Phase Lag Index Assessment of Functional Connectivity From Multi Channel EEG and MEG With - PDF document

PhaseLagIndex:AssessmentofFunctionalConnectivityFromMultiChannelEEGandMEGWithDiminishedBiasFromCommonSourcesCornelisJ.Stam,GuidoNolte,andAndreasDaffertshoferDepartmentofClinicalNeurophysiology,VUUniversityMedicalCenter,Amsterdam,TheNetherlandsHumanMotorControlSection,NINDS,NationalInstitutesofHealth,Bethesda,MarylandFraunhoferInstitute,Kekulestrae7,Berlin,GermanyInstituteforFundamentalandClinicalMovementSciences,VU,Amsterdam,TheNetherlands Toaddresstheproblemofvolumeconductionandactivereferenceelectrodesintheassessmentoffunctionalconnectivity,weproposeanovelmeasuretoquantifyphasesynchroniza-tion,thephaselagindex(PLI),andcompareitsperformancetothewell-knownphasecoherence(PC),andtotheimaginarycomponentofcoherency(IC).ThePLIisameasureoftheasymmetryofthedistributionofphasedifferencesbetweentwosignals.TheperformanceofPLI,PC,andICwasexaminedin(i)amodelof64globallycoupledoscil-lators,(ii)anEEGwithanabsenceseizure,(iii)anEEGdatasetof15Alzheimerpatientsand13con-trolsubjects,and(iv)twoMEGdatasets.PLIandPCweremoresensitivethanICtoincreasinglevelsoftruesynchronizationinthemodel.PCandICwerein”uencedstrongerthanPLIbyspuriouscorrelationsbecauseofcommonsources.Allmeasuresdetectedchangesinsynchronizationduringtheabsenceseizure.IncontrasttoPC,PLIandICwerebarelychangedbythechoiceofdifferentmontages.PLIandICweresuperiortoPCindetectingchangesinbetabandconnectivityinADpatients.Finally,PLIandICrevealedadiffer-entspatialpatternoffunctionalconnectivityinMEGdatathanPC.ThePLIperformedatleastaswellasthePCindetectingtruechangesinsynchronizationinmodelandrealdatabut,atthesametokenandlike-wisetheIC,itwasmuchlessaffectedbythein”uenceofcommonsourcesandactivereferenceelectrodes.HumBrainMapp28:1178…1193,2007.2007iley-iss,nc.Keywords:phaselagindex;phasesynchronization;coherence;volumeconduction;EEG;MEG;func-tionalconnectivity;absenceseizure;Alzheimersdisease Higherbrainfunctionsdependuponadelicatebalance Contractgrantsponsor:DutchScienceFoundation(NWO);Con-tractgrantnumber:52-04-344.*Correspondenceto:C.J.Stam,DepartmentofClinicalNeuro-physiology,VUUniversityMedicalCenter,P.O.Box7057,1007MBAmsterdam,TheNetherlands.E-mail:CJ.Stam@vumc.nlReceivedforpublication25April2006;Revised13July2006;Accepted21August2006DOI:10.1002/hbm.20346Publishedonline31January2007inWileyInterScience(www.interscience.wiley.com). apping28:1178…1193(2007) factorsdeterminetheorganizationofthesenetworks,andhowdoescommunicationinthesenetworkstakeplace?Withrespecttothelastquestion,thereisbynowampleevidencethatsynchronizationofneuralactivityconsti-tutesanimportantphysiologicalmechanismforfunc-tionalintegration[e.g.,Fries,2005;Singer,1999;Varelaetal.,2001].NeurophysiologicaltechniqueslikeEEGandMEGhaveahightemporalresolutionandarethusrathersuitableforidentifyingsynchronizationacrossfrequencybandsinlarge-scalefunctionalnetworks.Inencephalographicre-cordings,synchronizationisusuallyquanti“edvialinearmeasureslikecoherenceorvianonlinearmeasureslikethosebaseduponphasesynchronizationorgeneralizedsynchronization[Breakspear,2002;BreakspearandTerry,2002;Breakspearetal.,2004;Burns,2004;Nunezetal.,1997,1999;Peredaetal.,2005;Stam,2005,2006].Noticethatincontrasttotheneurophysiologicaltechniqueswithhightemporalresolution,fMRIoffersahigherspatialreso-lutionallowingforamoreaccurateidenti“cationofspe-ci“canatomicalareasconstitutingspeci“cnetworksrelatedtovarioustasksortotheso-calledrestingstate[Salvadoretal.,2005].Whenthestudyoffunctionalinteractionsisdirectedatidentifyingstatisticalinterdependenciesbetweenphysiologicaltimeseriesrecordedfromdifferentbrainareas,thisisreferredtoasfunctionalconnectivity[Fingelkurtsetal.,2005;Leeetal.,2003].Moreambitiousapproachesattempttoidentifycausalinteractionsfromapriorinetworkmodelsthatare“ttedtothedata[e.g.,Fris-ton,2002].Despitetheconsiderablesuccessoftheseapproachesincharacterizingnormalanddisruptednetworksinthebrainrelatedtonormalcognitionandvariousneuropsychiatricdisorders,furtherprogressishamperedby(amongstothers)methodologicallimitations.Assuch,fMRI-basedmethodssufferfromalimitedtimeresolution,whichisnottrivialtoovercomesinceitresultsfromtherecordedmetabolism.Neurophysiologicalmethodssufferfromthefactthatnouniquerelationexistsbetweentimeseriesrecordedfromthescalpandactivesourcesinthebrain.Timeseriesthatarerecordedfromnearbyelectrodesorsensorsareverylikelytopickupactivityfromthesame,i.e.common,sources,whichgivesrisetospuriouscorrela-tionsbetweenthesetimeseries;thisistheproblemofvol-umeconduction.AverymuchrelatedproblemuniquetoEEGisthatoftheactivereferenceelectrode.SuchanactivereferenceelectrodewillcontributesimilarcomponentstoEEGsignalsrecordedatdifferentelectrodes,therebyyield-ingafakecorrelation.Nunezetal.[1997]haveshownhowvolumeconductionanddifferenttypesofreferenceelec-trodemayaffectestimationsofcoherence.Inline,Guevaraandcoworkers[2005]haverecentlystudiedhowanactivereferenceelectrodecanalsoseriouslydisturbestimationsofphasesynchronizationinEEG.Twoprimaryapproacheshavebeensuggestedtodealwithwhatwewillcalltheproblemofcommonsources(referringbothtovolumeconductionaswellasactiveref-erenceelectrodes)First,severalgroupshaveattemptedtoreconstructasuitablesourcespace,whichcansubse-quentlybeusedasabasistodeterminefunctionalinterac-tions[Amoretal.,2005;Davidetal.,2002;Grossetal.,2001;Hadjipapasetal.,2005;Lehmannetal.,2006;Tassetal.,2003].Whiletheseapproachesareappealing,becausetheyallowforstudyingfunctionalinteractionsbetweenwell-speci“edanatomicalregions,theyalsoentailproblems.First,thereisnouniquechoiceforasourcemodelandeachchoiceisultimatelyarbitrary.Differentchoicesforasourcemodelpotentiallyaffecttheresultsoftheanalysisoffunctionalinteractions.Second,theassump-tionsofsomesourcemodels„forinstance:indecencyofeachofthesources„mayinterferewiththestatisticalinterdependenciesbetweenthesources[Hadjipapasetal.,2005].Thisproblemmaybeparticularlyawkwardinthecaseofstronglyinteractingsources.Asecondapproachtotheproblemofcommonsourcestriestoidentifyinformationinthecorrelationstructurebetweentwotimeseriesthatisunlikelytobeexplainedbycommonsources.Nunezetal.