1 Introduction MANY EFFORTS have been made to investigate the mechanisms of the alpha rhythm observed in electroencephalogram (LEG) and magnetoencephalogram (MEG) COHEN, 1968; CHAPMAN et al., 1984; LEHMANN et al., 1987; HE et al., 1988; 1989; LEHMANN and MICHEL, 1989). It is most interesting and meaningful to estimate, from the surface electromagnetic information, the neural sources in the brain generating the alpha waves (the dominant component of the spontaneous LEG) because this provides a good way of observing the status of the brain in the in vivo condition. The determination of the equivalent electrical sources fitting the observed surface field pattern is believed to lead to a better understanding of the brain function and enhancement of clinical diagnostics. Although the neural sources of the L E G and the M E G are not located in a few points, the dipole model is still a very useful tool in constructing an equivalent representation of the locations, strengths and orientations of the neural sources. A recent study shows the mathematical dipole model is adequate to

Correspondence should be addressed to Dr He at address 2. First received 3rd July 1990 and in final form 20th February 1991

9 IFMBE: 1992

324

describe a realistic generator of human brain activity (DE MUNCK et al., 1988). As the equivalent dipole model fits some realistic generators in the brain very well, the equivalent dipole model is widely adopted in approximating the generators of the L E G and MEG, especially for evoked potentials and fields (GRANDORI, 1982; KOSUGI et al., 1985; WOOD et al., 1985; HE et al., 1987; HOMMA et al., 1987). On the other hand, few reports have been made about the equivalent dipole solution for the spontaneous alpha activity. CHAPMAN et al. (1984) reported their effort to localise the generators of the normal magnetic alpha activity using two current dipoles. However, because of the lack of a multichannel M E G measurement system at the time, a single-channel S Q U I D was used and the relative covariance technique was used to make up the surface M E G map in their study, where the L E G signal measured on the vertex was taken as a reference. For this reason, only a pair of dipoles which correspond to alpha activity with a strong correlation to the L E G signal on the vertex, averaged over 2min, was obtained. HE et al. (1988) investigated the equivalent single-moving dipole solution of the L E G alpha activity by means of simultaneous measurement at 21 sites on the scalp. LEHMANN and MICHEL (1989) localised the single-dipole solution corresponding to a specific frequency map which was built up by F F T analysis. The purpose of the present study is to investigate the

Medical & Biological Engineering & Computing

May 1992

equivalent sources generating the spontaneous scalprecorded alpha activity by means of the multiple equivalent dipole fit technique. A method has been developed by means of which either a single dipole or two dipoles can be estimated, wherein the realistic geometry of the head is taken into consideration. The equivalent dipoles were obtained by minimising the least-squared error between the observed E E G potentials and the dipole-generated potentials on the scalp, with the locations of the dipoles variable (moving dipole model) or fixed (fixed dipole model) over time. 2 Method 2.1 Equivalent dipole fittings and numerical treatment The E E G equivalent dipoles are defined as dipoles best fitting the observed potential distribution on the scalp in the sense of the least-squared error. Because there is generally no knowledge available about the locations of the neural sources for a noninvasive observed potential distribution, the locations of the equivalent dipoles determined by inverse dipole estimation are of great interest, and this method is also called the 'dipole localisation method'. The term 'equivalent dipole' usually describes the equivalent moving dipole estimated by varying the location, orientation and magnitude of the dipole, so as to obtain a least-squared error between the observed potentials and the dipole-generated potentials. Most efforts in E E G dipole localisation have been concerned with calculating the equivalent moving dipole. The equivalent moving dipole solution can be obtained by minimising the squared difference sum at a given time t:

S(t)

