March8, 1999 23:55

Proceedings of 3icipeThird International Conference on Inverse Problems in Engineering

June 13-18, 1999, Port Ludlow, Washington, USA

3icipe/??

INDEPENDENT COMPONENT ANALYSIS FOR EEG SOURCE LOCALIZATION INREALISTIC HEAD MODELS

Leonid ZhukovCenter for Scientific Computing

and ImagingDepartment of Computer Science

University of UtahSalt Lake City, Utah 84112

Email: zhukov@cs.utah.edu

David WeinsteinCenter for Scientific Computing

and ImagingDepartment of Computer Science

University of UtahSalt Lake City, Utah 84112Email: dmw@cs.utah.edu

Chris JohnsonCenter for Scientific Computing

and ImagingDepartment of Computer Science

University of UtahSalt Lake City, Utah 84112

Email: crj@cs.utah.edu

ABSTRACTEstimatingthe locationanddistribution of electriccurrent

sourceswithin the brain from electroencephalographic(EEG)recordingsis anill-posedinverseproblem.The ill-posednatureof theinverseEEGproblemis due,in part,to thelackof auniquesolutionsuchthatdifferentconfigurationsof sourcescangener-ateidenticalexternalelectricfields. Additionally, theexistenceof a small numberof scalpmeasurementsincreasesthe under-determinednatureof thisproblemsuchthatthesolutiondoesnotdependcontinuouslyon thedata.

In this paperwe considera spatio-temporalmodel, takingadvantageof the entireEEG time seriesto reducetheextentoftheconfigurationspacewemustevaluate.Weapplytherecentlyderived infomaxalgorithmfor performingIndependentCompo-nentAnalysis(ICA) onthetime-dependentEEGdata.Thisalgo-rithm separatesmultichannelEEGdatainto activationmapsdueto temporallyindependentstationarysources.For every activa-tion map we performa sourcelocalizationprocedure,lookingonly for a singledipolepermap,thusdramaticallyreducingthesearchcomplexity. An addedbenefitof our ICA preprocessingstepis that we obtainan a priori estimationof the numberofindependentsourcesproducingthemeasuredsignal.

INTRODUCTIONElectroencephalography(EEG) is a techniquefor the non-

invasivecharacterizationof brainfunction. Scalpelectricpoten-

tial distributionsareadirectconsequenceof internalelectriccur-rentsassociatedwith neuronsfiring andcanbemeasuredat dis-creterecordingsiteson thescalpsurfaceover aperiodof time.

Estimationof thelocationanddistributionof currentsourceswithin the brain from the potentialrecordingon the scalp(i.e.,sourcelocalization)requiresthesolutionof an inverseproblem.This problemis ill-posed in the Hadamardsense(Hadamard,1902), suchthat the solution is not necessarilyunique. Phys-ically, this is a consequenceof the linear superpositionof theelectric field. Specifically, different internal sourceconfigura-tions canprovide identicalexternalelectromagneticfields. Ad-ditionally, onlyafinite numberof measurementof scalppotentialareavailable,increasingtheill-posednessof theproblem.

There exist several different approachesto solving thesourcelocalizationproblem. Initially, most of thesewere im-plementedon anon-realisticsphericalmodelof thehead.Thosemethodswhichprovedpromisingwerethenextendedto work onrealistic geometry. One of the most generalmethodsinvolvesstartingfrom someinitial distributedestimateof thesourceandthenrecursively enhancingthestrengthof someof the solutionelements,while decreasingthestrengthof therestof thesolutionelementsuntil they becomezero.In theend,onlyasmallnumberof elementswill remainnonzero,yielding a localizedsolution.This methodis implementedin suchalgorithmsasLORETTA(Pascual-Marqui,1994)andFOCUSS(Gorodnitsky, 1995).

Anotherapproachincorporatesa priori assumptionsaboutsourcesandtheir locationsin themodel.Electriccurrentdipoles

1 Copyright 1999by ASME

areusuallyusedassources,provided that the regionsof activa-tions are relatively focused(Nunez,1981). Although a singledipolemodelis themostwidely usedmodel,it hasbeendemon-stratedthat a multiple dipolemodelis requiredto accountfor acomplex field distributionon thesurfaceof thehead.Themulti-plesignalcharacterizationalgorithm,MUSIC, andits extension,RAP-MUSIC, usesubspaceprojectionsto find multiple dipolesources(Mosher, 1996).

