfunctionalconnectivityhavebeenexploredextensivelyandappliedtomanyexperimentallyrecordedneuraldatasets( Allumetal. 1982 ; Buckneretal. 2009 ; Malsburg 1994 ; Rogersetal. 2007 ; Uddinetal. 2009 ; Valdes-Sosaetal. 2005 ).Timeseriesinferencetechniquesformeasuringeffectiveconnectivity,ontheotherhand,whilegrowinginpopularity,haveonlyrecentlygainedacceptanceasatechniqueforanalyzingneuraltimeseries.Infact,manyofthesemeasureshaveonlybeendevelopedorproposedforuseonneuraldatawithinthepasttwodecades( BaccalaandSameshima 2001 ; Baccalaetal. 1998 ; KaminskiandBlinowska 1991 ).Ofthethreebroadapproachestodeterminingeffectiveconnectivity(timeseriesinference,statisticalmodelingandphysicalstimulation/ablation)onlytimeseriesinferencetechniquesareconsidereddata-drivenanddonotrequireeitherapredenedanatomicalmodelofthesystemunderstudy,asstatisticalmodelingtechniquesrequire,orphysicalinterventioninthenervoussystem,asstimulation/ablationexperimentsrequire.Suchalowbarriertoentryhasmadethesedata-driventimeseriesinferencetechniquesverypopularfordeterminingeffectiveconnectivity.However,beingfairlynewtechniquesontheneurosciencescene,theirlimitationshavenotyetbeenfullyexplored.Forthisreason,mystudyfocusesonanalyzingseveraltimeseriesinferencetechniquesformeasuringeffectiveconnectivity.ThisclassoftechniquesencompassesawidevarietyofmethodsincludinglinearapproachessuchasGrangercausality( Geweke 1982 1984 ; Granger 1969 ),partialdirectedcoherence(PDC)( Baccalaand 15

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Sameshima 2001 ),andthedirectedtransferfunction(DTF)( KaminskiandBlinowska 1991 ),nonlinearapproachesdesignedfornarrow-bandoscillatorydatasuchasphasedynamicsmodeling(PDM)( RosenblumandPikovsky 2001 ),aswellasnonlinearapproachessuchaskernel-Grangercausality( Marinazzoetal. 2008 )andapproachesbasedoninformationtheorysuchastransferentropy( Schreiber 2000 )andconditionalindependence( SethandPrincipe 2010 ).Whileanargumenthasbeenmadethatthenonlinearnatureofneuraldatarequiresnonlineartechniquesformeasuringeffectiveconnectivity( Marinazzoetal. 2008 ; Schreiber 2000 ),lineartechniqueshavebeenwidelyappliedtoexperimentaldata( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ; Havliceketal. 2010 ; Liaoetal. 2010a ; Satoetal. 2009 )and,furthermore,havebeenshowncomputationallytoproducecorrectresultsevenonnonlineardata( Winterhalderetal. 2005 2007 ).Becauseofthis,andinlightoftheirpopularity,Iexaminetheapplicabilityofseverallinearandonesemi-nonlineartimeseriesinferencetechniqueformeasuringeffectiveconnectivity. 1.3AssumptionsAllofthetechniquesIconsiderherearebasedontheconceptoftemporalprecedenceandassuch,relyonseveralkeyassumptions.Therstconcernsthefactthatthesetechniquesoperateindiscretetime,whichmeansthatthesamplingofnaturallycontinuoussystemdynamicsmustoccuratsucharateastocapturethetimescalesoverwhichcausalinteractionsaremanifest( Florinetal. 2010 ; Granger 1969 ).Second,experimentallyrecordedtimeseriesshouldaccuratelypreservethetemporalrelationshipsoftheunderlyingsystem.Forthisreason,nonuniformdelaysintheresponseorrecordingofsystemvariablescanbedetrimentaltocorrectlydistinguishingcauseandeffect( Friston 2009a ; Sporns 2010 ).Finally,thesetechniquesassumethatthesetofvariablesmeasuredforanalysisisbothcompleteandnon-redundant,i.e.,allpotentiallycausalvariablesaremeasuredandnoneofthesevariablescontainduplicate,redundantinformation( Granger 1969 ; Seth 2005 ).Thissensitivitytounmeasured 16

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variables,alsocalledthehiddenunitproblem,isalimitationofvirtuallyalltimeseriesanalysistoolsandhasbeendiscussedatlengthelsewhere( Guoetal. 2008 ).Itisassumedinthefollowingthattheseassumptionsholdforthedataunderconsideration.Evenwiththepreviousassumptionsinplace,thetechniquesconsideredhereformeasuringeffectiveconnectivityhavebeenshowntoproduceincorrectresultswhenthetimeseriesbeinganalyzedarehighlysynchronized( Bezruchkoetal. 2003 ; Smirnovetal. 2007 )orhavedifferentnoisecharacteristics( Alboetal. 2004 ; Nalatoreetal. 2007 ; Winterhalderetal. 2005 ).Nevertheless,itisstandardpracticetoapplythesetechniquestotimeserieswithnoregardforwhetherthedatafallintooneoftheseproblematiccategories.Onereasonforthisisthatwhiletheselimitationshavebeenobservedintheory,littleworkhasbeendonetodenetheseconstraintsastheyapplytoexperimentaltimeseries.TheclosestexampleofsuchworkisthatdonebySmirnovetal.( Smirnovetal. 2007 )whocomparetheapplicationoftwotimeseriesinferencetechniques,PDCandPDM,toseveraldifferentneuronaloscillatormodels.TheirresultisalooseupperboundonthelevelofsynchronizationbetweentwocoupledneuronaloscillatorsabovewhichPDCandPDMgenerallygivesincorrectresults.Smirnovetal.'sstudywaslimitedinscope,however,sinceitonlydelineatedboundsonsynchronizationinthecontextofnarrow-bandneuraloscillators( Smirnovetal. 2007 ).Systematicstudiesdelineatingboundsnotonlyonsynchronizationbutothertimeseriesfeaturesinthecontextofvarioustypesofneuraltimeseriesisstilllacking. 1.4KeyContributionsItisthereforethegoalofmystudyto1)investigatethelimitationsofseveraltimeseriesinferencetechniquesastheyareusedtodetermineeffectiveconnectivityinavarietyofexperimentalneuraltimeseriesand2)proposenoveltechniquesormodicationstoexistingtechniquesthatovercometheobservedlimitations.Thespeciccontributionsofmyworkareasfollows: 17

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Contribution1:Iinvestigatetheeffectsofdifferencesinnoisevariancebetweentimeseriesonseveraltimeseriesinferencetechniques.Inlinewithpreviousndings,IdemonstratethatPDCishighlyaffectedbynoisevariancedifferences( Winterhalderetal. 2005 ).IalsoobservesimilarbehaviorfromthePDMtechnique,andingwhichhasnotbeenpreviouslydescribedintheliterature.AsamorerobustalternativetoPDM,IproposeamodiedPDMtechniquewhichItermgeneralizedPDM(GPDM)duetoitssimilaritytoamodiedPDCtechniqueknownasgeneralizedPDC(GPDC).Finally,usingasimulatedexample,IdemonstratethatGPDMisindeedlessaffectedbynoisevariancedifferencesbetweentimeseriesthanitspredecessor,PDM.Contribution2:Idevelopamethodtoassesstheabilityofvarioustimeseriesinferencetechniquestoaccuratelydeterminethetrueeffectiveconnectivityofagivennetworkofcoupled,narrow-bandneuronaloscillators.Theoutputsofsuchoscillatorsarerepresentativeofexperimentalrecordingsonthescaleofsingleneurons.Mymethodisbasedondecisiontreeclassierswhicharetrainedusingseveraltimeseriesfeaturesthatcanbecomputedsolelyfromexperimentallyrecordeddata.Ishowthattheclassiersconstructedinmystudyprovideaframeworkfordeterminingwhetheraparticulartimeseriesinferencetechniqueislikelytoproduceincorrectresultswhenappliedtoagiventimeseries.Asaresult,mymethodcanbeappliedbyuserswithoutanypriorknowledgeofnetworkstructureordynamicssinceallthetimeseriesfeaturesusedtodetermineapplicabilitycanbemeasureddirectlyfromthetimeseriesthemselves.Contribution3:Iinvestigatetheaccuracyofseveraltimeseriesinferencetechniquesastheyapplytoeffectiveconnectivitydeterminationinnetworksofneuralmassmodels(NMMs).TheseNMMsarerepresentativeofexperimentalrecordings,suchasEEGs,thataremadeatthescaleoflargeneuronalpopulations.Specically,IconsidertheaccuracyofGC,GPDCandPDMwhenappliedtoanNMMnetworkintroducedbyZavagliaetal.( Zavaglia 2008 ).Ibeginwithasimplecasewhereeach 18

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neuronalpopulationintheNMMnetworkexhibitsrhythmicactivityatasingledominantfrequency.UsingthisNMMnetwork,IshowthatGPDCisabletoaccuratelyrecoverthetrueconnectivityofthemodelwithhighprobability.Next,Iconsiderthecaseofinterconnectedneuronalregionsofinterest(ROIs),eachofwhichcontainsseveralNMMswithdifferentdominantrhythmicfrequencies( Ursinoetal. 2007 ; Zavagliaetal. 2006 ).ThismodelismorecomplexsinceeachROIexhibitsactivityoverarangeoffrequencyvalues.Inthiscase,Ishowthatnoneoftheaforementionedtimeseriesinferencetechniquesaccuratelyrecoversthetrueconnectivityofthemodel.InordertoaddresstheobservedinadequaciesofthetimeseriesinferencetechniquesthatIconsidered,IproposeanewtechniquebasedonGPDMandtheempiricalmodedecomposition(EMD)thatiscapableofaccuratelyrecoveringeffectiveconnectivityinanROI-basedNMM.Inthefollowing,IdescribetheapplicationofmyproposedtechniqueanddemonstrateitseffectivenessonROI-basedNMMs. 19

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CHAPTER2GENERALMETHODSTheprimarygoalofmyworkistoinvestigatethelimitationsofseveraltimeseriesinferencetechniquesastheyareusedtoanalyzeeffectiveconnectivityinavarietyofexperimentalneuraldatasets.Thetimeseriesinferencetechniquesconsideredhereincludethreelinearmethods:Grangercausality(GC),partialdirectedcoherence(PDC),andgeneralizedpartialdirectedcoherence(GPDC)aswellasonenonlinearmethod:phasedynamicsmodeling(PDM).Inthischapter,Idescribehowthesevarioustechniquesareappliedtotimeseries.Thisdescriptionincludesthemathematicalformulationsofthetechniques,examplesoftheiruse,andthedeterminationofsignicancelevels.Asmotivationfordescribingalltechniques,IbeginbyassumingthatIhavesimultaneouslyrecordedtimeseriesmeasurementsfromnnodes,whereeachnoderepresentstheobjectsofstudyandcanbeeitherasingleneuron,apopulationofneurons,oranentirebrainregiondependingonthescaleatwhichrecordinghastakenplace.ThesentimeseriesmeasurementscanberepresentedinvectorformasX(t)=[x1(t),x2(t),,xn(t)]T.Thegoalofapplyingatimeseriesinferencetechniquetothisdataistoestimatetheeffectiveconnectivitybetweenthesenodes,i.e.,determinewhethernodeiwithrecordedtimeseriesxi(t)causallyinuencesnodejwithrecordedtimeseriesxj(t).Thelineartimeseriesinferencetechniquesconsideredinthischapterareactuallyformulatedforthegeneralclassofstochasticprocesses.Tomaintaingenerality,IusethetermstochasticprocesswhendiscussingtheformulationofthesetechniquesandIusethetermtimeserieswhendiscussingtheapplicationofthesetechniquestoexperimentaltimeseriesdata.ThelineartechniquesthatIconsiderherearebasedontheconceptofcausalityoriginallyproposedbyNorbertWiener.Wienerreasonedthatifavariable,xi(t),is 20

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causallyinuencinganothervariablexj(t),thenthepredictionofxi(t)basedonpastvaluesofxj(t)andxi(t)willbemoreaccuratethanthatbasedonpastvaluesofxj(t)alone( Wiener 1956 ).ItwasnotuntilsomeyearslaterthatCliveGranger,workingintheeldofeconomics,rstformalizedthisconceptofcausalityinthecontextoflinearstochasticprocesses( Granger 1980 ).Asaresult,thetermWiener-GrangercausalityorsimplyGrangercausalityisnowusedtorefertothisconceptualizationofcausality.IusethetermcausalityheretorefertothisconceptofGrangercausality,whichisdistinguishedfromotherphilosophicaldenitionsofcausalitythathavebeenproposedintheliterature( Zellner 1979 ).Informalizinghistechnique,Grangerproposedusingautoregressive(AR)modelstoquantifytheimprovementgainedbyaddingxi(t)totheregressionofxj(t)( Granger 1980 ).FollowingGranger'soriginalformulation,threeofthefourtimeseriesinferencetechniquesconsideredhere,GC,PDC,andGPDC,requireX(t)torstbettoanARmodel.Thus,beforeIoutlinethemathematicalformulationsofthesetechniques,IbeginwithanoverviewofARmodelconstruction. 2.1AutoregressiveModelingAnautoregressive(AR)modelisalinearregressionformulathatisusedtopredictfuturevaluesofastochasticprocessbasedonpastvaluesoftheprocess.Consideringasinglestationary,zeromeanstochasticprocess,x(t),witht=1,,N,thebasicformulaofaunivariateARmodelofx(t)is( Boxetal. 2008 ; Hamilton 1994 ):x(t)=pXk=1a(k)x(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+(t), (2)wherea(1),,a(p)arethemodelcoefcientsand(t)isawhitenoiseprocesswhichaccountsfortheresidualmodelerror.Themodelorder,denotedbyp,isindicativeofthenumberoftimelagsinthemodel.Forinstance,ifthepresentvalueofthestochasticprocess,x(t),dependsonitspastthreevalues,x(t)]TJ /F7 11.955 Tf 12.4 0 Td[(1),x(t)]TJ /F7 11.955 Tf 12.4 0 Td[(2),andx(t)]TJ /F7 11.955 Tf 12.4 0 Td[(3),thenp=3. 21

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FittinganappropriateARmodeltoastochasticprocessrequiresthespecicationofthemodelcoefcients,a(1),,a(p),thestandarddeviationofthenoiseprocess,denotedas,andthemodelorder,p.Valuesfora(1),,a(p)andcanbeobtainedbyrstmultiplyingbothsidesofEq.( 2 )byx(t)]TJ /F5 11.955 Tf 12.48 0 Td[(m)andtakingtheexpectedvalue.TheresultisthesetofYule-Walkerequations( Boxetal. 2008 ):(m)=pXk=1a(k)(m)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+2m,0, (2)form0where(m)=E[x(t)x(t)]TJ /F5 11.955 Tf 12.62 0 Td[(m)]istheautocorrelationfunctionofx(t)andm,0istheKroneckerdeltafunctionwhichtakesthevaluezeroeverywhereexceptatm=0whereitisequaltoone.Sincem=0,,p,Eq.( 2 )yieldsatotalofp+1equationswhichcanbeusedtosolveforthepmodelcoefcientsplusthestandarddeviationofthenoiseprocess,.TheYule-Walkerestimatesofa(1),,a(p)andareobtainedbyreplacingthetheoreticalautocorrelations,(m),inEq.( 2 )byestimatedautocorrelations,^(m),whichcanbeobtainedfromtheobservedx(t).Determininganoptimalorder,p,foranARmodelrequiresbalancingthedesiretominimizemodelerrorwiththedesireformodelparsimony.Modelerror,reectedbythevarianceofthenoiseterm,monotonicallydecreasesasmodelorderincreases.Atthesametime,increasingthemodelorderwilleventuallyleadtoovertting,whichinturnleadstoinaccurateestimatesforthemodelcoefcients.Inordertoresolvethistrade-off,severalcriteriahavebeendevelopedformodelorderselectionwhichincorporatebothaminimizationofthenoisevarianceandapenaltyforlargemodelorders.OneexampleisAkaike'sinformationcriterion(AIC)whichisdenedas( Akaike 1974 ):AIC(p)=ln(2(p))+2p N. (2)AnotherisAkaike'snalpredictionerror(FPE),denedas( Akaike 1969 ):FPE(p)=2(p)N+p+1 N)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1. (2) 22

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AthirdistheSchwarzBayesianinformationcriterion(BIC)( Schwarz 1978 ):BIC(p)=ln(2(p))+pln(N) N. (2)Alloftheseformulasareincreasingfunctionsof2(p)andincreasingfunctionsofp.Thevalueofpatwhicheachformulaisminimizedisconsideredtheoptimalmodelorderbasedontheparticularcriterioninquestion. 2.1.1VectorAutoregressiveModelingTheARformulapresentedinEq.( 2 )canalsobegeneralizedtocapturetheevolutionandinterdependenciesbetweenmultiplestochasticprocesses.Amodelinthisgeneralizedformisreferredtoasavectorautoregressive(VAR)model.AVARmodelofthestochasticprocessvector,X(t),isconstructedbyincludingforeachprocess,xi(t),anequationdescribingitsevolutionbasedonitsownpreviousvaluesalongwiththepreviousvaluesofallotherprocessesinthemodel.AssumingthateachstochasticprocessinX(t)iszeromeanandjointlystationary,theVARmodelofX(t)takesthefollowingform( Boxetal. 2008 ; Hamilton 1994 ; Lutkepohl 2005 ):X(t)=pXk=1A(k)X(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+(t). (2)ThisVARmodelisn-dimensionalandoforderp.Thepcoefcientmatrices,A(1),,A(p),areeachofdimensionnn.AssumingthatX(t)isadequatelyrepresentedbythisVARmodeloforderp,theterm(t),whichrepresentstheresidualmodelerror,isann1vectorofmutuallyuncorrelatedwhitenoiseprocesseswithcovariancematrix.AsintheunivariateARmodelcase,modelparametersaresolvedforbymultiplyingbothsidesofEq.( 2 )byXT(t)]TJ /F5 11.955 Tf 12.02 0 Td[(k)andtakingtheexpectedvalue.Theresultistheset 23

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ofYule-Walkermatrixequations:)]TJ /F7 11.955 Tf 6.94 0 Td[((m)=8>>><>>>:pPk=1A(k))]TJ /F7 11.955 Tf 6.94 0 Td[((m)]TJ /F5 11.955 Tf 11.96 0 Td[(k),ifm>0,pPk=1A(k))]TJ /F7 11.955 Tf 6.94 0 Td[(()]TJ /F5 11.955 Tf 9.29 0 Td[(k)+,ifm=0, (2)where)]TJ /F7 11.955 Tf 6.94 0 Td[((m)=E[X(t)XT(t)]TJ /F5 11.955 Tf 12.14 0 Td[(m)]istheautocorrelationfunctionofX(t).Thecoefcientmatrix,A(k),andthecovariancematrix,,ofthenoiseterm,areobtainedbysolvingEq.( 2 )viatheLevinson,WigginsandRobinson(LWR)algorithm( Morettin 1984 ; Whittle 1963 ).Modelorderisagainchosenbyminimizingoneofthefollowingmodelorderselectioncriterion( Lutkepohl 1985 ):AIC(p)=ln[det((p))]+2n2p N, (2)FPE(p)=det((p))N+np+1 N)]TJ /F5 11.955 Tf 11.96 0 Td[(np)]TJ /F7 11.955 Tf 11.96 0 Td[(1n, (2)BIC(p)=ln[det((p))]+n2pln(N) N, (2)wheredet()denotesdeterminant.Othermodelorderselectioncriterioncanalsobeusedforthispurpose( Lutkepohl 1985 ). 2.1.2SpectralPropertiesofVARModelsOnceaVARmodelisadequatelyestimatedforX(t)inthetimedomain,thismodelcanbeusedtoderivepropertiesofX(t)inthespectraldomain.Spectralfeaturesarederivedfromthecoefcientmatrix,A(k),andthenoisecovariancematrix,,oftheVARmodel.Specically,( Gardner 1988 ):A(f)=I)]TJ /F9 7.97 Tf 18.27 15.21 Td[(pXk=1A(k)e)]TJ /F8 7.97 Tf 6.59 0 Td[(2jfk (2) 24

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whereIisthen-dimensionalidentitymatrix,j=p )]TJ /F7 11.955 Tf 9.3 0 Td[(1andfisnormalizedfrequencyintheinterval[-0.5,0.5].Thetransformedcoefcientmatrix,A(f),isannnmatrixthatisnowafunctionoffrequency.Fromthetransformedcoefcientmatrixandthenoisecovariancematrix,,oftheVARmodel,onecanalsoobtainthetransfermatrix,denotedH(f),andthespectralmatrix,S(f),ofX(t).Thetransferfunctionmatrixwithnoise(t)asinputandX(t)asoutputisdenedas( Gardner 1988 ):H(f)=A)]TJ /F8 7.97 Tf 6.59 0 Td[(1(f), (2)whereA(f)isdenedinEq.( 2 ).Thespectralmatrixcanbeobtainedusingthefollowingformula( Boxetal. 2008 ):S(f)=H(f)H(f), (2)where*denotescomplexconjugatetranspose.Theserelationshipsformthebasisforspectraldomaintimeseriesinferencetechniques. 2.2GrangerCausality 2.2.1PairwiseGrangerCausalityPairwiseGrangercausality(GC)isatimeseriesinferencetechniquethatcanbeusedtomeasuretheeffectiveconnectivitybetweentwostochasticprocesses,denotedasxi(t)andxj(t).Thistechnique,sinceitisbasedonlinearARmodelsisitselflinearandparametric. 2.2.1.1TimedomainpairwiseGrangercausalityAssumingthatxi(t)andxj(t)arezeromean,individuallyandjointlystationarystochasticprocesses,univariateARmodelscanbettoeachprocesssuchthat:xi(t)=piXk=1a(k)xi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+i,1(t), (2) 25