= Ilum(t, r) - ua(t, r)ll 2

(1)

where u,, and un are column vectors consisting of the measured potentials and the potentials generated by M trial dipoles in N observation points r on the scalp, and S is the Euclidean distance between u m and un. This inverse estimation procedure consists of two steps. One step is the forward calculation of dipole-generated potentials u d on the scalp, and the other is a trial-and-error procedure in which S is minimised. If the scalp potentials can be given analytically, the calculation of S will be straightforward. When the realistic head geometry is considered, ua cannot be calculated analytically but can be calculated numerically. It has been shown that the geometry of the head plays an important role in the analysis of the brain potential field and E E G dipole localisation (WITWER et al., 1972; HE et al., 1987; MEIJS et al., 1988). An algorithm has been developed, by means of the boundary-element method, to evaluate u d efficiently and estimate the equivalent moving dipoles in a reasonable time, where the realistic geometry of the head has been taken into account: From lead field theory, the electric potential u(r) at an observation point r on the scalp is given by

u(r) = [" l(r, R) " J(R) dv

(2)

.Iv

where I is the lead field and J is the source current density at point R. In the case of a dipolar source, Jdv becomes the dipole component p(R) and eqn. 2 becomes

utr) = ltr, R) " p(R)

Medical & Biological Engineering & Computing

cpuv = - f v V 2 U / ( r - rp) dV + fsD, U/(r - rv) ds

(5)

-- fsUD,(1/(r - rv) ) ds

where up is the potential at the observation point rp, and c v is a coefficient which can be obtained by simple calculation (BREBBIA and WALKER, 1980). From the quasi-static field formulation the electric potential field u in the head volume V is governed by Poisson's equation (PLONSEY, 1969) and the boundary condition that the normal component of the electric current on the scalp S vanishes. Applying these to eqn. 5, the dipole-generated potential will be expressed as a sum of a double-layer potential and a volumetric potential. It is inconvenient to deal with the volumetric integral in the inverse dipole estimation. To eliminate this volumetric integral on the right-hand side of eqn. 5, the potential u is decomposed into a sum of a potential generated by the primary dipole sources (ut) and a potential generated by the secondary sources (u2) which are induced by the inhomogeneities (in the present analysis, uz represents the secondary sources induced in the boundary between the head and the air) in the infinite medium (AoKI et al., 1987). Replacing u with u2 in eqn. 5, eqn. 5 becomes

cvu2, v = - fsU2D,(1/(r-- rp)) ds - fsD, ul/(r - rv) ds

(6) Note that the potential uz becomes a sum of a single-layer potential and a double-layer potential. The discrete version of eqn. 6 can be obtained at as many points of the scalp as needed, and can be written as (7)

Hu2 + G q = 0

where u 2 and q ( = D , Ul) are column vectors of N s rows, and Ns is the number of nodes on the scalp. H and G are matrices Which depend on the geometry of the head, the discretisation procedure and the numerical calculation method adopted. Solving eqn. 7, the potential generated by dipoles in the brain is finally given by Ud = U~ -- (tPI-I)- ~Gq = (L t + L2)" P = L " p

(3)

Given the observation points r = (r~, r2, ..., ru) on the scalp and the locations of trial dipoles R = (Ra, R2 . . . . . RM), the electrical potentials generated by M dipoles on N electrodes are given by the following matrix equation: u(r) = L(r, R) "p(R)

Note that the lead field matrix L is mathematically identical to the transfer coefficient matrix between the trial dipole components and the electrical potentials on the scalp. L is a function of the locations of the dipoles, observation positions, geometry, characteristics of the medium and other factors. The problem to be solved is how to calculate the lead field or transfer coefficient numerically and efficiently in the inverse dipole estimation. In the present analysis, the head was assumed to be a linear, isotropic and homogeneous conductor. Although the effects of inhomogeneities will not be discussed here, these inhomogeneities will not greatly affect the significance of the present, strongly encouraging results regarding the determination of equivalent sources of the spontaneous E E G alpha activity; the detailed consideration of inhomogeneities will be considered in future work. The following integral equation can be obtained by using the second Green's theorem

(8)

where

(4)

May 1992

L1,

i =

[(r i --

Rl)/(r i -- RI) 3. . . . . (ri -- RM)/(r i - - RM)3] i = l . . . . . N~ (9)

