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Monte Carlo Simulation Studies of EEG and

MEG Localization Accuracy

Arthur K. Liu, Anders M. Dale, and John W. Belliveau*

Massachusetts General Hospital, NMR Center, Charlestown, Massachusetts

?

?

Abstract: Both electroencephalography (EEG) and magnetoencephalography (MEG) are currently used to

localize brain activity. The accuracy of source localization depends on numerous factors, including the

specific inverse approach and source model, fundamental differences in EEG and MEG data, and the

accuracy of the volume conductor model of the head (i.e., the forward model). Using Monte Carlo

simulations, this study removes the effect of forward model errors and theoretically compares the use

of EEG alone, MEG alone, and combined EEG/MEG data sets for source localization. Here, we use a

linear estimation inverse approach with a distributed source model and a realistic forward head

model. We evaluated its accuracy using the crosstalk and point spread metrics. The crosstalk metric

for a specified location on the cortex describes the amount of activity incorrectly localized onto that

location from other locations. The point spread metric provides the complementary measure: for that

same location, the point spread describes the mis-localization of activity from that specified location

to other locations in the brain. We also propose and examine the utility of a “noise sensitivity

normalized” inverse operator. Given our particular forward and inverse models, our results show

that 1) surprisingly, EEG localization is more accurate than MEG localization for the same number of

sensors averaged over many source locations and orientations; 2) as expected, combining EEG with

MEG produces the best accuracy for the same total number of sensors; 3) the noise sensitivity

normalized inverse operator improves the spatial resolution relative to the standard linear estimation

operator; and 4) use of an a priori fMRI constraint universally reduces both crosstalk and point

spread. Hum. Brain Mapping 16:47–62, 2002.

© 2002 Wiley-Liss, Inc.

Key words: linear estimation; Bayesian linear inverse; weighted minimum norm; forward problem;

inverse problem; fMRI; brain; human

?

?

INTRODUCTION

The electromagnetic inverse problem for the human

brain is to determine the neural source distribution

that gives rise to external electromagnetic potentials

and fields, measured by electroencephalography

(EEG) and magnetoencephalography (MEG), respec-

tively. There is great interest in the assessment of the

relative accuracy of EEG and MEG for source localiza-

tion. The accuracy of a solution to the inverse problem

Contract grant sponsor: Human Frontier Science Program; Contract

grant sponsor: National Foundation for Functional Brain Imaging;

Contract grant sponsor: Whitaker Foundation; Contract grant spon-

sor: National Institutes of Health (NINDS, NIMH, NCI, NCRR);

Contract grant number: R01-NS37462, R01-NS39581, RO1-RR13609,

P41-RR14075.

*Correspondence to: John Belliveau, Massachusetts General Hospi-

tal, NMR Center, Building 149, 13th Street, Charlestown, MA 02129.

E-mail: jack@nmr.mgh.harvard.edu

Received for publication 6 December 2000; accepted 9 November

2001

? Human Brain Mapping 16:47–62(2002) ?

DOI 10.1002/hbm.10024

Published online xx Month 2001

© 2002 Wiley-Liss, Inc.

Page 2

using EEG and MEG data depends on numerous fac-

tors, however, including the particular inverse ap-

proach and source model, fundamental differences in

EEG and MEG data, and the accuracy of the volume

conductor model of the head (i.e., the forward model).

There has been some debate over the relative accu-

racy of EEG or MEG based source localization. Exper-

imental EEG studies using phantoms [Henderson et

al., 1975] and implanted electrodes in epilepsy pa-

tients [Smith et al., 1983, 1985] have been reported.

Those studies found a localization accuracy of 10 mm

for the phantoms to 20 mm in the patients. In compar-

ison, MEG studies reported localization accuracy of 3

mm in spherical phantoms [Barth et al., 1986; Hansen

et al., 1988; Janday and Swithenby, 1987; Yamamoto et

al., 1988] and 4–8 mm for skull phantoms [Barth et al.,

1986; Janday and Swithenby, 1987; Weinberger et al.,

1986; Yamamoto et al., 1988]. Based on these results, it

has been commonly assumed that MEG localization

accuracy is far superior to that of EEG.

There were theoretical reasons, however, to believe

that the MEG and EEG accuracy should be compara-

ble. In an attempt to directly address this controversy,

both MEG and EEG measurements were made while

generating current dipoles from implanted electrodes

in an epilepsy patient [Cohen et al., 1990]. A sinusoi-

dal stimulus waveform was used to remove the spike

artifacts that likely contributed to the poor EEG local-

ization performance of the previous EEG measure-

ments [Smith et al., 1983, 1985]. The average MEG and

EEG localization errors for dipoles with sufficiently

good signal to noise were found to be 8 mm and 10

mm, respectively [Cohen et al., 1990]. These results

suggested that MEG and EEG provide comparable

accuracy.

Unfortunately, further experimental data have not

necessarily clarified this issue of relative accuracy.

Phantom studies have reported localization accuracy

for EEG of 7–8 mm [Leahy et al., 1998] and for MEG of

2–4 mm [Gharib et al., 1995; Leahy et al., 1998; Men-

ninghaus et al., 1994]. The localization superiority of

MEG over EEG is less obvious in data from measure-

ments made in patients. Using data generated from

artificial current dipoles implanted in epilepsy pa-

tients the localization accuracy of EEG was 10–17 mm

[Cuffin et al., 1991; Cuffin, 1996; Krings et al., 1999]

with the best EEG accuracy in one patient of 1–4 mm

[Cuffin, 1996]. This is compared to a localization ac-

curacy of 17 mm for MEG measurements of artificial

dipoles generated from implanted subdural strips

[Balish et al., 1991]. Other studies have estimated ac-

curacy by comparing lesion data (e.g., tumor, epilep-

togenic focus) in epileptic patients with the non-inva-

sive location estimates from EEG [Diekmann et al.,

1998; Herrendorf et al., 2000; Ko et al., 1998; Krings et

al., 1998; Nakasato et al., 1994] or MEG [Diekmann et

al., 1998; Ko et al., 1998; Mikuni et al., 1997; Nakasato

et al., 1994; Sutherling et al., 1987, 1988a,b; Stefan et al.,

1994; Tiihonen et al., 1990]. Similar to the results of the

artificial current dipoles, the EEG and MEG accuracy

were comparable (ranging from 10–20 mm).