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xj(t)=pjXk=1b(k)xj(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+j,1(t), (2)wherevar(i,1(t))=2iandvar(j,1(t))=2j.SettingX(t)=[xi(t),xj(t)]T,abivariateVARmodelcanalsobettoX(t)suchthat:X(t)=pXk=1A(k)X(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+(t), (2)where(t)=[i,2(t),j,2(t)]T.Thecovariancematrixof(t),denotedby,canbeexpandedas:=0B@iiijjijj1CA, (2)whereii=var(i,2),jj=var(j,2),andij=ji=cov(i,2,j,2).Ifxi(t)andxj(t)areindependent,ij=0,ii=2i,andjj=2j.FromEq.( 2 ),thevalueof2i(t)measurestheaccuracyoftheARpredictionofxi(t)basedonitsownpreviousvalues,whilethevalueofiirepresentstheaccuracyoftheVARpredictionofxi(t)basedonthepreviousvaluesofbothxi(t)andxj(t).AccordingtoGranger,thiscomparisongivesrisetothefollowingquanticationofcausalityfromxj(t)toxi(t)( Dingetal. 2006 ; Granger 1980 ):gcj!i=ln2i ii, (2)andsimilarlyfromxi(t)toxj(t):gci!j=ln2j jj. (2)Ifgcj!i=0,thenthereisnocausalityfromxj(t)toxi(t),whileifgcj!i>0,thencausalityfromxj(t)toxi(t)issaidtoexist.Thesameholdstrueforgci!j. 26

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2.2.1.2SpectraldomainpairwiseGrangercausalityPairwiseGC,presentedinthetimedomainintheprevioussection,canalsobeextendedtothespectraldomaininordertoobtainestimatesofthecausalrelationshipbetweenxi(t)andxj(t)atdifferentfrequencyvalues.DoingsorequiresrstrewritingtheVARmodelttoxi(t)andxj(t)inEq.( 2 ),intermsofthelagoperator,LkX(t)=X(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)whichgives(t)= A(0))]TJ /F9 7.97 Tf 18.27 15.21 Td[(pXk=1A(k)Lk!X(t), (2)whereA(0)=Iisthe2-dimensionalidentitymatrix.FouriertransformingbothsidesofEq.( 2 )thenleadstoA(f)X(f)=E(f). (2)RecallingthatH(f)=A)]TJ /F8 7.97 Tf 6.58 0 Td[(1(f)(cf.Eq.( 2 )),Eq.( 2 )canbewrittenasX(f)=H(f)E(f). (2)Eq.( 2 )canbeexpandedto0B@xi(f)xj(f)1CA=0B@Hii(f)Hij(f)Hji(f)Hjj(f)1CA0B@Ei(f)Ej(f)1CA. (2)ThespectralmatrixofX(t)fromEq.( 2 )canalsobeexpandedinthiswayas0B@Sii(f)Sij(f)Sji(f)Sjj(f)1CA=0B@Hii(f)Hij(f)Hji(f)Hjj(f)1CA0B@iiijjijj1CA0B@Hii(f)Hij(f)Hji(f)Hjj(f)1CA (2)wherethetermsSii(f)andSjj(f)representautospectraandSij(f)andSji(f)representcrossspectra.AccordingtoEq.( 2 ),theautospectrumofxi(t)is( Gardner 1988 )Sii(f)=Hii(f)iiHii(f)+2ijRe(Hii(f)Hij(f))+Hij(f)jjHij(f). (2) 27

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Whenxi(t)andxj(t)areindependent,ij=0,andtheautospectrumofxi(t)simpliestotwoterms.ThersttermisconsideredtheintrinsiccontributiontoSii(f)sinceitdependsonlyonHii(f)andthevarianceofi,2,whilethesecondtermisviewedasthecausalcontributionsinceitinvolvesHij(f)andthevarianceofj,2,whichcomesfromtheVARmodelequationforxj(t).Whenij6=0,itbecomesdifculttoseparatetheintrinsicandcausalcontributionstotheautospectrumofxi(t).Gewekeintroducedatransformation,however,thateliminatesthecrossterminEq.( 2 )( Geweke 1982 ).Thistransformation,callednormalization,consistsofleft-multiplyingbothsidesofEq.( 2 )bythematrixP=0B@10)]TJ /F8 7.97 Tf 10.49 6.11 Td[(ij ii11CA, (2)andtransformingtheresulttothefrequencydomainviaFouriertransform.Thisgives~A(f)X(f)=~E(f). (2)WritingEq.( 2 )intermsoftheexpandedtransferfunctionmatrixgives0B@xi(f)xj(f)1CA=0B@~Hii(f)Hij(f)~Hji(f)Hjj(f)1CA0B@Ei(f)~Ej(f)1CA (2)where~Ej(f)=Ej(f))]TJ /F8 7.97 Tf 13.17 6.11 Td[(ij iiEi(f),~Hii(f)=Hii(f)+ij iiHij(f),and~Hji(f)=Hji(f)+ij iiHii(f).Inthiscase,thecovariancematrixofthenoiseterms,Ei(f)and~Ej(f),is~=0B@ii00~jj1CA, (2)where~jj=jj)]TJ /F8 7.97 Tf 13.15 7.48 Td[(2ij ii.UsingEq.( 2 ),thenewautospectrumofxi(t)isSii(f)=~Hii(f)ii~Hii(f)+Hij(f)~jjHij(f), (2) 28

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withthersttermontherighthandsideinterpretedastheintrinsicpowerandthesecondtermasthecausalpowerduetoxj(t).AssuggestedbyGeweke,pairwiseGCinthespectraldomainisthendenedasthenaturallogarithmoftheratiooftotaltointrinsicpower( Geweke 1982 ):GCj!i(f)=lnjSii(f)j j~Hii(f)ii~Hii(f)j=lnjSii(f)j jSii(f))]TJ /F7 11.955 Tf 11.96 0 Td[((jj)]TJ /F8 7.97 Tf 13.15 7.55 Td[(2ij ii)jHij(f)j2j, (2)wherejjisabsolutevalue.Accordingtothisdenition,ifthecausalpowertermiszero,thenGCj!i(f)=0.Also,foraxedvalueoftotalpower,asthecausalpowertermincreases,GCj!i(f)increases.Thus,thisformulationaccuratelyreectstheintuitivemeaningofthetermcausality.InordertoobtainspectralGCfromxi(t)toxj(t),atransformationmatrixoftheformP=0B@1)]TJ /F8 7.97 Tf 10.59 6.11 Td[(ij jj011CA, (2)isagainleft-multipliedonbothsidesofEq.( 2 ).PerformingananalysissimilartotheoneaboveresultsinGCfromxi(t)toxj(t):GCi!j(f)=lnjSjj(f)j j~Hjj(f)jj~Hjj(f)j=lnjSjj(f)j jSjj(f))]TJ /F7 11.955 Tf 11.96 0 Td[((ii)]TJ /F8 7.97 Tf 13.24 7.54 Td[(2ij jj)jHji(f)j2j, (2)GewekedemonstratedthatthetimedomainGCmeasureistheoreticallyrelatedtothespectraldomainmeasureaccordingtogci!jZ1=2)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2GCi!j(f)df, (2)where,inmanycases,theequalityholds( Geweke 1982 ). 29

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2.2.1.3ExampleapplicationsConsidertwotimeseries,xi(t)andxj(t)generatedaccordingtothefollowingstochasticARmodelxi(t)=(t) (2)xj(t)=xi(t)]TJ /F7 11.955 Tf 11.96 0 Td[(1)+(t),where(t)and(t)arewhitenoiseprocesseswithzeromeanandvariances,2,2,respectively.Clearly,xj(t)iscausallyinuencedbyxi(t),whichcanbewrittenasi!j.Setting=1and=0.2,Isimulatedthissystemforatotalof5,000timepoints.Theresultingtimeseries,xi(t)andxj(t),areshowninFig. 2-1 .Then,makingnoassumptionsabouttheprocessesgeneratingxi(t)andxj(t),IusedtheformulationpresentedabovetocalculatebothtimedomainandspectraldomainpairwiseGC.Forthisexampleandallofthefollowingexamplesinthischapter,IusedtheARtMatlabpackagetotARandVARmodelstothetimeseriesalongwithBICtoselecttheoptimalmodelorder( NeumaierandSchneider 2001 ; SchneiderandNeumaier 2001 ; Schwarz 1978 ).Also,inordertoguaranteezeromeantimeseries,Isubtractedthetemporalmeanofeachtimeseriesfromitself.TimedomainpairwiseGCforthissystemisgcj!i=ln1.026 1.026=0gci!j=ln1.067 0.039=3.309.ThisresultdemonstratesthattimedomainGCaccuratelycapturesthefactthatcausalityowsunidirectionallyfromxi(t)toxj(t).SpectraldomainpairwiseGCforthissystemispresentedgraphicallyinFig. 2-2 sinceitisafunctionoffrequency.Itisassumedthatthesamplingrateofbothtimeseriesis1Hz.Fromthisgure,themaximumGCacrossallfrequenciesis 30

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MAX(GCj!i(f))=0,MAX(GCi!j(f))=3.576.Thus,causalityfromxi(t)toxj(t)isaccuratelycapturedinthefrequencydomainaswell.Furthermore,Icanverifynumericallythatgcj!i=0Z1=2)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2GCj!i(f)df=0gci!j=3.309Z1=2)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2GCi!j(f)df=3.230.Asanexampleofanonlinearsystem,considertwoseriesxi(t)andxj(t)generatedbystochasticvanderPoloscillators( vanderPol 1922 )xi(t)=(1)]TJ /F5 11.955 Tf 11.95 0 Td[(x2i)_xi)]TJ /F11 11.955 Tf 11.96 0 Td[(!2ixi+ii(t)+"ij(xj)]TJ /F5 11.955 Tf 11.96 0 Td[(xi) (2)xj(t)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(x2j)_xj)]TJ /F11 11.955 Tf 11.95 0 Td[(!2jxj+jj(t)+"ji(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xj),wherei(t)andj(t)arebothGaussiandistributedwhitenoiseprocesses.Iset=5,!i=1.5,!j=1.48andi=j=1.5.Thecouplingstrengthfromxi(t)toxj(t)iscontrolledby"jiandcouplingfromxj(t)toxi(t),by"ij.Iset"ji=0.4and"ij=0sothatthereisunidirectionalcouplingfromxi(t)toxj(t).Isimulatedthissystemforatotalof30,000timepointsatarateof10Hz.Inthisexampleandforallfuturestochasticdifferentialequationsinmywork,IemployedtheEuler-Maruyamamethodtondanapproximatenumericalsolutiontothestochasticdifferentialequation( KloedenandPlaten 1999 ).Inthiscase,thestepsizewassettot=0.005.Theresultingtimeseries,xi(t)andxj(t),areshowninFig. 2-3 .Makingnoassumptionsabouttheprocessesgeneratingxi(t)andxj(t),IusedtheformulationpresentedabovetocalculatebothtimedomainandspectraldomainpairwiseGC.TimedomainpairwiseGCforthissystemisgcj!i=ln0.166 0.166=0gci!j=ln0.025 0.013=0.181. 31

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FromtheseresultsitispossibletoconcludethattimedomainpairwiseGCcorrectlyidentiestheunidirectionalinteractionfromxi(t)toxj(t).SpectraldomainpairwiseGCforthissystemispresentedgraphicallyinFig. 2-4 .Fromthisgure,themaximumGCacrossallfrequenciesisMAX(GCj!i(f))=0.037,MAX(GCi!j(f))=4.687.Inthiscase,thenonlinearlyofthesystemcannotbefullycapturedbythelinearARandVARmodelsttothetimeseries.ThisleadstosomeamountoferrorinthecalculationofGCj!i(f),whichisnolongeridenticallyzero.Nevertheless,itisstillquantitativelymuchlowerthanGCi!j(f)anditispossibletoconcludethatpairwise,spectraldomainGCcorrectlyidentiestheunidirectionalinteractionfromxi(t)toxj(t).Thisresulthighlightstheneedforsomeformofsignicancetesting,wherebyitcanbedeterminedwhetherthevaluesofGCj!i(f)andGCi!j(f)aresignicantlydifferentfromzero.SuchtestingisdescribedinSection 2.6 2.2.2ConditionalGrangerCausalityConditionalGCisatimeseriesinferencetechniquethatcanbeusedtomeasuretheeffectiveconnectivitybetweenthreestochasticprocesses,denotedasxi(t),xj(t),andxh(t).Apairwiseanalysiscanalsobeusedinthiscasebyconsideringonlythepairwiseinteractionsbetweenprocesses,however,thisapproachhassomeinherentlimitationswhichareillustratedinSection 2.2.2.4 .Mostimportantly,conditionalGChastheabilitytoresolvewhetheraninteractionbetweentwoprocessesisdirectorismediatedbyathirdprocess.However,likepairwiseGC,thistechnique,sinceitisbasedonlinearVARmodelsisitselflinearandparametric.Thetechnique,asdescribedinthefollowing,canalsobegeneralizedtothreesetsofstochasticprocesses. 2.2.2.1TimedomainconditionalGrangercausalityAssumingthatxi(t),xj(t),andxh(t)arezeromean,individuallyandjointlystationarystochasticprocesses,abivariateVARmodelcanbettoxi(t)andxh(t) 32

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sothatxi(t)=pXk=1Aii,2(k)xi(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+pXk=1Aih,2(k)xh(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+ei,2(t) (2)xh(t)=pXk=1Ahi,2(k)xi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+pXk=1Ahh,2(k)xh(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+eh,2(t),wherethecovariancematrixofthenoisetermsis2=0B@ii,2ih,2hi,2hh,21CA (2).Next,atrivariateVARmodelcanbettoxi(t),xj(t),andxh(t)suchthatxi(t)=pXk=1Aii,3(k)xi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+pXk=1Aij,3(k)xj(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+pXk=1Aih,3(k)xh(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+ei,3(t)xj(t)=pXk=1Aji,3(k)xi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+pXk=1Ajj,3(k)xj(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+pXk=1Ajh,3(k)xh(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+ej,3(t)xh(t)=pXk=1Ahi,3(k)xi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+pXk=1Ahj,3(k)xj(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+pXk=1Ahh,3(k)xh(t)]TJ /F5 11.955 Tf 11.95 0 Td[(k)+eh,3(t), (2)wherethecovariancematrixofthenoisetermsis3=0BBBB@ii,3ij,3ih,3ji,3jj,3jh,3hi,3hj,3hh,31CCCCA (2).Then,timedomainGCfromxj(t)toxi(t)conditionedonxh(t)isdenedasgcj!ijh=lnii,2 ii,3, (2)withconditionalGCfortheotherinteractionssimilarlydened( Dingetal. 2006 ; Geweke 1984 ). 33

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Whencausalityfromxj(t)toxi(t)ismediatedentirelythoughxh(t),Aij,3(k)=0leavingii,3=ii,2.Thus,gcj!ijh=0,indicatingthatthereisnodirectcausalityfromxj(t)toxi(t).Ontheotherhand,whensomedirectcausalityfromxj(t)toxi(t)doesexist,addingxj(t)totheVARmodelofxi(t)reducesthevarianceoftheerrorterms,leadingtoii,30.Asthesecasesillustrate,thisdenitionofconditionalGCisinaccordwiththeintuitivemeaningofdirectcausality. 2.2.2.2SpectraldomainconditionalGrangercausalityFormulatingconditionalGCinthespectraldomaininvolvesrstnormalizingthebivariateVARmodelofEq.( 2 )byleft-multiplyingbothsidesoftheequationbythePmatrixinEq.( 2 ).Transformingtheresulttothefrequencydomaingives0B@xi(f)xh(f)1CA=0B@Gii(f)Gih(f)Ghi(f)Ghh(f)1CA0B@Ei,2(f)Eh,2(f)1CA, (2)where,duetonormalization,cov(Ei,2(f),Eh,2(f))=0.SincenormalizationdoesnotchangethevarianceofEi,2(f),itisstillthecasethatvar(Ei,2(f))=ii,2.Next,thetrivariateVARmodelofEq.( 2 )mustalsobenormalized.Thisisdonebyleft-multiplyingbothsidesoftheequationbythematrixP=P2P1( Dingetal. 2006 ; Geweke 1984 )whereP1=0BBBB@100)]TJ /F7 11.955 Tf 9.3 0 Td[(ji,3)]TJ /F8 7.97 Tf 6.58 0 Td[(1ii,310)]TJ /F7 11.955 Tf 9.3 0 Td[(hi,3)]TJ /F8 7.97 Tf 6.58 0 Td[(1ii,3011CCCCA,P2=0BBBB@1000100)]TJ /F8 7.97 Tf 10.5 7.21 Td[(hj,3)]TJ /F8 7.97 Tf 6.58 0 Td[(hi,3)]TJ /F16 5.978 Tf 5.75 0 Td[(1ii,3ij,3 jj,3)]TJ /F8 7.97 Tf 6.58 0 Td[(ji,3)]TJ /F16 5.978 Tf 5.76 0 Td[(1ii,3ij,311CCCCA. (2)Aftertransformingtheresulttothefrequencydomain,oneobtains0BBBB@xi(f)xj(f)xh(f)1CCCCA=0BBBB@Hii(f)Hij(f)Hih(f)Hji(f)Hjj(f)Hjh(f)Hhi(f)Hhj(f)Hhh(f)1CCCCA0BBBB@Ei,3(f)Ej,3(f)Eh,3(f)1CCCCA. (2) 34

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Thecovariancematrixofthenoiseterms,Ei,3(f),Ej,3(f),andEh,3(f),is^3=0BBBB@^ii,3000^jj,3000^hh,31CCCCA. (2)AkeyequalityprovenbyGewekeisthefollowing( Geweke 1984 ).:GCj!ijh(f)=GCjeh,2!ei,2(f), (2)whereeh,2andei,2representthetimeseries,eh,2(t)andei,2(t),thatresultfromtakingtheinverseFouriertransformofEh,2(f)andEi,2(f),respectively.ThisequalitymeansthatGCfromxj(t)toxi(t)conditionedonxh(t)isequivalenttounconditionalGCfromxj(t)andeh,2(t)toei,2(t).Thislatter,unconditionalGCcanbeformulatedbyrstcomputingtheautospectrumofei,2(t)andthendecomposingitintoanintrinsictermandacausalterm,similartoEqs.( 2 2 ).Theautospectrumofei,2(t)canbecomputedbyrstassumingthatxi(f)andxh(f)inEq.( 2 )areequivalenttoxi(f)andxh(f)inEq.( 2 ).Then,Eq.( 2 )andEq.( 2 )canbecombinedtoyield0BBBB@Ei,2(f)xj(f)Eh,2(f)1CCCCA=0BBBB@Gii(f)0Gih(f)010Ghi(f)0Ghh(f)1CCCCA)]TJ /F8 7.97 Tf 6.58 0 Td[(10BBBB@Hii(f)Hij(f)Hih(f)Hji(f)Hjj(f)Hjh(f)Hhi(f)Hhj(f)Hhh(f)1CCCCA0BBBB@Ei,3(f)Ej,3(f)Eh,3(f)1CCCCA (2)=0BBBB@Qii(f)Qij(f)Qih(f)Qji(f)Qjj(f)Qjh(f)Qhi(f)Qhj(f)Qhh(f)1CCCCA0BBBB@Ei,3(f)Ej,3(f)Eh,3(f)1CCCCA,whereQ(f)=G)]TJ /F8 7.97 Tf 6.59 0 Td[(1(f)H(f).WithQ(f)asthetransferfunctionand^3asthecovariancematrix,theautospectrumofei,2(t)canbecomputedas( Gardner 1988 )Sei,2ei,2(f)=Qii(f)^ii,3Qii(f)+Qij(f)^jj,3Qij(f)+Qih(f)^hh,3Qih(f). (2) 35

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Thersttermontherighthandsideofthisequationisconsideredtheintrinsicpowerandtheremainingtwotermsarethecombinedcausalityfromxj(t)andeh,2(t).WithGCdenedasthenaturallogarithmoftheratiooftotaltointrinsicpower,GCfromxj(t)andeh,2(t)toei,2(t)is( Dingetal. 2006 ; Geweke 1984 )GCjeh,2!ei,2(f)=lnjSei,2ei,2(f)j jQii(f)^ii,3Qii(f)j. (2)Becauseei,2(t)isawhitenoiseprocess,itsspectrum,Sei,2ei,2(f)isequaltoitsvariance,ii,2.Thus,thenalexpressionforconditionalGCfromxj(t)toxi(t)conditionedonxh(t)isGCj!ijh(f)=lnii,2 jQii(f)^ii,3Qii(f)j, (2)withconditionalGCfortheotherinteractionssimilarlydened.Onceagain,itwasshownbyGeweke( Geweke 1984 )thatgcj!ijhZ1=2)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2GCj!ijh(f)df. (2) 2.2.2.3PartitionmatrixGrangercausalityAkeyassumptionintheabovederivationofconditionalGCisthatxi(f)andxh(f)inEq.( 2 )andEq.( 2 )areequivalent.Whiletheoreticallythisholdtrue,itmaybehardtosatisfynumericallywhenusingrealdataduetoerrorsinestimation( Chenetal. 2004 ).Chenetal.demonstratedthatsucherrorscanleadtospectralGCthatcontainsnegativevalues,whichhavenointerpretationintermsofcausality,and/orseveralartifactualpeaksacrossfrequency.Chenetal.alsointroducedamethodforresolvingthisissuewhichtheytermedthepartitionmatrixtechnique( Chenetal. 2004 ).Thepartitionmatrixtechniqueavoidsviolatingtheassumptionthatxi(f)andxh(f)inEq.( 2 )andEq.( 2 )areequivalentbyeliminatingthebivariateVARmodelinEq.( 2 )altogether.Initsplace,thistechniquepartitionsthetrivariateVARmodelinEq.( 2 )intoabivariateexpressionforxi(f)andxh(f).Inthisway,itisguaranteed 36