L 2 = - ( H t H ) - 1Gw

(10)

325

W i =

[n/(r

i --

R1) 3 -- 3(n

"(r

i -

R,))(r

i --

R,)/(r

i -

R , ) 5,

9.. n/(r i -- RM)3 -- 3(n" (r, -- RM))(r i -- R~t)/(r i -- RM)5]

i = 1. . . . . N ,

(I1)

Note that L 1 and w are available analytically; and the matrix (/-P/-/)-IG is independent of the primary sources and needs to be calculated only once for a specific subject. Eqn. 8 was solved under the imposed boundary condition that the sum of surface potentials vanishes. In the fixed dipole model, the locations of the dipoles are given in advance and only the orientations and magnitudes of the dipoles are variable. Because of the linear relationship of the dipole moments to the electric potentials (eqn. 4), it is much simpler to estimate the fixed dipoles from the viewpoint of calculation. The problems are how to determine the locations of fixed dipoles, and how to describe the sensitivity of the results to the fixed locations. In the present analysis, the locations of fixed dipoles were taken from the averaged locations of equivalent moving dipoles. Because of limited dimensionality of the observed potential distribution on the scalp (KAVANAGH et al., 1976) and the effects of noise in measurement on the inverse solution (OKAMOTO et al., 1983), the equivalent dipole models have generally been restricted to one or two moving dipoles in E E G dipole localisation. In the present analysis, the singlemoving-dipole (SMD) fit, the two-moving-dipole (TMD) fit, the single-fixed-dipole (SFD) fit, and the two-fixeddipole (TFD) fit have been performed to investigate the behaviours of the equivalent sources of the EEG alpha activity. The resistance of the head was assumed to be 500f~cm (GEDDES and BAKER, 1967) in estimating the dipole moment. It is time-consuming to calculate coefficients of the matrices H and G in eqn. 8, although they need to be calculated only once for a specific subject and can be stored in computer auxiliary memory. One can find details of how to calculate H and G in BREBBIA and WALKER (1980). In the present analysis, the triangle boundary element and linear interpolation function were adopted. The scalp was divided into 300-400 nodes; the number of nodes was determined by compromise between accuracy and computing time needed. The realistic head geometry of subjects was taken from axial MRI-CT scans which were taken at 10 mm intervals. The peripheral co-ordinates of the cross-sections were digitised on a digitiser and the three-dimensional head model was constructed. The electrode positions on the scalp were also localised with the digitiser. The origin of the head model was set in the mid-

Fig. 1

326

Example of a reconstructed three-dimensional head model used in evaluating the electrical potential field in the head. The geometrical information was taken from axial N M R - C T scans which were taken at 10 mm intervals. The surface of the head was divided into 293 nodes and 597 triangles. The reconstruction of the model was performed in a Sun 4/260 workstation

point of a line connecting the ears. The x, y, and z-axes are taken longitudinally, laterally, and vertically, respectively. Fig. 1 shows an example of a reconstructed threedimensional head model for a subject. The accuracy of numerical analysis has been assessed by using a spherical conductor model in which an analytical solution is available. Fig. 2 shows the inverse estimation error against the eccentricity of the location of a dipole. Considering other factors affecting the accuracy of the inverse solution, such as noise in measurement and modelling error for medium and source, it is suggested that 300-400 nodes are suitable for the inverse dipole estima0.10

o Loc_dev-162x

0.08

9 Loc_dev_ 162z {3 Loc_dev_642x 9 Loc-dev_ 642

/9 /

(D

c 0"06 .o o

_~ 0.04 o

0-02

o-.2

-

0.4

0.6

o.8

~io

0.6

0.8

1-0

s

2.0

o Mom_dev_i62x 9 Mom-dev_ 162z [3 M o m _ d e v _

1-8

642x

II Mom_dev_642z

o

1.6

E o E

1"4 1.2

o Q.

5

1.o: 0.8

=

=

--

..