[1997]proposedtosubtracttherandomcoherencefromthemeasuredcoherencetoobtainareducedcoherence,whichislessin”uencedbyvolumeconductioneffects.ThisapproachwasrecentlyappliedbyBarryetal.[2005].Computationofpartialco-herenceisanotherapproachtodiminisheffectsofcommonreferencesandvolumeconduction[Mimaetal.,2000].Nolteandcoworkers[2004]havearguedthattheimagi-narycomponentofcoherencyisanindexofcorrelations,whichcannotbecausedbycommonsources„recallthatcoherencyisthecomplex-valued,normalizedcross-spectraldensitywhilecoherenceisgivenasitsmodulo[seeEq.(7)below].Themagnitudeofthisimaginarycomponent,how-ever,isstillnotanidealmeasureofthestrengthoftheinteractionssinceitdependsonboththeamplitudesofthesignalsandthemagnitudeofthephasedelay.Inarecentstudytheimaginarypartofcoherencywaslessusefulthanthecoherenceindemonstratingexperimentaleffects[Wheatonetal.,2005].Inthepresentarticle,weintroduceanalternativemea-sureofstatisticalinterdependenciesbetweentimeseries,whichre”ectsthestrengthofthecouplingbutisexpectedtobelesssensitivetothein”uenceofcommonsourcesandamplitudeeffects.Themeasure,thephaselagindex(PLI)isbasedupontheideathattheexistenceofaconsist-ent,nonzerophaselagbetweentwotimesseriescannotbeexplainedbyvolumeconductionfromasinglestrongsourceand,therefore,renderstrueinteractionsbetweentheunderlyingsystemsratherlikely.Suchconsistent,non-zerophaselagscanbedeterminedfromtheasymmetryofthedistributionofinstantaneousphasedifferencesbetweentwosignals.WeinvestigatetheperformanceofthePLIandcompareittoaclassicalmeasureofphasesynchronization[phasecoherence:Mormannetal.,2000]aswellastheaforementionedimaginarycomponentofcoherencyproposedbyNolteetal.[2004]inawell-knownmodelofcoupledoscillators,EEGwithanabsenceseizure, ssessmentof r1179r adatasetofEEGsofAlzheimerpatientsandsubjectswithsubjectivememorycomplaintsandtwoMEGdatasets.METHODSSignalAnalysisPhasesynchronizationandmeanphasecoherenceTheconceptofphasesynchronization(forchaoticoscil-lators)wasextensivelydiscussedbyRosenblumandcow-orkers[1996].Inbrief,rigorousphaselockingbetweentwosystemsrequiresthattheirphasedifferenceisconstant,whiletheweakerconceptofphaseentrainmentintroducedbyRosenblumetal.[1996]onlyrequiresthatthephasedif-ferenceremainsbounded(theboundhastobesmallerthan2).Ifarethephasesoftwotimeseries,isthephasedifferenceorrelativephase,thegen-someintegers)phasesynchroni-zationcanbefoundas:holds.Usingthisde“nition,phasesynchronizationcanbedeterminedfornoisy,nonstationary,andchaoticsignals.Intheremainderofthisarticle,werestrictourselvestothe(isofrequency)casewith1,thatis,Tocomputethephasesynchronization,itisnecessarytoknowtheinstantaneousphaseofthetwosignalsinvolved.ThiscanberealizedusingtheanalyticalsignalbasedontheHilberttransform[theapproachwithwaveletspro-videssimilarresults:Bruns,2004].Theanalyticalsignal)iscomplex-valuedwith)arealtimeseriesanditscorrespondingHilberttransform:TheHilberttransformof)isobtainedviaintegrationasfollows(seealsoAppendixB) pPVZ1
x referstotheCauchyprincipalvalue.TheHilberttransform(3)isrelatedtotheoriginalsignalbya[1/2]phaseshiftthatdoesnotalterthespectraldistribution(itcanbecomputedbyperformingaFouriertransform,shift-ingallthephasesby[1/2],followedbyaninverseFou-riertransform).FromEq.(2),boththeinstantaneousam-)andtheinstantaneousphase)canbecom-putedby:






























Following(1)fromtheinstantaneousphaseoftwosig-nals,thephasedifferenceorrelativephase)iscom-putedasafunctionoftime.Infact,thereareseveralmethodstodeterminewhetherthisphasedifferenceisbounded.Here,weusethenotionofphasecoherence(PC)describedbyMormannetal.[2000].Thisnotionbasicallyresemblestheconventionalstatisticsforcircular(ordirec-tional)data[e.g.,Mardia,1972].Instantaneousphasedif-ferencesareprojectedontheunitcircle,andthelengthoftheaverageresultantvectoriscomputedvia: arediscretetime-stepsandisthenumberofsamples.Inthecaseofperfectphaselocking(5)yieldswhereasinthecaseofarandomdistributionofphasesontheunitycircleRwilltendtozero.Notethatbyconstruc-isinsensitivetotheamplitudesofthesignalsandonlydependsuponthephaserelationsbetweenthetwosignals,thus,contrastsmoreconventionalcoherence.de“nedin(5)re”ectsbothzerophaselagaswellasnonzerophaselagcouplingofthephasesbetweentwoThephaselagindexThemajoraimofintroducingthePLIistoobtainreli-ableestimatesofphasesynchronizationthatareinvariantagainstthepresenceofcommonsources(volumeconduc-tionand/oractivereferenceelectrodesinthecaseofEEG).Aswillbeexplainedbelowthecentralideaistodiscardphasedifferencesthatcenteraround0mod.Onewaytorealizethisistode“neanasymmetryindexforthedistri-butionofphasedifferences,whenthedistributioniscen-teredaroundaphasedifferenceofzero.Ifnophasecou-plingexistsbetweentwotimeseries,thenthisdistributionisexpectedtobe”at.Anydeviationfromthis”atdistribu-tionindicatesphasesynchronization.Forexample,thisfactisemployedbyphasesynchronizationmeasuresthatarederivedfromtheShannoninformationentropyofthephasedifferencedistribution[Rosenblumetal.,1996;Tassetal.,1998].AmoredetailedmathematicalanalysisoftheideasunderlyingthePLIcanbefoundinanappendixtothisarticle.Asymmetryofthephasedifferencedistributionmeansthatthelikelihoodthatthephasedifferencewillbeintheinterval0isdifferentfromthelikelihoodthatitwillbeintheinterval0.Thisasymmetryimpliesthepresenceofaconsistent,nonzerophasediffer-ence(lag)betweenthetwotimeseries.Theexistenceofsuchaphasedifferenceortimelag,however,cannotbeexplainedbythein”uencesofvolumeconductionfromasinglestrongsourceoranactivereference,sincethesein”uencesareeffectivelyinstantaneous.Thedistributionisexpectedtobesymmetricwhenitis”at(nocoupling),orifthemedianphasedifferenceisequaltoorcentersaroundavalueof0mod(in”uenceofstrongcommonsource/activereference;pleasenotethatamedianphase