Generally, in experimental studies it is difficult to

separate the effect of errors in the head model from

localization errors due to inherent differences between

EEG and MEG. Specifically, there are fundamental

differences between the forward solution accuracy re-

quired by EEG and MEG, with MEG requiring a sim-

pler model [Hamalainen and Sarvas, 1989; Meijs et al.,

1987, 1989]. Therefore, one would expect better accu-

racy for experimental MEG data using the more accu-

rate MEG head model. Through the use of modeling

studies, it is possible to examine the relative accuracy

of EEG and MEG by using the same forward solution

to generate both the synthetic external EEG/MEG

measurements and the resulting inverse solution. In

other words, modeling studies can examine EEG and

MEG localization accuracy unbiased by possible inac-

curacies in the forward model that may differentially

affect localization of experimental EEG or MEG mea-

surements. Even the modeling studies, however, have

been equivocal. Some modeling work found MEG to

be more accurate than EEG [Murro et al., 1995; Stok,

1987], whereas others found EEG and MEG accuracy

to be comparable [Malmivuo et al., 1997], or EEG

accuracy better than MEG [Mosher et al., 1993; Pas-

cual-Marqui and Biscay-Lirio, 1993].

Clearly, the data from phantoms, patients, and the-

oretical studies give conflicting evidence for the rela-

tive accuracy of EEG and MEG. There are numerous

confounding factors in the interpretation of all of these

data. For example, experimental measurements in

phantoms and living human heads may reflect the

higher accuracy of the MEG forward solution, mea-

surement errors, or differences in signal to noise. Also,

many of the modeling studies used spherical head

models with differences between EEG and MEG sen-

sor sampling.

Here, we specifically examined inherent differences

in EEG and MEG data by using a single realistic head

model for the both the forward and inverse computa-

tions. In other words, we have removed the effect of

forward model errors from our analysis. For these

model studies, we used 1) a linear estimation tech-

nique; 2) a distributed source model; 3) a realistic

forward head model; and 4) similar EEG and MEG

sensor placement. In addition, we present a noise sen-

?Liu et al.?

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sitivity normalized inverse operator that is based on

the linear estimation approach. Monte Carlo modeling

studies (sampling over numerous source locations,

size and orientations) were used to determine the

theoretical limits (i.e., assuming no errors in the head

model) of EEG and MEG localization using our par-

ticular source model and inverse procedure. Localiza-

tion estimates were computed using MEG or EEG

data, both separately and combined. The effect of sen-

sor sampling density (i.e., number of sensors) was also

examined. In addition, as it is becoming more com-

mon to utilize other spatial information, such as func-

tional magnetic resonance imaging (fMRI), we also

modeled the effect of fMRI spatial constraints on the

relative accuracy of EEG and MEG source localization

[Liu et al., 1998].

Previous ECD localization accuracy studies typi-

cally report error in millimeters between known and

modeled dipole locations. Unfortunately, it is difficult

to characterize localization accuracy in terms of a dis-

tance error for the linear estimation approach using a

distributed source model because of the extended na-

ture of the localization estimates. It is more appropri-

ate to define localization accuracy and resolution in

terms of metrics that include the spatial aspect of the

extended source distributions. Therefore, we quanti-

fied localization accuracy using a crosstalk metric [Liu

et al., 1998] and a point spread metric that are speci-

fied by the resolution matrix [Grave de Peralta Me-

nendez et al., 1996, 1997; Grave de Peralta-Menendez

and Gonzalez-Andino, 1998; Lutkenhoner and Grave

de Peralta Menendez, 1997; Menke, 1989]. These two

metrics provide complementary accuracy information.

They specify from where and to where activity is in-

correctly localized. The crosstalk metric for a specified

location on the cortex describes the amount of activity

incorrectly localized onto that location from other lo-

cations. The point spread metric provides the comple-

mentary measure: for that same location, the point

spread describes the mis-localization of activity from

that specified location to other locations in the brain.

Lower crosstalk and point spread values indicate

higher localization accuracy.

The crosstalk and point spread metrics are also use-

ful beyond the context of these modeling studies.

When analyzing experimental data, one can calculate

crosstalk and point spread maps that are only based

upon the inverse operator and forward solution, and

are independent of the actual experimental data. For a

given estimated distribution of cortical activity, these

maps can aid in determining the confidence of the

spatiotemporal estimates. For example, if one were to

estimate the activity at a location that had a focal

crosstalk and point spread map, one would be more

confident that this estimate was, in fact, correct.

METHODS

Forward solution

The realistic boundary element method (BEM) was

adapted for calculating both the EEG and MEG for-

ward solutions [de Munck, 1992; Oostendorp and van

Oosterom, 1989]. Both forward solution computations

require the locations of all possible sources, the sensor

locations, and the sensor orientations (for MEG only).

For this analysis, we restricted all sources to be within

the cortex of the brain. Therefore, by construction, we

did not analyze non-cortical structures, such as the

cerebellum and basal ganglia. Each possible cortical

source was represented by a current dipole oriented

normal to the cortical surface, i.e., both the location and

orientation were constrained by the cortical surface.