Finally, there is a group of algorithmsthat utilize a timecourseof dipole activationsas temporalconstraintsfor the so-lution. This classof methodsincludesthe multi-startdownhillmethod,the geneticalgorithmand the taboosearch(Harrison,1996).

In this paperwe proposea new approachto theproblemofsourcelocalizationfor the inverseEEG problem. Our solutionconsistsof two steps. First, we preprocessthe time dependentdata,usingthe IndependentComponentAnalysis(ICA) (Com-mon,1994)signalprocessingtechnique.The resultof the pre-processingis asetof time-seriessignalsateachelectrode,whereeachtime-seriescorrespondsto an independentsourcein themodel.Thenumberof differentmapscreatedby theICA is equalto the numberof temporallyindependent,stationarysourcesintheproblem.To localizeeachof theseindependentsources,wesolve a separatesourcelocalizationproblem. Specifically, foreachindependentcomponent,wechooseaninstantin time fromthesignalandemploy adownhill simplex searchmethod(Nedler,1965)to determinethedipolewhichbestaccountsfor thatpartic-ular component’s contribution of themeasuredpotentialsat theelectrodes.

In our study we use simulateddata obtainedby placingdipoles in the brain in positionscorrespondingto physiologicphenomena.Wechoseto incorporatethreephysiologicallyplau-siblesources:thefirst in thetemporallobe(correspondingto anepilepticfocus),thesecondin theoccipital lobe(correspondingto observedvisualevokedresponse(ERP)studies),andthethirdin the frontal lobe (correspondingto languageprocessing).Foreachof thesesources,weusedatimesignalfrom aclinical studyto definetheir magnitudesover time. Thatis, we placethethreecurrentdipolesinsideour finite elementmodel,andfor eachin-stantin time, we projecttherealisticERP-lengthactivationsig-nalsonto32 clinically measuredscalpelectrodepositions.Theelectrodepositionsareshown in Figure1. Projectingthesourcesontotheelectrodesrequiresthesolutionof a forwardproblem.

FORWARD PROBLEMThe EEG forwardproblemcanbe statedasfollows: given

position and activationsof dipole currentsources,the geome-try andelectricalconductivity of thedifferentregionswithin thehead,calculatethe distribution of the electric potentialon thesurfaceof thehead(scalp).Mathematically, this problemcanbedescribedby Poisson’s equationfor electricalconduction(Plon-

Figure 1. TRIANGULATED SCALP SURFACE WITH 32 ELECTRODES.

THE ELECTRODES HAVE BEEN COLOR-MAPPED TO INDICATE OR-

DER: THEY ARE COLORED FROM BLUE TO RED AS THE CHANNEL

NUMBER INCREASES.

sey, 1995):

∇ σ∇Φ ∑ Is in Ω (1)

andboundaryconditions

σ∇Φ n 0 on ΓΩ (2)

whereσ is aconductivity tensorandIs arecurrentdipolesplacedwithin the headwhich modelelectriccurrentsources.The un-known Φ is theelectricpotentialcreatedin theheadby thedis-tributionof currentfrom thedipolesources.

To solve Poisson’s equationnumerically, we startedwiththe constructionof a computationalmodel. The realistic headgeometrywasobtainedfrom MRI data,wherethe volumewassegmentedand eachtissuematerialwas labeledin the under-lying voxels (Wells, 1994). The segmentedheadvolume wasthen tetrahedralizedvia a meshgeneratorwhich preserved theclassificationwhenmappingfrom voxels to elements(Schmidt,1995).For eachtissueclassification,we assignedaconductivitytensorfrom the literature(Foster, 1989). A cut-throughof theclassifiedmeshis shown in Figure2. We then usedthe finiteelementmethod(FEM) to computea solutionwithin the entire

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Figure 2. CUT-THROUGH OF THE TETRAHEDRAL MESH, WITH EL-

EMENTS COLORED ACCORDING TO CONDUCTIVITY CLASSIFICA-

TION. GREEN ELEMENTS CORRESPOND TO SKIN, BLUE TO SKULL,

YELLOW TO CEREBRO-SPINAL FLUID, PURPLE TO GRAY MATTER,

AND BLUE TO WHITE MATTER.

volumedomain(Jin, 1993). The FEM hasthe advantagethatwe are able to placecurrentsourcesin any location (not onlyon themeshnodesasin thefinite differencemethod)by simplyre-tetrahedralizingthesurroundingvolumewith a Watson-stylealgorithm(Watson,1981)after insertingthesources.Our headmodelconsistedof approximately768,000elementsand164,000nodes.The solutionof the resultingsparselinear system(con-taining approximately2,000,000non-zeroesentries)was com-putedusinga parallelconjugategradient(CG) methodandre-quiredapproximately12secondsof wall-clocktimeona14pro-cessorSGI Power Onyx with 195 MHz MIPS R10000proces-sors. Thesolutionto a singledipolesourceforwardproblemisvisualizedin Figure3. In this image,wedisplayanequipotentialsurfacein wire frame,indicatethedipole locationwith redandblue spheres,cut-throughthe initial MRI datawith orthogonalplanes,and renderthe surfacepotentialmapof the bioelectricfield on thecroppedscalpsurface.

