Improved Localization of Cortical Activity By Combining EEG and...

AbstractWe describe a comprehensive linear ap-

proach to the problem of imaging brain activitywith high temporal as well as spatial resolutionbased on combining EEG and MEG data withanatomical constraints derived from MRI im-ages. The “ inverse problem” of estimating thedistribution of dipole strengths over the corti-cal surface is highly underdetermined, evengiven closely spaced EEG and MEG record-ings. We have obtained much better solutionsto this problem by explicitly incorporating bothlocal cortical orientation as well as spatial co-variance of sources and sensors into our for-mulation.

An explicit polygonal model of the corti-cal manifold is first constructed as follows: (1)slice data in three orthogonal planes of section(needle-shaped voxels) are combined with alinear deblurring technique to make a singlehigh-resolution 3-D image (cubic voxels), (2)the image is recursively flood-filled to deter-mine the topology of the gray-white matter bor-der, and (3) the resulting continuous surface isrefined by relaxing it against the original 3-Dgray-scale image using a deformable templatemethod, which is also used to computationallyflatten the cortex for easier viewing. The ex-plicit solution to an error minimization formu-lation of an optimal inverse linear operator(for a particular cortical manifold, sensor

placement, noise and prior source covariance)gives rise to a compact expression that is prac-tically computable for hundreds of sensors andthousands of sources. The inverse solution canthen be weighted for a particular (averaged)event using the sensor covariance for thatevent. Model studies suggest that we may beable to localize multiple cortical sources withspatial resolution as good as PET with thistechnique, while retaining a much more finegrained picture of activity over time.

INTRODUCTIONOver the past few decades a variety of

techniques for non-invasively measuring brainactivity have been developed. Each of thesetechniques has important and unique advantag-es, but also significant limitations. For exam-ple, the positron-emission tomography (PET)technique using labeled water to detect bloodflow has good (~cm), uniform spatial resolu-tion, but relatively poor (~10s of sec) temporalresolution. Several recently developed mag-netic resonance imaging (MRI) techniques—measuring blood volume changes with a con-trast agent (Belliveau et al., 1991) and measur-ing hemoglobin oxygenation via its effects onnearby water (Ogawa et al., 1992)—promisesomewhat better spatial and temporal resolu-tion. As with PET, however, the indirect con-

Improved Localization of Cortical ActivityBy Combining EEG and MEG with MRI

Cortical Surface Reconstruction

ANDERS DALE AND MARTIN SERENO

2 ANDERS DALE AND MARTIN SERENO

nection between the neural activity and its mea-sured metabolic consequences conceals thefine (subsecond) structure of the underlyingneural events.

A widely used technique with better(~msec) temporal resolution is electroencepha-lography (EEG), which measures the potentialdifference between various locations on thescalp. A number of interesting correlations be-tween features of the measured waveforms andvarious aspects of attention, memory, and lin-guistic tasks have been discovered (see e.g.,Luck et al., 1990; van Petten & Kutas, 1991;Neville et al., 1991). The temporal resolutionof this technique is essentially limited only bythe time scale of the biological processes pro-ducing the potentials. The spatial resolution,however, is limited by several factors. Oneproblem is that activity in a small region of thebrain—especially if it is located deep inside thehead—can produce potentials that are spreadrather widely across the scalp, strongly over-lapping potentials produced by other sources.

Closely related to EEG is magnetoen-cephalography (MEG), which measures minutefluctuations in the magnetic field outside thehead using extremely sensitive (SQUID) sen-sors (see e.g., Wood et al., 1985; Hari & Lou-nasmaa, 1989; Pantev et al., 1990; Wood et al.,1990). The EEG and MEG are fundamentallyrelated through Maxwell’s equations to the dis-tribution of dipole moment throughout thebrain and head and hence have similar temporalresolution. However, the MEG has the advan-tage of being less affected by head inhomoge-neities, and somewhat less smeared out spatial-ly by skull impedance than the EEG. On theother hand, a weakness of the MEG is its rela-tive insensitivity to deep or radially orientedsources, making it effectively blind to certainpatterns of activity in the brain that would pro-duce an observable EEG.

The so-called forward problem of calcu-lating the electric and magnetic fields outsidethe head, given the current distribution insidehead and the conductive properties of the headand brain, is a well-defined problem of electro-statics (Nunez, 1981). By contrast, the so--

called inverse problem of finding the distribu-tion of currents inside the head, based on elec-tric and magnetic recordings outside the head,is fundamentally ill-posed—that is, it has nounique solution. For any set of measurementsoutside the head, there are infinitely many cur-rent distributions inside the head that are com-patible with those recordings. Although com-bining both electric and magnetic data aboutthe same event reduces the space of indistin-guishable solutions, additional constraints areneeded in order to make the problem solutionunique in a principled way. Additional con-straints come from assumptions about likelycurrent source distributions and statistics, sen-sor statistics, and information from other activ-ity imaging techniques like PET or functionalMRI.

In the following we will present a singleframework for combining data from: (1) EEGand MEG recordings (and PET or functionalMRI, if available), (2) cortical surface recon-structions based on MRI images, (3) prior as-sumptions about typical spatial distributions ofbrain activity, and (4) information about cova-riance of the sensors for a particular (averaged)event. Our primary goals are to retain a linearapproach, but constrain it so that the ill-posed-ness of the inverse problem is greatly reduced.A particularly insidious type of ill-posedness iswhen sources cancel each other, leading toequivalent solutions that are qualitatively verydifferent. Our studies suggest that the ill-pos-edness that remains is usually benign; nearbysources may not be resolved, but the qualitativestructure of the solution is preserved. By solv-ing directly onto the cortical manifold, it ismuch easier to assess and view solutions, espe-cially after the cortex has been partially “ inflat-ed” (PET or functional MRI data by themselvescould also advantageously be viewed this way).