The computation of the MEG forward solution has

been shown to only require the inner skull boundary

to achieve an accurate solution [Hamalainen and Sar-

vas, 1989; Meijs et al., 1987, 1989]. The EEG forward

solution computation requires the specification of

boundaries between brain and skull, skull and scalp,

scalp and air, and the relative conductivities of each of

those regions. We assumed conductivity ratios of

1:0.0125:1 for brain:skull:scalp [Cuffin, 1990].

The conductivity boundaries required for computa-

tion of the EEG and MEG forward solutions were

automatically reconstructed from a high-resolution

T1-weighted 3D MRI using our previously described

technique [Dale et al., 1999; Dale and Sereno, 1993;

Fischl et al., 1999]. The realistic surfaces used in our

calculations are shown in Figure 1. Each of the con-

ductivity boundaries was represented by 642 vertices.

The cortical surface was initially tessellated with about

150,000 vertices per hemisphere. For the inverse com-

putation, the cortical surface was decimated to ap-

proximately 3,000 dipoles per hemisphere, which is

roughly equivalent to 1 dipole every 10 mm along the

cortical surface.

The computation of the forward solution also re-

quires specification of the EEG electrode or MEG sen-

sor locations. We began with a realistic sensor descrip-

tion of 122 MEG sensors [Knuutila et al., 1993], which

is the same configuration that was used in our previ-

ous modeling study [Liu et al., 1998]. The 122 MEG

sensors are placed at 61 discrete locations, with two

orthogonal planar gradiometers at each location. The

61 locations were subsampled to 30 locations. Both the

61 and 30 locations were distributed over the entire

?EEG and MEG Localization Accuracy?

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head. For the two sets of locations, we created three

types of MEG sensor configurations: magnetometer,

radial gradiometer, and two orthogonal planar gradi-

ometers. To minimize effects from sampling differ-

ences, the EEG sensor locations were determined by

projecting the 61 and 30 MEG locations onto the outer

skin. We also modeled the effects of using various

combinations of MEG and EEG sensor configurations.

The MEG sensor locations are shown in Figure 2,

overlaid on an axial maximum intensity projection of

the T1-weighted MRI. The EEG sensor locations are

shown similarly in Figure 3.

For the combined EEG/MEG sensor configurations,

the gain matrix (A) contains both the EEG and MEG

forward solutions that were calculated separately. The

dimensions of the combined gain matrix are [(number

of EEG sensors plus number of MEG sensors) ? (num-

ber of dipoles)].

Inverse operator

The linear inverse operator used here can be de-

rived in various ways. In the appendix, we detail four

different derivations: 1) minimization of expected er-

ror [Dale and Sereno, 1993]; 2) Bayesian formulation

[Gelb, 1974; Phillips et al., 1997a,b]; 3) Tichonov reg-

ularization [Tichonov and Arsenin, 1977]; and 4) gen-

eralized Wiener filtering [Deutsch, 1965; Smith, 1992;

Sekihara and Scholz, 1995, 1996]. Assuming appropri-

ate initial conditions, all derivations result in the fol-

lowing (or equivalent) expression for the linear in-

verse operator:

W ? RAT?ARAT? C?? 1. (1)

Crosstalk metric

To quantify one aspect of the accuracy of the linear

estimation technique, we used a crosstalk metric [Liu

et al., 1998], which is similar to the averaging kernel of

the Backus-Gilbert method [Backus and Gilbert, 1970].

The crosstalk metric describes the sensitivity of the

estimate at a specified location to activity at other

locations. A location with lower crosstalk is less biased

by activity at other locations, and provides a more

accurate estimate for activity at that location. A more

formal description of the crosstalk metric follows.

The estimated source strength (s ˆi) at each location i

can be written as a weighted sum of the actual source

strengths at all locations, plus a noise contribution.

Figure 1.

EEG/MEG forward solution surfaces. The top figures show the

three head surfaces used in the calculation of the forward solu-

tions. The bottom figures are the left and right cortical surfaces

used to determine the locations and orientations of the sources.

The MEG forward solution was computed using only the inner

skull surface, whereas the EEG forward solution requires all three

boundaries.

Figure 2.

MEG sensor locations. Two different sets of locations.

Figure 3.

EEG sensor locations. Two different sets of locations.

?Liu et al.?

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This is due to the linearity of both the forward solution

and this inverse operator. More formally,

s ˆi? wix

(2)

? wi?A˜s ? n?

(3)

? wi??

??

j

j

a ˜jsj? n?

(4)

?wia ˜j?sj? win

(5)

where Wiis the ith row of W, and a ˜jis the jth column of

A˜(i.e., the “true” lead field including orientation in-

formation at location j). Depending on the particular

type of mis-specification being examined through

model studies, A (the forward model used in the

calculation of W) and A˜(the forward model with no

errors) may or may not be equivalent. Note, that when

determining the crosstalk metric for experimental

data, A and A˜will always be the same. The first term

in equation (5) is the sum of the activity (sj) at every

location j, weighted by the scalar wia ˜j. The second

term reflects the noise contribution to the estimated

activity at location i.

An explicit expression for the relative sensitivity of

the estimate for a given location (i) to activity coming

from other locations (j) is desired. A crosstalk metric

(?) is defined as follows:

?ij

2???WA˜?ij?2

??WA˜?ii?2??wia ˜j?2

?wia ˜i?2

(6)

where WA˜is the resolution matrix [Grave de Peralta

Menendez et al., 1996, 1997; Grave de Peralta-Menen-

dez and Gonzalez-Andino, 1998; Lutkenhoner and

Grave de Peralta Menendez, 1997; Menke, 1989].

By comparing equations (5) and (6), one can see that

the crosstalk metric ?ijdescribes the sensitivity (or

weighting) of the estimate at location i to activity at

location j relative to activity at location i. A crosstalk

value of 0% means that the estimated activity at loca-

tion i is completely insensitive to activity at location j.

A crosstalk value of 100% means that the estimated

activity at location i is equally sensitive to activity at

locations i and j. For any particular location, the

crosstalk from all other locations can be calculated. For

some particular location i, this computation corre-

sponds to the ith row of the resolution matrix (WA).