..

i

0'2

o'.a eccentricity

b

Fig. 2

Numerical accuracy of the present analysis method. A unit conductor sphere was used to evaluate the numerical error caused by the boundary element integral and the nonlinear minimisation procedure. The potential on the surface of the sphere generated by an electrical current dipole was calculated analytically to give a 'true" or "real' potential distribution. Then, the dipole was inversely estimated by means of the boundary element technique and the simplex minimising method. The criterion of the convergence in the nonlinear minimisation procedure was set as the size of the simplex, with a numerical value of O.O0]. In the boundary element analysis, the surface of the sphere was divided uniformly into different numbers of nodes. The numerical errors for different numbers of nodes have been compared, and the results for 162 nodes and 642 nodes are plotted. The distance between the original dipole and the estimated equivalent dipole is defined as the dipole location error and shown in (a) ; the ratio of the moment of the equivalent dipole to that of the original dipole is defined as the dipole moment ratio and shown in (b). The horizontal axis refers to the eccentricity of the dipole location in the sphere. It ts seen that the errors increase when the eccentricity of the dipole location increases. The errors of the dipoles perpendicular to the surface of the sphere (162z, 642z) are larger than those of the dipoles parallel to the surface of the sphere (162x, 642x). Considering the most eccentric area of the cortex has an eccentricity of about 0"86, it is suggested that 300-400 nodes are used in the boundary element analysis to obtain sufficient accuracy

Medical & Biological Engineering & Computing

May 1992

tion when the present algorithm is used. The trial-anderror procedure was performed with the aid of the simplex method (KOWALIK and OSBORNE, 1968). The criterion of convergence in the simplex method was taken to be the size of the simplex, which was set to 1.0mm in the E E G alpha source analysis. 2.2 Experimental procedure and data analysis E E G measurements were performed with 21 silver disk electrodes arranged on the scalp according to the international 10-20 system. Right ear reference was used as earlinked reference will distort the electric field of the head. Three healthy young subjects, one female and two males, aged from 21 to 31 years, were examined. The subjects were seated comfortably in a reclining position inside an electrically shielded room. Two minutes of E E G with eyes closed were collected by means of a multichannel bioelectric amplifier with a bandwidth setting of 0.5-30Hz. The gain of the amplifier was set to 20 000. The signals were A/D converted in a sampling time of 8 ms. The alpha component was then obtained through a digital filter with a passband of 8-13 Hz. Because there is no electrically silent point in the head, the mean value of the 21 measured potentials was subtracted from the potentials at every time point to obtain a reference-free measurement of the surface potential distribution. The corresponding mathematical operation was also performed on the dipole-generated potentials in the numerical calculation. Fig. 3 shows a segment of the filtered alpha waves on 21 sites for a period of 4096 ms for a young healthy male subject. As an objective measure of the equivalent dipole approximation, a parameter called 'dipolar proximity', (DP) has been introduced as DP(t) = (1 - Ilu.,(t, r)

-

Ueq(t ,

r)ll2/llu,,(t, r)[I2)

x 100 per cent

Fpl

(12)

where Ueq(t , r) is a column vector with rank of N generated by estimated equivalent dipoles at time point t. The DP shows the goodness of fit of the estimated equivalent dipoles. The equivalent dipole solutions were estimated at each time point for several alpha cycles to see the behaviour of the equivalent dipole solutions within an alpha cycle. The locations and orientations of the estimated dipoles were stable in half of an alpha cycle, whereas the magnitudes of the dipoles varied over time corresponding to the variation of strength of the alpha activity. Data compression has been performed in the present analysis, because of the large amount of data, by introducing the spatial variance of the potential (SVP) distribution N

SVP(t) = ~, (um, i(t )

-u

.....