Several components of the current ap-proach to the inverse problem have been con-sidered individually by other authors (Nunez,1981; Scherg, 1989; Ioannides, Bolton, & Clar-ke, 1990; Smith et al., 1990; Wood et al., 1990;Mosher, Lewis, & Leahy, 1992; George et al.,in press; Greenblatt, personal communication).

COMBINING EEG AND MEG WITH MRI 3

By integrating multiple constraints into a uni-tary framework, however, we have been able toobtain much better behaved solutions thanthose obtained with any technique used by it-self. Model studies suggest that we may beable to localize multiple cortical sources withspatial resolution comparable to PET or func-tional MRI while retaining a fine-grained pic-ture of activity over time.

A LINEAR APPROACH TO THE IN-VERSE PROBLEM

In the typical frequency range of neuralelectric activity of less than a few hundred Hz,the electric and magnetic fields of the brain canbe well accounted for by the quasi-static case ofMaxwell’s equations—that is, magnetic induc-tion and capacitive effects are negligible (Nun-ez, 1981). As has been noted previously by nu-merous authors, this results in a simple linearrelationship between the electric and magneticrecordings, and the components of dipole mo-ment at any location in the brain. More precise-ly, if we divide the brain volume into N/3 smallvolume elements and approximate the local di-pole moment within each volume element withits decomposition onto three orthogonal com-ponents, we get

(1)

or in matrix form

(2)

where vi is the potential at the ith electrode rel-

ative to a point at infinity, and sj is the strengthof the jth dipole component. The ith row of theE matrix specifies the lead field of the ith elec-trode, i.e. how the potential at the ith electrodevaries with the strength of each dipole compo-nent. The sum in Equation 1 ranges over allthree dipole components of all volume ele-ments. Similarly, the jth column of E specifiesthe gain vector for the jth dipole component, i.e.how much the measurement at each electrode

vi ei jsjj

N

∑=

v Es=

varies with the strength of the jth component.The coefficients in E are in general complicat-ed non-linear functions of the electrode loca-tions, and the shape and electrical properties ofthe head (see Appendix A).

For the magnetic recordings we have

(3)

or in matrix form

(4)

where mi is the component of the magneticfield along the orientation of the ith magneticsensor. The columns of the matrix B specifythe magnetic gain vector of each dipole compo-nent.

Note that Equations 2 and 4 can be com-bined into one equation expressing the linearrelationship between each dipole componentstrength and the composite electric and mag-netic recordings:

, where , (5)

More generally, if we assume some additivenoise at the sensors, we get,

(6)

where n is a zero-mean random vector1.

Inverse SolutionThe inverse problem can be stated as one

of finding the distribution of dipole strength sgiven recording data x. Clearly, if the varianceof the noise is non-zero, there will exist nowell-defined solution to this problem. Also,since the rank of A is always less than or equalto the number of sensors, there will exist infi-nitely many indistinguishable solutions when-ever the number of unknowns (dipole compo-nents), exceeds the number of knowns (sensorlocations). However, if a priori informationexists about the statistical distribution of dipole

mi bi jsjj

N

∑=

m Bs=

x As= x vm

= A EB

=

x As n+=


moment and sensor noise, the inverse problemcan be stated in terms of statistical estimationtheory. In the linear case, this corresponds tofinding the linear operator that minimizes theexpected difference between the estimated andthe correct solution. More specifically, the ex-pected error ErrW can be defined as

(7)

where W is a linear operator that maps a re-cording vector x into an estimated solution vec-tor . If we assume that both the noise vectorn and the dipole strength vector s are normallydistributed with zero mean and covariance ma-trices C and R, respectively, Equation 7 be-comes

(8)

(9)

, where

(10)

(11)

(12)

This expression can be explicitly minimized bytaking the gradient, setting it to zero and solv-ing for W. This yields an optimal linear esti-mator,

(13)

The expression for the optimal inverselinear operator W given in (13) can be shown tobe equivalent to the so called minimum-normsolution (Tikhonov & Arsenin, 1977; Ha-malainen & Ilmoniemi, 1984), provided the co-variance matrices C and R are proportional tothe identity matrix. This corresponds to the as-sumption that both the noise at each sensor andthe strength of each dipole are independent and

ErrW Wx s−2〈 〉=

ŝ

ErrW W As n+( ) s−2〈 〉=

WA I−( ) s Wn+ 2〈 〉=

M s Wn+ 2〈 〉=M WA I−=

M s 2〈 〉= Wn 2〈 〉+

Tr M RMT( )= Tr WCWT( )+

W RAT ARAT C+( )1−

=

of equal variance. An advantage of the presentformulation is that any empirical observationsor reasonable assumptions about the second or-der statistics of the sensor noise and the dipolestrengths can be explicitly incorporated to con-strain the solution.

It is also worth pointing out that evaluat-ing W from Equation 13 only requires inver-sion of a matrix square in the number ofknowns (sensors), rather than square in thenumber of unknowns (dipole components).This is important since the time required to in-vert an N by N matrix is proportional to N3, andthe number of sensors will typically be muchsmaller than the large number of dipole compo-nents (~10,000) that are required to accuratelytile the cortical mantle (see below). The onlypotentially time consuming part of evaluatingW is the matrix multiplication with R, which inthe worst case will take time proportional to thesquare of the number of dipole components.However, if we conservatively make no a prio-ri assumptions about long-range correlations,then the R matrix will be very sparse, and thememory and time needed for calculating Wwill increase more or less linearly with thenumber of unknowns.