We refer to this spatial representation of the crosstalk

metric as the “crosstalk map” for the specified loca-

tion.

Both the crosstalk metric and the crosstalk map are

specified for a given source. To simplify the represen-

tation of the crosstalk at all locations, we define an

average crosstalk map (ACM). For each location on

the cortical surface we compute the average of the

crosstalks between the specified location (i) location

and all other locations (j) on the surface:

ACM ??

j

?ij

2

j

.(7)

Point spread metric

Closely related to the crosstalk metric is the point

spread metric. The point spread for a location i de-

scribes the sensitivity of the estimates at other loca-

tions j to activity at location i. A location with lower

point spread has a smaller spatial extent. The point

spread metric (?) is defined as:

?ij

2???WA˜?ji?2

??WA˜?ii?2??wja ˜i?2

?wia ˜i?2

(8)

where WA˜is the resolution matrix. The point spread

map, more commonly known as the point spread

function (PSF), for a given location i corresponds to

the ith column of the crosstalk matrix (WA). Similar to

the average crosstalk map, we define the average PSF

maps (APSF). For each location we average the point

spread between the specified location (i) location and

all other locations (j) on the surface:

APSFi??

j

?ij

2

j

.(9)

One can see that crosstalk and point spread are closely

related. The crosstalk map and the point spread map

correspond to the rows and columns, respectively, of

the resolution matrix WA. In the linear estimation

framework examined up to this point, we can show

that the resolution matrix is symmetric, and therefore

the crosstalk map and the PSF for a given location are

equivalent. The resolution matrix is given by:

WA ? RAT?ARAT? C?? 1A. (10)

The transpose of the resolution matrix is:

?EEG and MEG Localization Accuracy?

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?WA?T? ?RAT?ARAT? C?? 1A?T

(11)

? AT??ARAT? C?? 1?TART.(12)

The term inside the parentheses is also symmetric

(since the noise covariance matrix, C, is symmetric):

?ARAT? C?T? ARTAT? CT? ARAT? C. (13)

The source covariance matrix (R) is also symmetric, so

we can rewrite equation (12):

? AT?ARAT? C?? 1AR. (14)

Because the term, AT(ARAT? C)?1A, and R are both

symmetric, we have:

AT?ARAT? C)?1ART? RAT?ARAT? C?? 1A

(15)

thus, demonstrating that the resolution matrix is sym-

metric.

The relationship between the crosstalk metric and

the point spread metric can be more easily seen graph-

ically (Fig. 4). The location of interest (i.e., location i) is

marked in green. The arrows indicate the “direction”

of mis-specification, or where activity is mis-localized.

The crosstalk metric describes the activity from other

locations that is mis-localized onto the location of

interest (arrows point towards location i). Conversely,

the point spread metric describes activity that is mis-

localized from location i (arrows point away from

location i).

Given that the crosstalk map and the point spread

function are equivalent for the linear estimation oper-

ator, our discussion applies equally to both. Various

suggested improvements to the linear estimation tech-

nique, however, will result in a non-symmetric reso-

lution matrix. For example, the Backus-Gilbert method

[Backus and Gilbert, 1970; Grave de Peralta Menendez

et al., 1996; Grave de Peralta Menendez and Gonzalez

Andino, 1999] that explicitly minimizes the crosstalk,

will result in some sort of tradeoff with the point

spread function. Our noise normalized linear inverse,

presented below, also yields a non-symmetric resolu-

tion matrix.

Noise sensitivity normalization

Similar to the statistical analysis of functional MRI,

we are primarily interested in locations whose activity

(i.e., “signal”) is significantly larger than the noise.

Therefore, we propose to normalize each row of the

inverse operator based on the noise sensitivity of the

inverse operator at that location. Locations that have

low noise sensitivity are given a larger weighting than

those locations with high noise sensitivity. We can

estimate the noise sensitivity by projecting the noise

covariance estimate into the inverse operator. The new

inverse operator will be pre-multiplied by a diagonal

noise sensitivity matrix (D), square in the number of

dipoles, where each diagonal element is:

Figure 4.

Mis-localized activity specified by the Crosstalk and the Point Spread metrics. The arrows indicate the “direction” of mis-specification.

?Liu et al.?

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Dii?

1

diagi??WCWT?.(16)

The noise sensitivity normalized inverse is now:

Wns_norm? DW.(17)

The resulting activity estimates will now resemble an

F-statistic, instead of an activation power [Dale et al.,

2000]. Our new activity estimates, s ˆi

location i, which we refer to as “noise sensitivity nor-

malized estimates”, are:

ns_norm, at each

s ˆi

ns_norm? ?Wns_normx?i? ?DWx?i??

?WxxTWT?i

?WCWT?i

. ?18?

For these model studies, we have assumed Gaussian,

white noise, so the noise covariance matrix (C) is a

multiple of the identity matrix. In this particular case,

this noise sensitivity normalization corresponds to

normalizing the rows of the inverse operator by the

norm of the row:

Wi

ns_norm?

Wi

?Wi

orig

orig?.(19)

We note that because we are scaling each row of W by

a single value, the rows of the resolution matrix are

simply scaled by that same value. Therefore our

crosstalk metric remains unchanged with this noise

sensitivity normalization. The PSF, however, will be

affected since the scaling of each column of W is not

uniform.

Monte Carlo simulations

As pointed out by numerous authors [e.g., Fuchs et

al., 1998; Hari et al., 1988; Liu et al., 1998; Murro et al.,

1995; Supek and Aine, 1993], localization accuracy is

highly dependent on the location of the source. There-

fore, to better approximate realistic data which can

occur anywhere in the brain, our simulations use a

large random sampling of source locations to provide

an average estimate of localization accuracy.