(t))2/N

(13)

i=1

where urn, i(t) and u . . . . . (t) represent the potential of the ith electrode and the averaged potential over N electrode positions, respectively, at time point t. In the following analysis, the equivalent dipole fits were performed at time points when the S V P is at local maximum, except for those indicated. 3 Results

3.1 Equivalent single and two moving dipole solutions The SMD model and the T M D model have been used to approximate the equivalent sources of human E E G alpha activity in the present study, and the results were compared. Table 1 summarises the SMD and T M D solutions estimated at time points corresponding to peaks of' the SVP. The data shown in Table 1 are mean values over a period of 2min of E E G for subjects S 1 (31 years old, male), $2 (21 years old, female), and $3 (25 years old, male. Note that the average dipolar proximities (DPs), which

F8

Fp2

~

T3

F3

~

T4

F4 c3

Fpz P3

F~

P4

Cz

o,

F7

~

i

ls

I

Fig. 3 Segment of alpha waves recorded in 21 sites on the scalp according to the international 10 20 electrode configuration. The passband of the filter was set from 8 Hz to 13 Hz. Subject S 1 , male, 31 years old Medical & Biological Engineering & Computing

May 1992

327

Table 1 Moving dipole solutions for human EEG alpha activity on peaks of the SVP

Subject

S1

S2

S3

SMD

Proximity, per cent Moment, n A ' m Depth, mm

91.2 70.8 48.0

94.0 119.4 47.0

92-7 121.2 61.0

TMD

Proximity, per cent Distance, mm

96.9 39.0

98-4 37"0

97.2 33'0

1 Moment, nA-m Depth, mm

36.2 46.0

45-5 42-0

55.2 50.0

2 Moment, n A ' m Depth, mm

39'9 44.0

35.3 44.0

62.4 56-0

the goodness of fit of the dipole model, were 92.6 per cent for the S M D and 97.5 per cent for the TMD. The present results reveal that the inverse moving dipole model can account for the E E G alpha field on the scalp very well and the T M D model is an especially good fit to the alpha activity observed on the scalp. The dipole moment of the SMD solution was strong with an order of 100nA "m. The moments for T M D ranged from 3 5 . 3 n A ' m to 62.4nA-m. The depth beneath the scalp of the inverse equivalent dipole ranges from 4 2 m m to 61mm, and the S M D is approximately 4 - 5 m m deeper than the TMD. The distance between the two equivalent dipoles in the T M D fitting was approximately 3 4 cm. Fig. 4a shows an example of the SMD solutions tor the

subject $1. The horizontal and frontal views of the SMD at time points corresponding to peaks of the SVP between two vertical lines on the top frame are plotted; the SVP at each time point and the D P at peaks of the SVP are also plotted. In Fig. 6a, the T M D solutions estimated from the same recorded potentials are plotted. It is seen that the SMDs moved from the activated area in the left hemi-

50, E E O O t--

represent

o.

o_ 100%

i

100% I ~

? oo

-5( 0 Fig. 5

100

200 t,ms

300

400

Co-ordinates of dipole locations for the SMD solutions on the peaks of the S V P in an epoch as shown in Fig. 4a Ox Vy ~z Q.. u3

>~

t/3

100*

100%

JL_

s n

0%

o%

t3_ C3

0%

top ~

0%

top

front front

front

I

50mm

I 50mm'

0

Fig. 6 a

Fig. 4

328

b

Horizontal and frontal views of (a) the SMO and (b) the SFD solutions for subject S x during a period between two vertical lines in the top frames. The SVP wave and the DPs on the peaks of the SVP (with a range from 0 to 100 per cent) are also plotted. Dark circles refer to dipole locations and arrows refer to the dipole moments. The dipole locations are also linked by solid lines. The SMD moved from the left hemisphere to the right hemisphere; its orientation alternated along the longitudinal direction. The location of the SFD was determined by averaging locations of the estimated SMD over a period of 2 min, and it was near the medial cross-section. Note that the DP became lower by fixing the location of the equivalent dipole in comparison with the SMD solution