Error PredictionAn important advantage of the linear esti-

mation approach to the inverse problem is thatit is possible to quantify the influence of sensornoise and activity of other dipoles on estimateddipole strengths. More precisely, the ith row ofthe matrix M = WA − I specifies how much aunit of dipole strength at each dipole locationwould contribute to the estimation error of theith dipole. Consequently, the expected squarederror of the strength of the ith dipole due to ac-tivity of other dipoles is given by M iRM i

T,where M i is the i

th row of M , and R is the co-variance matrix of the sources. Similarly, theith row of the matrix W specifies how much aunit of noise at each sensor contributes to theestimation error of the ith dipole strength. Theexpected squared estimation error for the ith di-pole due to noise is given by W iCW i

T, whereW i is the i

th row of W and C is the covariance


matrix of the noise.Such expressions for the likely estimation

errors can be quite useful for quantifying confi-dence intervals for hypothesis testing, as wellas for designing sensor configurations whichoptimize the estimation accuracy in some re-gion of interest. Similar measures are difficultto obtain for iterative non-linear approaches tothe inverse problem without explicit, computa-tionally intractable searches for alternate solu-tions.

CONSTRAINING THE INVERSE SOLU-TION

The inclusion of electric and magneticdata in a single formulation constrains the solu-tion to the inverse problem since these two re-cording techniques often yield complementaryinformation (see Model Studies below). Nev-ertheless, many equivalent solutions will re-main, even in the presence of a single source,and it is necessary to add additional constraintsin the form of a priori information about likelysolutions. Ideally, we would like to avoid arbi-trary a priori constraints—such as having todecide how many source dipoles the solutionwill contain (cf. Scherg, 1989). In the follow-ing, we describe how more biologically plausi-ble constraints can be incorporated into the lin-ear estimation approach outlined above. Ourgoal is to constrain our solutions while retain-ing a relatively ’automatic’ procedure in whichthe user is spared sensitive, yet arbitrary deci-sions.

Using Anatomical ConstraintsA crucial way to reduce the ambiguity of

the inverse problem is to incorporate anatomi-cal constraints explicitly into the solution (seealso Wood et al., 1990; George et al., 1992).We can consider in the forward solution onlythose dipole locations and orientations that areconsistent with the anatomical data.

A common assumption is that much ofthe EEG and MEG observable at a distance isproduced by currents flowing in the apical den-drites of cortical pyramidal cells. Because ofthe columnar organization of the cortex, the re-

sulting local dipole moment would be orientedperpendicularly to the cortical surface. Subdu-ral and intracortical recordings of field poten-tials at varying distances from an activated cor-tical locus are consistent with this picture (seee.g., Mitzdorf, 1987; Dagnelie, Spekreijse, &van Dijk, 1989; Barth & Di, 1990), in general,having revealed substantial vertical, but littlelocal horizontal variation in potential. Thus, ifthe shape of the cortical sheet is known, the lo-cations and orientations of cortical sources canbe constrained by dividing the sheet into patch-es that are sufficiently small so that a dipole inthe center of a patch accounts for any distribu-tion of dipole moment within the patch. The in-verse problem then reduces to estimating thescalar distribution of dipole strength over theoriented cortical patches. This should be com-pared to unconstrained situation where wewould have to solve onto the orthogonal triplesof “ regional dipoles” distributed throughout thevolume of the forebrain (see e.g., Smith et al.,1990); for a given number of dipoles, the solu-tion is not only less constrained, but muchcoarser.

It is important to note that the EEG andMEG may be generated by activity in subcorti-cal structures. In order to localize such activitycorrectly, the model must include dipole com-ponents in these locations as well as in corticalones. Since subcortical sources are generallylocated much further away from the EEG andMEG sensors than are the cortical sources, thediscretization of these regions can be coarser.Some of these structures are laminated andcontain cells with elongated dendrites perpen-dicular to the laminae (e.g., the medial superiorolive). In structures without clearly elongatedcellular morphology, one “ regional” dipole tri-ple in the center of each nucleus may be suffi-cient to account for any distribution of currentflow within it.

Using the Assumed Source CovarianceAnother useful type of constraint on the

inverse problem comes from a priori informa-tion about correlation between the dipolestrength at different locations. For instance, it


is probably reasonable to assume that activitiesin two neighboring patches of cortex are notcompletely independent, but somewhat posi-tively correlated. If the correlation betweenany two cortical patches is known, the priorsource covariance matrix R is given by

(14)

where σi2 is the variance of the strength of theith dipole, and Corr(i,j) is the correlation be-tween the strengths of the ith and the jth dipoles.The actual correlation of dipole strength as afunction of distance on the cortical surfacecould be estimated by invasive recordings onanimals or human patients.

Note that if the dipoles are assumed a pri-ori to be completely independent (Corr(i,j) = 0,if i ≠ j), and have the same variance (σi2 = σj2),then the method reduces exactly to the mini-mum-norm approach mentioned above.

Using the Observed Sensor CovarianceEven after incorporating the constraints

described above, however, localized sourcesstill tend to be smeared out by the inverse solu-tion. Denser sensor arrays help for superficialsources, but deep sources are often displaced tothe surface and spread over several gyri (seeModel Studies below). An additional powerfulconstraint on the inverse solution that we nowturn to comes from considering the entire timecourse of the electric and magnetic recordings,rather than just a single time point.

A commonly made assumption is that re-cordings throughout an epoch are caused by ac-tivity in a limited number of locations in thebrain, each represented by a single dipole withfixed orientation. For the sake of analysis, it isuseful to make the following additional as-sumptions: (1) the activity of each of the, say,k locations is not completely correlated withthe activity in any of the other locations, (2) thegain vectors of the active locations are linearlyindependent, (3) the sensor noise is additiveand white with constant variance σ2, i.e. C =σ2I . The sensor covariance matrix

Ri j σiσjCor r i j,( )=

, (15)

(16)

where the summations range over all active di-poles, has a singular value decomposition giv-en by

(17)

, (18)

The first k column vectors U1..Uk of U form anorthonormal basis for the so-called signal sub-space spanned by the k linearly independentgain vectors for the active locations, andUk+1..UN form an orthonormal basis for theso-called noise subspace, defined as the or-thogonal complement of the signal subspace.Each eigenvalue λi specifies the component ofsensor covariance in the direction of the corre-sponding eigenvector.