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thatxi(f)andxh(f)areequivalentinthebivariateandtrivariateexpressions.Thebivariateexpressionforxi(f)andxh(f),constructedbypartitioningEq.( 2 )is0B@xi(f)xh(f)1CA=0B@Hii(f)Hih(f)Hhi(f)Hhh(f)1CA0B@Ei,2(f)Eh,2(f)1CA, (2)whereitcanbeshownthat0B@Ei,2(f)Eh,2(f)1CA=0B@Ei,2(f)Eh,2(f)1CA+0B@Hii(f)Hih(f)Hhi(f)Hhh(f)1CA)]TJ /F8 7.97 Tf 6.59 0 Td[(10B@Hij(f)Hhj(f)1CAEj,2(f). (2)CondensingthetwoHmatricesintheaboveequationintoasingletermgives0B@Hij(f)Hhj(f)1CA=0B@Hii(f)Hih(f)Hhi(f)Hhh(f)1CA)]TJ /F8 7.97 Tf 6.59 0 Td[(10B@Hij(f)Hhj(f)1CA. (2)Then,thecovariancematrixofthenoiseterms,Ei,2(f)andEh,2(f),canbefoundusingEq.( 2 )andis(f)=0B@ii,3ih,3hi,3hh,31CA+0B@Hij(f)Hhj(f)1CAij,3hj,3+0B@ij,3hj,31CAHij(f)Hhj(f)+jj,30B@Hij(f)Hhj(f)1CAHij(f)Hhj(f), (2)withthevaluescomingfromEq.( 2 ).DuetotheH(f)terms,thecross-termsofthiscovariancematrixarenolongerrealbutsatisfytherelationshipih,3(f)=hi,3(f).NormalizingEq.( 2 )inordertomakethenoiseterms,Ei,2(f)andEh,2(f),independentnowrequiresthefollowingtransformationmatrix( Chenetal. 2004 )P=0B@10)]TJ /F8 7.97 Tf 11.48 7.55 Td[(ih,3(f) ii,311CA. (2) 37

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Afterapplyingthistransformationmatrix,thetransferfunctionofEq.( 2 )isG(f)=0B@Hii(f)Hih(f)Hhi(f)Hhh(f)1CA0B@10)]TJ /F8 7.97 Tf 11.48 7.54 Td[(ih,3(f) ii,311CA)]TJ /F8 7.97 Tf 6.58 0 Td[(1. (2)AsinEq.( 2 ),G(f)canbeexpandedtoobtainQ(f)=G)]TJ /F8 7.97 Tf 6.59 0 Td[(1(f)H(f).Now,ii,2=ii,3,yielding( Chenetal. 2004 ; Dingetal. 2006 )GCj!ijh(f)=lnii,3 jQii(f)^ii,3Qii(f)j. (2) 2.2.2.4ExampleapplicationsConditionalGCrequiresatleastthreetimeseriesbeavailableforanalysis.Therefore,inordertodemonstrateitsapplication,consideranextendedVARmodelofthestochasticARmodelpresentedinEq.( 2 )xi(t)=(t) (2)xj(t)=xi(t)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+(t)xh(t)=xh(t)]TJ /F7 11.955 Tf 11.96 0 Td[(1)+xi(t)]TJ /F7 11.955 Tf 11.96 0 Td[(2)+"(t),where(t),(t),and"(t)arewhitenoiseprocesseswithzeromeanandvariances,2,2,2",respectively,andjj<1isaparameterofthesystem.Inthiscase,xi(t)iscausallyinuencingbothxj(t)andxh(t),butwithdifferentdelays.Setting=1,=0.2,"=0.3,and=0.5,Isimulatedthissystemforatotalof5,000timepoints.TheresultingtimeseriesweresimilartothoseinFig. 2-1 .Then,makingnoassumptionsabouttheprocessesgeneratingthetimeseries,IusedtheformulationpresentedabovetocalculatebothtimedomainandspectraldomainconditionalGC.Ibeginbyperformingapairwiseanalysisonthesystem,usingpairwisetimedomainGCtodeterminetheinteractionbetweeneachpairoftimeseries.Thisresultsinthe 38

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followinggci!j=ln1.028 0.041=3.222,gcj!i=ln0.995 0.995=0gch!j=ln1.028 1.028=0,gcj!h=ln1.091 0.1312=2.118gci!h=ln1.091 0.090=2.495,gch!i=ln0.995 0.995=0.Whilethecausalityfromxi(t)toxj(t)andfromxi(t)toxh(t)iscorrectlycaptured,erroneouscausalityfromxj(t)toxh(t)isalsocapturedduetothecommoninuenceofxi(t)onbothxj(t)andxh(t)withdifferenttimedelays.ThisexampleillustratesonelimitationofusingpairwiseGCanalysisonasystemcontainingmorethantwotimeseries.Next,IappliedconditionalGCtothesystem.ThetimedomainconditionalGCforthissystemisgci!jjh=ln1.029 0.041=3.222,gcj!ijh=ln0.996 0.996=0gch!jji=ln0.041 0.041=0,gcj!hji=ln0.090 0.090=0gci!hjj=ln0.131 0.090=0.377,gch!ijj=ln0.996 0.996=0ThisresultdemonstratesthattimedomainconditionalGCaccuratelycapturesthefactthatcausalityowsonlyfromxi(t)toxj(t)andfromxi(t)toxh(t).Theerroneousconnectionfromxj(t)toxh(t)thatwaspresentinthepairwiseanalysisiscorrectlyeliminatedbythetechnique.Inthespectraldomain,IappliedconditionalGCbutdidnotemploythepartitionmatrixmethod.Fig. 2-5 illustratestheresults.Inthiscase,GCi!jjh(f)appearscorrect,butbothGCh!ijj(f)andGCh!jji(f)takeonnegativevaluesforwhichthereisnocausalinterpretation.Uponfurtherinspectionofthesystem,theproblemliesinthefactthatxi(f)andxj(f),whichshouldbeequivalentaccordingtoEq.( 2 )andEq.( 2 ),arenotequal. 39

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Toresolvethisissue,IappliedthepartitionmatrixmethodandobtainedtheresultsinFig. 2-6 .Fromthisgure,themaximumconditionalGCacrossallfrequenciesfromxi(t)toxj(t)andfromxi(t)toxh(t)isMAX(GCi!jjh(f))=3.363andMAX(GCi!hjj(f))=3.912.ThemaximumconditionalGCacrossallremaininginteractionsandfrequencieswas0.056.Ignoringthissmallamountoferrorthatcreptintothecalculation,causalityonlyfromxi(t)toxj(t)andfromxi(t)toxh(t)isaccuratelycapturedinthefrequencydomainusingthepartitionmatrixapproach.Unfortunately,usingthepartitionmatrixmethodcausestheresultstoviolateGeweke'sinequalityfromEq.( 2 )( Geweke 1984 )whichcanbeseennumericallygci!jjh=3.222
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abouttheprocessesgeneratingxi(t),xj(t)andxh(t),IusedtheformulationpresentedabovetocalculatebothtimedomainandspectraldomainconditionalGC.TimedomainconditionalGCforthissystemisgci!jjh=ln0.254 0.227=0.112,gcj!ijh=ln0.167 0.167=0gch!jji=ln0.222 0.227=)]TJ /F7 11.955 Tf 9.3 0 Td[(0.022,gcj!hji=ln0.201 0.200=0gci!hjj=ln0.236 0.200=0.166,gch!ijj=ln0.167 0.167=0.005Again,thenonlinearlyofthesystemcannotbefullycapturedbythelinearARandVARmodelsttothetimeseries.Thisleadstosomeamountoferrorinthecalculationofgch!jjiandgch!ijjwhicharenolongeridenticallyzero.Nevertheless,thesevaluesarequantitativelymuchlowerthangci!jjhandgci!hjj,makingitpossibletoconcludethatGCcorrectlyidentiestheunidirectionalinteractionfromxi(t)toxj(t)andfromxi(t)toxh(t).SpectraldomainconditionalGCforthissystem,computedusingthepartitionmatrixmethod,ispresentedgraphicallyinFig. 2-7 .ThemaximumGCacrossallfrequenciesforthetwocausalinteractionsactuallypresentinthesystemisMAX(GCi!jjh(f))=6.72andMAX(GCi!hjj(f))=6.57.Incomparison,themaximumGCacrossallfrequenciesandacrossallinteractionsnotpresentisthesystemis2.88,whichcorrespondstoMAX(GCh!ijj(f)).Whilesignicancetestingisneededinordertoconrmthatthisvalueisnotsignicant,thesequantitativeresultssuggestthatspectraldomainconditionalGCaccuratelycapturesthecausalityinthissystem. 2.3PartialDirectedCoherencePartialdirectedcoherence(PDC)wassuggestedbyBaccalaandSameshimaasatimeseriesinferencetechniquefordeterminingthedirectionalityofinformationowbetweenmultivariatestochasticprocessesinthespectraldomain( BaccalaandSameshima 2001 ).Inageneralsense,thistechniquecanbeconsideredanothermeasureofGCsinceitreliesontheconceptoftemporalprecedenceandrestsonmanyofthesameassumptionsasthepairwiseandconditionalGCmeasurespreviously 41

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discussedinthischapter.LiketheseotherGCmeasures,applyingPDCtotimeseriesdatashouldenableonetodeterminetheeffectiveconnectivitybetweenthenodesfromwhichthetimeserieswererecorded.However,PDCisonlyformulatedinthespectraldomainandhasnotimedomainequivalent.Moreover,PDCisinherentlyamultivariatemeasurethatcanbeappliedtodatasetswithanynumberoftimeseries.ThisisincontrasttopairwiseGCwhichisdesignedforbivariatetimeseriesandforwhichadifferentformulation,namelyconditionalGC,mustbeusedwhenthenumberoftimeseriesisgreaterthantwo.SincethebasisforPDCisVARmodeling,itisconsideredalinearandparametrictechnique.ThePDCformulationthatfollowsisbasedon BaccalaandSameshima ( 2001 ).ConsideravectorofstochasticprocessesX(t)=[x1(t),x2(t),,xn(t)]Tthatarezeromeanandjointlystationary.AVARmodelttoX(t)hastheformX(t)=pXk=1A(k)X(t)]TJ /F5 11.955 Tf 11.96 0 Td[(k)+(t). (2)withthecovariancematrixofthenoiseterms,(t),denotedby.Thecoefcientmatrix,A(k),isannnmatrixwhereAji(k)denotestheVARmodelcoefcientrepresentingprocessxi(t)'scontributiontoprocessxj(t).TheformulationofPDCrestsontheobservationthatcausalityfromxi(t)toxj(t)shouldbereectedinthecoefcientmatrix,Aji(k),whichshouldbenonzero.ThisreasoningisextendedtotheFouriertransformofthecoefcientmatrix,A(k),(cf.Eq.( 2 )),whichisA(f).BaccalaandSameshimasuggestnormalizingeachcoefcientterminordertoarriveatthefollowingdenitionofPDCfromxi(t)toxj(t)( BaccalaandSameshima 2001 )PDCi!j=jAji(f)j p Ai(f)Ai(f) (2)whereAi(f)istheithcolumnoftheA(f)matrix.Directcausalityfromprocessxi(t)toprocessxj(t)isinferredbyanon-zerovalueofPDCi!j,whichisnormalizedbetweenzeroandone.Intuitively,thisdenitioncapturestherelativeinuenceofxi(t)onxj(t) 42

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comparedtotheinuenceofxi(t)onallotherprocessesinthesystem.PDCfromxj(t)toxi(t)isanalogouslydened.AnimportantdifferencebetweenPDCandthepairwiseandconditionalGCmeasurespresentedpreviouslyisthatPDCdoesnotincludetermsfromtheVARmodel'snoisecovariancematrixinitsformulation.Thefollowingexampledemonstratesalimitationthatarisesfromthisfact.ConsiderthreeindependentGaussianwhitenoiseprocesses,xi(t)=ii(t),xj(t)=jj(t),andxh(t)=hh(t)withvariancesi=1,j=h=500.TheresultsofaPDCanalysisofthissystemareshowninFig. 2.6 .Whilethereisnoactualcausalityinthesystem,PDCincorrectlyidentiescausalityfromxi(t)toxj(t)andfromxi(t)toxh(t).Thisfalsedetectionofcausalityisaresultofthedifferenceinvariancebetweenxi(t),xj(t),andxh(t).PDCincorrectlyidentiescausalityfromtheprocesswithlowvariancetotheprocesseswithmuchhighervariances.ThisiscausedbyerroneousVARmodelparameterestimateswhichareduetothevariancedifferencesinthesystem( Winterhalderetal. 2005 ).Forinstance,intheexampleabove,thecoefcientsjAij(f)j,jAhj(f)j,jAih(f)j,andjAij(f)jdonotexceedvaluesontheorderof10)]TJ /F8 7.97 Tf 6.59 0 Td[(3.Incontrast,MAX(jAji(f)j)=3.9andMAX(jAhi(f)j)=7.5.SincethevaluesofAji(f)andAhi(f)areseveralordersofmagnitudegreaterthanalloftheothercoefcientvaluesinthemodel,PDCerroneouslyidentiescausalityfromxi(t)toxj(t)andfromxi(t)toxh(t).GeneralizedPDC,whichisdiscussedinthenextsection,wasformulatedtoremedythislimitationofPDC. 2.3.1GeneralizedPartialDirectedCoherenceBaccalaetal.introducedgeneralizedPDC(GPDC)inordertoovercomethelimitationofPDCdiscussedabove( Baccalaetal. 2007 ).GPDCincludestermsfromtheVARmodel'snoisecovariancematrixinitsformulation,effectuallyaccountingforerrorsintheestimationofthecoefcientmatrixthatarisewhentherearelargedifferencesinvarianceinthesystem. 43

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GPDCfromxi(t)toxj(t)isdenedas( Baccalaetal. 2007 )GPDCi!j=1 iijAji(f)j s nPk=11 2kkAki(f)Aki(f), (2)whereiiandkkrepresentthevariancesofthenoisetermsintheVARmodel'sequationsforxi(t)andxk(t),respectively.Directcausalityfromprocessxi(t)toprocessxj(t)isinferredbyanon-zerovalueofGPDCi!j,whichisnormalizedbetweenzeroandone.TheintuitionbehindGPDCasameasureofcausalityisthesameasthatforPDC.GPDCfromxj(t)toprocessxi(t)isformulatedinananalogousfashion.AgainconsideringtheexampleofthreeindependentGaussianwhitenoiseprocesses,xi(t)=2ii(t),xj(t)=2jj(t),andxh(t)=2hh(t)withthesameparametersettingsasabove,itisshowninFig. 2-9 thatGPDCcorrectlyidentiesnocausalityinthesystem( Winterhalderetal. 2005 ). 2.3.2ExampleApplicationsConsiderthesystemdescribedinEq.( 2 ).TheresultswhenPDCandGPDCareappliedtothissystemareshowninFigs. 2-10 and 2-11 ,respectively.Fromtheseresults,itisclearthatbothmethodsareabletocorrectlyidentifycausalityinthesystem,specicallythatfromxi(t)toxj(t)andfromxi(t)toxh(t).EventhoughthevanderPolsystemisanonlinearone,linearmethodssuchasPDCandGPDCareabletocorrectlyidentifythecausalinteractionsinthesystem. 2.4PhaseDynamicsModelingPhasedynamicsmodeling(PDM)wasintroducedbyRosenblumandPikovskyasatimeseriesinferencetechniquedesignedspecicallyfordetectingthedirectionalityofinteractionsbetweenweaklycoupled,narrow-bandoscillators( RosenblumandPikovsky 2001 ).Whilethisclassofapplicablesystemsseemsrestrictive,narrow-bandoscillatorsarethetopicofawiderangeofstudiesastheyarerepresentativeofmanynaturalphenomena( Pikovskyetal. 2002 ).LikethepreviouslydescribedGC 44

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techniques,applyingPDMtotimeseriesofweaklycoupledoscillatorsshouldenableonetodeterminetheeffectiveconnectivitybetweentheoscillatorsfromwhichthetimeserieswererecorded.Whilerstformulatedasapairwisetechnique,PDMwaslaterextendedtoanalyzemultivariatetimeseriesaswell( SmirnovandBezruchko 2009 ).Inaddition,PDMisstrictlyatimedomainmeasureandhasnospectraldomainequivalent.AkeyfeatureofPDMisthefactthatitutilizestimeseriesofinstantaneousphaseratherthantheactualtimeseriessignalrecordedfromtheoscillatorsthemselves.Thesetimeseriesofinstantaneousphase,whichwedenoteas(t),canbeobtainedbytakingtheHilberttransformoftheoriginaltimeseriesoftheoscillator( Pikovskyetal. 2002 ).Forexample,givenatimeseriess(t),theHilberttransformofs(t)isdenedassH(t)=)]TJ /F8 7.97 Tf 6.59 0 Td[(1P.V.Z1s() t)]TJ /F11 11.955 Tf 11.96 0 Td[(d, (2)whereP.V.meansthattheintegralistakeninthesenseoftheCauchyprincipalvalue.Then,ananalyticsignal,(t),whichisacomplexfunctionoftime,isdenedby(t)=s(t)+isH(t)=A(t)ei(t). (2)Theinstantaneousamplitude,A(t),andtheinstantaneousphase,(t),ofs(t)aredenedaccordingtothisanalyticfunction.Whilethereareothertechniquesfordeningtheinstantaneousphaseofasignal( Pikovskyetal. 2002 ),IusetheHilberttransform.ThegeneralapproachofPDMissimilartothattakenbyGCandPDC.Amodelisrstttothetimeseriesofinstantaneousphasesandthecoefcientsofthismodelareusedtocalculatecausality.However,unlikeGCandPDCwhichttimeseriestolinearVARmodels,PDMtsthetimeseriestofunctionsofsineandcosine.Sincetheseresultingfunctionsarenonlinearbyvirtueofthefactthatsineandcosinearenonlinear,PDMcanbeconsideredanonlineartechnique.Thenonlinearitycapturedbyfunctions 45

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ofsineandcosine,however,islimitedandthusithasbeenshownthatPDMisonlycapableofcapturingsomespecicformsofnonlinearity( Smirnovetal. 2005 ).AnotherdistinguishingfeatureofPDMisthat,unlikethelinearmethodsGCandPDC,itisapplicabletodeterministictimeseries.Thisusecaseisgenerallyonlyconsideredintheory,however,sincetimeseriesrecordedfromphysicalprocesseswillnearlyalwayscontainsomenoise.ThePDMformulationthatfollowsisbasedonworkbyRosenblumetal.andSmirnovetal.( RosenblumandPikovsky 2001 ; SmirnovandBezruchko 2009 ). 2.4.1PairwisePhaseDynamicModelingLetxiandxjbetwosignalsrecordedfromtwoweaklycoupledoscillatorsandleti(t)andj(t)bethecorrespondinginstantaneousphasesasdenedabove.Itcanbeassumedthatthephaseincrementi(t+))]TJ /F11 11.955 Tf 12.63 0 Td[(i(t)isgeneratedbysomeunknowntwo-dimensionalnoisymapsuchthat i(t+))]TJ /F11 11.955 Tf 11.96 0 Td[(i(t)=Fi(i(t),j(t))+i(t).(2)Thevalueoftheconstanthasbeenshowntoberelativelyunimportantandistypicallysettoonebasicoscillationperiodofoscillatori.Therandomprocess,,isofzeromeanandthefunctionFi(i(t),j(t))isatrigonometricpolynomialoftheform( RosenblumandPikovsky 2001 )Fi(i(t),j(t))=Xm,`[am,`cos(mi(t)+`j(t))+bm,`sin(mi(t)+`j(t))]. (2)ThecoefcientsofEq.( 2 ),am,`andbm,`,areestimatedusingaleastsquaresttotheinstantaneousphaseincrementtimeseries,i(t+))]TJ /F11 11.955 Tf 12.23 0 Td[(i(t),andtheconstants,mand`determinetheorderofthefunctionFi(i(t),j(t)).Ahigher-orderfunctionisabletocapturemorenonlinearityinthesystembutcanalsoleadtomodelovertting.SincethirdorderfunctionshavebeenfoundtobesuitableforPDManalysis,Ichosem3and`3( RosenblumandPikovsky 2001 ). 46

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ThestrengthofcausalityfromoscillatorjtooscillatoriisdeterminedbythedegreeofdependenceofFionj(t),aquantitythatcanbemeasuredbythesensitivityofFitoj(t).Thus,PDMfromj!iisdenedas( RosenblumandPikovsky 2001 ) PDMj!i=s 1 22Z20Z20@Fi(i(t),j(t)) @j2didj,(2)where@Fi(i(t),j(t))=@j(t)isthesensitivityofFi(i(t),j(t))toj(t).Itcanbeshownthat( SmirnovandBezruchko 2003 )PDMj!i=s Xm,``2(a2m,`+b2m,`). (2)AvalueofPDMj!ithatissignicantlygreaterthanzerogenerallyindicatescausalityfromoscillatorjtooscillatori.Causalityfromoscillatoritooscillatorj,PDMi!j,isdenedanalogously. 2.4.2MultivariatePhaseDynamicModelingTheformulaforPDMpresentedaboveisapairwisemeasuresinceitisconstructedfortwoinstantaneousphasetimeseries,namelyi(t)andj(t).Extendingthismeasuretomultivariatetimeseriesingeneralrequiresrstcomputingtheinstantaneousphaseofalloscillatorsbeingconsidered.Assumingthattherearenoscillatorsinasystem,theseinstantaneousphasetimeseriescanbedenotedas1(t),,n(t).Asinthepairwisecase,thephaseincrementofi(t)canbemodeledbythefunction i(t+))]TJ /F11 11.955 Tf 11.96 0 Td[(i(t)=Fi(1(t),,n(t))+i(t),(2)whereFiisnowafunctionofallinstantaneousphasetimeseriesinthesystemandi(t)andaredenedastheywereinthepairwisecase.ThefunctionFiisnowatrigonometricpolynomialoftheform( SmirnovandBezruchko 2009 )Fi(1(t),,n(t))=i,0+nXj=1(j6=i)[i,jcos(j(t))]TJ /F11 11.955 Tf 11.96 0 Td[(i(t))+i,jsin(j(t))]TJ /F11 11.955 Tf 11.95 0 Td[(i(t))]. (2) 47