tametal. r1180r differenceof0moddoesnotimplythatthemodeormodesofthedistributionhavetobeequalto0mod).Itisthelattercaseinwhichconventionalmeasuresofphasesynchronizationprovideshighvalues,whereasthepro-posedindexyieldslowones.Anindexoftheasymmetryofthephasedifferencedistributioncanbeobtainedfromatimeseriesofphasedifferences...inthefollowingway¼jhÞijðThePLIrangesbetween0and1:01.APLIofzeroindicateseithernocouplingorcouplingwithaphasedifferencecenteredaround0mod.APLIof1indicatesperfectphaselockingatavalueofdifferentfrom0.Thestrongerthisnonzerophaselockingis,thelargerPLIwillbe.NotethatPLIdoesnolongerindicate,whichofthetwosignalsisleadinginphase.Wheneverneeded,however,thisinformationcanbeeasilyrecovered,forinstance,byomittingtheabsolutevaluein(6).TodeterminewhetherPLIissigni“cantlylargerthanzero,onemayintroducesurrogatedata[see,e.g.,appendixAinPeredaetal.,2005].Inbrief,onehastocomputePLIforboththeoriginaltimeseriesasetofsurrogatedatathatmatchtheoriginaldatabutlackanycorrelationsbetweenchannels(e.g.,byshiftingeachchannelbysomerandomphase).ThedifferencesbetweenPLIoforiginalandsurro-gatedatayield-scoresthatsuf“cetode“nesigni“canceTheimaginarypartofcoherencyThecomplexcoherencybetweentwotimeseriescanbede“nedasthecrossspectrumdividedbytheproductofthetwopowerspectra.Itsmeanoverallfrequenciescanalternativelybecomputedviathemeanovertimeofthecorrespondinganalyticalsignalslike:
















aretheamplitudesofthetwotimese-ries,andistheinstantaneousphasedifferencebetween(theHilberttransformsof)thetwotimeseries.Theabso-lutevalueofcoherency,typicallyreferredtoascoherence,isboundedbetween0and1.Theimaginarypartofcoher-ency(IC)isgivenby:


















Animportantpropertyoftheimaginarypartofthecoherencyisthatits(nonvanishing)“nitevaluecannotbecausedbyalinearmixingofuncorrelatedsources(volumeconduction)andthusre”ectstrueinteractionsofthesour-cesunderlyingthetwotimeseries[Nolteetal.,2004],seealsoAppendixA.However,the(absolutevalueofthe)imaginarypartisnotyetausefulmeasureofcouplingsinceitdependsuponthestrengthofthecouplingaswellasthemagnitudeofthephasedifference.KuramotoModelTostudythein”uenceofcommonsourcesontheabilityofPC,PLI,andICtodetectrealchangesinsynchroniza-tion,weusedawell-studiedmodelofgloballycoupledlimit-cycleoscillatorsthathasoriginallybeendescribedbyKuramoto[1975].Anexcellentoverviewofthecurrentstateofresearchonthatmodel(orclassofmodels)canbefoundinStrogatz[2000]andforabriefintroductionplac-ingitinawidercontextofresearchoncomplexnetworkscanbefoundinStrogatz[2001].Themodeldescribesthephasedynamicsofalargenet-workofgloballycoupledlimit-cycleoscillators.Thephasedynamicsaregivenbythefollowingdifferential duidt¼viþ Inthisequation,denotesthephaseofthethoscilla-tor,whichhasthenaturalfrequency,andisthestrengthoftheconnectionsbetweentheoscillators.Thus,thephaseevolutionofeachoscillatorisdeterminedbyitsnaturalfrequencyandtheaveragein”uenceofallotheroscillators.ThenaturalfrequenciesaretypicallycollectedfromaLorentziandistributioncenteredaround.ThatLorentziandistributionisgivenby: Usually,thegloballevelofsynchronizationinthesys-temofoscillatorsattimecanbedescribedbyanorder),whichisde“nedasfollows: isaveragedovertime,itisabbreviatedasNotethecloserelationbetweenandthePCde“nedin(5).Intheabsenceofsynchronization,vanishes(andwhenalloscillatorsareperfectlyphase-locked,then1holds(pleasenotethatthisparameterdescribeszerophaselagsynchronization).Kuramotoshowedthatthemodeldisplaysaphasetransitionfromadesynchronizedtoapartiallysynchronizedstateatacriticalvalue De“nition(6)requiresthatthephasedifferenceisboundedinthe.If,incontrast,phasesarede“nedintheinterval0,then(6)shouldbemodi“edto¼jh ssessmentof r1181r .When,thesystemisnotsynchronized,and0(inthelimitof).When,asingleclusterofsynchronizedoscillatorsemerges,whichgrowsforincreasing.For,theorderparametergivenby:
















IfthenaturalfrequenciesoftheoscillatorsaretakenfromtheLorentziandistributiondescribedin(7),thenthecriticalconnectionstrengthisgivenby:Thephasetransitionisthussolelydeterminedbythewidthofthedistributionofthenaturalfrequencies.Finally,the”uctuationin)dependsupon.Thestand-arddeviationof)ismaximalat,andlowerforaswellas.Inotherwords,thevariabilityoftheglobalsynchronizationlevelismaximalatthephaseModelsimulationsForthesimulationofthemodel,weusedasystemof64oscillators.Althoughtheoreticallyanin“nitenum-berofoscillatorsisnecessaryfortheanalyticalresultstohold,ithasbeenshownthatwithonly64,themodelcanbereadilyusedtoexplainvariousempiricalresults[Kissetal.,2002].Foreachoscillator,Eq.(9)wasnumeri-callyintegratedwithatimestepof2ms(correspondingtoasamplefrequency500Hz,thesameasthesamplefre-quencyoftheEEGdatasetsdescribedbelow).Inallsimu-lations,theinitial5,000iterationswerediscardedtoelimi-natetransients.Thestateofoscillatorattimewasgiven„notethatweusedaconstantamplitudethatwasequalacrossoscillators.TheresultingamplitudetimeseriesoftheoscillatorswereusedtocreatetheEEGtimeseriesofthemodel.Weperformedthreeseriesofsimulationswithmeanfre-quency10Hz(alphaband)andadistributionwidthof1each.Fromthetimeseriesof64oscillators,timese-riesof64EEGchannelswerecreatedwithdifferentdegreesofoverlap.Thevoltage)ofthethEEGchan-nelattimewasrelatedtothestate)ofthethoscilla-torattime