Horizontal and frontal views of (a) the TMD and (b) the TFD solutions for subject S 1 during the same epoch of the SVP as shown in Fig. 4. The locations of the TMD in each hemisphere are linked by solid lines. Note that very high DPs between the observed alpha field and the TMDgenerated field have been obtained. This shows the TMD model can excellently account for the alpha activity observed on the scalp. The estimated TMDs were separately located in each hemisphere around an activated area. The locations of the TFD were determined by averaging corresponding locations of the TMD in each hemisphere over a period of 2min. In contrast to the SFD solution, the orientations of the TFD were directed almost along one line during the whole epoch of the SVP. The dipolar proximities on the peaks of the SVP were high enough when the SVP took a large value (i.e. for high alpha activity), but became low for low alpha activity

Medical & Biological Engineering & Computing

May 1992

sphere, where the T M D s were located, to the area in the right hemisphere, pointed in opposite directions longitudinally; their averaged location was positioned near the middle cross-section of the brain. Fig. 5 shows the x, y and z-co-ordinates of the SMDs in Fig. 4a as functions of time. It is seen that the SMDs were located at about the same levels in the x- and z-directions but travelled along the

y-direction. On the other hand, the T M D s were located in both hemispheres and around an activated area in each hemisphere. The co-ordinates of the T M D s in Fig. 6a are plotted as functions of time in Fig. 7. Fig. 7a shows the traces of the equivalent dipole in the right hemisphere and Fig. 7b shows those in the left hemisphere. The orientations of the

E 5 E

%

~ .

O

E

c~0%. . . .

t

o

t

0

0

-5C

o

2;0

3;o

o

Fig. 9

0

.E 0

I

-5C

o

260

360

t,ms

b

Fig. 7

Co-ordinates of dipole locations for the T M D solutions on the peaks of the SVP in an epoch as shown in Fig, 6a. (a) shows the dipole locations for the equivalent dipoles located at the right hemisphere, (b) shows the dipoles located at the left hemisphere 0 x ~ y ~ z

1-0

0-5 E i1/

0

k3 bq

:~ -0.5

-1.0

T M D were different from those of the SMD, but pointed in similar directions during an epoch of the SVP. The dynamic behaviours of the equivalent dipole moments are shown in Fig. 8, which depicts the projections of the dipole moments of T M D s in the right hemisphere on peaks of the SVP in the x-direction. The original data are shown in Fig. 3. It is seen that the strength of the equivalent dipoles shows a similar rhythm to the original alpha rhythm. Note also that the dipolar proximity (DP) for the T M D solution corresponding to a large amplitude of the SVP was extremely high. Fig. 9 shows the T M D solutions at each sampling point around a peak of the SVP for (a) subject S 2 and (b) subject $1. 3.2 Equivalent single and two f i x e d dipole solutions In the present fixed-dipole analysis, the averaged locations of the S M D and the T M D over a period of 2 m i n were used. Table 2 summarises the mean values of S F D / T F D fits over a period of 2min of E E G at time points corresponding to peaks of the SVP. Note that the depth of the dipoles and distance between the two dipoles in the T F D fit are the same as those in the moving dipole fit. Compared with results of the moving dipole fit shown in Table 1, the D P is reduced for both the SFD and the T F D fits. While the T F D fit is still reasonable in the sense of its high DP, the S F D fit gives rise to a low DP. The magnitudes of the SFD and the T F D fits were of the same Table 2 Fixed dipole solutions for human EEG alpha activity on peaks of the SVP

0

1000

2000

3000

z~O00

t,ms

Fig. 8

a b Horizontal, frontal and sagittal views of the T M D solutions estimated at each sampled time point around a peak of the SVPs as shown by two vertical lines in the top right frame. The SVP and DP waves are plotted on each sampled time point. (a) subject $2, female; (b) subject $1, male

Variation of the equivalent dipole moment estimated from the EEG alpha waves shown in Fig. 3. The plot shows the amplitude of the x-component (P~) of the equivalent dipole located at the right hemisphere where the T M D model was applied. The inverse calculations were performed on peaks of the S V P over a period of 4096 ms. The same variations were observed for other components of the TMDs as well as for other dipole models. It is seen that variation of strength of the equivalent dipole represents variation of the alpha activity