Either 5, 10, or 20 sources were randomly located on

the cortical surface, each with varying volumetric ex-

tent (1 cm or 2 cm diameter). The random selection

ensures no systemic location bias in these model stud-

ies. The numbers and extents of sources were chosen

to represent experimentally realistic regions of brain

activity that might be seen with a given cognitive task.

The diagonal elements of R (the a priori source covari-

ance estimates Rii? ?i

values correspond to fMRI weightings of 0% (equiva-

lent to minimum norm) and 90%, respectively. Previ-

ous modeling studies suggest that an fMRI weighting

of 90% represents a reasonable compromise between

separation of activity from correctly localized sources

(by fMRI) and minimization of error due to missing

fMRI sources [Liu et al., 1998].

We made no a priori assumptions about source

correlation. Therefore, the off-diagonal elements of R

were set to zero, i.e., Rij? 0 for i ? j. It should be noted

that this does not force the sources to be uncorrelated

or orthogonal in time. Noise was assumed to be addi-

tive, Gaussian, uniform, and spatially uncorrelated.

More precisely, C ? Itr(ARAT)/n

SNR2

tity matrix, tr is the trace of a square matrix, SNR is the

assumed rms signal-to-noise ratio, and n is the num-

ber of sensors. Here, a conservative SNR of 10 was

assumed.

In these model studies, the number of MEG sensors

was assumed to be either 30, 60 (30 locations with two

orthogonal planar gradiometers), 61 or 122 (61 loca-

tions with two orthogonal planar gradiometers), with

the smaller numbers of sensors subsampled from the

complete 122 channel description to still give full head

coverage at a sparser sampling. For the EEG sensors,

30 or 61 sensors were distributed over the entire head.

To minimize any sampling differences between MEG

and EEG, the locations of the EEG sensors were based

on the corresponding MEG sensor configuration pro-

jected down onto the outer skin surface. In addition,

all combinations of MEG and EEG sensors were stud-

ied. For each of the sensor configurations, the crosstalk

was averaged over the different number and extent of

sources.

For comparison of the noise sensitivity normalized

inverse, the average crosstalk map and the average

PSF were computed at each source location. No fMRI

weighting was used, thus removing the need for

Monte Carlo simulations.

One additional set of modeling studies was per-

formed here. In computing the EEG forward solution,

conductivity ratios, not exact conductivities, of the

different head regions were used. If the actual conduc-

tivities differ from the assumed conductivities, even

though the conductivity ratios are correctly estimated,

then the computed source activities using EEG will

not be in the same units as the computed source

activities using MEG. This effectively introduces a

scaling factor between the EEG and MEG forward

solutions [Fuchs et al., 1998]. We modeled the effect of

2) were set to 1 or 0.01. These

, where I is the iden-

?EEG and MEG Localization Accuracy?

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mis-estimating the scaling factor between EEG and

MEG forward solutions (corresponding to the mis-

estimate of the actual conductivities) from 0.2–5. The

scaling factor was incorporated into the combined

EEG/MEG forward solution by multiplying those

rows corresponding to EEG sensors by the factor. This

scaled gain matrix was used in the computation of the

inverse operator. The correct gain matrix (i.e., a scaling

factor of 1) was used for the crosstalk metric compu-

tation. More explicitly, if we represent the scaled for-

ward solution by Asand the true forward solution by

A˜, the crosstalk matrix is given by:

WsA˜? RAs

T?AsRAs

T? C?? 1A˜.(20)

For this portion of the model studies, we assumed 30

EEG electrodes and 30 MEG radial gradiometers, and

an fMRI weighting of 90%.

RESULTS

Figures 5 and 8 show the average crosstalk vs. sen-

sor configuration. No fMRI weighting (equivalent to

minimum norm) was used in Figure 5. An fMRI

weighting of 90% was used in Figure 8. As a reminder,

the crosstalk metric specifies the amount of variance

from another location that is incorrectly mapped to the

location of interest.

Increasing the number of sensors for both MEG and

EEG decreases the average crosstalk (i.e., improves

average localization accuracy). There is little differ-

ence between equal numbers of magnetometers, radial

gradiometers or planar gradiometers. The average

crosstalk for the EEG configurations is smaller than

that for the same number of MEG sensors. Perhaps

most importantly, there is a large decrease in crosstalk

with the combined sensor configurations.

The average crosstalk map and average PSF map for

three different sensor configurations (60 MEG planar

gradiometers, 61 EEG sensors and the combined 61

EEG/ 60 MEG sensors) are shown on the inflated

cortical surface in Figure 6. No fMRI weighting was

used. As expected, the crosstalk and PSF maps are

equivalent. For the same number of sensors, EEG

(7.7% ? 6.1%) has lower average crosstalk than MEG

(17.3% ? 37.0%). This difference is largely due to the

very large crosstalk for MEG measurements of the

deep or radial sources. The combined EEG/MEG

sensor configuration provides the lowest average

crosstalk. There is large spatial variability in the

crosstalk map, especially for the MEG sensors, dem-

onstrating that the crosstalk map and point spread are

highly dependent of the cortical location of the source.

For both MEG and EEG, the crosstalk is larger in the

depths of the sulci than on the gyri, with the largest

crosstalk in the insula, inferior frontal and superior

temporal cortex. The MEG average crosstalk maps

also demonstrate the orientation dependence of MEG.

For MEG, source locations on the crowns of gyri that

are largely radial in orientation have very high aver-

age crosstalk (e.g., see white arrow).

The average crosstalk and average PSF maps for

three different sensor configurations (60 MEG planar

gradiometers, 61 EEG sensors and the combined 61

EEG/ 60 MEG sensors) using the noise sensitivity

normalized inverse operator are shown in Figure 7.