determinesthecontributionofmultiplesourcestoeachEEGchannel.ThenumberofsharedoscillatorsforconsecutiveEEGchannelswas2.Simulationswereper-formedfor4,and8,forvaluesofingfrom0to8,instepsof0.5.Foreachvalueof10trialsweredone,andtheresultingtimeseriesof64channelsand4,096samplesweresubjectedtosynchroniza-tionanalysis.Thesynchronizationanalysisinvolvedcom-putationofthePC,PLI,andICforallpossiblecombina-tionsofEEGchannels.AbsenceEEGThein”uenceofdifferentmontagesontheabilityofPC,PLI,andICtotrackchangesinsynchronizationwasinves-tigatedwithanEEGrecordcontainingaclassicalabsenceseizurewithgeneralized3Hzspikewavedischarges.TheEEGwasrecordedwiththeBrainlab(R)digitalEEGsys-tem(OSG,Rumst,Belgium).TheEEGwasrecordedfrom21tinelectrodespositionedaccordingtothe10…20system7,81,2,Fz,Cz,andPz)againstanaveragereference(includingallchannelsexceptFpandA).ECGwasrecordedinaseparatechannel.Electrodeimpedancewasbelow5kOhm.Filtersettingsduringrecordingweretimeconstant1s,high-passcut-offfrequency70Hz,samplefrequency500Hz,andA-Dprecision16bit.Elevenconsecutiveepochsof4,096samples(8.19s)wereselectedofflineandconvertedtoASCII.Thisseriesofcon-secutiveepochscontainedpreictal(epoch1…5),ictal(epoch6and7),andpostictal(epoch8…11)EEG.ReformattingandanalysesofthisEEGwererealizedwiththeDIGEEGXPsoftwaredevelopedatthedepartment.Thefol-lowingmontageswerestudied:(1)averagereference(includingall21channelsexceptFpandA);(2)linkedearelectrodesA;(3)source(localaveragecomputedfromthe3or4surroundingelectrodes);(4)bipolarshortdistanceanteriortoposteriorchains;(5)Cz.ForeachofthesemontagesPC,PLI,andICwerecomputedforallpossiblepairsofEEGchannelsforeach4,096samplesepochafteroff-linedigital“lteringbetween0.5and48Hz.Fromthis,anoverallmeansynchronizationaswellassubaveragesofintra-andinter-hemisphericshortandlongdistanceswerecomputed.AlzheimerandControlEEGsThenextdatasetinvolvedareanalysisofEEGsrecordedin28subjects,15withadiagnosisofprobableAlzheimersdisease(4males;meanage69.6years;S.D.7.9;range54…77);and13controlsubjectswithonlysubjec-tivememorycomplaints(SC;6males;meanage70.6years;S.D.7.7;range:57…78).MeanMMSEscoreoftheAlzheimerpatientswas21.4.8(S.D.4.0;range15…28);meanMMSEscoreoftheSCsubjectswas28.4(S.D.1.1;range27…30).Thisdatasetwaspreviouslyanalyzedwiththesynchronizationlikelihoodandgraphtheoreticalmeas-ures,andisknowntodisplayalossofbetabandconnec-tivityintheADgroup[Stametal.,2006].EEGrecordingandsettingswereexactlythesameasfortheabsenceEEGdescribedabove.Fromallrecordings,4epochsof4,096samples(8.19s)werestoredasASCII“lesforfurtheranal- tametal. r1182r ysis.Afteroff-linedigital“lteringbetween13and30Hz,thePC,PLI,andICweredeterminedfromallepochsandallchannelpairs,andaveragedoverthefourepochs.Furtheraveragingwasperformedtoobtainthetotallevelofsynchronization,andsubaveragesforshortandlongdistancesforintra-andinter-hemisphericelectrodepairsasfollows:(1)intrahemisphericshort(meanof:Fp2-F4,F4-C4,C4-P4,P4-O2,F8-T4,T4-T6,Fp1-F3,F3-C3,C3-P3,P3-O1,F7-T3,T3-T5);(2)intrahemisphericlong(meanof:F8-T6,Fp2-C4,C4-O2,Fp1-C3,C3-O1,F7-T5);(3)interhemi-sphericshort(meanof:Fp2-Fp1,F8-F4,F4-F3,F3-F7,T4-C4,C4-C3,C3-T3,T6-P4,P4-P3,P3-T5,O2-O1);(4)interhe-misphericlong(meanofF8-F7,T4-T3,T6-T5).MEGDataToillustratethein”uenceofvolumeconductiononthespatialpatternofMEG,recordingsoftwohealthymalesubjects(takenfromthecontrolgroupofonongoingAlz-heimerstudy)wereanalyzed.Magnetic“eldswererecordedwhilesubjectswereseatedinsideamagneticallyshieldedroom(VacuumschmelzeGmbH,Hanau,Germany)usinga151-channelwhole-headMEGsystem(CTFSys-tems,PortCoquitlam,BC,Canada).Athird-ordersoftwaregradientwasusedwitharecordingpassbandof0.25…125Hz.Fieldsweremeasuredduringano-task,eyes-closedcondition.Atthebeginningandattheendofeachrecord-ing,theheadpositionrelativetothecoordinatesystemofthehelmetwasrecordedbyleadingsmallalternatingcur-rentsthroughthreeheadpositioncoilsattachedtotheleftandrightpre-auricularpointsandthenasiononthesub-jectshead.Headpositionchangesduringarecordingcon-ditionupto1.5cmwereaccepted.DuringtheMEGre-cording,thepatientswereinstructedtoclosetheireyestoreduceartifactsignalsbecauseofeyemovements.Forthepresentanalyses,149ofthe151channelscouldbeused.MEGrecordingswereconvertedtoASCII“lesanddown-sampledfrom625to312.5Hz.FromtheseASCII“les,artifactfreeepochsof4,096samples(13.083s)wereselectedbyvisualinspectionand“lteredinthealphaband(8…13Hz).RESULTSKuramotoModelResultsfortheKuramotomodelaresummarizedinFig-ures1…3.Figure1showsthemeanPC,averagedoverallpairsofthe64simulatedEEGchannels,asafunctionofcouplingstrengthanddegreeofoverlap(numberofoscillatorscontributingtoeachEEGchannel).Inthecaseofnooverlap,PCstayedatrelativelylowlevelsforlowerthan2.From2onwards,therewasasuddenandstrongincreaseofPC,whichleveledofforhighvalues(recallthatweused1,thatis,2).ThisbehaviorofPCisincloseagreementwiththeanalyticalresultsforthemodel,thatis,theknownbifurcationtoatthecriticallevelof2.Whentheoverlap Figure1.Meanphasecoherence(PC,averagedoverallpossiblepairsof64modeledEEGchannels)asafunctionofcouplingstrengthintheKuramotomodelwith64oscillatorsasafunctionofover-lapbetweensubsequentEEGchannels(CS:commonsources,rangingfrom0to16).Allresultsaretheaverageto10trials.The“rst5,000samplesofeachtrialwereignored.Epochlengthforeachtrialwas4,096samples.Meanfrequencyoftheoscilla-torsinthemodelwas10Hz,thewidthoftheLorentzdistribu-1.Samplefrequencywas500Hz.Theseparametersyieldacriticalvalueof ssessmentof r1183r