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Onceagain,thecoefcients,i,jandi,j,areestimatedusingaleastsquaresttotheinstantaneousphasetimeseries.MultivariatePDMisthendenedas( SmirnovandBezruchko 2009 )PDMj!i=1 2(2i,j+2i,j). (2)AvalueofPDMj!ithatissignicantlygreaterthanzerogenerallyindicatescausalityfromoscillatorjtooscillatori.Inthiscase,Eq.( 2 )isoforderonesotheformulaforPDMissimplied.However,higher-ordertermscanbeincorporatedintoEq.( 2 )ifnecessary( SmirnovandBezruchko 2009 ).InboththepairwiseandmultivariateformulationsofPDM,itisassumedthattheinstantaneousphasetimeseries,1(t),,n(t),areindependent.ThisisanecessaryassumptionforttingthephaseincrementstothefunctionsFi(i(t),j(t)))andFi(1(t),,n(t)))inEqs.( 2 2 )( RosenblumandPikovsky 2001 ). 2.4.3ExampleApplicationsInthissection,IwilldemonstratetheapplicationofmultivariatePDMtotwoexamplesystems.TherstistheVARmodeldescribedinEq.( 2 )andthesecondisthesystemofvanderPoloscillatorsdescribedinEq.( 2 ).First,IappliedPDMtothetrivariateVARmodelinEq.( 2 ).TheresultswereasfollowsPDMi!j=1.920,PDMi!h=1.341PDMj!i=1.116,PDMj!h=1.178PDMh!i=0.534,PDMh!j=0.586.ThevaluesofPDMj!iandPDMj!harenearlyashighasthevaluesofPDMi!jandPDMi!h,indicatingthatPDMmostlikelyidentiesthefollowingcausalinteractions:xi(t)toxj(t),xi(t)toxh(t),xj(t)toxi(t)andxj(t)toxh(t).Sincetheonlytruecausalinteractionsinthesystemarefromxi(t)toxj(t)andfromxi(t)toxh(t),PDMerroneously 48

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identiescausalityfromxj(t)toxi(t)andfromxj(t)toxh(t).Thisisnotunexpectedsince,fromthediscussionabove,PDMisdesignedforsystemsofweaklycoupledoscillatorsandthelinearVARmodelconsideredinthisexampledoesnotbehavelikeanoscillator.Next,IappliedPDMtothetrivariatevanderPolmodelinEq.( 2 ).UnlikethelinearVARmodelexample,vanderPoloscillatorsshouldbeamenabletoPDManalysis.TheresultswereasfollowsPDMi!j=0.902,PDMi!h=0.482PDMj!i=0.015,PDMj!h=0.012PDMh!i=0.015,PDMh!j=0.027.InthiscasePDMi!jandPDMi!hareanorderofmagnitudegreaterthanthePDMvaluesoftheotherinteractions.Thus,PDMlikelyidentiescausalityfromxi(t)toxj(t)andfromxi(t)toxh(t)anddoesnoterroneouslyidentifynonexistentinteractions. 2.5RecentAdvancesandApplicationsOfthetimeseriesinferencetechniquesdiscussedinthischapter,GCisthemostwidelyappliedtoneuraltimeseries.Despiteitsknownlimitations,pairwiseGCisoftenusedtoanalyzetimeseriesfrommulti-channelfMRIandEEGrecordings( Cadotteetal. 2010 ; Goebeletal. 2003 ; Roebroecketal. 2005 ).Morerecentstudies,however,haveappliedconditionalGCtosuchmultivariatedatasets( Liaoetal. 2010b ; Zhouetal. 2011 ).AnotherrecentlyproposedapproachutilizesprincipalcomponentanalysistorstreducethedimensionalityofmultivariatetimeseriespriortotheapplicationofconditionalGC( Zhouetal. 2009 ).Thisapproachgreatlydecreasesthecomputationalcostofperformingtheanalysis.PDChasalsobeenappliedtoanumberofneuraltimeseriesdatasetsincludingthosefromfMRIandEEGrecordings( Satoetal. 2009 ; Tropinietal. 2011 ).However,GPDCasproposedbyBaccalaetal.,( Baccalaetal. 2007 )hasyettobeappliedto 49

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recordedneuraltimeseries.Schelteretal.proposedanapproachsimilartoGPDC,whichtheytermedrenormalizedPDC,andappliedthistechniquetoEEGdatarecordedfrompatientssufferingfromParkinsoniantremors( Schelteretal. 2009 ).Becauseitisrestrictedtotheanalysisofnarrow-bandoscillatorydata,PDMhasnotyetgainedwidespreadpopularityasatimeseriesinferencetechniqueinneuroscience.ManyofthestudiesinvestigatingPDMusesimulateddatasets( Bezruchkoetal. 2003 ; Kralemannetal. 2011 ; RosenblumandPikovsky 2001 ).However,afewstudieshaveconsideredtheuseofPDMtodetermineeffectiveconnectivityintherespiratorysystem( Rosenblumetal. 2002 )andinParkinsoniantremors( Smirnovetal. 2008 ). 2.6SignicanceTestingAswasdemonstratedinSections 2.2.1.3 and 2.2.2.4 ,whenlineartimeseriesinferencetechniquesareappliedinthetimedomaintosystemsthatareinherentlylinearandautoregressive,theirresultsareusuallyexact.Forinstance,inEq.( 2 ),thereisnocausalityfromtimeseriesxj(t)toxi(t)andgcj!i=0.Ontheotherhand,thereiscausalityfromxi(t)toxj(t)andgci!j>0.Whensuchlineartechniquesareappliedtononlineartimeseries,however,theirresultstendtobeapproximatesincethelinearmodelstheyarebasedoncannotfullycapturethenonlinearnatureofthesystemunderstudy.Evenwhenthesetechniquesareappliedtothegeneralclassoflinearsystems,theymaynotbeexactduetotheapproximatenatureofVARmodeltting.UsingspectraldomaintechniquesfurtherconfoundsthisproblemsincethespectralmatrixobtainedviaaVARmodeltisalsoanestimationofthetruespectrum.ThissameargumentcanbeextendedtoPDMsinceitcannotfullycaptureallformsofnonlinearitypresentinagivensystemofoscillatorsandthettingofinstantaneousphasetimeseriestoapredenedmodelisapproximate.Therefore,itmustbedeterminedwhethercausalityestimatesobtainedbythesetechniquesaresignicantlygreaterthanzero,inwhichcasecausalityislikelytoexist. 50

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Onestrategyfordeterminingwhetheracausalityestimateobtainedthroughthesetechniquesissignicantlygreaterthanzeroistocompareittoanullcase.Herethenullcaseconsistsofasystemwithnocausalitybetweenitsnodes.Byapplyingeachtechniquetothisnullcase,itispossibletoobtainsignicancethresholdsabovewhichthecausalityestimatesareconsideredsignicantlygreaterthanzero( Fisher 1925 ).Inmywork,Iconstructnullcasesviasurrogateanalysis( PrichardandTheiler 1994 ; SchreiberandSchmitz 2000 ; Timmer 2000 ).Foragivensetoftimeseriesunderstudy,Igenerate1,000surrogatesetsoftimeseriesbywindowingthetimeseriesofeachnodeintodiscreteblocksandthenrandomlyshufingtheseblocks.Thismaintainstheenergycontentofeachnode'stimeseriesbutdestroysthetemporalrelationshipbetweentimeseriesintheset( Kaminskietal. 2001 ).Inthisway,Iobtain1,000setsoftimeserieseachofwhichrepresentsthenullcase.ThenIapplyeachtimeseriesinferencetechniquetothisnullcaseand,usingap-valueofp=0.05,determineasignicancethresholdforeachtechnique.InthecaseofspectraldomainGC,PDCandGPDC,Iconsideronlythemaximumcausalityestimateacrossallfrequencies( BlairandKarniski 1993 ; Chenetal. 2006 ).SurrogatetestingtodeterminesignicancelevelswasnecessaryinthecaseofspectralGCandGPDCsincenoanalyticalboundsonsignicancehavebeenderivedforthesetechniques.WhileanalyticalboundsdoexistforPDCandPDM( Schelteretal. 2006 ),IchosetousesurrogatetestingforthesemeasuresaswellinordertomaketheresultsmorecomparabletothoseobtainedbyGCandGPDC.Inaddition,thereissomeindicationthatnumericalmethodsperformbetteronnitedatasetsthananalyticalboundsderivedasymptotically( DavidsonandMacKinnon 2003 ; MacKinnon 2006 ). 51

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Figure2-1. Timeseriesxi(t),inblue(top),andxj(t),inred(bottom),generatedfromtwoAR,stochasticprocesses.Inthiscase,xi(t)iscausallyinuencingxj(t). 52

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Figure2-2. Theresultsobtainedfromapplyingspectraldomain,pairwiseGCtotwoexampletimeseriesgeneratedfromstochastic,ARmodels.GCfromxi(t)toxj(t)isinblueandGCfromxj(t)toxi(t)isinred.SinceGCi!j(f)isalwaysgreaterthanGCj!i(f),whichisidenticallyzero,GCcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t). 53

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Figure2-3. Timeseriesxi(t),inblue(top),andxj(t),inred(bottom),generatedfromtwostochasticvanderPoloscillators.Inthiscase,xi(t)iscausallyinuencingxj(t). 54

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Figure2-4. Theresultsobtainedfromapplyingspectraldomain,pairwiseGCtotwoexampletimeseriesgeneratedfromstochastic,vanderPoloscillators.GCfromxi(t)toxj(t)isinblueandGCfromxj(t)toxi(t)isinred.SinceGCi!j(f)isalwaysgreaterthanGCj!i(f),whichisnearlyzero,GCcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t). 55

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Figure2-5. Theresultsobtainedfromapplyingspectraldomain,conditionalGCwithoutthepartitionmatrixmethodtothreeexampletimeseriesgeneratedfromstochastic,ARmodels.GCfromxi(t)toxj(t)conditionedonxh(t)isinblue,fromxi(t)toxh(t)conditionedonxj(t)isinred,xh(t)toxi(t)conditionedonxj(t)isinyellow,andxh(t)toxj(t)conditionedonxi(t)isingreen.GCfromxh(t)toxi(t)conditionedonxj(t)andfromxh(t)toxj(t)conditionedonxi(t)takesonnegativevalueswhichhavenomeaningfulinterpretationintermsofcausality.Thus,thisillustratestheneedforthepartitionmatrixmethod. 56

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Figure2-6. Theresultsobtainedfromapplyingspectraldomain,conditionalGCwiththepartitionmatrixmethodtothreeexampletimeseriesgeneratedfromstochastic,ARmodels.GCfromxi(t)toxj(t)conditionedonxh(t)isinblue,fromxi(t)toxh(t)conditionedonxj(t)isinred,andtheremainingfourinteractions,whicharenearlyidenticaltooneanother,areinyellow.GCwiththepartitionmatrixmethodcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t)andfromxi(t)toxh(t)anddoesnoterroneouslyidentifyanycausalinteractionsinthesystemthatarenotpresent. 57

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Figure2-7. Theresultsobtainedfromapplyingspectraldomain,conditionalGCwiththepartitionmatrixmethodtothreeexampletimeseriesgeneratedfromstochastic,vanderPoloscillators.ThemaximumvaluesofGCi!jjh(f)andGCi!hjj(f)aremuchgreaterthanthemaximumGC(f)valuesoftheotherinteractions.Thus,GCwiththepartitionmatrixmethodcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t)andfromxi(t)toxh(t). 58

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Figure2-8. TheresultsofaPDCanalysisofthreewhitenoiseprocesses.Processxi(t)hasamuchlowervariancethaneitherxj(t)orxh(t)andbecauseofthis,PDCincorrectlyidentiescausalityfromxi(t)toxj(t)andxh(t),showninthetoptwoplots. 59

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Figure2-9. TheresultsofaGPDCanalysisofthreewhitenoiseprocesses.UnlikePDC,GPDCcorrectlyidentiesnocausalityinthesystem. 60

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Figure2-10. TheresultsofaPDCanalysisofthreeexampletimeseriesgeneratedfromstochastic,vanderPoloscillators.ThemaximumvaluesofPDCi!j(f)andPDCi!h(f)aremuchgreaterthanthemaximumPDC(f)valuesoftheotherinteractions.Thus,PDCcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t)andfromxi(t)toxh(t). 61

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Figure2-11. TheresultsofaGPDCanalysisofthreeexampletimeseriesgeneratedfromstochastic,vanderPoloscillators.ThemaximumvaluesofGPDCi!j(f)andGPDCi!h(f)aremuchgreaterthanthemaximumGPDC(f)valuesoftheotherinteractions.Thus,GPDCcapturesthecorrectcausalityinthesystem,whichisfromxi(t)toxj(t)andfromxi(t)toxh(t). 62

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CHAPTER3CHARACTERIZATIONANDMITIGATIONOFTHEEFFECTSOFNOISEVARIANCEONTIMESERIESINFERENCETECHNIQUESNoiseisubiquitousinbiologicalsystems( Samoilovetal. 2006 ),withthenervoussystembeingnoexception( Faisaletal. 2008 ).Itisthereforeimportanttothoroughlyunderstandthateffectsofnoiseontimeseriesinferencestechniquesusedtodetermineeffectiveconnectivityinneuraltimeseriesrecordings.Noisecanbeendogenoustoaneuralsystembeinganalyzed,arising,forexample,fromthestochasticnatureofcellularmachineryorexogenous,suchasmeasurementnoisewhichmaybeaddedtosignalsbyrecordingdevices.Inthischapter,IinvestigatetheeffectsofnoiseonthetimeseriesinferencestechniquesdiscussedinChapter 2 .Specically,Iconsiderendogenousnoiseandexploretheimpactthatdifferencesinthevarianceofnoisebetweentimeserieshasonthesetechniques.Indthatpartialdirectedcoherence(PDC)andphasedynamicsmodeling(PDM)arebothhighlyaffectedbythesenoisevariancedifferences.WhilethisobservationhasbeenmadepreviouslyinregardstoPDC( Winterhalderetal. 2005 ),thisisthersttimeithasbeennotedasalimitationofPDM.Inlightofthisnding,IproposeamodiedPDMtechnique,whichIcallgeneralizedPDM(GPDM),thatisalongthesamelinesasapreviouslysuggestedmodicationofPDC,knownasgeneralizedPDC(GPDC)( Baccalaetal. 2007 ).UnlikePDM,GPDMisunaffectedbynoisevariancedifferences.IdemonstratetheapplicationofGPDMonanexamplesystemofcoupledoscillatorsandshowthatitisindeedimmunetonoisevariancedifferencesbetweentimeseries. 3.1NoiseVarianceandLinearTimeSeriesInferenceTechniques 3.1.1TheImpactofNoiseVarianceonLinearTimeSeriesInferenceTechniquesAsdemonstratedinSection 2.3 ,thelineartimeseriesinferencetechniquePDCishighlyaffectedbydifferencesinnoisevariancebetweentimeseries.Forinstance,giventwotimeseries,PDCislikelytoidentifycausalityfromthetimeserieswithalower 63

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noisevariancetothetimeserieswithahighernoisevarianceregardlessofthetrueeffectiveconnectivityofthesystem( Winterhalderetal. 2005 ).ThistendencyisduetothefactthatPDCdoesnotincludeinitsformulationthenoisevariancetermsofthelinearVARmodelrstttothetimeseriesdata,asexplainedinSection 2.3 .WhilethisledPDCtoerroneouslyidentifycausalityintheexamplegiveninSection 2.3 ,noisevariancedifferencesdonotinvariablyresultinPDCincorrectlyidentifyingcausality.Atrivialexampleofthisiswhencausalityinasystemisactuallydirectedfromalownoisevariancetimeseriestoahighernoisevariancetimeseries.Inthiscase,evenifPDCisaffectedthenoisevariancedifference,itsresultisnotincorrectsincecausalityfromthelowtohighnoisevariancetimeseriesisphysicallypresentinthesystem.Evenoutsideofthistrivialcase,however,theabovestatementholdstrue.AsIdemonstrateinthefollowing,theprobabilitythatPDCincorrectlyidentiescausalityfromalownoisevariancetimeseriestoahighernoisevariancetimeseriesdependsonthemagnitudeofthenoisevariancedifferenceaswellasthestrengthofthecausalconnectionfromthehighertothelowernoisevariancetimeseries.Considertwotimeseries,xi(t)andxj(t),suchthatxi(t)=xj(t)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+ii(t)xj(t)=jj(t),wherei(t)andj(t)areGaussianwhitenoiseprocesseswithzeromeanandastandarddeviationofone.Inaddition,var(i(t))=2iandvar(j(t))=2j.When>0,thetruecausalityinthissystemisfromxj(t)toxi(t).Furthermore,giventhatij,thissystemdoesnotrepresentthetrivialcasewherecausalityowsfromthelowtothehighnoisevariancetimeseries.Withthissysteminmind,onemayask,underwhatconditionsdoesPDCincorrectlyidentifycausalityfromxi(t)toxj(t)?Inordertoanswerthisquestion,IsimulatedtheabovesystemforarangeofandjvaluesandcomputedtheprobabilitythatPDCincorrectlyidentiedcausality 64

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fromxi(t)toxj(t).Specically,Iseti=1whilevaryingintherange[0,1]andjintherange[1,500].Foreachparametersetting,Isimulatedthesystem1,000timesandappliedPDCtoeachinstance.Then,byaveragingoverthe1,000samplesofeachsystem,IestimatedtheprobabilitythatPDCcorrectlyidentiedcausalityfromxj(t)toxi(t)aswellastheprobabilitythatitincorrectlyidentiedcausalityfromxi(t)toxj(t).Aprobabilisticanalysiswasnecessaryduetothestochasticnatureofthetimeseries.TheresultsarepresentedinFig. 3-1 .TheprobabilitythatPDCincorrectlyidentiescausalityfromxi(t)toxj(t),whichIdenoteasP(PDCi!j),isafunctionofbothandj.Asincreases,P(PDCi!j)decreasesandasjincreases,P(PDCi!j)increases.Sinceiisxedatone,anincreaseinjisalsoanincreaseinthemagnitudeofthenoisevariancedifferencebetweenxi(t)andxj(t),namelyji)]TJ /F11 11.955 Tf 12.5 0 Td[(jj.Inallcases,aslongas>0,PDCalwayscorrectlyidentiescausalityfromxj(t)toxi(t).Thus,theseresultsshowthattheprobabilitythatPDCincorrectlyidentiescausalityfromalownoisevariancetimeseriestoahighnoisevariancetimeseriesdependsonboth,thestrengthofcausalinuencefromthehightothelownoisevariancetimeseries,aswellasonjj)]TJ /F11 11.955 Tf 10.11 0 Td[(ij,themagnitudeofthenoisevariancedifferencebetweenthetimeseries.Asthestrengthofcausalinuencefromthehightothelownoisevariancetimeseriesincreases,PDCislesslikelytoincorrectlyidentifycausalityandasthemagnitudeofthenoisevariancedifferencebetweenthetimeseriesincreases,PDCismorelikelytoincorrectlyidentifycausality. 3.1.2MitigatingtheImpactofNoiseVarianceonLinearTimeSeriesInferenceTechniquesOnetechniqueformitigatingtheimpactofnoisevarianceonPDCistheGPDCtechniquedescribedinSection 2.3.1 .GPDCincorporatesthevarianceofthettedVARmodel'snoisetermsintotheoriginalPDCformulation,effectivelyaccountingforanynoisevariancedifferencebetweenthetimeseries(cf.Eq.( 2 ))( Baccalaetal. 65

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2007 ; Winterhalderetal. 2005 ).WiththismodicationtoPDCinplace,GPDCisnotaffectedbydifferencesinthenoisevariancebetweentimeseries.Infact,repeatingthesameanalysisthatwasperformedaboveonGPDCdemonstratesthatGPDCdoesnotincorrectlyidentifycausalityfromalownoisevariancetimeseriestoahighnoisevariancetimeserieswhenthiscausalityisnotphysicallypresentinthesystem.TheresultsfromthisanalysisarepresentedinFig. 3-2 .GPDCalwaysidentiesthetruecausalityfromxj(t)toxi(t)when>0andhasaverylowprobabilityofincorrectlyidentifyingcausalityfromxi(t)toxj(t),regardlessofthevalueoforjj)]TJ /F11 11.955 Tf 11.95 0 Td[(ij.Grangercausality(GC)isalsoimmunetotheeffectsofnoisevariancedifference,aresultofthefactthatittooincorporatesthevarianceofttedVARmodel'snoiseterms( Dingetal. 2006 ).TheresultsoftheaboveanalysisperformedonGCarepresentedinFig. 3-3 andareidenticaltotheGPDCresults.Thus,theconclusionsmadeaboveinregardstoGPDCareequallyapplicabletoGC.Asthisexampleillustrates,theimpactofnoisevarianceonlineartimeseriesinferencetechniquescanbemitigatedbyincludingintheirformulationthenoisevariancetermsofthelinearVARmodelttothetimeseriesdata.Inthefollowingsections,IusethisobservationtoproposeanewtechniqueformitigatingtheeffectsofnoisevarianceonPDM,anonlineartimeseriesinferencetechnique. 3.2NoiseVarianceandPhaseDynamicsModeling 3.2.1TheImpactofNoiseVarianceonPhaseDynamicsModelingLikePDC,PDMdoesnotincorporateinitsformulationanyinformationregardingthenoisetermsofthemodelthatisrstttothedata.ThiscanbeseeninEq.( 2 )whichdescribesthemodelthatisusedinthePDMtechniquetotthetimeseriesofinstantaneousphaseincrements.Thenoisetermi(t)presentinthepreviousequation,Eq.( 2 ),isnotpresentinEq.( 2 ),noristhistermincludedinthenaldenitionofPDMgiveninEq.( 2 ).IshowinthefollowingthatthisfactresultsinPDMbeinghighlyaffectedbydifferencesinnoisevariancebetweentimeseriesmuchlikePDC. 66