The noise subspace projection ηi of a gainvector Ai, which can be written

, (19)

vanishes for true dipole locations. It remains fi-nite for locations whose gain vectors do not lieentirely within the signal subspace. The loca-tions of the true dipoles can thus be estimated,

based on the peaks in a plot of as a func-

tion of location, which is essentially the ideabehind the MUSIC algorithm (Mosher, Lewis,& Leahy, 1992).

One limitation of this approach is that itrequires a clear-cut separation between signalspace and noise space. Since the eigenvalues of

D xxT〈 〉=

σ2I σiσjCor r i j,( ) A iA jT

j∑

i∑+=

D UΛUT=

U1 … UN

λ1 0 00 … 00 0 λN

U1 … UNT

=

η iA i

T Uk 1+ … UN Uk 1+ … UNTA i

A iTA i

=

1η i


the sensor covariance matrix typically decreasemore or less smoothly, the choice of eigenvaluethreshold is somewhat arbitrary. One way toavoid this problem is by using a more gradednotion of noise subspace and signal subspace.For instance, by weighting the projection of thegain vectors onto every eigenvector Ui of D bythe reciprocal of the corresponding eigenvalueλi, a measure can be obtained which is large forany gain vector which has a significant compo-nent in a direction of low sensor covariance(“noise subspace” component), without requir-ing an explicit eigenvalue threshold. More pre-cisely, a new measure ξi can be defined as

(20)

Note that ξi converges to ηi as

and .

This measure can then be incorporatedinto the linear estimation framework as some-thing similar to an a priori variance estimate2

for the ith dipole as

(21)

where f is a continuous, non-decreasing func-tion. As before, information about correlationbetween dipole component strengths of neigh-boring locations can be coded into the estimat-ed source covariance matrix R by

(22)

Note that if the condition number of D isclose to unity (i.e. all eigenvalues equal), thenall Rii are also equal and this method essential-ly reduces to the minimum-norm-like approachdiscussed above. However, if the largest andsmallest eigenvalues are significantly different,as is usually the case, this method will assignlow a priori variance estimates to dipole com-ponents with significant “noise-space” projec-

ξ iA i

TUΛ 1− UTA iA i

TA i=

λ1…k ∞→

λk 1…N+ 1→

Ri i f1ξ i

( )=

Ri j Ri iRj j= Corr i j,( )( )

tions, essentially eliminating many of the loca-tions in the brain from consideration. In themodel studies that follow, we have chosen f(x)= x. However, the localization of deep, point--like sources can be further improved by choos-ing an f(x) that pushes small arguments closerto zero.

Although the sensor covariance matrix Dcan not be measured directly, it can be approx-imated by

, (23)

where x1...xn are the recording vectors at n dif-ferent times. With extended epochs of activity,it may be preferable to calculate a new set ofRii’ s for each of a series of sub-epochs to helptease apart nearby sources, since different com-binations of sources may be active in differentsub-epochs. Note that Equation 20 applies onlyto dipoles whose orientation is known. Howev-er, it can be extended to handle “ regional di-poles” in a manner similar to that developed inMosher, Lewis, and Leahy (1992).

Using PET InformationAlthough activity imaging techniques

like PET and functional MRI may provide littleinformation about the fine-grained temporal se-quence of brain activity, they do provide infor-mation about average brain activity with rela-tively high and uniform spatial resolution. Itmay be reasonable to assume that regions in thebrain that show increased activity using meta-bolic techniques are also ones that are on theaverage more electrically active over time.Thus, a simple way to incorporate this data intothe framework outlined above is to make theprior variance estimate for a location in thebrain an increasing function of the PET or func-tional MRI values at that location. It wouldclearly be preferable to have a more precise,empirically-based model of how the processesthat affect PET and functional MRI signals(e.g. cerebral blood flow or hemoglobin oxy-genation) are related to the current dipole dis-tribution of the EEG and MEG.

D̂ 1n

x1 … xn x1 … xnT

=


FINDING THE CORTICAL SURFACEFor the approach described above to be

practically useful, the shape of the corticalsheet (and the location of possible subcorticalsources) must be known. Since the precise ge-ometry of the cortical manifold varies substan-tially among different people, it is essential tobe able to reconstruct the cortical sheet of eachsubject from non-invasive imaging techniques,like MRI. This poses two daunting challenges:(1) the MRI data has to have sufficient spatialresolution in all directions to resolve all the sul-ci and gyri, while also providing sufficient con-trast between the relevant tissue types, (2) acomputationally tractable algorithm has to bedeveloped for automatically constructing awire-frame representation of the cortical sheetbased on the MRI data.

Three-Dimensional MRI ReconstructionWith conventional two-dimensional

MRI, it is possible to obtain images with excel-lent contrast between most relevant tissuetypes, like cortical gray and white matter, cere-brospinal fluid, skull, and scalp with an in--plane resolution of better than 1 mm. However,the resulting 2-D sections are usually relativelythick (e.g., 3-6 mm). Thus, the resolution in thedirection perpendicular to the plane of sectionis much poorer than within the plane—individ-ual volume elements (voxels) are elongated.This causes problems whenever the corticalsurface deviates from being nearly perpendicu-lar to the slice plane; single voxels will then av-erage gray and white matter together, generat-ing a smeared image of the cortical mantle.