betweenEEGchannelswasmodi“edfrom0to8clearchangesinPCwerefound.First,theentirecurvewasshiftedtowardahigherlevelforallvaluesof.Second,therelativeincreaseinPCstartedatlowervaluesthantheanalyticallyexpectedvalueof2.IncreasingthelevelofoverlapbetweenEEGchannelsfrom8to16showedanupwarddisplacementofthecurve,butonlyforvaluesof2.5.TherelativeincreaseofPCstartedatevenlowervaluesof.Thus,whilePCwassensitivetotruechangesintheconnectionstrength,itwasalsoquitesensitivetospuriousin”uencesofcommonsources,whichchangedboththeabsolutevaluesaswellasthequalitativebehaviorofPCasafunctionofTheresultsforPLIaredepictedinFigure2.Intheidealcasewithoutcommonsources,PLIshowedlowvaluesfor,andincreasingvaluesforhigherasexpectedfromtheory.ComparedtoPC,however,PLIstartedtoincreaseatsomewhatlowervaluesof.Addingthein”u-enceofcommonsourcesincreasedPLIvaluesslightlyfor2.5anddecreasedthePLIvaluesfor�2.5.Forveryhighvaluesof,PLIunderestimatedthetruelevelofcou-pling.Therewasnocleardifferencebetweenanoverlapof8or16oscillators.Thus,PLIalsoshowedtheexpectedincreaseasafunctionofbutcomparedwithPC,itwaslesssensitivetothespuriousin”uenceofcommonsources.Finally,theresultsforICareshowninFigure3.Intheabsenceofvolumeconduction,ICstartedtoincreasefor,butneverreachedvaluesmuchhigherthanIC0.2evenforveryhighcouplingstrength(notethattheupperboundforICis1).Thatis,ICsystematicallyunder- Figure2.Meanphaselagindex(PLI,aver-agedoverallpossiblepairsof64modeledEEGchannels)asafunctionofcouplingstrengthintheKuramotomodel.Param-etersareidenticalwithFigure1. Figure3.Meanabsolutevalueoftheimaginarypartofcoherency(IC,averagedoverallpossiblepairsof64modeledEEGchannels)asafunctionofcouplingstrengthintheKuramotomodel.ForparametersseeFigure1. tametal. r1184r estimatedthetruecouplingstrengthinthemodel.Forincreasingeffectsofvolumeconduction,ICincreasedfor2.5anddecreasedfor�2.5.Hence,theeffectsofsimulatedvolumeconductionfurthercompromisedthemodestsensitivityofICtoincreaseincouplingstrength.EEGandMEGRecordingAbsenceEEGThetransitionbetweeninter-ictaltoictalEEGisshowninFigure4.TheresultsfortheabsenceEEGaregiveninFigures5…7.Figure5showsthemeanPCaveragedoverallpossibleelectrodepairsforeachoftheepochsandforvariousdifferentmontages.ByandlargePCstayedroughlyconstantatabaselinelevelduringthe“rst5non-seizureepochs.Afterthat,wefoundasuddenincreaseinPCinepoch6and7,whichcontainedthegeneralizedspike-and-wavedischarges.Inthe“nalepochs(8…11),PCdecreasedtothebaselinelevel.AscanbeseeninFigure5,themontageshadquiteanin”uenceonPCvalues.Thelowestvalueswerefoundforthesource(localaverage)derivation.Slightlyhighervalueswereobtainedforthebipolarderivation.ValuesofPCwereevenhigherforaveragereferenceandthelinkedearsderivation.Therethelinkedearsderivationshowedthestrongestrelativeincreasebyafactorof2.5duringthesei-zureascomparedtobaseline.ThehighestvaluesduringbaselineaswellasthelowestvaluesduringtheseizurewereobtainedwiththeCzreference;relativeincreasewaslessthanafactorof1.5.Overall,thetypeofreferencestronglyin”uencedtheabsolutevaluesofPCaswellastherelativeincreaseduringtheseizure.TheresultsforPLIaredepictedinFigure6.Duringthe“vepreseizureepochs,PLIstayedmoreorlessconstantatalowlevelslightlyabove0.1.Duringtheseizureepochs6and7,aclearincreasecouldbefound.Thisincreasewasfollowedbyanimmediatedecreaseinepochs8…11.Duringthepreseizureepochs,PLIvalueswerehardlyin”uencedbydifferentderivations.Duringtheseizuredifferencesdid Figure4.DetailofEEGrecordingwithabsenceseizureconsistingof3Hzgeneralizedspike-and-slowwavedischarges.Averagereference,“ltersettings:highpass0.5Hzandlowpass48Hz.Verticalbluebarsindicate1sintervals.[Color“gurecanbeviewedintheonlineissue,whichisavailableatwww.interscience.wiley.com.] Figure5.MeanPC(averagedoverallpairsof21channels)fordifferentmontages(average,source,mastoids,bipolar,andCz).Eachepochhasalengthof4,096samples(8.18s).Epochno.6and7correspondtotheseizure. ssessmentof r1185r emerge:PLIvalueswerelowfortheCzderivation,inter-mediatefortheaveragereferenceandthelinkedearsref-erence,andhighestforthebipolarandsourcederivation.TherelativeincreaseinPLIduringtheseizurewasroughlyafactorof3fortheworstreference(Cz)andafac-torof5forthebest(source).Intheearlypostseizureepochs8and9,PLIvalueswerestillhigherthaninthepre-seizureepochs,especiallyfortheaveragereferenceandthelinkedearsreference.Overall,PLIundoubtedlyshowedincreasesduringtheseizureepochsandwaslessin”uencedbythedifferentmontagesthanPC,althoughdifferencescouldstillbeseenduringtheseizure.Expressedasrelativeincrease(synchronizationduringsei-zurecomparedtobaseline)PLIperformedbetterthanPCforallmontages.TheresultsforICareshowninFigure7.Forthepresei-zureepochs,theICofthedifferentmontages”uctuatedaround0.04.Afterthat,wefoundaclearincreaseduringtheseizuresepochs(6…7),whichwasfollowedbyagrad-ualdecreaseinthepostictalepochs.Thedifferentmon-tageshadonlyasmalleffectonICforthepre-andpost-ictalepochs,butduringtheseizurethereweredifferences:ICwasrelativelylowforCz(relativeincreasefactor2)andhighforthebipolarmontage(relativeincreasefactor3.5).Theothermontagesshowedintermediatevalues.Thus,fordetectingarelativeincreaseinsynchronizationfrompreseizuretoseizureepochs,ICperformedbetterthanPCandonlyslightlyworsethanthePLI.AlzheimerandcontrolEEGResultsofthesynchronizationanalysisoftheAlzheimerandcontrolEEGsareshowninFigures8…10.AveragePCinthebetabandwaslowerinAlzheimerpatientsthanincontrols(0.023;Fig.8).Analysesofsubaveragesforlongandshortdistancesandintra/interhemisphericelec-trodepairsdidnotrevealsigni“cantdifferences,althoughtherewasanalmostsigni“cant(0.054)decreaseinshortdistanceintrahemisphericPCintheAlzheimergroup.ResultsforPLIaregiveninFigure9.TheaveragePLIinthebetabandwassigni“cantlylowerintheAlzhei-mergroupcomparedwithcontrols(0.009).FurtheranalysisrevealedthatPLIvaluesforbothshort(andlongdistance(0.016)intrahemisphericelectrode Figure7.MeanIC(averagedoverallpairsof21channels)fordifferentmontages(average,source,mastoids,bipolar,andCz).Eachepochhasalengthof4,096samples(8.18s).Epochno.6and7correspondtotheseizure. Figure8.MeanPCfor15subjectswithAlzheimersdiseaseand13controlsubjectswithsubjectivememorycomplaints.Errorbarsindicatestandarddeviations.Resultsaretheaverageoffourepochs(aver-agereference,epochlength4,096samples,21channels,digitally“lteredbetween13and30Hz,samplefrequency500Hz).Total:averageofallpairsof21channels;intra_s:averageofallshort,intrahemisphericelectrodepairs;intra_l:averageofalllongintra-hemisphericelectrodepairs;inter_s:averageofallshortinterhe-misphericelectrodepairs;inter_l:averageofalllonginterhemi-sphericelectrodepairs.Detailsofthespeci“celectrodepairsmakingupthefoursubaveragescanbefoundinthemethods