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ConsiderthesystemoftwocoupledvanderPoloscillatorsfromEq.( 2 )with!i=0.5,!j=0.97,"ij=0and"ji=0.01sothatthereiscouplingonlyfromxi(t)toxj(t).Byxingj=0.01andvaryingi,IcanobservetheeffectsofnoisevariancedifferencesonthePDMtechnique.Thetrivialcasewherecausalityactuallyowsfromthelownoisevariancetothehighnoisevariancetimeseriesisavoidedbyensuringthatij.IperformedthisexperimentbysimulatingthecoupledvanderPolsystemfordifferentvaluesofiintherange[0.01,20]andapplyingthePDMtechniquetotheresultingtimeseries.TheresultsarepresentedinFig. 3-4 ,whereanyPDMvalueabovethesignicancethreshold(indicatedbytheblackdottedline)issignicant.Theseresultsshowthatwheni10,PDMincorrectlyidentiescausalityfromthelownoisevariancetimeseries,xj(t),tothehighnoisevariancetimeseries,xi(t).Inallcases,regardlessofthevalueofi,PDMcorrectlyidentiesthetruecausalityfromxi(t)toxj(t).TheseresultsdemonstratethatPDMishighlyaffectedbydifferencesinnoisevariancebetweentimeseries.Specically,ifthemagnitudeofthenoisevariancedifferencebetweentimeseriesislargeenough(inthiscaseji)]TJ /F11 11.955 Tf 12.58 0 Td[(jjj10)]TJ /F7 11.955 Tf 12.58 0 Td[(0.01j=9.99),PDMwillidentifycausalityfromthetimeserieswithalowernoisevariancetothetimeserieswithahighernoisevarianceregardlessofthetrueeffectiveconnectivityofthesystem. 3.2.2ANewTechniqueforMitigatingtheImpactofNoiseVarianceonPhaseDynamicsModelingInthecaseoflineartimeseriesinferencetechniques,theeffectsofnoisevariancedifferencesaremitigatedbyincludingintheformulationofthesetechniquesthenoisevariancetermsfromthelinearVARmodelttothetimeseriesdata.Inasimilarfashion,IproposeamodicationofthePDMtechniquewhichservestomitigatethenegativeimpactofnoisevariancedifferences,whichIcallgeneralizedPDM(GPDM).ConsidertheformulationofthepairwisePDMtechniquepresentedin 67

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Eqs.( 2 2 2 2 ).Denotetheestimatedfunction,Fi(i(t),j(t)),obtainedbydoingaleastsquaresttotheinstantaneousphaseincrementtimeseries,i(t+))]TJ /F11 11.955 Tf -458.7 -23.91 Td[(i(t),as^Fi(i(t),j(t)).Computetheerrorof^Fi(i(t),j(t))asi(t)=^Fi(i(t),j(t)))]TJ /F7 11.955 Tf 11.96 0 Td[((i(t+))]TJ /F11 11.955 Tf 11.95 0 Td[(i(t)). (3)Then,letibethesamplestandarddeviationofi(t),computedasi=vuut 1 N)]TJ /F7 11.955 Tf 11.96 0 Td[(1NXt=1(i(t))]TJ /F7 11.955 Tf 12.77 0 Td[(i)2 (3)wherei=1 NPNt=1i(t)isthesamplemeanandNisthetotalnumberofdatapointsini(t)andj(t).GPDMisthendenedasGPDMj!i=1 is Xm,``2(a2m,`+b2m,`). (3)FollowingthemultivariateformulationofPDMinEq.( 2 ),multivariateGPDMcanbedenedasGPDMj!i=1 2i(2i,j+2i,j). (3)IappliedGPDMtothesystemoftwocoupledvanderPoloscillatorsdescribedabove.TheresultsarepresentedinFig. 3-4 ,whereanyGPDMvalueabovethesignicancethresholdindicatedbytheblackdottedlineissignicant.Theseresultsdemonstratethat,inallcases,regardlessofthedifferencebetweeniandj,GPDMcorrectlyidentiescausalityfromxi(t)toxj(t)anddoesnotincorrectlyidentifycausalityfromxj(t)toxi(t).Thus,likeGPDC,GPDMdoesnotincorrectlyidentifycausalityfromalownoisevariancetimeseriestoahighnoisevariancetimeserieswhenthiscausalityisnotphysicallypresentinthesystem. 68

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Figure3-1. TheprobabilitythatPDCincorrectlyidentiescausalityfromtimeseriesxi(t)toxj(t)(topplot)andtheprobabilityPDCcorrectlyidentiescausalityfromxj(t)toxi(t)(bottomplot)arebothpresentedasafunctionof,thestrengthofthecausalconnectionfromxj(t)toxi(t),andj,thenoisevarianceofxj(t).Sincethenoisevarianceofxi(t)isxedati=1,jalsorepresentsthemagnitudeofthenoisevariancedifferencebetweenthetimeseries,ji)]TJ /F11 11.955 Tf 11.96 0 Td[(jj.PDChasahighprobabilityofincorrectlyidentifyingcausalityfromxi(t)toxj(t)whenislowandji)]TJ /F11 11.955 Tf 11.96 0 Td[(jjishigh. 69

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Figure3-2. TheprobabilitythatGPDCincorrectlyidentiescausalityfromtimeseriesxi(t)toxj(t)(topplot)andtheprobabilityGPDCcorrectlyidentiescausalityfromxj(t)toxi(t)(bottomplot)arebothpresentedasafunctionof,thestrengthofthecausalconnectionfromxj(t)toxi(t),andj,thenoisevarianceofxj(t).Sincethenoisevarianceofxi(t)isxedati=1,jalsorepresentsthemagnitudeofthenoisevariancedifferencebetweenthetimeseries,ji)]TJ /F11 11.955 Tf 11.96 0 Td[(jj.GPDChasalowprobabilityofincorrectlyidentifyingcausalityfromxi(t)toxj(t)regardlessoforji)]TJ /F11 11.955 Tf 11.95 0 Td[(jj. 70

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Figure3-3. TheprobabilitythatGCincorrectlyidentiescausalityfromtimeseriesxi(t)toxj(t)(topplot)andtheprobabilityGCcorrectlyidentiescausalityfromxj(t)toxi(t)(bottomplot)arebothpresentedasafunctionof,thestrengthofthecausalconnectionfromxj(t)toxi(t),andj,thenoisevarianceofxj(t).Sincethenoisevarianceofxi(t)isxedati=1,jalsorepresentsthemagnitudeofthenoisevariancedifferencebetweenthetimeseries,ji)]TJ /F11 11.955 Tf 11.96 0 Td[(jj.GChasalowprobabilityofincorrectlyidentifyingcausalityfromxi(t)toxj(t)regardlessoforji)]TJ /F11 11.955 Tf 11.96 0 Td[(jj. 71

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Figure3-4. TheresultsofaPDManalysisoftwocoupledvanderPoloscillatorsarepresentedwiththePDMvaluesfromoscillatoritoj(topplot)andfromoscillatorjtoi(bottomplot)asafunctionofthenoisevarianceofi,i.Sincethenoisevarianceofjisxedatj=0.01,ialsorepresentsthemagnitudeofthenoisevariancedifferencebetweentheoscillators,ji)]TJ /F11 11.955 Tf 11.95 0 Td[(jj.Theblackdottedlineindicatesthesignicancethresholdof0.024.Anyvalueabovethisthresholdindicatesasigniantcausalinteraction.PDMincorrectlyidentiescausalityfromoscillatorjtoiwhenji)]TJ /F11 11.955 Tf 11.96 0 Td[(jjj10)]TJ /F7 11.955 Tf 11.95 0 Td[(0.01j=9.99.PDMalwaysidentiesthetruecausalityfromitojregardlessofji)]TJ /F11 11.955 Tf 11.96 0 Td[(jj. 72

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Figure3-5. TheresultsofaGPDManalysisoftwocoupledvanderPoloscillatorsarepresentedwiththeGPDMvaluesfromoscillatoritoj(topplot)andfromoscillatorjtoi(bottomplot)asafunctionofthenoisevarianceofi,i.Sincethenoisevarianceofjisxedatj=0.01,ialsorepresentsthemagnitudeofthenoisevariancedifferencebetweentheoscillators,ji)]TJ /F11 11.955 Tf 11.95 0 Td[(jj.Theblackdottedlineindicatesthesignicancethresholdof0.15.Anyvalueabovethisthresholdindicatesasigniantcausalinteraction.GPDMneverincorrectlyidentiescausalityfromoscillatorjtoiandalwaysidentiesthetruecausalityfromitojregardlessofji)]TJ /F11 11.955 Tf 11.95 0 Td[(jj. 73

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CHAPTER4DETECTINGEFFECTIVECONNECTIVITYINNETWORKSOFCOUPLEDNEURONALOSCILLATORSTheimportanceofneuronaloscillatornetworksincoordinatingbrainfunctionhasbeenwell-establishedinnumerousstudies.Atthesingle-celllevel,specializedcircuitsofneuronaloscillatorshavebeenimplicatedinmemoryformation,visualprocessing,learning,andavarietyofothermotorandcognitivetasks( Buzsaki 2006 ; BuzsakiandDraguhn 2004 ; Deisteretal. 2009 ; Fries 2009 ).Ofparticularinterestistheabilityofneuronalnetworkstosynchronizetheirringactivity,sincethisbehaviorhasbeenlinkedtopathologicalbrainstatessuchasepilepsy( UhlhaasandSinger 2010 )andParkinson'sdisease( Hammondetal. 2007 ).Threekeycomponentsofneuronalnetworksthatplayacentralroleintheemergenceofpathologicalsynchronization,are:(a)thephysicalpresenceofcouplingbetweenneuronaloscillators,referredtoasstructuralconnectivity,(b)thestatisticaldependencebetweenneuronaloscillators,referredtoasfunctionalconnectivity,and(c)thecausaleffectsofoneneuronaloscillatoronanother,referredtoaseffectiveconnectivity( Spornsetal. 2004 ).TwopopulartimeseriesinferencetechniquesfordeterminingeffectiveconnectivityincoupledoscillatornetworksincludeGrangercausality(GC)( Geweke 1982 1984 ; Granger 1969 )andphasedynamicsmodeling(PDM)( RosenblumandPikovsky 2001 ).Anumberofrecentstudieshavefocusedoncomparingtheabilityofthesetechniquestocorrectlyidentifyeffectiveconnectivityinagivennetworkofcoupledoscillators( Lungarellaetal. 2007 ; Smirnovetal. 2007 ; Winterhalderetal. 2005 ).However,asystematicstudyoftheapplicabilitydomainsofthesetechniquestoneuronalnetworks,whichtakesintoaccountthecouplingstrength,thedegreeofnoiseinthenetwork,thelevelofneuronalnetworksynchronyandthetypeofsynapticinteraction(chemicalvs.electrical)isstilllacking.Inthischapter,IfocusontheseissueswithinthecontextofasimplenetworkofcoupledMorris-Lecar(ML)neuronaloscillators( MorrisandLecar 1981 ). 74

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IbeginbyconsideringthecaseoftwounidirectionallycoupledMLneuronsunderthreedifferentcouplingscenarios:1)lineardiffusivecouplingwhichmodelsanelectricalsynapseorgapjunction( Bennett 1997 ; ChowandKopell 2000 ),2)nonlinear,fastthresholdmodulationcouplingwhichmodelsafasttimescalechemicalsynapse( SomersandKopell 1993 ),and3)nonlinearthresholdcouplingthatmodelsaslowtimescalechemicalsynapse( Abarbaneletal. 2003 ).Foreachcouplingscheme,sampletimeseriesdatasetsaregeneratedbyvaryingthecouplingstrengths,noiselevels,andintrinsicringfrequenciesofthecoupledneuronaloscillators.Theresultisalargeensembleoftimeseriesdatasetsfromwhichtheregionsofapplicabilityofseveraltechniquesusedtoestimateeffectiveconnectivitycanbeinvestigated.Theapplicabilitydomainsofthesevarioustechniquesarethendeterminedusingdecisiontreeclassiersthataretrainedandtestedonthesetimeseriesdatasets.Theresultisametaheuristicapproachthatcanbeappliedbypractitionerswithoutanypriorknowledgeofnetworkstructureordynamicssinceallthetimeseriesfeaturesusedtodetermineapplicabilitycanbemeasureddirectlyfromthetimeseriesthemselves.Formystudy,Iconsiderthelinearspectralmethods:GC(Eqs.( 2 2 2 ))( Geweke 1982 1984 ; Granger 1969 ),partialdirectedcoherence(PDC)(Eq.( 2 ))( BaccalaandSameshima 2001 )andamodiedversionofPDCknownasthegeneralizedPDC(GPDC)(Eq.( 2 ))( Baccalaetal. 2007 ; Winterhalderetal. 2005 ).IalsoincludePDMinmyanalysissinceitisspecicallydesignedfornarrow-bandinteractingoscillatorsandcanhandleamoderateamountofnonlinearityinthedata(Eqs.( 2 2 ))( RosenblumandPikovsky 2001 ).Therestofthischapterisorganizedasfollows.First,Idescribetheneuronaloscillatormodelandnetworksusedtogeneratesampledatasets.Next,Ioutlinethemethodologyusedforgeneratingthetestdataandtheconstructionoftheclassiers.Theseclassiersarethenusedtoassesstheapplicabilityofthesetechniquesfor 75

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estimatingtheeffectiveconnectivityinthegivennetwork.Finally,Ipresenttheresults,adiscussionofmystudyandasummaryofthechapter. 4.1Neuron,SynapseandNetworkModels 4.1.1NeuronModelInthisstudy,IusetheMLspikingneuronmodelasageneralexampleofaminimal,conductance-basedneuronaloscillator( Izhikevich 2007 ; MorrisandLecar 1981 ).Thismodelisdescribedbythefollowingequations:Cdv dt=)]TJ /F5 11.955 Tf 9.3 0 Td[(gCam1(v)(v)]TJ /F5 11.955 Tf 11.96 0 Td[(VCa))]TJ /F5 11.955 Tf 11.95 0 Td[(gKw(v)]TJ /F5 11.955 Tf 11.96 0 Td[(VK))]TJ /F5 11.955 Tf 11.95 0 Td[(gL(v)]TJ /F5 11.955 Tf 11.95 0 Td[(VL)+Idc+Inoise+Isynapse (4)dw dt=w1(v))]TJ /F5 11.955 Tf 11.95 0 Td[(w 1(v)whereInoise=DIsynapse=kf(vpost,vpre)m1(v)=0.51+tanhv)]TJ /F5 11.955 Tf 11.95 0 Td[(V1 V2w1(v)=0.51+tanhv)]TJ /F5 11.955 Tf 11.96 0 Td[(V3 V41(v)=coshv)]TJ /F5 11.955 Tf 11.96 0 Td[(V3 2V4)]TJ /F8 7.97 Tf 6.59 0 Td[(1Herevrepresentsthetransmembranevoltageofaneuron,wrepresentstheactionofthepotassiumcurrent,Idcistheexternalinputcurrent,InoiseistheinputcurrentduetonoiseandIsynapseistheinputcurrentfromsynapticallyconnectedneurons.IntheequationforInoise,isGaussianwhitenoisewithzeromean,unitvarianceandDisthenoiseintensity.IntheequationforIsynapse,kisthestrengthofcouplingandthefunction,f(vpost,vpre),representsthecouplingtypewherevpreisthemembranevoltageofthepresynapticneuronandvpostisthemembranevoltageofthepostsynapticneuron.Thisfunctionischangedinordertomodeldifferentrealisticsynapses,specicallylineardiffusivecoupling,fastthresholdmodulationcoupling,andnonlinearsynapticthreshold 76

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coupling.Theremainingparametersaresetasfollows:V1=)]TJ /F7 11.955 Tf 9.29 0 Td[(1.2mV,V2=18mV,V3=12mV,V4=17.4mV,VL=)]TJ /F7 11.955 Tf 9.3 0 Td[(60mV,VK=)]TJ /F7 11.955 Tf 9.3 0 Td[(84mV,VCa=120mV,gL=2S/cm2,gK=8S/cm2,gCa=4S/cm2,=1 15s)]TJ /F8 7.97 Tf 6.59 0 Td[(1,andC=20F/cm2.Fortheseparameters,themodelexhibitsTypeIexcitability,whichmeansthatoscillationscanoccurwitharbitrarilylowfrequencyascurrentisinjectedintotheneuron( Ermentrout 1996 ). 4.1.2SynapseModelsThefollowingthreerealisticsynapticcouplingmodelsareconsidered: 1. Lineardiffusivecouplingwhichmodelsanelectricalsynapseorgapjunction( Bennett 1997 ; ChowandKopell 2000 ):f(vpost,vpre)=vpre)]TJ /F5 11.955 Tf 11.96 0 Td[(vpost (4) 2. Nonlinear,fastthresholdmodulationcouplingwhichmodelsasimpliedchemicalsynapse( SomersandKopell 1993 ):f(vpost,vpre)=0.51+tanhvpre)]TJ /F5 11.955 Tf 11.96 0 Td[(Vth Vslope(vpost)]TJ /F5 11.955 Tf 11.95 0 Td[(Vrev) (4)whereVth=0mV,Vslope=1mVandVrevrepresentsthesynapticpotential,whichissetto-84mVtomimictheeffectofinhibitorycoupling. 3. Nonlinear,thresholdcouplingwhichmodelsachemicalsynapseandaccountsforthedynamicsofthereleaseandabsorptionofneurotransmittersinthesynapticcleft( Abarbaneletal. 2003 ; Talathietal. 2010 ):f(vpost)=S(t)(vpost)]TJ /F5 11.955 Tf 11.95 0 Td[(Vrev)) (4)dS dt=S1(vpre(t)(t)))]TJ /F5 11.955 Tf 11.95 0 Td[(S(t) [S0)]TJ /F5 11.955 Tf 11.96 0 Td[(S1(vpre(t)(t))]whereVrevisdenedasinEq.( 4 ).(t)=Pi(t)]TJ /F5 11.955 Tf 12.15 0 Td[(ti)((ti+R))]TJ /F5 11.955 Tf 12.15 0 Td[(t)where()istheHeavisidestepfunctionandtiisthetimeoftheithpresynapticneuronalspike.Thetimescaleconstantgoverningreceptorbindingisrepresentedbyandissetto7.9ms,S0=1.013,andR=(S0)]TJ /F7 11.955 Tf 12.79 0 Td[(1).Finally,S1isthesigmoidalfunctiondenedasS1(v)=0.5(1+tanh(120(v)]TJ /F7 11.955 Tf 11.96 0 Td[(0.1))). 4.1.3NetworkModelIconsidersimplenetworkscomposedofeithertwoorthreecoupledMLneuronaloscillatorsinthiswork.Therstnetworkiscomprisedoftwounidirectionallycoupled 77

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neuronmodelswhosenetworkstructureisillustratedinFig. 4-1 A.Thesecondisanensembleof12different3-neuronnetworks,wherenetworksareconstructedbypermutingthepresenceandabsenceofthevariousconnectionsinFig. 4-1 B. 4.2ClassierConstructionMethodology 4.2.1DenitionofTimeSeriesFeaturesIbeginbydeningthetimeseriesfeaturesusedtocharacterizethedatasetsinvestigatedinthisstudy.Ithendescribehowthesefeaturesareusedtoobtainthedecisiontreesthatdeterminetheapplicabilitydomainsofthevarioustechniquesusedtoestimatingeffectiveconnectivity. 1. Phasecoherence(PC):isametricthatcanbeusedasameasureofsynchronybetweentwooscillators.Giventwotimeserieswithinstantaneousphase,i(t)andj(t),PCcanbedenedas( Mormannetal. 2000 ):PC(i,j)=p hcos(i(t))]TJ /F11 11.955 Tf 11.96 0 Td[(j(t))i2+hsin(i(t))]TJ /F11 11.955 Tf 11.95 0 Td[(j(t))i2 (4)wherehidenotestimeaveraging.Thismeasureissymmetricini(t)andj(t)andPC(i,j)=1whenthereiscompletephasesynchronybetweentheoscillators,i.e.,wheni(t))]TJ /F11 11.955 Tf 11.96 0 Td[(j(t)=constant. 2. Coefcientofvariationofinterspikeinterval(CV):isameasureofdynamicalnoiseintheneuronaloscillator.Givenneuronalmembranepotential,onecanestimatetheinter-spikeinterval(ISI)asthetimeintervalbetweenconsecutivecrossingsofthemembranepotentialwithsomethreshold.FromthedistributionofISIs,oneobtainsthemeanISIandthevariance2ISI.CVisthendenedas( Perkeletal. 1967 ):CV=ISI ISI (4)ACVofoneindicatesaPoissonprocesswhereallspikesoccurindependentlywhileaCVofzeroindicatesaseriesofperfectlyperiodicspikes.ThefollowingpairwisemeasureofCVisintroducedtoaccountforagivenpairofneurons:CVmean(i,j)=1 2(CVi+CVj) (4)CVdiff(i,j)=(CVi)]TJ /F1 11.955 Tf 11.96 0 Td[(CVj) 78

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3. Firingfrequency(FF):isobtainedasFF=1 ISI (4)Againintroducingthefollowingpair-wisemeasureforFFFFmean(i,j)=(FFi)]TJ /F1 11.955 Tf 11.95 0 Td[(FFj) max(FFi,FFj) (4)ThetimeseriesfeaturesgiveninEqs.( 4 4 4 )canbecalculatedfromthemembranevoltagetimeseriesofaneuronaloscillator.Thisisincontrasttomodelparameterssuchask,D,andIdcwhichareoftenhiddeninexperimentaldatasets.Previousstudieshavefocusedeitheronndingregionsofmodelparameterspacewherethetimeseriesinferencetechniquesforestimatingeffectiveconnectivityfail( Kayseretal. 2009 ; Lungarellaetal. 2007 ; Winterhalderetal. 2007 ),oronnon-dynamicalfeaturesofadatasetsuchassignal-to-noiseratioordatasetlengththatinuencetheaccuracyofagiventechniqueforestimatingeffectiveconnectivity( Astoletal. 2007 ; SmirnovandAndrzejak 2005 ).Unlikethesepastworks,mystudypresentsresultsinthespaceofthetimeseriesfeaturesoutlinedabove,makingitmoreapplicableforpractitionerswhowishtoapplythetimeseriesinferencetechniquesconsideredheretoestimateeffectiveconnectivity. 4.2.2DecisionTreeConstructionThefollowingisanoverviewoftheprocedureadoptedtoconstructdecisiontreesusingthetimeseriesfeaturesdescribedinsection 4.2.1 1. Generateanensembleoftimeseriesdatasetsbysystematicallyvaryingtheparameters,k,D,andIdcoftheMLneuronsinEq.( 4 )whicharecoupledaccordingtoFig. 4-1 A.RepeatthisforeachofthethreesynapsetypespresentedinSection 4.1.2 2. CalculatethetimeseriesfeaturesdenedinSection 4.2.1 foreachdataset. 3. EstimateeffectiveconnectivityusingGC,PDC,GPDC,andPDMforeachtimeseries. 79