Using so-called volume acquisition tech-niques, it is theoretically possible to achieveresolutions of 1 mm in all directions. Unfortu-nately, current volume acquisition protocolsare inherently less flexible than the protocolspossible with 2-D scans (since each pulse ex-cites the entire volume of the brain, interleav-ing is not possible, restricting protocols tosmaller flip angles and shorter TR values). Onthe standard MRI scanner available to us forthis study, the contrast between cortical gray

and white matter possible with an optimal 2-Dinversion recovery (IR) protocol was far supe-rior to that possible using volume acquisition.Since the tessellation of the cortex depends ona clear gray/white matter distinction (see be-low), we had to find a way to overcome the“partial-voluming” problem.

We have developed a method for combin-ing three orthogonal (coronal, sagittal, horizon-tal) series of conventional, moderately thicksections into a single volumetric data set withthe same high (i.e., subslice) resolution in allthree directions. The method is based on thesimple observation that each pixel in a typical2-D scan represents a weighted average of thesignal emitted from an elongated rectangularprism of tissue (pixel x-size by pixel y-size byslice thickness). By combining data from dif-ferent directions it is possible to estimate thesignal emitted from cubic voxels of smallersize using a linear estimation technique verysimilar to that described above for currentsource localization (see Appendix B).

A major advantage of this technique isthat any 2-D acquisition protocol can be used,including inversion recovery (IR) protocols forT1 weighting, and spin-echo (SE) protocolswith long repetition times for proton-densityand T2 weighting. By combining spatially reg-istered 3-D data sets made with different proto-cols, it is possible to simultaneously classify allmajor tissue types, which is not possible usingany single scan type (cf. Buxton & Greensite,1991). Thus, we can retain optimal grey/whitematter contrast (crucial for cortical surface re-construction) while still being able to distin-guish gray and white matter from skull, skin,and cerebrospinal fluid (necessary for automat-ic skull removal).

Figure 1A shows a stack of 6 mm thickcoronal T1 weighted (inversion-recovery) slic-es of the brain (see Methods). The resolutionwithin the section plane is obviously much bet-ter than in the anterior-posterior direction. The3-D reconstruction resulting from combiningthe sagittal and horizontal slice series with thecoronal series is shown in Figure 1B. The res-olution in the anterior-posterior direction is


Figure 1. Three-dimensional MRI reconstruction. Coronal sections (6 mm thick) from an inversion recovery protocol are merely stacked in A. In B, the coronal series has been combined with a horizontal and a sagittal series using a linear deblurring technique to give an image with uniformly high resolution. C and D illustrate the same deblurring technique applied to proton density and T2-weighted images. The contrast between the gray and white matter is much reduced in comparison to the T1-weighted inversion recovery image.


greatly improved at only a small cost to the in--plane resolution. Note that this image couldhave been sectioned in any other (non-orthogo-nal) plane without a loss in resolution. Thethree-dimensional reconstruction of the protondensity and T2 weighted data sets are shown inFigures 1C and 1D, for comparison. Clearly,the contrast between the gray and white matteris most striking in the T1 image.

“Shrink-Wrap” Surface ReconstructionA very realistic-appearing image of the

cortex can be generated by displaying stackedsections using interslice interpolation andtransparency (see e.g., Damasio and Frank,1991). Such an image, however, cannot be di-rectly used to constrain the orientations ofsource dipoles. For this, we need to construct awireframe model that explicitly recovers the to-pology of the cortical sheet. A typical approachto this problem has been to manually trace theoutline of the cortex in series of 2-D sections,and then use some heuristic algorithm (or apracticed hand) to connect the contours in eachsection into a continuous surface. The mainproblems with this approach are: (1) it requiresconsiderable manual work for each subject, (2)it has trouble with sulci or gyri that are parallelto the plane of section, (3) the topology of theresulting surface may be incorrect, especiallywhen contours in each section have been madecontinuous for computational reasons (e.g., insections where the temporal lobe is ’de-tached’ ), and (4) the resulting surface is diffi-cult or impossible to unfold accurately.

The method we have developed for re-constructing the cortical surface largely over-comes these problems by adopting an automat-ic deformable template algorithm (see e.g.,Yuille, 1991). The basic idea behind this meth-od is to start with a simple surface with the cor-rect topology—e.g, a circle in 2-D, or a spheri-cal shell in 3-D—and then gradually deformthe shape of the surface by rubber-sheet trans-formations to conform to the cortical sheet.The location of each vertex of the surface is up-dated iteratively according to elastic “ forces”between neighboring vertices, and repulsive

and attractive forces along the local surfacenormal depending on the MRI data at the ver-tex. A nice feature of this technique is that allcomputations needed in the “shrink-wrapping”process are local. The motion of each vertexcan be calculated based on local informationabout neighboring vertices, and local MRI data.The more global topological constraint is en-forced implicitly by the connectivity of the ver-tices (see Appendix C).

In order to speed up convergence of the3-D “shrink-wrap” (and to avoid the computa-tional expense of determining whether the sur-face has passed through itself at each timestep), an initial estimate of the boundary be-tween the cortical gray and white matter wasobtained using a three-stage flood-filling algo-rithm. The white matter of the brain, as classi-fied by MRI data, is initially filled in 3-D fromone or more seed locations inside the whitematter. Then, a second fill of the volume out-side the volume filled by the initial fill is per-formed to eliminate internal holes. Finally, thevolume inside the volume filled by the second,external fill is itself filled, to eliminate externalislands. The result is a connected volume rep-resentation of the white matter. A single,closed tessellation of the white matter surfacecan then be constructed from the faces of filledvoxels bordering unfilled voxels. Figure 2Ashows the result of “shrink-wrapping” the ini-tial tessellation of the flood-filled white matteragainst the MRI data to smooth it. The local re-pulsive criterion has been set so that the com-puted surface settles near the surface of thegray matter.