Figure6.MeanPLI(averagedoverallpairsof21channels)fordifferentmontages(average,source,mastoids,bipolar,andCz).Eachepochhasalengthof4,096samples(8.18s).Epochno.6and7correspondtotheseizure. tametal. r1186r pairswerelowerintheAlzheimergroup.ResultsforICareshowninFigure10.TheaverageICwassigni“cantlylowerinAlzheimerpatients(0.002).Furtheranalysisshowedthatthiswasduetoasigni“cantlylowerICinAlzheimerpatientsforshortintrahemisphericdistances0.005)andlonginterhemisphericdistances(0.013).Overall,PLIandICwerebetterindistinguishingbetweenAlzheimerpatientsandcontrolsthanPC.Also,PCshowedacleardropfromshorttolongdistances,whichwaslesspronouncedforICandvirtuallyabsentforMEGdataMEGdataoftwohealthysubjectswereanalyzedtoillustratethein”uenceofvolumeconductiononspatialpatternsoffunctionalconnectivity.Resultsaresummar-izedinFigure11.Inbothsubjects,thehighestvaluesforthe8…13HzPCintheno-task,eyes-closedstateshowedacharacteristicpatternwithaclearpredominanceofsmalldistancesandavirtualabsenceoflongdistances.Incon-trast,theothermeasures(PLIandIC)displayedadifferentpattern.ForsubjectA(upperrow),PLIshowedthestrong-estcorrelationsbetweenaclusterofchannelsaboverighttemporal/occipitalareasandanumberofstrongleftcen-traltorighttemporal/occipitalcorrelations.TheIChadasimilarspatialpatternaswellasanumberofleftandrightfrontotemporalcorrelations.ForsubjectB(lowerrow),PLIshowedstrongcorrelationsradiatingfromoccipitalregionstotemporalandfrontalregionsaswellasleft/rightcorrelationsovertheposteriorareas.TheICshowedasomewhatsimilarpatternbutwithmorerelativelyshortdistancecorrelationsovertherighttemporalarea.DISCUSSIONWehaveintroducedthePLIasanovelmeasureofphasesynchronizationexploitingtheasymmetryofthedistributionofinstantaneousphasedifferencesbetweentwosignals.InnumericalsimulationsoftheKuramotomodel,PLIincreasedasafunctionofcouplingstrengthcontrastingICandwaslesssensitivetovolumeconductionthanPC.InEEGabsencedata,PCwasmoresensitivetomontageeffectsthanbothPLIandIC.PLIandICper-formedbetterindetectinglossofEEGbetabandconnec-tivityinAlzheimerpatientscomparedwithcontrols.Finally,thespatialpatternofMEGalphabandconnectiv-itybaseduponPCwasdifferentformthepatternsbaseduponeitherPLIorIC,whichwerequitesimilartoeachWeusedtheKuramotomodelofgloballycoupledoscil-latorstostudytheeffectsofchangesintruesynchroniza-tionandvolumeconductiononPC,PLI,andICfortwomajorreasons:(i)theoscillatorsmaypresentanaturalmodelforoscillatoryEEGorMEGactivity;(ii)thebehav-iorofthemodelisverywellstudiedand,e.g.,theonsetofsynchronizationasafunctionofcouplingstrengthisexactlyknown[Strogatz,2000].Wemodeledvolumecon-ductionquitesimplisticallybyallowingformorethanasingleoscillatortocontributetoeachsimulatedEEGchan-nel.Whilethisconstructionstronglyexaggeratedeffectsofvolumeconduction,itallowedfortestingthebehaviorofthevariousmeasuresunderextremeconditions.Noticethatmodelingrealisticsourcesinavolumeconductorisbeyondthescopeofthepresentpaper.Also,useofamorebiologicallyinspiredmodeloftheEEGwouldhavethedisadvantagethatinsuchmodelstheexactrelationbetweenchangesincouplingstrengthandsynchronizationisnotanalyticallyaccessible.Ourmodelsimulationsshowedthat,asexpected,bothPCandPLIrespondedtoincreasesinthecouplingstrengthintheformofasuddenincreaseatthebifurcation.WhilethisresultforPCappearsobvious,itunderlinesthePLIscapacities,althoughPLIiscon-structedtojustdetectnonzerophaselagcoupling.Thatis,PLIisabletodetectsynchronizationintheKuramotomodelwithmoderatecouplingstrength.Withveryhigh Figure10.MeanICfor15subjectswithAlzheimersdiseaseand13controlsubjectswithsubjectivememorycomplaints;cf.Figure8. Figure9.MeanPLIfor15subjectswithAlzheimersdiseaseand13con-trolsubjectswithsubjectivememorycomplaints;cf.Figure8. ssessmentof r1187r valuesof,themeanphasedifferencebetweenalltheoscillatorsvanishes,whichmightexplainwhythePLIdoesnotreachavalueofone.ComparedwithPLI,ICper-formedmuchworse,probablybecauseitsimplyre”ectsthesmallvalueofthemeanphasedifferenceevenforrela-tivelysmallvaluesof.ThemodelshowedthatPCisstronglyin”uencedbythesimulatedvolumeconductioneffects,althoughitstillincreaseswithincreasesincouplingstrength.ThissuggeststhatabsolutevaluesofPCcannotbeinterpretedinthecontextof(unknown)in”uencesofvolumeconduction,whilechangesinPCbetweenexperi-mentalconditionsand/orgroupscouldstillre”ectchangesincoupling.However,ifthevolumeconductioneffectsalsochangeasafunctionofconditionorgroup,thenthisconclusionisnolongervalid.AlthoughPLIisnotimmunetothevolumeconduction,theseeffectsareclearlysmallerthanforPC,especiallyformoderatelyhighvaluesofThisreadilysuggeststhatforallpracticalpurposesPLImightbeamorereliablemeasureoftruesynchroniza-tionthanPC.TheICwasclearlyin”uencedbythesimu-latedvolumeconductioninthemodel,especiallyforhighvaluesof.Thisin”uencemightbecausedbya(relative)decreaseofcoherencysimaginarycomponentinthecaseofasimultaneousincreaseinthevalueoftherealcompo-nent„thelatterwillincreaseifthezerophaselagcouplinginthedataincreases.Thus,whiletheexistenceofanimag-inarycomponentcannotbeexplainedbyvolumeconduc-tion,itsvaluecanstillbein”uencedbyit.Whilebeingusefulforstudyingcertainfeaturesofcou-plingmeasuresunderwell-controlledcircumstances,mod-elingcannotpredicttheextenttowhichthesemeasureswillperformwithexperimentaldata.Toillustratetheirperformance,westudiedtheparadigmaticcaseofstronglyincreasedsynchronization(EEGinabsenceseizure)andanexampleofafairlysubtledecreaseofsynchronizationandspatialconnectivitypatternsinMEG.Intheabsencedata,allthreemeasuresshowedanincreaseduringtheseizure.Notethatincreasedsynchronizationduringabsenceseiz-uresisawell-knownphenomenon,whichshouldberepro-ducedbyanyusefulmeasureofsynchronization[Amor