No fMRI weighting was used. The crosstalk maps are

unchanged from Figure 6. The PSF maps are greatly

different, however, especially for 60 MEG planar gra-

diometers alone. There is greater spatial uniformity

and no PSFs are greater than 20% with the noise

sensitivity normalized inverse. The average PSF over

all locations is lower for the normalized inverse, with

the largest gain occurring for MEG sensors alone (70%

lower).

The general trend of the results using fMRI weight-

ing (Fig. 8) is similar to that without fMRI weighting

(Figure 5). The addition of fMRI information results in

lower crosstalk for all sensor configurations. However,

it should be noted that this improvement is for those

sources that are correctly specified by fMRI. In cases

Figure 5.

Average crosstalk vs. sensor number; no fMRI weighting. The

sensor configuration consisted of 30 magnetometers (30m), 30

radial gradiometers (30r), 60 orthogonal planar gradiometers

(30p), 61 magnetometers (61m), 61 radial gradiometers (61r), 122

orthogonal planar gradiometers (61p), 30 EEG sensors, 61 EEG

sensors, and all possible combinations of MEG and EEG sensors.

The crosstalk values were averaged over a range of source num-

ber (5, 10, or 20 sources) and extent (1 cm or 2 cm in diameter).

?Liu et al.?

? 54 ?

Page 9

where fMRI has mis-specified the source, crosstalk

increases relative to the fMRI unweighted solution

shown in Figure 5 [Liu et al., 1998]. Increasing the

number of sensors, regardless of the type, improves

localization accuracy. The localization accuracy for the

same number of sensors is only slightly better with

EEG. The average crosstalk for the combined EEG and

MEG sensors is lower than either EEG or MEG alone,

with 61 EEG and 122 MEG (61p) sensors providing the

lowest crosstalk of the different sensor configurations

studied.

The observed improvement in localization accuracy

obtained by combining EEG and MEG data assumes

that one knows the proper scaling factor between the

two forward solutions. If this scaling factor is un-

known, additional errors can arise. Figure 9 demon-

strates the effect of mis-specifying the EEG/MEG scal-

ing factor.

An increasing discrepancy between the EEG and

MEG forward solution (modeled by a deviation of the

EEG/MEG scaling factor from unity) results in in-

creasingly larger average crosstalk.

DISCUSSION

Assuming equal SNR, changes in the sensor config-

uration produce similar results independent of the use

of fMRI weighting. Not surprisingly, increasing sensor

number, regardless of the type of MEG sensor, results

in improved localization accuracy. Comparing MEG

to EEG for the same number of sensors, we see that, on

average, EEG is superior to MEG. In fact, when using

no fMRI weighting, 30 EEG sensors (16.1%) are com-

parable to 61 MEG magnetometers (17.6%) or radial

gradiometers (17.4%). The superiority of EEG over

MEG results from the greater depth and orientation

sensitivity of MEG. For EEG sensors alone, there are

no average crosstalk values greater than 40%. In com-

parison, for 60 MEG planar gradiometers, 12% of the

sources have average crosstalk values greater than

40%. These sources fall into two categories: deep cor-

tical sources and sources that are largely radial in

orientation. Although the fMRI weighting constraint

reduces crosstalk for properly detected sources in all

cases, the relative difference between EEG and MEG

Figure 6.

Average crosstalk map and av-

erage point spread function

map at each location for three

different sensor configurations.

The average crosstalk or point

spread for each location is en-

coded in color. Full red equals

5% and full yellow equals 25%

crosstalk or point spread. Be-

low each map is the histogram

of the number of dipoles with

average crosstalk/point spread

valueswithina5%bin.NofMRI

weighting was used.

?EEG and MEG Localization Accuracy?

? 55 ?

Page 10

are less pronounced with fMRI weighting. When in-

cluding the fMRI constraint, the crosstalk is largely

defined by the spatial priors provided by the fMRI.

Subsequently, the inverse operator is less sensitive to

depth and orientation.

The lowest crosstalk is achieved by combining EEG

and MEG sensors. Using 61 EEG sensors and 122 MEG

planar gradiometers together results in one-third of

the average crosstalk of 30 MEG sensors alone. The

largest reductions in crosstalk (a decrease of ?50% in

crosstalk) are seen when going from 0 to 30 EEG

sensors in addition to any number of MEG sensors.

The next largest decreases in crosstalk occur with the

addition of 30 MEG sensors (magnetometers or radial

gradiometers) to any number of EEG sensors. Once

both EEG and MEG sensors are included, there are

only small decreases in crosstalk. This last result sug-

gests an important experimental consideration. In-

creasing MEG sensors does not increase the setup time

for the experiment, whereas the placement of numer-

ous EEG sensors can be extremely time consuming.

Significant gains in localization accuracy can be

achieved simply by placing a small number of EEG

channels to be recorded simultaneously with a large

number of MEG channels. Clearly a combined EEG/

MEG approach is superior to using either technique

independently, regardless of whether fMRI informa-

tion is included.

Our results compared the average crosstalk to the

average PSF over all locations. Using the standard

linear estimation inverse operator, the crosstalk map

and the PSF are equivalent (i.e., the resolution matrix

is symmetric). This equivalence will not necessarily be

true when using other inverse operators. In fact, the

crosstalk map and PSF reflect two different aspects of

localization accuracy. The crosstalk metric for a spec-

ified location on the cortex describes the amount of

activity incorrectly localized onto that location from

other locations, whereas the point spread metric pro-

vides the complementary measure: for that same loca-

tion, the point spread describes the mis-localization of

activity from that specified location to other locations

in the brain. If the crosstalk map for a location showed

only a single point at that same location, i.e., a delta

function, the estimate at that location would com-

pletely reflect activity at that location. Conversely, if

the PSF for a given location was a single point at that

same location, a point source would be spatially local-

ized as a point source.