Flattening the CortexA straightforward adaptation of the tech-

nique described above can be used to computa-tionally flatten the cortical sheet. The surfaceis relaxed towards minimal surface tension byincluding only the local elastic forces—i.e., byfreeing it from the MRI data. The algorithmwill then gradually unfold the cortex while pre-serving its topology and minimizing local geo-metric distortions. Figures 2B, 2C, and 2Dshow snapshots of the cortical surface during


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the flattening process (after 30, 200, and 1000iterations; eventually the surface will approacha sphere). In these figures, locations on the ini-tial folded surface with large positive curvature(~sulci) are colored red, while locations withlarge negative curvature (~gyri) are stainedgreen. Notice how some of the major sulci(e.g., the superior temporal sulcus) that are hardto distinguish from minor ones in the initialfolded cortex have become much more con-spicuous after a partial flattening. This con-firms our experience in physically flatteningthe cortex of post-mortem specimens of the hu-man occipital lobe (Sereno, McDonald, &Allman, 1988; Sereno & Allman, 1991).

A particular advantage of this partiallyflattened representation is that the cortex re-tains its global shape, appearing as if it has beengently inflated. This effectively exposes hid-den sulcal cortex without the rigors of a com-plete flattening to a plane, which requires cutsin order to relax the surface, and which is muchharder to interpret (see e.g., Jouandet et al.,1989). The partially flattened images are quitereminiscent of a macaque brain. Solutions dis-played on such an “ inflated” representation aremuch easier to parse. This representation couldeasily be adapted to display patterns of activitydetected by other techniques like PET or func-tional MRI; the folded cortical model wouldsimply be colored with the activity data andthen inflated.

MODEL STUDIESIn the following we present some results

of applying the source localization techniquedescribed above to simulated EEG and MEGdata. In the 2-D studies the cortical contour wasapproximated by about 400 vertices and wascomputed from a coronal MRI image using the“shrink-wrapping” algorithm described above.Similarly, the 3-D studies were done on the po-lygonal representation of the folded cortical sur-face, with dipoles at the vertices oriented alongthe locally estimated surface normal. The orig-inal 150,000 vertices of the surface tessellationwere evenly subsampled to about 10,000 di-poles, for a spacing between adjacent dipoles of

about 4 mm. The forward transformation matrix A

was computed based on the locations of thesensors and the location and orientation of eachdipole, using the equations for the EEG andMEG given in Appendix A. The results of the2-D studies are shown on a slightly tilted shad-ed 3-D rendering of the “shrink-wrap” solutiononto a coronal MRI section, with EEG elec-trodes and MEG sensors shown as small planarpatches in a concentric circular arrangement.The inner circle represents the EEG electrodesand the outer circle represents the MEG sen-sors. For the 3-D studies a geodesic arrange-ment of 61 electric and 61 magnetic sensorswas assumed, resulting in an approximatelyuniform distribution of sensors over the head,for a spacing between sensors of about 40 mm.The sensor coordinates were calculated basedon the vertices of a half icosahedron subdividedwith a frequency of 3 (see Kenner, 1976). Thearrangement was scaled according to the radiusof the head, as estimated by half the distancefrom the nasion to the inion, and rotated so asto align the equator of the arrangement with aline from the nasion to the inion, based on MRIdata. Note that this spherical approximation tothe head tends to overestimate the distancefrom sensors on the lateral parts of the head tothe brain, since the head is typically not quite aswide as it is long.

The dipole strengths and recording val-ues are color coded. Positive values are indi-cated by red, while negative values are indicat-ed by green. The magnitude is coded by thesaturation of the color; gray indicates that thevalue is close to zero, saturated red indicates alarge positive value, and saturated green indi-cates a large negative value.

Single Radial or Tangential DipoleFigure 3A shows the location of a single

superficial, nearly radial dipole. Figures 3B,3C and 3D show the corresponding linear in-verse solution using electric data alone, mag-netic data alone, and both kinds of data togeth-er. Notice that the solution based on magneticdata alone is much worse than that based on


Figure 5. Deep dipole. A radial source in the insula (A) illustrates a well-known problem with minimum-norm-like solutions. Even with both electric and magnetic information, deep focal activity is interpreted as superficial spread-out activity (B). Dramatic improvement in the solution results when temporal information is taken into account (D). We have assumed white noise and a signal to noise ratio of 3:1. The corresponding calculated variance estimates Rii for each dipole are shown in C.

Figure 4. A single superficial, nearly tangential dipole source is shown in A. The linear inverse solutions for electric data alone (B), magnetic data alone (C), and both kinds of data (D) now show that the solution based on magnetic data alone is much better than that based on electric data alone.

Figure 3. A single superficial, nearly radial dipole source is shown in A. The forward solutions are shown on the sensors, and the linear inverse solution on the cortical ribbon for electric data alone (B), magnetic data alone (C), and both kinds of data (D). The dipole strengths and recording values are color coded and the inner and outer circles represent the EEG and MEG sensors. Positive values are indicated by red, while negative values are indicated by green. The solution based on magnetic data alone is much worse than that based on electric data alone.


electric data alone.Figure 4A shows the location of a single su-

perficial, nearly tangential dipole. Figures 4B,4C and 4D show the corresponding linear in-verse solution again for electric data alone,magnetic data alone, and both kinds of data to-gether. The solution based on magnetic dataalone is now much better than that based onelectric data alone.

Interestingly, even though the MEG by it-self is essentially blind to radial sources, it canhelp rescue radial sources when combined withEEG. This is because it can rule out incorrectalternative interpretations of the EEG data. Forexample, a radial dipole near a tangential di-pole generates a scalp current distribution sim-ilar to that produced by a single, laterallydisplaced radial dipole; adding MEG data picksout the tangential dipole, thus restoring the ra-dial dipole to its rightful location.