Figure11.Illustrationofthespatialdistributionofthestrongestcorrela-tionsbetweenpairsofMEGchannelsusingeitherphasecoher-ence(PC,leftcolumn),thephaselagindex(PLI,middlecolumn)ortheimaginarypartofcoherency(IC,rightcolumn).Dataarecollectedfromtwodifferenthealthysubjects(upperrowandlowerrow).Inallmaps,onlycorrelationsabovethresholdaredisplayed.Thethresholdwaschosensuchthatsuf“cientlymanyconnectionswerevisibletoallowforaproperevaluationofthespatialpatternofsuprathresholdconnections.Eyesclosed,notaskMEG(samplefrequency312.5Hz;“ltersettings:8…13Hz).Epochlength4,096samples(13.083s). tametal. r1188r etal.,2005;Mormannetal.,2000].ForEEG,however,differentmontagesandactivereferenceelectrodesmaystronglyin”uencetheoutcomeofestimatedsynchroniza-tionbetweenthechannels[Guevaraetal.,2005;Lachauxetal.,1999;Nunezetal.,1997].Thein”uenceofmontagewasquiteclearforthePC,wherebothpreictal,ictal,andpostictalvalueswereaffected.ThePCvalueswerelowestforthesourcederiva-tionandhighestfortheCzreference.Thesourcederiva-tionperformedbestintermsoftherelativeincreaseduringtheseizure.TheseresultsagreewithstudiesoncoherencebyNunezandcoworker[1997].NotethatMimaetal.[2000]alreadysuggestedtherelativesuperiorityofthesourcederivationforestimatingtruecoupling.IncontrasttoPC,bothPLIandICwerelesssensitivetoin”uencesofmontage.Inthepre-andpost-ictalphases,montagehadalmostnoeffectbutduringtheseizurewefounddifferen-ces.There,thesourcemontageperformedbest,especiallyforPLIshowingarelativeincreasewhencomparedwithpreictallevelsbyafactorof5(comparedwithamaximumincreasebyafactorof3.5forIC).Thus,evenwithPLIandIC,thechoiceofmontagestillhasanimpactbutperform-anceintermsofdetectingchangesinlevelsofsynchroni-zationisclearlyincreasedwhencomparedwithPC.TheAlzheimerdatasetwasusedtostudythesensitivityofthethreemeasuresindetectingsubtlechangesinbetabandcoupling.Suchchangeswerealreadydemonstratedforthisdatasetinanotherstudy[Stametal.,2006].ThePCshowedamoderatelysigni“cantoveralllossofbetabandsynchronizationbutcouldnotfurtherdifferentiatethisgroupeffectinlong/shortdistanceofintra-/inter-hemisphericcomponents.Ofinterest,PCwasconsistentlyhigherforshortdistancesimplyingvolumeconductioneffects.Incontrast,bothPLIandICshowedmoresigni“-cantgroupdifferencesandrevealedmoredetailsofthetypesofconnectionscontributingtothisgroupdifference.Also,especiallyforPLI,therewasalmostnodifferencebetweenshortandlongdistances,whichsuggestsadimin-ishedin”uenceofvolumeconduction.Animportantcon-clusionthatcanbedrawnfromthisdatasetisthateveninthenoisybetaband,whichshowsonlyweakcouplingandsmallgroupdifferences,itispossibletodetectnonzerophaselagsynchronization.Theexistenceofnonzerophaselagcouplinghasalreadybeenshownattheneuronallevel[Roelfsemaetal.,1997]andinintracranialrecordings[Tallon-Baudryetal.,2001].Whilezerophaselagcouplingcouldbeduetobothvolumeconduction/activereferenceelectrodesandtruecoupling,nonzerophaselagcouplingismorelikelytore”ecttruecouplingofunderlyingsour-ces.Thus,theexistenceofthistypeofcouplinginrestingstatebrainactivityandthefactthatitischangedinaneu-rologicaldisorderareofconsiderabletheoreticalinterest.InthestudyofThatcheretal.[2005],couplingwithasmallphaselagbetweenfrontalEEGchannelswastheEEGmeasuremoststronglycorrelatedtointelligence.Finally,westudiedwhethervolumeconductioneffectsinMEGmightre”ectthespatialpatternsofestimatedfunctionalconnectivity.Recently,Langheimetal.[2006]describedsuchpatternsinsomedetail.ThepatternofalphabandconnectivitybaseduponPC,displayedbyshowingsensorpairswithaPCaboveacertainthresholdasatwo-dimensionalgraph,showedsomesimilaritytothepatternsinthepaperbyLangheimandcoworkers.However,bothPLIandICshowedacompletelydifferentspatialpattern.Remarkably,PLIandICpatternswerequitesimilartoeachotherinbothsubjects.Thecompari-sonbetweenthePCpatternontheonehandandthePLIandICpatternsontheotherhandclearlyrevealedthatthePCpatternwasdominatedbylocalconnectionsbetweenadjacentsensors.SuchlocalconnectionswereabsentinPLIandICpatterns,whichweredominatedbylongdis-tanceinteractions.Thisresultsuggeststhat,forMEG,PCestimatesfornearbychannelswerestronglyin”uencedbyvolumeconductionandthatthisin”uencewasdiminishedinthecaseofPLIandIC.Quantifying(thestrengthof)interactionbyPLI,andsimi-larlybyIC,onecertainlyacceptstherisktomisslinearbutfunctionallymeaningfulinteractions,which,inprinciple,mightbeexpressedinnearzerophasecoherence.Here,wewouldliketostressthatthispotentialomissionisdeliberateand,whilerealizingthattheremaininginformationmightbeincomplete,PLI(andIC)areclearlyfreeofanyartefactsofvolumeconductions.Webelievethatthelatterarethebyfarmostfrequentcauseformisinterpretationofmoregen-eralmeasuresofinteraction.Wemustadmit,however,thattheobviousquestionHowmuchdowemiss?canyetnotbeansweredasit,aboveall,dependsonthespeci“cnatureofthesystemunderstudy.Nolteandcolleagues[2004]haveshownthatanonvan-ishingimaginarycomponentofcoherencycannotbeexplainedbyvolumeconduction.Sucharigorousstate-mentyetawaitstobeprovenforPLI,althoughthecorrela-tionstructureoftheanalyticalsignal(whichformsthebasisforourphasede“nition)doesindicatecertainsymmetriesofthecorrespondingphasedistribution(seeAppendixB).Furthermore,Guevaraandcoworkersrecentlyexpressedtheirconcernsaboutphasesynchronizationwithlags[Guevaraetal.,2005].Lachauxetal.statedthatAnothercommonassumptionisthatthephasedifferencebetweenelectrodesshouldbezeroincaseofconductionsynchrony.Thisisusuallyfalse[Lachauxetal.,2005:page202].Thus,thefactthatPLIisonlysensitivetophasesynchroni-zationwithanonzerophaselagisnoguaranteethatitwillnotbeaffectedbyvolumeconduction.Ourresultssuggest,however,thatitmaybesigni“cantlylesssensitive