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4. Trainaclassierforagiventimeseriesinferencetechniquebysettingthetimeseriesfeaturesaspredictorsandtheoutputgeneratedbythechosentechniqueastheresponse. 5. Testtheclassiersbygeneratinganewensembleof3-neurontimeseriesdatasets(Fig. 4-1 B)andevaluatingclassierperformanceonthisnewdata.Forstep1,atotalof1,000timeseriesdatasetsweregeneratedforeachsynapsetypebysystematicallyvaryingkofneuron2intherange[1e-4,0.5],Dofbothneuronsintherange[0,80],andIdcofbothneuronsintherange[40,85].Onlyunidirectionalcouplingswereconsidered,sok=0forneuron1.AtotalofN=51,000datapointsweregeneratedforeachtimeserieswiththerst1,000datapointsthrownouttoremovetransients.Eq.( 4 )wassolvednumericallyviathestochasticEuler-Maruyamamethodwithastepsizeoft=0.005( KloedenandPlaten 1999 ).Timeseriesfeatureswerecalculatedinstep2asdenedinSection 4.2.1 .Instantaneousphase,whichisneededtocalculatephasecoherence,wasobtainedviaHilberttransformofthevoltagetimeseries(cf.Eq.( 2 ))( Pikovskyetal. 2002 ).Thefourtechniquesfordeterminingeffectiveconnectivitywereappliedtothedatasetsinstep3.Theclassiersinstep4wereconstructedusingMatlab'sclassregtreefunctionwhichtrainsadecisiontreebasedonagivensetofpredictorsandresponses(alsoreferredtoasfeaturesandcategories,respectively)( Breimanetal. 1984 ; Hastieetal. 2001 ).Foreachmethodandcouplingtype,adecisiontreewasconstructed.Inthiswork,thepredictorsarethetimeseriesfeaturesdenedinSection 4.2.1 includingPC(i,j),CVmean(i,j),abs(CVdiff(i,j)),andabs(FFdiff(i,j))andtheresponsesaretheoutputsgeneratedbythefourtechniquesinstep3.AbsolutevaluesofCVdiff(i,j)andFFdiff(i,j)areusedfortheclassiersince,inanexperimentalsetting,thereisnoaprioriorderingofneurons.Asanexample,theGC/linearcouplingdecisiontreewasconstructedbysupplyingtheclassregtreefunctionwiththePC(i,j),CVmean(i,j),abs(CVdiff(i,j)),and 80

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abs(FFdiff(i,j))valuesofall1,000linearcouplingdatasetsaspredictors.Inaddition,thefunctionwasgiventhecorrespondingGCresultastheresponseforeachdataset.Responseswerelabeledaseither`1'ifGCdetectedthecorrectdirectionalityoftheinteractionor`0'ifGCfailedtodetectcorrectdirectionality.SimilardecisiontreeswereconstructedforPDC,GPDCandPDM.Decisiontreeswereusedinthisworkbecausetheyhadthelowestmisclassicationerrorcomparedtootherclassicationmethodssuchaslinearorquadraticdiscriminantanalysis,NaiveBayes,andsupportvectormachineswhenappliedtothesimulateddatasets(seeTable 4-1 andSection 4.2.3 ).Theperformanceofthedecisiontreeclassierswasveriedusinga10-foldcross-validationscheme( Schaffer 1993 ).Instep5,Itestedtheapplicabilityofthedecisiontreeclassierstothegeneralclassofamulti-nodeneuronalnetworkbyrstgeneratinganewensembleof3-neurontimeseriesdatasetsaccordingtothenetworkstructureinFig. 4-1 B.ValuesfortheMLmodelparametersk,D,andIdcwererandomlyselectedwithintheaforementionedrangesand1,000newtimeseriesdatasetsweregeneratedforeachcouplingtypeusingthepreviouslydescribedmethodology.Thepresenceanddirectionallyofthepossiblecouplingsinthenetworkwassetrandomly.Next,Icalculatedasetofpredictorsfromthisnewdatathatcouldbeclassiedbythepreviouslyconstructeddecisiontrees.Onceagain,thetimeseriesfeaturesdenedinSection 4.2.1 wereusedaspredictors.Inthiscase,timeseriesfeatureswerecalculatedforeachnewtesttimeseriesdatasetbyconsideringonlypairwiseinteractions.Thus,eachtimeseriesdatasetconsistedofthreevaluesforeachtimeseriesfeature,i.e.,PC(1,2)capturesthephasecoherencebetweenneuron1andneuron2,PC(1,3)thephasecoherencebetweenneuron1and3,andPC(2,3)thephasecoherencebetweenneuron2and3,withtheremainingtimeseriesfeaturesspeciedinasimilarfashion.Forinstance,givenasingledataset,threepredictorswereconstructed:oneconsistingofthePC(i,j),CVmean(i,j),abs(CVdiff(i,j)),and 81

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abs(FFdiff(i,j))valuesbetweenneurons1and2,asecondconsistingofthesetimeseriesfeaturesforneurons1and3,andathirdwiththesetimeseriesfeaturesforneurons2and3.Withanensemblesizeof1,000,atotalof3,000predictorswasconstructed.Thedecisiontreeclassierswerethenappliedtothisnewsetoftestpredictors,yieldingasetofclassierresponses.Theseclassierresponses,labeledeither`1'or`0',indicatedwhetheraparticulartimeseriesinferencetechniquewaspredictedtobecorrectorincorrectbyitscorrespondingdecisiontree.Finally,IcalculatedtheactualresponsesobtainedwhenIappliedeachofthefourtechniques,GC,PDC,GPDCandPDM,tothenew3-nodenetworkdatasets.PDCandGPDCwereappliedasdescribedinEqs.( 2 2 )sincetheseformulasimplicitlyhandlenetworkswithmorethantwonodes.MultivariateformulationsofGCandPDMwereusedinordertoproperlyhandlethismultivariatedata.Specically,conditionalspectraldomainGCwiththepartitionmatrixmethodgivenbyEq.( 2 )wasusedalongwithmultivariatePDM,givenbyEq.( 2 ).Theresultsofthefourtechniquesfordeterminingeffectiveconnectivitywerelabeledaseither`1'or`0,'denotingthatthetechniquewaseithercorrectorincorrect,respectively.Theclassierresponseswerethencomparedwiththeactualresponsesofagiventimeseriesinferencetechniqueinordertocalculatethetree'saccuracy.TheexperimentalproceduredescribedaboveisillustratedinFig. 4-2 4.2.3ResultingDecisionTreesDecisiontreeswereconstructedforeachofthefourtechniquesfollowingthemethodologydescribedinSection 4.2.2 .Inasimilarfashion,Ialsoconstructedclassiersbasedonlinearandquadraticdiscriminantanalysis,NaiveBayes,andsupportvectormachines.Iused10-foldcross-validationoneachoftheresultingclassiersinordertoestimatethetesterror,whichistheexpectedmisclassicationerrorofaclassierwhenitisappliedtodatathatisindependentoftheoriginaltraining 82

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data( Schaffer 1993 ).Theresults,presentedinTable 4-1 ,showthatthedecisiontreeclassierhadthelowesttesterroroftheveclassiersinvestigated.Therefore,Ichosedecisiontreeclassiersforthisstudy.Afterconstructingdecisiontreesforeachcouplingtype,itwasfoundthatthebasicstructureofthedecisiontrees,suchasthefeaturesoneachbranchandtheirordering,wasthesameacrossallthreetypesofcouplingthatwereconsidered.However,thespecicfeaturevaluesoneachdecisionbranchpointdidvaryamongthedifferentcouplingtypesforeachtecnique.AnexampleofthisisshowninFig. 4-3 forGC.Ratherthanpresentseparatedecisiontreesforeachcouplingtype,Ichosetoaveragethefeaturevaluesatthesebranchpointsinordertogetacompositedecisiontreeforeachofthefourtechniquesthatcouldbeusedregardlessofthetypeofcouplingbetweenneuronaloscillators.Sinceapractitionermaynotknowtheexactformofcouplingbetweenneuronsinagivenexperimentaldataset,thisapproachisalsothemostapplicableinpractice.ThecompositetreesareshowninFig. 4-4 andwereusedtoderivethefollowingresults.Eachdecisiontreecontainsonlyasubsetoftheoriginalfourtimeseriesfeaturesusedaspredictorsforgeneratingthetrees.Thisprovidesinsightintothepredictivefeaturesofagivendatasetthatarecriticalforeachofthefourtechniquesintermsoftheirabilitytocorrectlydetermineeffectiveconnectivityinthenetwork.ThePDCtreeisthemostextremeexampleasitonlyrequiresPC(i,j)topredictwhetherornotthemethodwillwork.TheresultthatPDCislikelytofailwhenPC(i,j)0.69correspondstotheresultsobtainedinSmirnovetal.'sstudywhichindicatedthatPC(i,j)valuesabove0.75or,insomecases,above0.5weredetrimentaltoPDC( Smirnovetal. 2007 ).ThetreesconstructedforGCandGPDCarealmostidenticalduetothefactthatthesetechniquesnearlyalwayspredictthesameeffectiveconnectivityfor2-nodenetworks.Forinstance,outofthe1,000,2-nodetrainingdatasetsconsidered,GCand 83

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GPDCagreedonthedirectionalityofinteraction97.0%,97.4%,and97.8%ofthetimeforlinear,fastthresholdmodulation,andsynapticcoupling,respectively.Thus,forbothtechniques,thedecisiontreesindicatethatPC(i,j)andCVmean(i,j)arecriticalfactorsfordeterminingcorrecteffectiveconnectivity.ThefactthatCVmean(i,j)mustbeequalorgreaterthan0.022inthecaseofbothGCandGPDC,ismostlikelyareectionofthefactthatbothtechniquesrequiresomeamountofdynamicalnoisebepresentinthesysteminorderforthemtowork.TherequirementthatPC(i,j)mustbelessthan0.92or0.87forGCandGPDC,respectively,restrictsthelevelofsynchronizationthatcanexistbetweentwosignalsandisanexpectedrequirementbasedonpreviousstudiesofthesetechniques( Smirnovetal. 2007 ).Incontrasttotheothertechniquesconsidered,PDMisable,tosomeextent,todetecteffectiveconnectivityacrossdifferentfrequenciesduetothemandntermsinEq.( 2 )( SmirnovandBezruchko 2009 ).Infact,variationintheintrinsicringfrequenciesbetweenneuronaloscillatorsisbenecialtothetechniquebecauseitresultsina1(t)versus2(t)trajectorythatllsupthephasespace( RosenblumandPikovsky 2001 ).OnlyinthiscasecanthefunctionF(i(t),j(t))inEq.( 2 )beproperlyestimatedwithi(t)andj(t)asindependentvariables.ThisisillustratedinFig. 4-5 wherethe1(t)versus2(t)phasespacetrajectoriesfortwodatasets,onewithahighabs(FFdiff(i,j))andonewithalowabs(FFdiff(i,j)),areshown.DespitepreviousobservationsthatPDMaccuracyishighlydependentonPC(i,j)( Smirnovetal. 2007 ),myresultsindicatethatPC(i,j)isnotascriticalafactorindeterminingwhetherthetechniquewillfailasothermetricssuchasFFdiff(i,j)andCVmean(i,j).Thesensitivityandspecicityofeachdecisiontreewascalculatedasfollows:sensitivity=#oftruepositives #oftruepositives+#offalsenegatives (4) 84

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specicity=#oftruenegatives #oftruenegatives+#offalsepositives (4)wheretruepositivesinthiscaseoccurwhenthedecisiontreelabelsadatasetas`1'andthetimeseriesinferencetechniqueproducesacorrectresult,truenegativesoccurwhenthedecisiontreelabelsadatasetas`0'andthetechniqueproducesanincorrectresult,falsepositivesoccurwhenthedecisiontreelabelsadatasetas`1'andthegiventechniqueproducesanincorrectresult,andfalsenegativesoccurwhenthedecisiontreelabelsadatasetas`0'andthegiventechniqueproducesacorrectresult.Asensitivityof100%meansthedecisiontreeidentiesallcaseswhenthetechniqueiscorrectandaspecicityof100%meansthedecisiontreeidentiesallcaseswhenthetechniqueproducesincorrectresults.Compositetreesforeachtechniquewereusedalongwith10-foldcross-validationtocalculatesensitivityandspecicityvalues.TheresultsarepresentedinTable 4-2 .Inlightoftheseresults,Icannowcommentonthechoiceofusingonlyunidirectionalcouplinginthetrainingdataensemble.Whentimeseriesdatasetswithbidirectionalcouplingwereincludedinthetrainingensemble,thesensitivityofthedecisiontreesdroppedsignicantly.However,whenIapplieddecisiontreesthatwerebasedonlyonunidirectionalcouplingtodatasetswithbidirectionalcoupling,thesensitivitywasnotaffected.Thisindicatesthatthespaceoftimeseriesfeatureswhereunidirectionalcouplingcanbedetectedismuchmorerestrictivethanthoseregionswherebidirectionalcouplingcanbedetected.Thereasonforthisisthatbidirectionalcouplingbiasesthedecisiontreestowardhighvaluesofphasecoherencesince,regardlessofactualcouplingdirectionality,ahighphasecoherencevaluealmostalwaysresultsinthetimeseriesinferencetechniquespredictingbidirectionalcouplingbetweenneurons.Thus,ifIincludedbidirectionalconnectionsinthetrainingdata,thedecisiontreeswouldfavordatasetswithahighphasecoherencevalueeventhoughthisis 85

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generallydetrimentalfordetectingunidirectionalcoupling.AsaresultIdecidedtouseonlyunidirectionallycoupledMLnetworkdatasetstoconstructtheclassiers.Sincethisanalysiswasperformedonnetworksof2nodes,thequestionremainsastotheutilityofthedecisiontreesderivedaboveforanalyzingtheapplicabilityoftimeseriesinferencetechniquetolargernetworks.IaddressthisquestioninthenextsectionwhereIdemonstratetheutilityofmydecisiontreeclassiersinthecaseof3-nodenetworks. 4.2.4ExtensiontoLargerNetworksIappliedtheclassierdecisiontreesofFig. 4-4 totimeseriesdatasetsgeneratedfromanensembleof3-neuronnetworks.Agivendatasetfromthisensemblewasclassiedatotalofthreetimesbythedecisiontree,onceforeachofitsthreepairwiseinteractions,withthefollowingresults:adatasetcouldeitherhaveallofitspairwiseinteractionsclassiedas`1',twopairwiseinteractionsclassiedas`1'andoneas`0',onepairwiseinteractionclassiedas`1'andtwoas`0',orallthreeclassiedas`0.'Multivariatetimeseriesinferencetechniqueswereappliedtoeachdataset.Ithencalculatedthelikelihoodthatagiventechniquecorrectlyidentiedthedirectionalityofallinteractionsinthe3-neuronnetworkgiventhenumberofpairwisenetworkinteractionsclassiedas`1'bythedecisiontree.Forexample,considera3-neuronnetworkwithcouplingfromneuron1to2andfromneuron3to1.Giventhatonlytwoofthethreepairwiseinteractionsinthisnetworkwereclassiedbythedecisionas`1',Icalculatedthelikelihoodthatatechniquewillcorrectlyidentifythecorrectnetworkcouplingfromneuron1to2and3to1.TheselikelihoodsarepresentedinTable 4-3 .Compositetreesforeachmethodwereusedforclassication.Inorderforallofthefourtechniquesconsideredheretobecorrectwithprobabilitygreaterthan50%,amajorityofthepairwiseinteractionsinthenetworkmustbeclassiedbythedecisiontreeas`1.'ThisresultisshowninthethirdcolumnofTable 4-3 whichgivestheaccuracyofeachtechniquegiventhattwoofthethreepairwise 86

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interactionsare`1.'TheseresultsarenotsignicantlydifferentfromthesensitivityvaluesinTable 4-2 .Ontheotherhand,accuracypercentagesforthetechniqueswhenallthreepairwiseinteractionsareclassiedas`0'ina3-nodenetworkarepresentedintherstcolumnofTable 4-3 .Inthiscase,theprobabilityofanygiventechniquesucceedingislessthan20%.Forinstance,theprobabilitythatGCgivescorrectresultsinthiscaseisonly166.6%.GiventheseaccuraciesintherstcolumnofTable 4-3 ,noneofthesetechniquesshouldbeappliedifallpairwiseinteractionsinadatasetarecategorizedas`0'bythedecisiontreesince,withhighprobability,theresultswillinfactbeincorrect. 4.3RecommendationsforDecisionTreeClassierUseinNetworksofCoupledNeuronalOscillatorsNumerousstudieshaveappliedlineartimeseriesinferencetechniquessuchasGCandPDCornonlinearmethodssuchasPDMtoneuraldatasetsinanattempttogaininsightintothedirectionalityofinteractionsinthebrain( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ; Havliceketal. 2010 ; Liaoetal. 2010a ; Satoetal. 2009 ).Datasetsaregenerallyconsideredamenabletosuchananalysisaslongastheyarestationary.Theresultsofthisstudydemonstratethenecessityforanalyzingtimeseriesdatasetsbeyondjustdeterminingtheirstationarity.Synchronyisoneexampleofatimeseriesfeaturethatcanresultinthefailureofmanytechniquesandpreviousworkhasindicatedapproximateboundsonthedegreeofphasecoherencebetweentwooscillatorsabovewhichseveralofthesetechniquestendtofail( Bezruchkoetal. 2003 ; Smirnovetal. 2007 ).Iextendedtheseresultsbyreningthelimitsonphasecoherenceaswellasconsideringlimitsonothertimeseriesfeatures,suchasnoiseandringfrequency,forthreeofthemostwidelyusedlinearmethodsfordeterminingeffectiveconnectivity,GC,PDCandGPDC,aswellasPDM,whichcanhandlesomenonlinearities.Thisistherstsystematiccomparativestudyofthesefourtechniquesastheyapplytonarrow-bandoscillatorydata. 87

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Inaddition,Iconsideredseveralrealisticmodelsforneuronalsynapses.Theobservedconsistencyinthederiveddecisiontreesregardlessofcouplingtypedemonstratesthatthesendingscanbeappliedtotimeseriesdatasetsevenwhenthecouplingtypeisunknown.Importantly,thefeaturesIconsideredareobservablefromthetimeseriesoftheoscillatorsthemselveswhichmeansthatpractitionerscanreadilyapplytheseresultstotheirexperimentaldatasets.Thisisincontrasttostudieswhichinvestigatetheapplicabilityoftimeseriesinferencetechniquesintermsofmodelparameters,suchascouplingstrength,whicharenotasevidentfromatimeseriesalone( Kayseretal. 2009 ; Lungarellaetal. 2007 ; Winterhalderetal. 2007 ).Thecentralcontributionofthischapteristheintroductionofamethodologytoassesswhetheragiventechniquefordeterminingeffectiveconnectivitywillproducecorrectresultswhenappliedtoagivendatasetbasedontheobservable,dynamicalcharacteristicsofthedataset.Theproposedmethodologyconsistsofdecisiontreeclassiersthatcanbeappliedwithoutanypriorknowledgeofnetworkstructure.Whenapplyingtechniquesfordeterminingeffectiveconnectivitytosmallnetworksof2-3nodes,Irecommendthatpractitionersrstusethedecisiontreesderivedinthisworktodetermineaclassicationforeachofthepairwiseinteractionsintheirnetwork.Ifamajorityoftheseinteractionsareclassiedas`0'bythedecisiontreeforagiventechnique,thetechniqueshouldnotbeappliedsincetheresultswillmostlikelybeincorrect.Ifamajorityareclassiedas`1',ontheotherhand,thetechniquecanbeappliedwithanexpectedprobabilityofsuccessgreaterthan66%.Extendingthisresulttoarbitrarilylargenetworksisbeyondthescopeofmywork.However,resultsfrom3-nodenetworksstronglysuggestthatforanynetwork,regardlessofsize,ifallpairwiseinteractionsareclassiedas`0'bythedecisiontreeofagiventechnique,thenthetechniqueshouldnotbeappliedtothedatasinceresultswillbeincorrectwithhighprobability.Itislikelythecasethatanevenlessrestrictivestatementcanbemade,namelythatthesetechniqueswillhaveverylowsuccessratesaslongasamajorityof 88

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thepairwiseinteractionsinanetworkareclassiedas`0.'Icautionagainsttheuseofagiventechniquefordeterminingeffectiveconnectivityinsuchacase.Insummary,thefollowingkeyconclusionscanbedrawnfromthischapter: Thedecisiontreespresentedprovideamethodologyfordeterminingtheapplicabilityofagiventimeseriesinferencetechniquetoagivensetoftimeseriesdata. Thisdecisiontreemethodologycanbeappliedwithoutanypriorknowledgeofnetworkstructureordynamicssinceallthetimeseriesfeaturesusedtodetermineapplicabilitycanbemeasureddirectlyfromthetimeseriesthemselves. Inthecaseofsmall,2-3nodenetworks,myresultscanbeapplieddirectly. Forlargernetworks,Irecommendthatagiventechniquenotbeappliedifamajorityofitspairwiseinteractionsareclassiedas`0'bythedecisiontreeforthattechnique. IncontrasttoGC,PDCandGPDC,differencesinintrinsicringfrequenciesbetweentwotimeseriesisbenecialforPDMsinceitresultsinamoredispersedinstantaneousphasetrajectory. 89

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Table4-1. Averagetesterrorofvariousclassierstrainedonsimulateddatasets.Testerrorisaveragedoverthefourtimeseriesinferencetechniquesandthreecouplingtypes. ClassierTypeTesterror Lineardiscriminantanalysis35.5%Quadraticdiscriminantanalysis35.2%NaiveBayeswithkerneldensityestimation26.0%Supportvectormachinewiththirdorderpolynomialkernel26.6%Decisiontree24.7% Table4-2. Decisiontreesensitivityandspecicityvalueswith95%condenceintervalsgiveninparentheses. sensitivityspecicity GCTree72.0(7.7)%81.7(3.1)%GPDCTree74.5(7.9)%81.5(3.0)%PDCTree66.4(4.5)%81.7(3.9)%PDMTree80.5(4.4)%77.0(3.3)% Table4-3. Likelihoodthatagiventimeseriesinferencetechniquecorrectlyidentiescausalityinanetworkgiventhenumberofpairwiseinteractionsclassiedas`1'bythedecisiontree.Ninetyvepercentcondenceintervalsaregiveninparentheses. 0outof3`1'1outof3`1'2outof3`1' GC12(3.3)%20(3.5)%67(7.0)%GPDC12(4.4)%43(4.6)%66(8.3)%PDC16(6.6)%46(4.0)%74(3.9)%PDM18(5.7)%25(2.4)%84(5.4)% 90