Deep DipoleFigure 5A shows the location of a radial di-

pole located deep inside the insula. Figure 5Bshows the corresponding linear solution usingboth electrical and magnetic information. Thisillustrates the well-known problem that mini-mum-norm-like techniques tend to interpretdeep, focal activity as more spread-out, super-ficial activity.

Figure 5D shows the dramatic improve-ment in the solution made possible by takingtemporal information into account. In this sim-ple case with only a single dipole active, thesensor covariance matrix will be equal to theouter product of the gain vector of the active di-pole location with itself plus some multiple ofthe identity matrix (see Eq. 15). We have as-sumed white noise and a signal to noise ratio of3:1. The variance estimates Rii for each dipoleare shown in Figure 5C.

3-D StudiesThe locations of eight assumed sources are

illustrated on the unfolded cortex in Figure 6A.The estimated solution shown in Figure 6C (thesensors are omitted) demonstrates that thesewell-separated sources can all be distinguished.

The sensor covariance matrix was computedassuming correlation between each of thesources of 0.5, additive white noise, and a sig-nal to noise ratio of 10:1. Figure 6B shows thecorresponding calculated variance estimate Riifor each dipole. Nearby correlated sources willmerge, however, if they are close enough toeach other. In Figures 7A-C, the similarly--signed (green) source pair at the left are begin-ning to merge. Nearby sources with differentsigns can sometimes be distinguished at closerdistances (red-green pair at right), but willeventually cancel each other if they are closeenough (upper red-green pair).

It should also be noted that using sensor co-variance information does not always rescuedeep sources. If the magnitude of the signalproduced by a deep source is too weak com-pared to the noise, or if there are active superfi-cial sources which could produce a signal closeto that of the deep source, some of the activityof the deep source will be attributed to the su-perficial sources. This uncertainty in the esti-mate of the strength of a deep source can bedetected by analyzing the predicted estimationerror as described previously.

CONCLUSIONAll known techniques for imaging brain

activity have limitations in temporal and spatialresolution. Some of these limitations can beovercome by combining data obtained usingdifferent techniques in a way that takes advan-tage of the strengths of each technique to obtaina single optimal solution. Note that this is verydifferent from finding solutions using eachtechnique independently and then comparingthe results. If the solutions are different, it isnot clear which part of which solution to trust.

There are a number of advantages of thelinear approach over traditional approaches tothe problem of source localization and imag-ing.

(1) It provides a principled framework forcombining EEG and MEG recording data withinformation about cortical geometry from MRI.

(2) Since it makes no assumptions about thenumber of equivalent dipole sources, this tech-


Figure 7. Nearby sources. Correlated sources can merge if they are close together. The lower left pair of similarly-signed sources in A are beginning to merge in the variance estimates in B and in the solution in C; even nearer pairs can be distinguished if they are of opposite sign (lower red-green pair at right), but eventually cancel each other if close enough (upper red-green pair).

Figure 6. 3-D model studies. The locations of eight sources are displayed on the flattened cortex in A (all calculations were performed using the folded cortex). The solution using temporal information is shown in C (61 EEG and MEG sensors placed in a geodesic arrangement are omitted; the sensor covariance matrix was computed assuming between-source correlation of 0.5 and additive white noise with signal to noise ratio of 10; the 150,000 polygon surface was subsampled to about 10,000 dipoles for these solutions). The corresponding calculated variance estimates Rii for each dipole are shown in B.


nique is well suited to situations with many cor-related sources of cortical activity.

(3) By incorporating an improved soft con-straint based on the observed covariance ofsensors over time for a particular event, local-ization of deep sources is greatly improved,without disallowing spread-out solutions.

(4) Other sources of data like PET or func-tional MRI can be easily incorporated.

(5) The recovery of the topology of the cor-tical manifold makes it possible to convenient-ly view the solutions (as well as data from otheractivity imaging techniques) after gently “ in-flating” the brain, as well as allowing more ef-fective intersubject comparisons.

(6) The expected error of the solution canbe computed for different source configura-tions, providing a principled way to quantifystatistical significance of various hypotheses.

(7) The spatial resolution is comparable tothat of PET or functional MRI without sacrific-ing fine-grained temporal resolution.

METHODS

MRI imagesThe brain of the second author was scanned

on a GE SIGNA system with a 1.5T magnetand the ADVANTAGE software upgrade. Thescanning sequence consisted of three orthogo-nal T1/inversion recovery slice sequences(TR=2000, TI=708, TE=12) and three orthogo-nal T2/proton density spin-echo slice sequenc-es (TR=2000, TE=11, 70) for a total scan timeof 82 minutes. All scans used a 25 cm field--of-view, a matrix size of 256 x 192, and contig-uous 6 mm slices. Since the linear“de-voluming” of the images was quite sensi-tive to movement between successive images, aspecial effort was made to stabilize the head,and, in particular, to prevent rotation in the sag-ittal plane.

By evaluating the expression for absolutevalue of the inversion recovery signal strength(with Mathematica on the NeXT computer) fordifferent repetition (TR) and inversion (TI)times using T1 values for gray and white matter(Mitchell et al., 1984; Hyman et al., 1989; R.

Buxton, personal communication), it was pos-sible to search for the TI/TR combination giv-ing the best contrast between these two tissuetypes.

SoftwareThe major computational routines for the

project (linear de-voluming, tissue typing andskull removal, flood-filling, surface tessella-tion, geodesic sensor placement, corticalshrink-wrapping and inflation, forward solu-tion, inverse solution, sensor covarianceweighting) were written in C. Display softwarefor the 2-D slice and 3-D cortical surface imag-es was also written in C using Silicon GraphicsGL routines, which take advantage of fast hard-ware implementation of 3-D transformations,polygon filling, and lighting calculations onSilicon Graphics machines.