Alreadyaconductiondelayof2mswithinasystemof50mspe-riodisfairlylargeincommonlystudiedsystems,since,e.g.,ICcanbeaslargeassin(22ms/50ms)0.25.Hence,notonlythatourmeasuresareblindagainstlinearinteractions,theyalsoappearalmostblindtosymmetricsystemswherethedelayispresentbutnotdetectable.Althoughwebelievethatasigni“cantportionofbrainsystemsissubstantiallyasymmetric,weare,how-ever,notabletoprove,yet. ssessmentof r1189r tosucheffectsthanthecommonlyusedPC.Also,incon-trasttoIC,PLIisafairlysimplemeasureofphasesyn-chronizationthatiscloselyrelatedtoestimatesbasedonthephasedistributions(Shannon)entropy[e.g.,Tassetal.,1998].WehaveshownthatPLIperformsquitewellbothinamodelaswellasindifferenttypesofempiricaldata.Forthelatter,thePLIandICperformedequallywellandweresuperiortoPC.Inconclusion,wesuggestusingoneofthesemeasureswhenstudyingfunctionalconnectiv-itywithEEGorMEG,especiallywhenthisanalysisisbasedonsignalspace.APPENDIXAppendixA:ImaginaryPartofCoherencyTocomputethefrequencydependentcorrelationsbetweendifferentencephalographicchannels,weassumethat{)}isa“nitesetofstatisticallyindependentcom-monsourcesyieldingsignalsatchannelmintheformofalinearcombinationlikeStatisticalindependenceimpliesthesourcescross-spec-traldensities„forthesakeoflegibility,wehereomitanynormalizationandusethecross-spectraldensityratherthancoherency,cf.Eq.(7)„havetheformÞi¼1if0otherwiseinwhichdenotestheKronecker-delta.BytheuseofEq.(A2)onecanreadilyconcludethatthecross-spectralisrealsincewe“ndÞi¼Inwords,asetofuncorrelatedsourcesvolumecon-duction),eachofwhichbeingrecordedatchannelmwithweightingfactor,onlycausesareal-valuedcoherency(thecross-phasespectrumvanishesforphasesotherthan0,dependentonthesignofBythesamereasoning,butbeingabitmorerealistic,wepicktwodistinctsources,andassumethatthoseare(nonlinearly)correlated.Thus,wereplaceEq.(A2)byÞi¼ÞiþÞiðÞi¼hÞi2.UsingEq.(A4)yieldsforthecross-spectraldensitybetweenÞi¼ÞiþHence,volumeconductionmayalterthecoherency,whichisoriginallycausedbycorrelatedsourcesandonlybyashiftalongtherealaxis:thesourcedonotinterminglealongsomenontrivialdirection,althoughheretwoofthemarealreadycorrelated.Thatis,ifisreal-valued(i.e.,hasameanphaseat0or),thenanadditional,arbitrarynumberofuncorrelatedsourcesthatin“ltratetherecordingchannelsviavolumeconductioncannotrotatecoherencytowardanimaginarydirection.Putdifferently,aphasedistributionthatdoesnotpeakaround0orcannotbecausedbyvolumeAppendixB:CorrelationofAnalyticSignalsTocomparetheaboveresultswiththenumericalesti-matesbasedonsimulationandempiricaldata,weusethesamelineofreasoningbutstayinthetime-domainratherthan(Fourier-)transformingtoafrequencyrepresentation.Phasewillthennolongerbegivenbythecross-spectrumintheFourierdomainbutviatheHilbertphase.Indetail,wetakeforwhichweconstructtheanalyticalsignalusingtheHil-berttransform(i.e.theconvolutionwith½¼ t¼PVZ1 theintegralrepresentstheCauchyprincipalvalue(seealsoEq.(3).Thisconvolutionformstheimaginarypartoftheanalyticalsignalsthatwewriteas pPVZ1 aretheanalyticalsignalscorrespondingto(fortheequalityontheright-handside,weusedthelinearityofEq.(B1)anddissociativityofthecon-Forthesakeofsimplicity,wealwaysassumethatallsourceshavevanishingmean,i.e.Þi¼0.Further,weassumethatthesourcesareuncorrelatedbymeansof Equation(B4)formsamuchweakerassumptionthanEq.(A2)asisdisplaysonlytheabsenceoflinearcorrelationsratherthancom-pletestatisticalindependence. tametal. r1190r Þi¼Þi¼denotestheautocorrelationfunctionofsource.Thisreadilyyieldsvanishingcross-correlationsforthecorre-spondinganalyticalsignals,thatis,we“ndÞi¼ÞiðNoticethatwewritetheconjugatecomplexinbracketstoleavethede“nitionofthecorrelationfunctionofcom-plexsignalsopen,atleastforthetimebeing.Noticealsothatweareprimarilyinterestedintheinstantaneouscorre-lation,thatis,attheendofthedaywewillcomputethe0.Hence,weneedtocomputetheautocorrela-tionoftheanalyticsignal,forwhichweobtain(forthesakeoflegibilitywedroppedthe-notationinfrontoftheintegrals)Þi¼h 1p2Z11 skðsÞskðs0Þiðtþ

sÞðt
s0Þds0dsþ ipZ1 skðsÞskðtÞitþ

s The“rsttermontheright-handsideofEq.(B6)is“nite,whilethelastonescanceleachotheras skðsÞskðtÞitþ

sds¼Z1 kðs
tÞtþ

sds¼Z1 kðsÞ

sds¼
Z1 kðtþ

sÞt
sds1 which,withouttheconjugatecomplexform(+),triviallyyieldszeroforthelasttermsinEq.(B6)and.Forthecon-jugatecomplexform(),onecanexploitthesymmetryoftheautocorrelationfunctionbymeansofwhentakingthelimit0(seeabove).ForthemiddleterminEq.(B6)we“nd SkðsÞSkðs0Þiðtþ

sÞðt
s0Þds0ds¼Z11 ~xðs
s0Þðt
sþ
Þðt
s0Þds0ds¼Z11 Asannounced,weevaluatethelimitfor0,exploitthesymmetryof),and,ofcourse,weassumetheinte-grabilityoftheautocorrelationfunction)dividedbythatresultsinÞi¼ p2Z11 Insummary,wehaveÞi¼sothatthecross-correlationbetweenanalyticalsignalsatchannelsmandnbecomesÞi¼Importantly,Eq.(B11)yieldsonlyrealvaluessothatthephaseofthecross-correlationisalways0orinagree-mentwithEq.(A3).Thenextstepistoallowfortwosourcestobenontri-viallycorrelatedandtolookforpotentialeffectsofthepresentindependentsources.InlinewithEq.(A4),weassumethatÞi¼ThiscausestheautocorrelationassummarizedinEq.(B9)butalsogeneratesadditionalcross-termsoftheformÞi¼Consequently,Eq.(B11)canbereplacedbyÞi¼Inwords,conformwithEq.(A5),volumeconductionmaychangethe(zero-lagorinstantaneous)correlationbetweentwoanalyticalsignalsatchannelsmandn,whichisoriginallycausedbytwonontriviallycor-relatedsources,onlybyashiftalongtherealInfact,thiscorrelationstructuredoesnotimplythatasymmetricdistributionofthecorrespondingrelativeHilbertphase(i.e.peakingat0or)necessarilystayssymmetricinthepresenceofcommonsources,asthisalsodependsonthecorrespondingHilbertamplitudes,thesourcesindependencerendersacorrelatedimpactofamplitudesunlikely.Putdifferently,itisquitelikelythattheuncorrelatednessofsources(oreventheircompletestatisticalindependence)causesaninvarianceofthephasedistributionssymmetryagainstthepresenceofcommonsources.Arigorousproofforthis,admittedlysomewhathandwaving,argument,isyettobefound. ssessmentof r1191r 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