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Figure4-1. ThenetworkstructureofcoupledMLneuronaloscillatorsconsideredinthischapter.A)Neurons1and2areunidirectionallycoupledwithNeuron1drivingNeuron2.B)Neurons1,2and3arerandomlyconnectedwithdashedlinesindicatingpossiblecouplingbetweenneurons 91

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Figure4-2. Aowchartillustratingtheexperimentaldesignofmystudy.NumberscorrespondtothelistinSection 4.2.2 92

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Figure4-3. DecisiontreesconstructedforGCacrossthreedifferentcouplingtypesconsidered.Alltreessharethesamestructurebutvaryintheexactpredictorvaluesateachdecisionpoint. 93

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Figure4-4. Compositedecisiontreesforalltimeseriesinferencetechniques.Thesetreeswereconstructedbyaveragingthepredictorvaluesateachdecisionpointoverallcouplingtypesforeachtechnique. 94

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Figure4-5. Thetrajectoriesoftwoinstantaneousphases,1(t)versus2(t),fromdatasetswithlinearcouplingandA)abs(FF(i,j)diff)=2.4e-4,B)abs(FFdiff(i,j))=0.4.Alargerabs(FFdiff(i,j))valueresultsinthetrajectoryllingupmoreofthephasespace. 95

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CHAPTER5DETERMININGEFFECTIVECONNECTIVITYINNERUALMASSMODELNETWORKSInthepreviouschapter,Iconsideredtheproblemofeffectiveconnectivitydeterminationinsystemsofspikingoscillators.Suchoscillatorsarecommonlyusedtomodelbraindynamicsatthesingle-neuronscale( Izhikevich 2007 ; MorrisandLecar 1981 ).Inpractice,dataatthesingle-neuronscalecanberecordedbypatchclampingindividualcellsorthroughtheuseofmultielectrodearraysimplantedinvivo( Hamilletal. 1981 ; Potter 2001 ).Duetothespecializedandinvasivenatureofsuchrecordingtechniques,however,dataatthesingle-neuronscaleisnotaswidelyavailableaselectroencephalogram(EEG)orfunctionalmagneticresonanceimaging(fMRI)data.EEGsandfMRIsrecorddataattheneuronalpopulationscale,capturingthecombinedbehaviorofmillionsofindividualinteractingneuronsthatmakeupvariousregionsofthebrain( Kandeletal. 2000 ).Thenoninvasivenatureofthesetoolshasmadethempopularforstudyingthebrain'selectricalactivityandasaresult,thereisnoshortageofEEGandfMRIdataavailableforanalysis.NumerousstudieshavemadeuseofthisEEGandfMRIdatatoanalyzeeffectiveconnectivitybetweenbrainregions( Davidetal. 2008 ; Kiebeletal. 2009 ; LiandOuyang 2010 ; Ludtkeetal. 2010 ).Forinstance,timeseriesinferencetechniquesincludingGrangercausality(GC),partialdirectedcoherence(PDC),thedirectedtransferfunction(DTF),aswellasmanyothershaveallbeenappliedtoEEGandfMRIdatainanattempttouncoverthedirectedowofinformationbetweenregionsofinterestinthebrain( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ; Havliceketal. 2010 ; Liaoetal. 2010a ; Satoetal. 2009 ).However,fewstudieshaveexploredthevalidityofsuchtechniqueswhenappliedtononlinearandpotentiallybroadbandtimeseriesdatafromEEGandfMRIrecordings.Infact,somecriticismhasbeenleveledagainsttheapplicationofthesetechniquestofMRIdatasinceanfMRImeasuresthehemodynamic(bloodow)responsetoneuralactivityinthebrainratherthantheelectricalsignalsof 96

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theneuronsthemselves( Davidetal. 2008 ; Friston 2009b ).Becauseofthis,observedfMRIdatamayviolatetheassumptionthatthereareuniformresponseandrecordingdelaysinsystemvariables,onwhichthesetimeseriesinferencetechniquesarebased(cf.Chapter 1 ).Methodsforreducingtheeffectsofnon-uniformdelayscanbeemployedtomitigatethisproblembutarebeyondthescopeofmywork( Roebroecketal. 2005 ).Instead,inthischapter,IwillfocusontheapplicationoftimeseriesinferencetechniquestoEEGtimeseries.WhiletimeseriesinferencetechniqueshavebeenwidelyappliedtoexperimentallyrecordedEEGtimeseries,itisdifculttovalidatetheresultsobtainedsince1)thetruestructuralconnectivityofthebrainisrarelyknownwithabsolutecertaintyand2)thepossibleconnectivitycongurationsthatcanbeexploredarelimitedtothosethatareexperimentallyobserved.AmorerigorousapproachtovalidatingthesetechniquesinvolvessimulatingamodelofinterconnectedneuronalpopulationsthatiscapableofreproducingobservedEEGsignals.Becausethestrengthanddirectionalityofconnectionsinthismodelcanbesetbyhand,thetrueconnectivityofthemodelwouldbeknownpriortotheapplicationoftimeseriesinferencetechniques.Inthisway,theresultsofthesetechniquescouldbecomparedagainstknownconnectivity.Furthermore,thismodelwouldallowfortheexplorationofawiderangeofconnectivitycongurationssothatthetimeseriesinferencetechniquescouldbeevaluatedunderanumberoffeasiblescenarios.AssumingthatthemodelchosentoreplicatetheEEGsignalsisasatisfactoryrepresentationoftheneuronalpopulationsgeneratingtheEEGsignals,suchanapproachwouldservetodemonstratethevalidityofapplyingtimeseriesinferencetechniquestoEEGdata.Infact,severalsuchmodelsofneuronalpopulationsarealreadyinuse.Inthemid-1970s,LopesdaSilvaetal.andFreemanproposedmassactionmodelsofcorticalcolumnsinthebrain,whichwereshowntoreplicatesomeaspectsofEEGsignals( Freeman 1978 ; daSilvaetal. 1976 ).Thesemodelsweresubsequentlyimproved 97

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uponinworkbyJansenandRitandWendlingetal.andbecameknownasneuralmassmodels(NMMs)( JansenandRit 1995 ; Wendlingetal. 2000 ).NMMshavebeenusedtosimulateawiderangeofEEGfeatures,includingalpharhythms( JansenandRit 1995 ),synchronizationinthegammaband( SchillenandKonig 1994 ),andepilepticpatterns( Wendlingetal. 2002 ).Morerecently,NMMshavebeenshowncapableofreproducingtheentirespectrumofEEGsignals( DavidandFriston 2003 ).ThefactthatNMMsnotonlyproduceoutputthatisrepresentativeofrecordedEEGsignalsbutarealsobasedonthephysicalstructureofneuronalpopulationsinthebrain,makesthemexcellentcandidatesforinvestigatingtheapplicationoftimeseriesinferencetechniquestoEEGsforthepurposeofdeterminingeffectiveconnectivity.However,onlyafewstudieshaveconsideredNMMsforthispurpose.Forinstance,Astoletal.appliedseveraltimeseriesinferencetechniquesinthespectraldomainincludingthedirectedtransferfunction(DTF)anditsmodicationknownasthedirectDTF(dDTF)alongwithpartialdirectedcoherence(PDC)toNMMs( Astoletal. 2007 ).Thegoaloftheirworkwastodeterminetheeffectofsignal-to-noiseratio(SNR)anddatalength(DL)ontheeffectiveconnectivityestimatesproducedbythesetimeseriesinferencetechniques.However,sinceAstoletal.wereconcernedsolelywiththeeffectsofSNRandDL,theyonlyconsideredthemagnitudeoftheeffectiveconnectivityestimates.Forexample,toinvestigatetheimpactofSNR,theylookedatthemagnitudeoftheoutputproducedbyeachtechniqueasafunctionofSNRanddeterminedthatmagnitudeandSNRwereindeedcorrelated.Thisapproachdidnotconsider,however,theaccuracyofeachtechniquesincenothresholdsofsignicanceweretakenintoaccount.Thus,thisstudydidnotinvestigatethevalidityofthesetechniqueswhenappliedtoNMMs.Similarly,LiandOuyangcomparedtheapplicationofGrangercausality(GC)andtheirproposedmethodofpermutationconditionalmutualinformationtoNMMsbutonlyconsideredthemagnitudeoftheoutputproducedbythesetechniques( LiandOuyang 98

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2010 ).Thus,whiletheywerealsoabletodemonstratetherelativeeffectsofSNRandDLonthesetechniques,theycouldnotdrawconclusionsabouttheaccuracyofGCingeneralwhenappliedtoNMMs.AninvestigationintotheaccuracyoftimeseriesinferencetechniquesastheyapplytoeffectiveconnectivitydeterminationinNMMshasyettobeexploredintheliteraturebutisofcriticalimportancesincethesetechniquesareoftenusedtoanalyzeexperimentalEEGdata( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ).Furthermore,thefactthatmanyofthesetechniquesarelinear,includingGCandPDC,callsintoquestiontheirapplicabilitytononlinearEEGsignals.Evenanonlineartechniquesuchasphasedynamicsmodeling(PDM),maynotbeappropriatetoapplytoEEGdatasincePDMrequiresnarrow-bandinputandmanyEEGsignalsarespreadacrossarangeoffrequencies( RosenblumandPikovsky 2001 ).Inthefollowingstudy,IinvestigatetheaccuracyofseveraltimeseriesinferencetechniquesastheyapplytoeffectiveconnectivitydeterminationinNMMnetworks.Specically,IconsidertheaccuracyofGC,generalizedPDC(GPDC)andPDMwhenappliedtoanNMMnetworkintroducedbyZavagliaetal.( Zavaglia 2008 ),whichinturnisbasedonanNMMproposedbyWendlingetal.( Wendlingetal. 2000 ).IbeginwithasimplecasewhereeachneuronalpopulationintheNMMnetworkexhibitsrhythmicactivityatasingledominantfrequency.UsingthisNMMnetwork,IshowthatGPDCisabletoaccuratelyrecoverthetrueconnectivityofthemodelwithhighprobability.Next,Iconsiderthecaseofinterconnectedneuronalregionsofinterest(ROIs),eachofwhichcontainsseveralNMMswithdifferentdominantrhythmicfrequencies( Ursinoetal. 2007 ; Zavagliaetal. 2006 ).ThismodelismorecomplexsinceeachROIexhibitsrhythmicactivityoverarangeoffrequencyvalues.Inthiscase,Ishowthatnoneoftheaforementionedtimeseriesinferencetechniquesaccuratelyrecoversthetrueconnectivityofthemodel. 99

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InordertoaddresstheobservedinadequaciesofthetimeseriesinferencetechniquesthatIconsidered,IproposeanewtechniquebasedonPDMandtheempiricalmodedecomposition(EMD)thatiscapableofaccuratelyrecoveringeffectiveconnectivityinanROI-basedNMM.Inthefollowing,IdescribetheapplicationofmyproposedtechniqueanddemonstrateitseffectivenessonanROI-basedNMM. 5.1NeuralMassModelsInthefollowing,Iconsidertheneuralmassmodel(NMM)introducedbyZavagliaetal.( Zavaglia 2008 ),whichisamodiedversionoftheoneoriginallyproposedbyWendlingetal.( Wendlingetal. 2000 ).Thismodelconsistsoffourinterconnectedneuronalgroupsnamelypyramidalneurons,excitatoryinterneurons,inhibitoryinterneuronswithslowsynaptickinetics,andinhibitoryinterneuronswithfastsynaptickinetics.Neuronsarelumpedtogetherineachgroupanditisassumedthattheysharethesamemembranepotential.Therearenodendritesorintrinsicconductancesinthemodel,insteadlumpedrepresentationsofneuronsineachgroupcommunicateviatheaverageringrateoftheentiregroupofneurons.ThesinglepopulationNMMdescribedaboveisillustratedinFig. 5-1 .Eachneuronalgroupisrepresentedbyemboldenedrectangles.Thesignals,yi(i=0,1,2,3),owingintoeachoftheseneuronalgroupsrepresenttheaveragepostsynapticmembranepotentialofanotherneuronalgroup.Withineachgroup,thesemembranepotentialsarerstconvertedintoanaveragepresynapticspikedensityviaastaticsigmoidalfunction.Theresultingaveragepresynapticspikedensityisthenconvertedbackintopostsynapticmembranepotentialviaasecondorderlineartransferfunction.Thesetransferfunctionshaveimpulseresponseshe,hi,andhg,representingthesynapticeffectofexcitatory(pyramidalandexcitatoryinterneurons),slowinhibitoryinterneuronsandfastinhibitoryinterneurons,respectively.Thesystemofdifferentialequationsdescribingthismodelareasfollows: 100

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PyramidalNeuronsdy0(t) dt=y5(t)dy5(t) dt=Aa1z0(t))]TJ /F7 11.955 Tf 11.95 0 Td[(2a1y5(t))]TJ /F5 11.955 Tf 11.96 0 Td[(a21y0(t)z0(t)=2e0 1+er(s0)]TJ /F9 7.97 Tf 6.58 0 Td[(v0)v0(t)=C2y1(t))]TJ /F5 11.955 Tf 11.95 0 Td[(C4y2(t))]TJ /F5 11.955 Tf 11.96 0 Td[(C7y3(t)ExcitatoryInterneuronsdy1(t) dt=y6(t)dy6(t) dt=Aa1z1(t)+p(t) C2)]TJ /F7 11.955 Tf 11.95 0 Td[(2a1y6(t))]TJ /F5 11.955 Tf 11.96 0 Td[(a21y1(t)z1(t)=2e0 1+er(s0)]TJ /F9 7.97 Tf 6.58 0 Td[(v1)v1(t)=C1y0(t)SlowInhibitoryInterneuronsdy2(t) dt=y7(t)dy7(t) dt=Bb1z2(t))]TJ /F7 11.955 Tf 11.96 0 Td[(2b1y7(t))]TJ /F5 11.955 Tf 11.95 0 Td[(b21y2(t)z2(t)=2e0 1+er(s0)]TJ /F9 7.97 Tf 6.58 0 Td[(v2)v2(t)=C3y0(t)FastInhibitoryInterneuronsdy3(t) dt=y8(t)dy8(t) dt=Gg1z3(t))]TJ /F7 11.955 Tf 11.95 0 Td[(2g1y8(t))]TJ /F5 11.955 Tf 11.96 0 Td[(g21y3(t)z3(t)=2e0 1+er(s0)]TJ /F9 7.97 Tf 6.58 0 Td[(v3)v3(t)=C5y0(t))]TJ /F5 11.955 Tf 11.95 0 Td[(C6y2(t) (5) 101

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Intheaboveequations,vi(i=0,1,2,3),representstheaveragemembranepotentialsofthefourdifferentgroups.Thesesignalsaresuppliedasinputstothesigmoidfunctionswhichconvertthemintoaveragespikedensities,representedbyzi(i=0,1,2,3).Theaveragespikedensities,inturn,entersynapses,eachofwhichisdescribedbyanaveragegainfactorandatimeconstant.TheaveragegainsareA,B,andGandthecorrespondingtimeconstantsarea1,b1,andg1forexcitatory,slowinhibitoryandfastinhibitorysynapses,respectively.Thesynapseoutputsareagainpostsynapticmembranepotentialsrepresentedbyyi(i=0,1,2,3).InteractionsamongneuronsarecapturedbythesevenconnectivityconstantsCi(i=1,...7)whichrepresenttheaveragenumberofsynapticcontacts.Finally,p(t),aGaussianwhitenoisesignalwithmean,,andvariance2,representsallexogenousinputstothemodelincludingtheaveragedensityofafferentactionpotentialsandexcitationfromexternalsources.Byvaryingthesynaptickineticsofthevariousgroupsofneuronsinthismodel,onecanchangetheintrinsicringrhythmofthepopulationasawhole.Forexample,theparametersA,B,G,a1,b1,andg1ofthismodelcanbetunedinordertomaketheNMMasawholeexhibitanintrinsicringrhythmatalow,mediumorhighfrequency.Morespecically,Zavagliaetal.identiedvaluesfortheseparameterssuchthatthepopulationexhibitedeithertheta/alphabandrhythms(4-12Hz),betabandrhythms(12-30Hz),orgammabandrhythms(>30Hz)( Zavaglia 2008 ).TheseparametervaluesaregiveninTables 5-1 and 5-2 .ThepopulationscorrespondingtothesevariousrhythmsaredenotedasLFforlowfrequency,MFformediumfrequency,andHFforhighfrequency,respectively.Thepowerspectraldensity(PSD)ofanLF,MF,andHFpopulationisillustratedinFig. 5-2 .Zavagliaetal.usedtheseNMMstoinvestigatetheeffectsofdifferentconnectivitypatternsbetweenLF,MF,andHFpopulations( Zavaglia 2008 ).Specically,theyconsideredanetworkcomposedofthreeNMMpopulations,oneLFpopulation,oneMFpopulation,andoneHFpopulationwiththeconnectivitypatternsillustratedin 102

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Fig. 5-3 ,anddemonstratedthatthePSDofeachpopulationandthecoherencebetweenpopulationswerebothafunctionoftheunderlyingconnectivityofthenetwork.Next,theyusedthenetworkinFig. 5-3 Aandadjustedthestrengthsofcouplingbetweenthepopulationsinordertotthemodel'sPSDtothePSDofanEEGrecordedfromthecingulatedcortexoftestsubjects.Zavagliaetal.alsoextendedtheirmodeltoconsiderentireROIswithinthebrain( Zavagliaetal. 2006 ).Inthiscase,eachROIiscomposedofthreeNMMpopulations(oneLF,oneMFandoneHF)andcanbeconnectedtoanynumberofotherROIstoformanetwork.AsimplenetworkoftwocoupledROIsisillustratedinFig. 5-4 .BecauseeachROIiscomposedofthreedifferentpopulations,eachwithadifferentintrinsicringrhythm,theROIasawholecanexhibitrhythmicringacrossarangeoffrequencies.ThisisillustratedinFig. 5-5 whichshowsthePSDofasingleROI.Muchliketheirpreviousstudy,Ursinoetal.rstinvestigatedtheimpactthatdifferentROInetworkconnectivitypatternshadonthePSDofeachROI( Ursinoetal. 2007 ).Finally,usinganetworkofthreeROIs,theyadjustedthestrengthofvariousconnectionsinthenetworkinordertotthePSDofeachROItothePSDofEEGsrecordedfromseveraltestsubjectsperformingngermovingtasks.Ursinoetal.putthismethodforwardasatechniqueforestimatingeffectiveconnectivityofthebrainviaphysiologicalmodels( Ursinoetal. 2007 ).Determiningconnectivityinthiswayisamodel-drivenapproachsinceitreliesonanassumednetworkofROIs,witheachROIcomposedofseveralNMMpopulationspossessingparticularfrequencycharacteristics.Generaltimeseriesinferencetechniquesaremuchmoreversatilesincetheyarenotpredicatedonaparticularmodeloftheunderlyingsystem.Asofyet,however,theirefcacywhenappliedtononlinear,EEG-likedatahasnotbeenveried. 103

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5.2DeterminingEffectiveConnectivityinNetworksofNeuralMassModelPopulationsIbeganbyconsideringnetworksofNMMpopulationssimilartothatillustratedinFig. 5-3 ( Zavaglia 2008 ).Specically,thenetworksIconsideredwereeachcomposedofthreeNMMpopulations,oneLF,oneMFandoneHFpopulation,alongwithallpossiblecongurationsofeffectiveconnectivitybetweenthepopulations.Inordertoevaluatetheaccuracyofvarioustimeseriesinferencetechniqueswhenappliedtosuchnetworks,Irstsimulated1,000ofthesenetworkswiththepresenceanddirectionalityofcouplingbetweenpopulationschosenatrandom.1Bidirectionalconnectionswereallowedbutself-connectionswerenot.Foreachnetwork,IappliedthetimeseriesinferencetechniquesconditionalGC,GPDCandmultivariatePDMinordertoestimatetheeffectiveconnectivitybetweenNMMpopulations(refertoChapter 2 forimplementationdetailsofeachtechniquealongwithcalculationofsignicancethresholds).AnexampleofoneofthesenetworksalongwithoutputfromGC,GPDC,andPDMwhenappliedtothenetworkisshowninFig. 5-6 .Sincethetrueeffectiveconnectivityofeachnetworkwasknowninadvance,Iwasabletocalculatetheaccuracyofeachtechniqueacrossthe1,000simulatednetworks.Anaccuratetechniqueisonethata)identiesalleffectiveconnectionsthatarepresentinthenetworkandb)doesnotidentifyanyeffectiveconnectionsthatarenotinthenetwork.Thesefeaturescanbequantiedusingthefollowingstatisticaltermsastheyrelatetoidentifyingeffectiveconnectivityusingtimeseriesinferencetechniques: Truepositive-thetechniqueidentiesaneffectiveconnectionthatisactuallypresentinthenetwork 1ThestochasticdifferentialequationsgiveninEq. 5.1 weresolvedusingthestochasticEuler-Maruyamamethodwithastepsizeoft=0.001( KloedenandPlaten 1999 ). 104

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Truenegative-thetechniquedoesnotidentifyaneffectiveconnectionthatisnotactuallypresentinthenetwork Falsepositive-thetechniqueidentiesaneffectiveconnectionthatisnotactuallypresentinthenetwork Falsenegative-thetechniquedoesnotidentifyaneffectiveconnectionthatisactuallypresentinthenetworkUsingthesestatistics,sensitivityandspecicityvaluesforeachofthetechniquescanbecalculatedusingthefollowingformulas:sensitivity=#oftruepositives #oftruepositives+#offalsenegatives (5)specicity=#oftruenegatives #oftruenegatives+#offalsepositives (5)Asensitivityof100%meansthatthetechniqueidentiesalleffectiveconnectionsthatarepresentinthenetworkwhileaspecicityof100%meansthatthetechniquedoesnotidentifyanyeffectiveconnectionsthatarenotinthenetwork.SensitivityandspecicityvaluesforeachofthethreetechniquesarepresentedinTable 5-3 .ThehighspecicityandlowsensitivityofGCandPDMindicatesthatthesetechniquestendtorejectmanyconnectionsthatareactuallypresentinthesystem.Clearly,GPDCoutperformsbothofthesetechniquesintermsofbothsensitivityandspecicity,withvalues92%and90%,respectively.Furthermore,thesehighsensitivityandspecicityvaluesareagoodindicationthatGPDCisaviabletechniquefordeterminingeffectiveconnectivityinnetworksofNMMpopulationswheneachpopulationexhibitsasingledominantringfrequency.Severalreal-worldexamplesofsystemsexhibitingsingledominantfrequenciesincludebetabandoscillationsobservedintheprimarymotorcortexduringmotormaintenancebehaviorandthalamocorticalalphabandoscillationsobservedattheonsetofsleep( Brovellietal. 2004 ; Steriadeetal. 1993 ).SuchsystemsaregoodcandidatesforanalysisviatheGPDCtechnique. 105