In orderto obtaintime dependentdata,we assignedthedif-ferenttime activationsdescribedabove to thedipolesandcom-putedthe resultingprojectionon all electrodesasa functionoftime. Weconsidereda32electrodemodelfor thisstudy.

The solutionof the forward problemis needednot only toderivethesimulatedelectroderecordings,but alsolateronasthe

Figure 3. SOLUTION TO A SINGLE DIPOLE SOURCE FORWARD

PROBLEM. THE UNDERLYING MODEL IS SHOWN IN THE MRI

PLANES, THE DIPOLE SOURCE IS INDICATED WITH THE RED AND

BLUE SPHERES, AND THE ELECTRIC FIELD IS VISUALIZED BY

A CROPPED SCALP POTENTIAL MAPPING AND A WIRE-FRAME

EQUIPOTENTIAL ISOSURFACE.

iteratively appliedenginefor solvingtheinversesourcelocaliza-tion problem.

INVERSE PROBLEMThegeneralEEGinverseproblemcanbestatedasfollows:

given a time dependentsetof electricpotentialson the surfaceof theheadandtheassociatedpositionsof thosemeasurements,aswell asthegeometryandconductivity of thedifferentregionswithin the head,calculatethe locationsand magnitudesof theelectriccurrentsourceswithin the brain. Mathematically, it isan inversesourceproblemin termsof the primary electriccur-rent sourceswithin thebrain andcanbe describedby thesamePoisson’sequationastheforwardproblem:

∇ σ∇Φ Is in Ω (3)

but with adifferentsetof boundaryconditions:

σ∇Φ n 0 and Φ φ on ΓΩ (4)

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whereφ is theknown electrostaticpotentialon thesurfaceof theheadandIs areunknown currentsources.

The solution to this inverseproblemcanbe formulatedasfinding a leastsquaresfit of a setof currentdipolesto the ob-serveddatafor a singletime step,or minimizationwith respectto themodelparametersof thefollowing costfunction:

∑i

φi φi 2 σ2

i (5)

whereφi is thevalueof themeasuredelectricpotentialon the ith

electrodeandφi is the resultof the forwardmodelcomputationfor a particularchoiceof parameters;σi is a standarddeviationin theith channel.

To employ the above methodwe must solve the forwardproblemfor everypossibleconfigurationandnumberof dipoles.Eachdipolein themodelhas6 parameters:locationcoordinates(x, y, z), orientation(θ, φ) and time-dependentdipole strengthPt . Thenumberof dipolesis usuallydeterminedby iteratively

addingonedipoleata timeuntil a “reasonable”fit to thedatahasbeenfound. Even whenrestrictingthe locationof thedipole tothelatticesites,theconfigurationspaceis factorially large.Thisis a bottleneckof many localizationprocedures(Mosher, 1996;Harrison,1996).

In this paper, we usethedownhill simplex method(Nedler,1965) to find the minimum of the multidimensionalcost func-tion. In an N dimensionalspace,the simplex is a geometricalfigurethatconsistsof N+1 interconnectedvertices(for example,a tetrahedronin 3D). Thedownhill simplex methodminimizesafunctionby takinga seriesof steps,eachtime moving thepointin thesimplex away from wherethefunctionis largest.

Assumenow that we have somehow managedto filter thesignalson theelectrodes,suchthatweknow electrodepotentialsdueto everydipoleseparately. Thenfor everysetof electrodepo-tentialswe needto searchonly for onedipole,thusdramaticallyreducingtheconfigurationspace.Wewill discussthis usefulfil-teringtechniquein thenext section.