Medical & Biological Engineering & Computing

SFD

TFD

May 1992

Subject Proximity, per cent Moment, nA "m Depth, mm

S1 82.5 55-I 48.0

S2 88.1 76.2 47.0

S3 86.4 90-0 61.0

Proximity, per cent Distance, mm

92-0 39.0

94.4 37.0

92-8 33.0

1 Moment, n A ' m Depth, mm

38.4 46-0

43.4 42.0

50-8 50-0

2 Moment, nA-m Depth, mm

38.4 44.0

40.4 44.0

61.2 56.0 329

order of those of the moving dipole fits shown in Table 1. Figs. 4b and 6b show the SFD and TFD solutions for subject S t in the same epoch of the SVP in which the moving dipole solutions were estimated. It is seen that the D P representing the goodness of fit is lower than those of the moving dipole fit. Except for the epoch of the SVP with large magnitude, the SFD fit was poor. The TFD fit was good in the sense of having an extremely high DP. The DPs of the fixed-dipole fits show a strong correlation with the SVP wave, i.e. the global activity of the alpha rhythm. Excellent fit was obtained when the potential distribution on the scalp had a large spatial variance. 3.3 .Comparison of four models Fig. 10 shows the DPs of the SMD, TMD, SFD, and T F D in the segment corresponding to Fig. 3 for subject $1. The SVP is also plotted above the curves of DPs. Note svp

DPsMD

DPTMD

DPsF D

DPTFD

~ 0

LO96ms

Fig. 10 Segment of the spatial variance of potentials and the dipolar proximities for the four models. The curves correspond to the SVP and the DPs for the SMD, the TMD, the SFD, and the TFD, respectively, from top to bottom (DP 0-100 per cent). All the DPs at each sampled time point over 4096ms were calculated and plotted. The original EEG alpha data are shown in Fig. 3. It is found that the T M D model can almost account for the alpha activity observed on the scalp

measured

Fig. ll

330

SMD

that the inverse estimation was performed at each time point over this period. It is found that, in general, the T M D model best fits the EEG alpha field, while the T F D and SMD fits are good only when the SVP takes a large amplitude, i.e. when the global alpha activity is large. The same phenomenon was observed in other segments for subject $1 as well as the case for other two subjects. Fig. 11 shows topographies of the measured potential field on two peaks of the SVP in the epoch shown in Fig. 4, the TMD-, the TFD-, the SMD- and the SFD-generated fields. It is seen that the TMD-generated maps can represent the measurement maps extremely well, and the decreasing sequence of the goodness of fit is TMD > T F D > SMD > SFD. 4 Discussion

The present two-moving dipole solutions are very encouraging for estimating neural sources of the alpha activity using equivalent dipole models. CHAPMAN et al. (1984) reported that the averaged magnetic field pattern over 2 min obtained by using a single-channel SQUID and the relative covariance method can be approximated by bilateral current sources. In a previous study, we investigated the single-moving dipole solutions of the spontaneous human EEG alpha activity by means of simultaneous measurement of 21 potentials on the scalp (HE et al., 1988). Our results showed that the equivalent dipole is a useful model in the determination of equivalent electric sources of EEG alpha activity. LEHMANN and MICHEL (1989) reported later the single equivalent dipole solution of a spatial power map corresponding to 10.5Hz rhythm b y means of F F T analysis. The present two-moving dipole approach showed the EEG alpha activity on the scalp can be represented very well by two equivalent electrical current dipole sources with locations fluctuating in an activated area in each hemisphere. The two-dipole model is more successful in representing the sources of the alpha activity than the single-dipole model, as demonstrated by the observation of high dipolar proximity (DP) in the two-dipole model, and from the anatomical and physiological features of the brain. When the two-moving-dipole (TMD) model is used, the D P is 97-98 per cent, which gives an excellent fit between the measured potential field and the model-generated one. Note that the dipolar proximity defined in the present paper is different from the measure 'dipolarity' we used in the previous