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5.3DeterminingEffectiveConnectivityinNetworksofNeuralMassModelRegionsofInterest 5.3.1DeterminingEffectiveConnectivityViaGrangerCausality,PartialDirectedCoherenceandPhaseDynamicsModelingNext,IconsiderednetworkscomposedoftwointerconnectedNMMROIssuchasthatillustratedinFig. 5-4 ( Ursinoetal. 2007 ; Zavagliaetal. 2006 ).Similartomypreviousanalysis,Isimulated1,000ofthesenetworkswiththepresenceanddirectionalityofconnectionschosenatrandom.2Bidirectionalconnectionswereallowedbutself-connectionswerenot.IthenappliedconditionalGC,GPDCandmultivariatePDMtoeachofthesenetworksinordertoobtainestimatesofthenetworks'effectiveconnectivityandonceagaincalculatedeachtechnique'ssensitivityandspecicitystatistics.TheresultsarepresentedinTable 5-4 .Inthiscase,PDMistheleastaccurateofthethreetechniques.Itslowspecicityvaluemeansthatitoftenconcludesthataneffectiveconnectionispresentevenwhenitnotactuallyinthenetwork.PDM'sinaccuracyisanexpectedresultconsideringthatPDMisdesignedforuseonnarrow-bandsignalsandNMMROIsproducesignalsspreadacrossmultiplefrequencies.However,theseresultsalsoindicatethatGCandGPDCarenotreliablewhenappliedtosuchnetworks.Thelowspecicityvaluesofbothtechniquesindicatethattheyoftenincorrectlyidentifyaneffectiveconnectionaspresentwhenitisnotactuallyinthenetwork.ThiscallsintoquestionthevalidityofresultsobtainedwhenapplyingthesetechniquestogeneralEEGsignalscontainingoscillatoryactivityacrossarangeoffrequencies.Inthefollowing,Ipresentanovelalternativetothesetechniquesforuseonsuchsignals. 2ThestochasticdifferentialequationsgiveninEq. 5.1 weresolvedusingthestochasticEuler-Maruyamamethodwithastepsizeoft=0.001( KloedenandPlaten 1999 ). 106

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5.3.2EMD-GPDM:ANovelTechniqueforDeterminingEffectiveConnectivityinNetworksofNeuralMassModelRegionsofInterestThetimeseriesinferencetechniqueIintroduceherefordeterminingeffectiveconnectivityisbasedonPDMandempiricalmodedecomposition(EMD)( Huangetal. 1998 ).Byitself,PDMisnotapplicabletotimeseriesderivedfromNMMROIssince,asIdemonstratedabove,thespectralcharacteristicsofthesetimeseriesarenotnarrow-band.Inordertoovercomethislimitation,IuseEMDtorstseparateanNMMROItimeseriessignalintoseveralofitsconstituentsignalcomponentsknownasintrinsicmodefunctions(IMFs).DetailsontheextractionofIMFsusingEMDisgiveninSection 5.4 .AsaconsequenceofEMD,eachIMFisnarrow-bandandhasawell-denedinstantaneousphase.Thus,IMFsarewell-suitedtoPDManalysis.AsIdemonstratebelow,byapplyingPDMtotheseIMFsindividually,oneisabletodeterminetheeffectiveconnectivityofanetworkofNMMROIs.ItshouldbenotedthatinorderforPDMtoworkproperlyinthisapplication,amodicationtothetechniqueisneeded.ThismodiedPDMtechniqueisreferredtoasgeneralizedPDM(GPDM)andisdiscussedinmoredetailinChapter 3 .GPDMisessentialformyproposedapproachsinceIMFsvarywidelyintheamountofnoisetheycontain.TherstIMFsthatEMDextractsfromasignalcontainnearlyallofthenoisethatwasintheoriginalsignalwhilethelastfewIMFscontainalmostnonoise.Forinstance,letusconsiderIMFiandIMFjwhenivar(IMFj),wherevar()denotesvariance.ThisisillustratedinFig. 5-7 whichshowstwoIMFsextractedfromthetimeseriesofanNMMROI.ThevarianceofIMF4is2.08whilethevarianceofIMF9is0.162.Asaconsequence,whenPDMisappliedtoIMF4andIMF9,effectiveconnectivityisnearlyalwaysidentiedasgoingfromIMF9toIMF4,regardlessofthetrueeffectiveconnectivityinthesystem.Inthisexamplecase,thereisnoactualeffectiveconnectivitybetweenthesignalssincebothoriginatefromthesameNMMROI 107

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whichdoesnothaveaselfconnection.Nevertheless,PDMgivesavalueof0.77intheIMF9toIMF4directioncomparedtoavalueof0.005intheIMF4toIMF9direction.ThemechanismbehindPDM'sbiasofmisidentifyingeffectiveconnectivityascomingfromalessnoisysignalandgoingtoamorenoisysignalisgiveninChapter 3 .Inaddition,IintroducedinthischapteratechniquetermedGPDMwhichremovesthisbiasandisthereforeabletoidentifyeffectiveconnectivityacrossdifferentIMFsregardlessoftheirnoisedifferenceor,inotherwords,regardlessoftheorderinwhichtheywereextractedfromthesignal.BecausemynewtechniqueutilizesbothEMDandGPDMtoextracteffectiveconnectivityfromNMMROItimeseries,IrefertoitasEMD-GPDM.TheapplicationofEMD-GPDMtotwotimeseries,x(t)andy(t),involvesthefollowingsteps: 1. ApplyEMDtox(t)andy(t)inordertoobtainIMFsforeachtimeseries.AssumethetotalnumberofIMFsofx(t)isnandthetotalnumberofIMFsofy(t)ism.DenotetheseIMFsasIMFixandIMFjywhereiandjrangefrom1tonandm,respectively. 2. CalculatethePSDofeachIMF,denotedasPSDix(f)andPSDjy(f)fortimeseriesx(t)andy(t),respectively.ThenndthemaximumpowerofeachIMFasPSDmaxix=max(PSDix(f))orPSDmaxjy=max(PSDjy(f)). 3. FindtheIMFforeachtimeseriesthatsatises:^i=argmaxi(PSDmaxix)^j=argmaxj(PSDmaxjy)Inotherwords,foreachtimeseries,ndtheIMFwiththegreatestoverallpower. 4. UseGPDMtoestimatetheeffectiveconnectivitybetweenIMF^ixandIMF^jy. 5. IfGPDMfromIMF^ixtoIMF^jyissignicant,theneffectiveconnectivityfromx(t)toy(t)existsinthenetwork.Similarly,ifGPDMfromIMF^jytoIMF^ixissignicant,theneffectiveconnectivityfromy(t)tox(t)existsinthenetwork. 108

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AnexampleoftheapplicationofEMD-GPDMtotwoNMMROItimeseriesispresentedinFig. 5-8 .IappliedmyEMD-GPDMtechniquetothe1,000previouslysimulatedNMMROInetworksinordertodemonstrateitsaccuracy.TheresultingsensitivityandspecicitystatisticsoftheEMD-GPDMtechniquearepresentedinTable 5-5 .Withsensitivityandspecicityvaluesofatleast90%,itisclearthatEMD-GPDMcorrectlydetermineseffectiveconnectivityinanNMMROInetworkwithhighprobability.TheseresultsprovideastrongindicationthatEMD-GPDMisaviabletechniquefordeterminingeffectiveconnectivityinsystemswithtimeseriessimilartothosederivedfromnetworksofNMMROIs.SuchsystemsarecommonlyencounteredandincludealargefractionofrecordedEEGsignals.Thus,thistechniqueiswidelyapplicabletoalargenumberofrecordedEEGtimeseries.AsfurtherproofoftheneedforGPDMinstep4oftheEMD-GPDMtechnique,IrepeatedtheaboveanalysiswithPDMinplaceofGPDM.TheresultsarepresentedinthesecondrowofTable 5-5 .Fromthespecicityvalue,itcanbeseenthatPDMperformspoorlywhenusedinplaceofGPDMforthereasonsdiscussedabove.ThisprovidesaclearmotivationfortheuseoftheGPDMtechniquewhenconsideringnoisytimeseries. 5.4EmpiricalModeDecompositionEMDisatechniqueproposedbyHuangetal.foranalyzingnonlinear,non-stationaryand/orbroadbandsignals( Huangetal. 1998 ).Suchsignalshaveoftenposedaproblemfortraditionalharmonicandempiricalorthogonalfunction(EOF)analyses.Forinstance,theFouriertransformofanon-stationarysignalisill-denedsinceatraditionalFourieranalysisassumesthatthefrequencyofasignalisdenedbyaconstantsine/cosinefunctionthatspanstheentiredatalength.Fornon-stationarydata,however,thefrequencyofthesignalvariesovertime,andsothissine/cosinefunctionwillnotbeconstantovertheentiredatalength( Priestley 1967 ).Another 109

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exampleistheHilberttransformcomplexEOF.AccordingtoMerrieldandGuza( MerrieldandGuza 1990 ),thiscomplexEOFonlymakessenseifthefrequencydistributionoftheoriginalsignalisnarrowband(thoughthereisstillsomecontroversysurroundingthisissue).Thus,itisarguedthattheHilberttransformofabroadbandsignalisuninterpretable.EMDresolvesbothissuesdescribedabovefornon-stationaryandbroadbandsignalsbydecomposingasignalintoitsconstituentintrinsicmodefunctions(IMFs)( Huangetal. 1998 ).EachIMFmustsatisfytwoconditions:1)fortheentiredatalength,thenumberofextremaandthenumberofzerocrossingsmustbeeitherequalordifferbyatmostoneand2)atanypoint,themeanvalueoftheenvelopesdenedbythelocalmaximaandlocalminimamustbezero.Therstconditionissimilartoastandarddenitionofanarrowbandsignal,namelythattheexpectednumberofextremaandzerocrossingsmustbeequal.Thesecondconditionmeansthatlocally,themeanofeachIMFiszeroandthussymmetricwithrespecttothelocalzeromean.Sincethisconceptoflocalitycanbedifculttodene,especiallyfornon-stationarysignals,EMDusesthelocalmeanoftheenvelopesdenedbythelocalmaximaandlocalminimatoforcelocalsymmetry.ThetwoconditionsoutlinedaboveleadtotheEMDprocessforextractinganIMFfromasignal,x(t): 1. Identifyallthelocalextremaofx(t) 2. Connectallthelocalmaximbyacubicsplinelineanddenethisastheupperenvelope 3. ConnectallthelocalminimabyacubicsplinelineanddenethisasthelowerenvelopeDenotethemeanoftheupperandlowerenvelopesasm1.Thenthedifferencebetweenx(t)andm1istherstcomponent,h1.Inorderwords,x(t))]TJ /F5 11.955 Tf 12.64 0 Td[(m1=h1.Atthispoint,h1probablydoesnotqualifyasanIMFandsoasiftingprocesstakesplacewhich 110

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eliminatesridingwavesandmakesthewave-prolesmoresymmetric.Inthissiftingprocess,h1istreatedasthedatax(t)instep1aboveandtheremainingstepsarerepeatedresultinginh1)]TJ /F5 11.955 Tf 11.38 0 Td[(m11=h11.Siftingcontinuesktimesuntilh1kisanIMFatwhichpointitisdesignatedasc1=h1k.Becausethesiftingprocesssmoothsoutunevenamplitudesinanattempttomakethewave-prolessymmetric,carryingtheprocesstotheextremecouldeliminateallphysicallymeaningfulamplitudeuctuationsintheoriginalsignal.Thus,astoppingcriteriaisneededtoguaranteethatIMFsretaintheirphysicalrelevance.ThestoppingcriterionproposedbyHuangetal.( Huangetal. 1998 )issimilartotheCauchyconvergencetestandisdenedbyasumofthedifference(SD)asSDk=PTt=0jhk)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t))]TJ /F5 11.955 Tf 11.96 0 Td[(hk(t)j2 PTt=0h2k)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t) (5)Typically,whenSDkreachesavaluebetween0.2and0.3thesiftingprocessstops.Oncethesiftingprocessiscomplete,therstIMFobtainedisc1anditshouldcontaintheshortestperiodcomponentoftheoriginalsignal.Theprocesscontinuesbyremovingc1fromx(t)andndingthenextIMFinthedata.Specically,x(t))]TJ /F5 11.955 Tf 12.29 0 Td[(c1=r1,wheretheresiduer1isnowusedasinputtotheEMDprocedure.TheEMDprocedurenallystopswhentheresidue,rn,becomesamonotonicfunctionfromwhichnomoreIMFscanbeextracted.Atthispoint,nIMFshavebeenextractedandtheoriginalsignalx(t)canbewrittenasx(t)=nXj=1cj+rn (5)Thus,EMDformsacompletebasisfortheoriginalsignalx(t).Inaddition,Huangetal.observedthatlocally,anytwoIMFcomponentsshouldalsobeorthogonalforallpracticalpurposes( Huangetal. 1998 ).TheotherbenetsofEMDarethatisintuitiveandadaptivetoavarietyofdifferenttypesofsignals.However,beinganempiricalapproach,itdoesnotallowfortheoreticalanalysisandperformanceevaluation. 111

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Withinthecontextofeffectiveconnectivitydetermination,EMDisaperfectcomplementtothePDMtechniquesincePDMrequiresinputsignalsthatarenarrow-band( RosenblumandPikovsky 2001 ).DecomposingasignalviaEMDintoitsIMFsandthenapplyingPDMacrosstheseIMFsisanaturalextensionofthePDMtechnique,enablingittobeappliedtobroadbandsignals.TheEMD-GPDMtechniquethatIproposeinthischapterformalizesthisapproachandprovestobeanaccuratetechniquefordeterminingeffectiveconnectivityinbroadbandEEGsignals. 112

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Table5-1. Neuralmassmodelbasalparametervaluesthatareconsistentacrossallpopulationtypes ParameterValue C1135C2108C333.75C433.75C540.5C613.5C7108s06e02.5r0.56 Table5-2. Neuralmassmodelbasalparametervaluesthatdifferamongpopulationtypes ParameterLFValueMFValueHFValue A2.75.25.6B3.24.53.8G394375a1(s)]TJ /F8 7.97 Tf 6.59 0 Td[(1)4085110b1(s)]TJ /F8 7.97 Tf 6.58 0 Td[(1)203040g1(s)]TJ /F8 7.97 Tf 6.58 0 Td[(1)3003507902202020000 Table5-3. SensitivityandspecicityvaluesofGC,GPDC,andPDMwhenappliedtonetworksofthreecoupledNMMpopulations SensitivitySpecicity GC34%97%GPDC92%90%PDM20%79% Table5-4. SensitivityandspecicityvaluesofGC,GPDC,andPDMwhenappliedtonetworksoftwocoupledNMMROIs SensitivitySpecicity GC100%33%GPDC99%39%PDM76%15% 113

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Table5-5. SensitivityandspecicityvaluesofEMD-GPDMandEMD-PDMwhenappliedtonetworksoftwocoupledNMMROIs SensitivitySpecicity EMD-GPDM90%92%EMD-PDM79%0% Figure5-1. Asystemdiagramofasinglepopulationneuralmassmodel 114

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Figure5-2. Powerspectraldensitiesofneuralmassmodelpopulationswithlowintrinsicringfrequency(LF)between4-12Hz,mediumintrinsicringfrequency(MF)between12-30Hz,andhighintrinsicringfrequency(HF)greaterthan30Hz. 115

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Figure5-3. Twoneuralmassmodelpopulationnetworks,eachcomposedofonelowfrequency(LF)population,onemediumfrequency(MF)populationandonehighfrequency(HF)population 116

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Figure5-4. Twocoupledneuralmassmodelregionsofinterest.ThestrengthofconnectionfromROI1toROI2atlowfrequencyisdenotedbyWL12,atmediumfrequencybyWM12andathighfrequencybyWH12.Theinput,p(t)toeachpopulationisaGaussianwhitenoisesignal. Figure5-5. Powerspectraldensityofasingleneuralmassmodelregionofinterest 117

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Figure5-6. Anexampleneuralmassmodelpopulationnetworkandtheresultsobtainedfromtimeseriesinferencetechniqueswhenappliedtothenetwork.GCandGPDCoutputisafunctionoffrequency,soitispresentedinplotswithnormalizedfrequencyonthex-axis.PDMresultsarescalarvalueswhicharegiveninthetableatbottom.SignicancethresholdsareshownintheplotsforGCandGPDCasdashedlines.Signicanteffectiveconnectivityasdeterminedbyeachtechniqueishighlightedbygreyrectangles.GPDCproperlyidentieseffectiveconnectivityfromtheLFtotheMFpopulationandfromtheMFtotheHFpopulation.GCdoesnotidentifyanyoftheconnectionsandPDMidentiesthetwotrueconnectionsaswellasoneincorrectconnectionfromtheMFtotheLFpopulation. 118

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Figure5-7. AnexampleNMMROIsignalandtwoofitsconstituentIMFsextractedbyEMD.IMF4isextractedviaEMDbeforeIMF9andIMF4hasamuchlargervariancethanIMF9. 119

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Figure5-8. TheEMD-GPDMtechniqueappliedtoanexampleNMMROInetwork 120

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CHAPTER6CONCLUSIONSNumerousstudieshaveappliedlineartimeseriesinferencetechniquessuchasGrangercausality(GC)andpartialdirectedcoherence(PDC)ornonlineartechniquessuchasphasedynamicsmodeling(PDM)toneuraldatasetsinanattempttogaininsightintothedirectionalityofinteractionsinthebrain( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ; Havliceketal. 2010 ; Liaoetal. 2010a ; Satoetal. 2009 ).Datasetsaregenerallyconsideredamenabletosuchananalysisaslongastheyarestationary.Theresultsofmystudydemonstratethenecessityforanalyzingtimeseriesdatasetsbeyondjustdeterminingtheirstationarity.Forinstance,IdemonstratedthatnoisevariancedifferencesbetweentimeseriescanhaveasignicantimpactontheresultsofPDCandPDM.AmodiedPDCtechnique,termedgeneralizedPDC(GPDC)waspreviouslydevelopedasanoise-variance-immunealternativetoPDC.Inasimilarfashion,IproposedanddemonstratedtheuseofanovelPDMtechnique,whichIrefertoasgeneralizedPDM(GPDM),whichisalsoimmunetonoisevariancedifferences.Synchronyisanotherexampleofatimeseriesfeaturethatcanresultinthefailureofmanytechniquesandpreviousworkhasindicatedapproximateboundsonthedegreeofphasecoherencebetweentwooscillatorsabovewhichseveralofthesetechniquestendtofail( Bezruchkoetal. 2003 ; Smirnovetal. 2007 ).Iextendedtheseresultsbyreningthelimitsonphasecoherenceaswellasconsideringlimitsonothertimeseriesfeatures,suchasnoiseandringfrequency,forthreeofthemostwidelyusedlinearmethodsfordeterminingeffectiveconnectivity,GC,PDCandGPDC,aswellasPDM,whichcanhandlesomenonlinearities.Animportantcontributionofmyworkistheintroductionofamethodologytoassesswhetheragiventechniquefordeterminingeffectiveconnectivitywillproducecorrectresultswhenappliedtoagivendatasetbasedontheobservable,dynamical 121

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characteristicsofthedataset.Theproposedmethodologyconsistsofdecisiontreeclassiersthatcanbeappliedwithoutanypriorknowledgeofnetworkstructure.Idevelopedthismethodologyanddemonstrateditsusewithinthecontextofnarrow-bandneuronaloscillators,whicharerepresentativeofdataonthesingle-neuronscale.Furthermore,Iperformedamuch-neededinvestigationintotheapplicabilityoftimeseriesinferencetechniquestoneuralmassmodels(NMMs),whicharerepresentativeofelectroencephalographic(EEG)datacollectedatthescaleoflargeneuronalpopulations.WhilethesetechniquesareoftenusedtoanalyzeexperimentalEEGdata( Brovellietal. 2004 ; Cadotteetal. 2010 ; Chenetal. 2006 ),fewstudieshaveexploredthevalidityofthesetechniqueswhenappliedtosuchdata.IshowedthatwhileGPDCisaccuratewithhighprobabilitywhenappliedtoNMMsthatexhibitrhythmicactivityatasingledominantfrequency,noneofthetechniquesIconsideredwereaccuratewhenappliedtoNMMsthatexhibitrhythmicactivityoverarangeoffrequencyvalues.IintroducedanewtechniquewhichIcallEMD-GPDM,todetermineeffectiveconnectivityinsuchNMMs.Theempiricalmodedecomposition(EMD)aspectofthetechnique,whichdecomposesasignalintoconstituent,narrow-bandsignals,perfectlycomplementstheapplicationofGPDMsinceGPDMrequiresnarrow-bandsignals.MyproposedEMD-GPDMtechniqueprovestobeanaccuratemethodfordeterminingeffectiveconnectivityinNMMswithbroadbandcharacteristics,whicharerepresentativeofalargeclassofexperimentalEEGrecordings. 122

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BIOGRAPHICALSKETCH ErinBoykin,bornErinTaylor,wasraisedinOrlando,FloridaandgraduatedfromLakeHowellHighSchoolin2001.ShereceivedherB.S.incomputerengineeringfromtheUniversityofFloridain2005,graduatingcumlaude,andcontinuedontoearnherM.S.degreeinelectricalandcomputerengineeringatUFin2007.ShemarriedPatrickOscarBoykinin2009. 132