Typically, neither the crosstalk map nor the PSF

would be a map with only a single point. Importantly,

the spatial extent of these two maps can be used to

determine the confidence of the estimates. If there are

other active areas that overlap with the crosstalk map

for a given location, the activity at that location is

affected by those other areas. In that situation, the

Figure 7.

Average crosstalk and average

PSF maps for the noise sensi-

tivity normalized inverse oper-

ator. The average crosstalk or

point spread for each location

is encoded in color. Full red

equals5%andfullyellowequals

25% crosstalk or point spread.

Below each map is the histo-

gram of the number of dipoles

with average crosstalk/point

spread values within a 5% bin.

No fMRI weighting was used.

?Liu et al.?

? 56 ?

Page 11

estimate for that location will not only reflect activity

from that location, but will represent a weighted sum

of all the activity within the region defined by the

crosstalk map. If, on the other hand, there are no other

active areas within a crosstalk map, one can be confi-

dent that the estimated activity reflects the true activ-

ity. The PSF is easier to interpret than the crosstalk

map. The PSF for a location defines the spatial extent

of activity that would be localized for a point source at

that location. Any activity in the region defined by the

PSF cannot be separated from activity at the given

location.

Although the noise sensitivity normalized inverse

has no effect on the crosstalk map, there is significant

improvement in the point spread function, especially

for deep and radial sources for MEG sensors alone.

Therefore, we would expect to have more focal

sources with similar temporal accuracy as the unnor-

malized inverse operator. Note, that there is a disad-

vantage to using the noise sensitivity normalized in-

verse operator. Because each location has a different

normalization factor, direct amplitude comparisons

between the timecourses of different locations cannot

be made. One can, however, still make amplitude

comparisons for a given location across different task

paradigms.

Many empirical experiments are now collecting

EEG and MEG data simultaneously. Both types of

data are acquired in the hope of producing more ac-

curate source localization estimates. To maximally use

this combined information, the scaling factor between

the two different types of data must be known. Be-

cause the EEG forward solution can be calculated

using conductivity ratios instead of actual conductiv-

ities, there can be a discrepancy between EEG and

MEG with respect to the units of the estimated source

strengths. Thus, when combining the two techniques,

error is introduced if the EEG/MEG forward solution

scaling factor is mis-estimated. We find that even if the

conductivity ratios are known, the actual scaling fac-

tor between EEG and MEG needs to be known within

a factor of 2. If the scaling factor is mis-estimated by a

factor of 0.2 (or 5), the average crosstalk is over twice

as large as when the scaling factor is correctly deter-

mined.

Certain caveats apply to these results. First, to spe-

cifically evaluate the inverse procedure, we assumed

that there were no errors in the EEG and MEG for-

ward solutions. Currently, the MEG forward solution

is more accurate than the EEG forward solution, ow-

ing to the fact that the MEG forward solution requires

only the inner skull surface and does not depend on

the conductivities of the various tissue types in the

brain [Hamalainen and Sarvas, 1989; Meijs et al., 1987,

1989]. If errors in the head model are included, which

is likely to be the case with the currently available

head models, the EEG accuracy will worsen relative to

the MEG accuracy. It is likely, however, that future

Figure 8.

Average crosstalk vs. sensor number; 90% fMRI weighting. The

sensor configuration consisted of 30 magnetometers (30m), 30

radial gradiometers (30r), 60 orthogonal planar gradiometers

(30p), 61 magnetometers (61m), 61 radial gradiometers (61r), 122

orthogonal planar gradiometers (61p), 30 EEG sensors, 61 EEG

sensors, and all possible combinations of MEG and EEG sensors.

The crosstalk values were averaged over a range of source num-

ber (5, 10, or 20 sources) and extent (1 cm or 2 cm in diameter).

Figure 9.

Average crosstalk vs. EEG/MEG forward solution scaling. A sensor

configuration of 30 EEG sensors and 30 MEG radial gradiometers

was used. An fMRI weighting of 90% was used. The scaling factor

between the EEG and MEG forward solutions was varied from

0.2–5. A scaling factor of 1 assumes no error in the scaling

between EEG and MEG forward solutions. The crosstalk values

were averaged over a range of source number (5, 10, or 20

sources) and extent (1 cm or 2 cm in diameter).

?EEG and MEG Localization Accuracy?

? 57 ?

Page 12

work will greatly increase the accuracy of the EEG

head model, making this first caveat less relevant.

Second, we also assumed equal signal-to-noise ratios

for all sensor configurations. In cases where the SNR

differs (such as magnetometers versus radial gradio-

meters), localization using the configuration with the

best SNR will result in greater accuracy than is shown

here. Finally, for the randomly placed sources, we did

not restrict the source orientations (i.e., radial and

tangential sources were equally probably in the Monte

Carlo simulations). As discussed above, since MEG

poorly localizes radial versus tangential sources, some

of the predicted superiority of EEG is due to this

orientation dependence of MEG.

Recently, other simulations examining the combina-

tion of EEG and MEG measurements were presented

[Fuchs et al., 1998]. Fuchs et al. [1998] used a single

equivalent current dipole (ECD) inverse approach to

localize test dipoles in a spherical three-shell head

model. Because these authors used a different forward

model, inverse method and source model, our results

are not directly comparable. Similar results, however,

were obtained with respect to two findings: 1) increas-

ing sensor number decreases localization errors and 2)

a combination of EEG and MEG is better than either

modality alone.

Overall, these results demonstrate that both EEG

and MEG are useful technologies for the localization

of brain activity. The lowest crosstalk was achieved by

combining EEG and MEG data, providing motivation

for further development of both methodologies. A

large reduction in the PSF (i.e., increase in spatial

accuracy) was provided by using the noise sensitivity

normalized inverse operator. More accurate head

models will improve localization accuracy, but will

not eliminate the need for the acquisition of simulta-

neous EEG and MEG information.