INDEPENDENT COMPONENT ANALYSISIndependentcomponent analysis (ICA) is a statistical

methodfor transforminganobservedmultidimensionalrandomvectorinto componentsthatareasindependentfrom eachotheraspossible(Bell, 1995). Thealgorithmachievesthis by factor-ing themultivariateprobabilitydensityfunctionof theinput sig-nalsinto theproductof fy ∏i fyi

yi probabilitydensityfunc-

tions(p.d.f.) of everyindependentvariable.Thisfactorizationin-volvesmakingthemutualinformationbetweenvariables(chan-nels)go to zero,i.e., makingoutputsignalsthatarestatisticallyindependent.

TheICA processconsistsof two phases:thelearningphaseandtheprocessingphase.During thelearningphase,theICA al-

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Figure 4. SIMULATED DIPOLE SOURCES MAPPED ONTO 32 CHAN-

NELS (ELECTRODES). CHANNELS ARE NUMBERED LEFT TO RIGHT,

TOP TO BOTTOM. THE FIRST CHANNEL IS THE REFERNCE ELEC-

TRODE. THESE SIGNALS ARE THE INPUT DATA FOR THE ICA ALGO-

RITHM. THE LOCATIONS OF THESE 32 ELECTRODES ARE SHOWN

IN FIGURE 1.

gorithmfindsamatrixW, whichminimizestheKullback-Leiblerdivergencebetweenthemultivariateprobabilitydensityandthemarginal distributions(p.d.f) of transformedinput vectorsx

t

(Amari, 1996):

DW f

y log

fy

∏i fiyi dy (6)

where

yt g

W x t (7)

andg is somelogistic nonlineartransformfunctionwith a slopematchingthep.d.f. of theinput f

y .

The minimizationof this integral is achieved by usingthestochasticgradientdescentmethod(Makeig,1994),whereW isiteratively adjustedusingthedatavectorsfrom x

t by applying

4 Copyright 1999by ASME

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Figure 5. ICA ACTIVATION MAPS OBTAINED BY UNMIXING THE IN-

PUT SIGNALS. WE OBSERVE THAT THERE ARE ONLY THREE INDE-

PENDENT PATTERNS, INDICATING THE PRESENCE OF ONLY THREE

SEPARATE SIGNALS IN THE ORIGINAL DATA.

thefollowing rule:

Wk 1 Wk µk I

1 2 y Wk x T Wk (8)

whereµ is alearningrateandI is aunit matrix.For thenonlineartrnasformx

t y

t , weuse

yt 1

1 e W x t (9)

Thestoppingcriterionfor thelearningphasecanbe,for ex-ample,thevalueof thelearningrate.Westoppediterationswhenµ becamesmallerthan10 6, or, in otherwords,whenonconsec-utivestepstheunmixingmatrixW doesnotchangeby morethan10 6.

Thesecondphaseof theICA algorithmis theactualsourceseparation.Independentcomponents(activations)canbe com-putedby applyingtheunmixingmatrixW to theinitial data:

ut W x t (10)

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Figure 6. THE PROJECTION OF THE FIRST ACTIVATION MAP FROM

FIGURE 5 ONTO THE 32 ELECTRODES.

There are several assumptionsone needsto make about thesourcesin orderto effectively useICA algorithms:

- thesourcesmustbe independent(signalscomeform statis-tically independentbrainprocesses);

- thereis no delayin signalpropagationfrom thesourcestodetectors(conductingmediawithout delaysat sourcefre-quencies);

- themixtureis linear(Laplace’sequationis linear);- the numberof independentsignalsourcesdoesnot exceed

the numberof electrodes(we expect to have fewer strongsourcesthanour32electrodes).

ICA returnsthe sourceactivationsup to permutationandscale,becauseit operateson distribution functions,which do not de-pendon the relative strengthor order of the signals(this alsomeansthat the relative polarities of the obtainedsignalsaremeaningless).

In EEGexperiments,electricpotentialis measuredwith anarrayof electrodes(typically 32/64/128)positionedprimarily onthetophalf of thehead,asshown in Figure1. For studiesof thehumanvisual/auditorysystem(ERP studies),the dataare typ-ically sampledevery millisecondduring the interval of interestafterstimuluspresentation,andareaveragedover many trials to

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Figure 7. SCALP SURFACE POTENTIAL MAP DUE TO SEVERAL

DIPOLES, CORRESPONDING TO TIME T=160MS FROM THE SIG-

NALS SHOWN IN FIGURE 4.

remove backgroundnoise. For a given electrodeconfiguration,the time dependentdatacanbearrangedasa matrix, whereev-erycolumncorrespondsto thesampledtimeframeandeveryrowcorrespondsto a channel(electrode).For example,thedataob-tainedby 32 electrodesin 180mscanbesampledin 180framesandrepresentedasa matrix (32 180). For ICA purposes,thismatrix canbeconsideredasx

t , whereinsteadof a continuous

variablet we have sampledtime frames. After computingtheunmixingmatrixW, wecanseparatetheindependentsourcesig-nalsusing(10). Projectionof independentactivationmapsbackontotheelectrodearrayscanbedoneby:

xkt W 1

ki uit (11)

wherext is thesetof scalppotentialsdueto just the ith source.