APPENDIX A: FORWARD SOLUTIONIn the following we will present the actual

equations used in the model studies for calcu-lating the forward solutions for the EEG andMEG. The idealized assumptions made hereabout head geometry and electrical parametersare not essential for the general linear approachto the inverse problem, since these assumptionsaffect only the values in the coefficient matri-ces, E and B. These are simply treated as arbi-trary arguments in the inverse calculations.More realistic forward calculations, based onthe exact shape of the scalp, skull, and brain,can be done using finite element methods(FEM), in order to generate more realistic Eand B matrices. Quantitative estimates of theshape and thickness of the real scalp, skull, andconductive bone sutures are, of course, directlyavailable from the same MRI images used tofind the cortical manifold.

EEG CalculationsTo calculate the EEG, we assume a single

dipole in a 3-shell inhomogeneous sphericalconductor. The shell model consists of a ho-mogenous sphere of neural tissue with radiusr1, surrounded by a concentric spherical shell


of outside radius r2, representing the skull, andanother concentric spherical shell of outside ra-dius R representing the scalp. The scalp andthe neural tissue are assumed to have the sameconductivity σ, and the skull to have conduc-tivity σs. To simplify the calculations the co-ordinate system is assumed rotated so that thedipole lies on the z-axis, pointing in the x-zplane.

The potential recorded at location (α,β)on the scalp can be expressed as

(A.24)

(A.25)

where ξ = σs / σ, f1 = r1 / R and f2 = r2 / R.st and sr are the tangential and radial compo-nents of the dipole moment, or strength s. Pnand Pn

1 are the Legendre and the associatedLegendre polynomials, respectively (Ary,Klein and Fender, 1981).

MEG CalculationsMaking assumptions similar to those

made for the EEG calculations we get the fol-lowing expression for the radial component ofthe magnetic field h observed at location(r,α,β) outside the head

, (A.26)

where (A.27)

Somewhat more complicated expressions canbe found for the non-radial components of themagnetic field (Cuffin and Cohen, 1977). Notethat the magnetic field is independent of the ra-dial component sr of the dipole.

v α β,( )1

4πσ2n 1+

nb

ξ 2n 1+( ) 2

dn n 1+( )

n 1=

∞

∑=

nsrPn αcos stPn1 αcos βcos+( )×

dn n 1+( ) ξ n+( )nξ

n 1+1+( )=

1 ξ−( ) n 1+( ) ξ n+( ) f12n 1+ f2

2n 1+−( )+

n 1 ξ−( ) 2f1f2

2n 1+

−

h r α β, ,( )b α( )sin β( )sin st

4πr2γ3 2⁄=

γ 12b αcos

r− b

r( )

2

+=

APPENDIX B: ORTHOGONAL MRISLICE COMBINATION (“ DEVOLUM-ING”)

The intersection between any three or-thogonal MRI sections forms a cube, with sidesequal to the section thickness (typically severaltimes the in-plane pixel width). Let v denotethe vector of N3 unknown cubic volume ele-ments (voxels) inside this cube; and let c, h, ands denote vectors of N2 known picture elements(pixels) of the coronal, horizontal and sagittalsides of the cube, respectively. Then, if eachpixel in a section reflects the linearly weightedsum of MRI signal from the tissue throughoutthe thickness of the section, the following “ for-ward” equations hold:

, , , (B.1)

where C, H and S are the easily derived linearoperators that map N3 voxel values into N2 pix-el values in the coronal, horizontal and sagittalplanes, respectively. A least-squared-error es-timate of v is then given by

(B.2)

where R is a regularization matrix, which istypically set equal to a small multiple of theidentity matrix.

APPENDIX C: CORTICAL SURFACEREFINEMENT (“SHRINK-WRAPPING”)

In the “shrink-wrapping” procedure, theposition of each vertex is updated iterativelyusing a data term and smoothness term accord-ing to the following equations:

, where (C.1)

, (C.2)

c Cv= h Hv= s Sv=

v* CTC HTH STS R+ + +( ) 1− CTc HTh STs+ +=

pit 1+( ) pi

t( ) αF pit( )( )ni

t( )+=

β 1Ni

vi j,t( )

j ℵ i∈∑+

ni1Ni

ui ℵ i

1, ui ℵ i2,×( ) u

i ℵ i2, ui ℵ i3,×( )+(=

… ui ℵ i

Ni, ui ℵ i1,×( ) )+ +


research and graduate support.

Notes1. This does not address the problem of

non-additive “noise” arising from errors in the forward solution (coefficients of the A matrix). This noise can be reduced by using a more realistic forward solution (see Appendix A).

2. Leaving out the normalizing denominator in Equation 20 results in 1/ξi having the same units as σi2. However, this has the undesir-able effect of assigning very large a priori variance estimates for dipole locations with gain vectors of small magnitude (e.g., deep sources).

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AcknowledgmentsWe would like to thank David Amaral for

help and support in obtaining the MRI imagesand Marta Kutas, Chris Wood, Steven Hillyard,Richard Greenblatt, Rick Buxton, John George,Greg Simpson, Bill Loftus, John Allman,George Carman, Brandt Kehoe, Kechen Zhang,Eric Courchesne, Terri Jernigan, Carsten Lut-ken, and an anonymous reviewer for help anddiscussions. Supported by UCSD McDonnell--Pew Cognitive Neuroscience Center grants for

ui j,vi j,vi j,

=

vi j, pj pi−=

pit( )

ℵ i Niℵ i

j

ℵ iu

i ℵ ij, ui ℵ ij 1+,×

F pi( )

F p( ) γ IR p( ) IRGray−( )( )tanh=

αF pit( )( )ni

t( )

β 1Ni

vi j,t( )

j ℵ i∈∑


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