ACKNOWLEDGMENTS

We thank Drs. David Cohen and Denis Schwartz for

their many helpful comments. We also thank the

anonymous reviewers for their suggestions. This work

was conducted during the tenure of an Established

Investigatorship from the American Heart Association

to J.W.B.

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APPENDIX: INVERSE OPERATOR

DERIVATIONS

The linear inverse operator that we use here can be

derived in various ways. We detail four different der-

ivations: 1) minimization of expected error [Dale and

Sereno, 1993], 2) Bayesian formulation [Gelb, 1974;

Phillips et al., 1997a,b], 3) Tichonov regularization

[Tichonov and Arsenin, 1977], and 4) generalized Wie-

ner filtering [Deutsch, 1965; Smith, 1992; Sekihara and

Scholz, 1995, 1996]. All derivations arrive at equiva-

lent inverse operators, given certain initial conditions.

The minimization of expected error begins with a

set of measurements

x ? As ? n

(21)

where x is the measurement vector, A is the gain

matrix, s is the strength of each dipole component, and

n is the noise vector. One would like to calculate a

linear inverse operator W that minimizes the expected

difference between the estimated and the correct

source solution. The expected error can be defined as:

ErrW? ??Wx ? s?2?.(22)

Here we assume that both n and s are normally dis-

tributed with zero mean. Using their corresponding

covariance matrices C and R, the expected error can be

rewritten as:

ErrW? ??W?As ? n? ? s?2?

(23)

? ???WA ? I?s ? Wn?2?

(24)

? ??Ms ? Wn?2?

(25)

where M ? WA ? I

? ??Ms?2? ? ??Wn?2?

(26)

? tr?MRMT? ? tr?WCWT?

(27)

where tr(A) is the trace of A and is defined as the sum

of the diagonal entries. Re-expanding the expression

gives:

? tr?WARATWT? RATWT? WAR ? R?

? tr?WCWT?. (28)

This expression can be explicitly minimized by taking

the derivative with respect to W, setting it to zero and

solving for W.

0 ? 2WARAT? 2RAT? 2WC

(29)

Solving for W:

WARAT? WC ? RAT

(30)

W?ARAT? C? ? RAT. (31)

This yields the expression for the linear inverse oper-

ator:

W ? RAT?ARAT? C?? 1.(32)

The Bayesian linear inverse derivation begins with the

expression for conditional probability:

P?s?x? ?P?x?s?P?s?

P?x?

(33)

which one would like to maximize. Beginning with a

measurement vector x:

x ? As ? n

(34)

where A is the gain matrix, s is the strength of each

dipole component, and n is the noise vector. Assum-

ing both n and s are normally distributed with zero

mean and covariance matrices C and R, respectively,

one can rewrite P(x?s) and P(s):

?Liu et al.?

? 60 ?

Page 15

P?x?s??e? ?As ? x?TC? 1?As ? x?

(35)

P?s??e? sTR? 1s.(36)

This gives a simplified Bayesian expression:

max?P?s?x?? ? max?

?e? ?As ? x?TC? 1?As ? x???e? sTR? 1s?

P?x?

?

(37)

? max???As ? x?TC? 1?As ? x? ? sTR? 1s?

(38)

? min??As ? x?TC? 1?As ? x? ? sTR? 1s?

(39)

? min?sTATC? 1As ? sTATC? 1x ? xTC? 1As

? xTC? 1x ? sTR? 1s?. (40)

Taking the derivative with respect to s and setting it to

zero:

2ATC? 1As ? 2ATC? 1x ? 2R? 1s ? 0.(41)

Solving for s gives:

s ? ?ATC? 1A ? R? 1?? 1ATC? 1x ? Wx

(42)

which yields the expression for the Bayesian linear

operator

W ? ?ATC? 1A ? R? 1?? 1ATC? 1.(43)

The above Bayesian linear operator is very similar to

that derived using Tichonov regularization. Again,

one begins with a measurement vector x:

As ? x. (44)

A smoothing functional F is defined as:

F ? ?As ? x?2? ??Ms?2

(45)

where ? and M are added for regularization. To cal-

culate the operator, the smoothing functional is explic-

itly minimized (taking its derivative and setting it to

zero). Solving for s:

0 ? 2ATAs ? 2ATx ? 2?MTMs

(46)

?ATA ? ?MTM?s ? ATx

(47)

s ? ?ATA ? ?MTM?? 1ATx ? Wx

(48)

W ? ?ATA ? ?MTM?? 1AT.(49)

This is equivalent to the Bayesian linear operator

when C ? C?1? I and ?MTM ? R?1. Wiener filtering

(also known as the Kalman-Bucy method) filtering

uses an optimal linear filter to minimize the expected

error between the actual source (i.e., input) and the

estimated source (i.e., noisy output):

ErrW? ??Wx ? s?2?.(50)

The operator must satisfy the Wiener-Hopf equation:

?sx? W?x

(51)

where ?sx? ?sxT? and ?x? ?xxT?. Expanding the co-

variance terms gives:

?s?As ? n?T? ? W??As ? n??As ? n?T?

(52)

?ssTAT? snT? ? W?AssTAT? nsTAT? Asn ? nnT?.

(53)

Because the signal and noise are independent, the

signal-noise covariance terms (e.g., ?snT?) equal zero,

leaving:

?ssTAT? ? W?AssTAT? nnT?.(54)

Again, because the signal and noise are independent,

we can separate the terms on the right side:

?ssTAT? ? W??AssTAT? ? ?nnT??

(55)

RAT? W?ARAT? C?. (56)

Thus, the inverse operator is:

W ? RAT?ARAT? C?? 1.(57)

These particular inverse derivations are very general

and allow us to express many different kinds of in-

verse methods. For example, arbitrary basis functions

can be used in the inverse approach by constructing R

such that

R ? UUT

(58)

?EEG and MEG Localization Accuracy?

? 61 ?