As such,ICA allowsusto reconstructsurfacepotentialsthatwould exist dueto eachdipoleasif it weretheonly source.Forexample,if theoutputof ICA givesthreestrongactivationchan-nels,thatmeanswe will be looking for only threedipoles.Pro-jectingeachactivationmaponthescalpelectrodesgivesusthreedifferentmaps,eachwith atimesequenceof values.For eachac-tivationmap,wechooseonevaluefrom thetimesequence(fixedpointin time),andthenuseeachmapto localizeonedipoleusingthedownhill simplex method. The resultsof numericalexperi-mentsarepresentedin thenext section.

Figure 8. PROJECTION OF THE FIRST ICA COMPONENT ONTO THE

32 CHANNELS AT TIME T=160MS.

NUMERICAL SIMULATIONSWepreparedthesimulateddataasdescribedin theprevious

sections.Thetime dependentcourseof 200msfor all 32 chan-nelsis shown in Figure4. We alsoprovide a color mappedplotof the potentialson the surfaceof the headfor the time stepat160msin Figure7. As canbeseenin thisfigure,thedistributionof potentialsis creatednot by a singledipole, but by a config-uration of several dipoles. We perform the ICA procedureonthegiven time dependentEEGdataandthe resultingactivationmapsareshown in Figure5. Noticethatthereonly threedifferentactivationpatternspresented;therestareeitherredundantarees-sentiallynoise.Projectingthefirst activationon all 32 channels,wegetthesignalsshown in Figure6,whicharethepotentialsdueto thesingletemporallobedipole. Plotting thepotentialsagainfor thetimestepat 160msin Figure8, onecaneasilyrecognizethesurfacepotentialmapasresultingfrom theactivationasingledipolesource.

We cannow checktheaccuracy of the ICA decompositionby comparingit to theresultsof theforwardproblemsimulationrun with two of the threedipoles“turned off ”. BecauseICAdoesnot preserve scale,we usecorrelationcoefficientsas ourmetric for comparingthe potentialsat the electrodes.The setsof electrodepotentialsareviewedasvectorsin N-space(in ourcaseof 180 time steps,N 180) andthecosineof the “angle”betweenthemis calculatedby takingthedot-productof thetwovectorsafter they’ve beennormalized.For thekth channel,the

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correlationcoefficientwill be:

CCk xkt xk

t

xkt xk

t (12)

A valueof CCk 1 indicatesthat thesimulatedandICA recov-eredtimeseriesat thatelectrodeareidenticalupto ascalingfac-tor. The ICA error canthusbe cumulatively estimatedover allelectrodesover theentiretime sequence,by evaluatingtheroot-mean-square(RMS)differenceof CCk from 1 over all channels:

∑32

k 1

CCk 1 232

(13)

Evaluatedwith the above formula, our threeactivation projec-tions restoredtheoriginal (unmixed)potentialdistribution withRMSerrorsof 3%,4%and10%,respectively.

We then applied the simplex downhill methoddescribedabove and found the location of the dipole which minimizedthe cost function for eachICA map. The localizedtemporallobedipolewasfoundto beaccuratewithin 7 mm of theactualsource.We repeatedthis localizationprocedurefor theoccipitalandfrontal lobe dipolesandwereable to determinetheir posi-tionswith errorsof 9 and16mm,respectively.

CONCLUSIONSWe have presentedan algorithmthat reducesthe complex-

ity of localizingmultiple neuralsourcesby exploiting the time-dependenceof thedata.We have shown thaton a realisticheadmodel with simulatedEEG data, our algorithm is capableofcorrectly predictingthe numberof independentsourcesin themodel and reconstructingpotentialsdue to eachsourcesepa-rately. Thesepotentialmapscan thenbe successfullyusedbysourcelocalizationmethodsto independentlylocalize separatesources.

ACKNOWLEDGMENTThis work was supportedin part by the National Science

Foundation,theDepartmentof Energy andtheUtahStateCen-tersof ExcellenceProgram.

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