TMD

SFD

TFD

Topographies of the observed alpha field on the scalp and the model-generated ones at two peaks of the SVP during the epoch shown in Fig. 4. The measured, the SMD-generated, the TMD-generated, the TFD-generated, and the TFD-generated topographies are plotted from left to right. The spline interpolation method has been used to interpolate the potential values among potentials on the recording electrodes. The solid line and dotted line refer to the positive and negative potentials, respectively. It is seen again that the TMD model accounts well for the alpha activity observed on the scalp, and the sequence of the goodness of fit is the TMD, TFD, SMD and SFD, in decreasing order

Medical & Biological Engineering & Computing

May 1992

studies (HE et al., 1987; 1988). The reason we adopted the D P rather than the dipolarity is that the difference between the single-dipole fit and the two-dipole fit is small for the dipolarity but large enough for the DP. In other words, the D P is much more sensitive to the goodness of fit of the equivalent dipole model. Fig. 10 shows the difference between the singledipole fits and the two-dipole fits clearly. From the potential field pattern shown in Fig. 11, it can be seen that there are multiple sources in the brain generating the surface alpha field. The two-dipole-generated field pattern approximates the actual one better than the single-dipole-generated one. And the TMD-generated field pattern can almost account for the observed pattern. Although both locations and moments of the T M D model are variable, the T M D solutions have been estimated around an area in each hemisphere. Considering the two-hemisphere structure of the brain as well as the exellence of fit of the T M D model, it could be concluded that there are two dominant sources, one in each hemisphere, which can account for the EEG alpha field on the scalp. The moving dipole model has been used mostly in localisation of neural sources in the brain, because estimation of locations of the sources is one of the major goals of the research. GRANDORI (1982) investigated the equivalent source of the brain stem auditory evoked potentials using the SFD model. With respect to the neural sources generating the EEG alpha rhythm, it is still quantitatively unclear about the locations of the sources. Because there is variation from subject to subject, the averaged location of the estimated inverse moving dipole location for the same subject has been adopted as the location of the corresponding fixed dipole model. From Table 2, the depth of the averaged dipole location is from 4 cm to 6 cm for both the single-dipole approximation and the two-dipole

approximation. This result agreed with the results by CHAPMAN et aL (1984) who used an averaged magnetic field measurement technique. Although different signals (averaged magnetic field and spontaneous electrical potential field) and different techniques (in both data acquisition and processing) have been used, it is argued that the neural sources for the alpha activity can be well approximated by bilateral current sources in occipital regions at a depth of 4-6 cm from the scalp. The distance between the two current sources is also in agreement between the present study and the study performed by Chapman et al., although there is no restriction on dipole parameters in the present study. This suggests that the EEG and M E G alpha activities could come from the same neural origins. However, the present results show a little asymmetry in the two estimated equivalent dipole locations, which probably comes from a difference between the two hemispheres. Considering only a single SQUID was used in Chapman et al.'s study to measure the spontaneous alpha activity, the present results would provide higher resolution of the dynamic behaviour of the equivalent dipoles. Further exploration is necessary to investigate the source structure of both the hemisphere and interhemisphere relationships. LEHMANN et al. (1987) proposed the global field power as an identifier of the potential field, and pointed out the concept of segmentation of brain micro-status from the maximum value configuration. Actually, the SVP turns out to be identifiable with the square of the global field power. From Fig. 10 it is seen that the fixed dipole fit is good over a large amplitude epoch of the SVP, or when large alpha activity is observed. This suggests that the fixed-dipole model may provide useful information for the largeamplitude epoch of the SVP wave, when the activity is globally highly ordered in the brain. The present study

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Inversely recovered EEG alpha activity on 21 electrodes from the original data shown in Fig. 3 by employing the T M D model. The inverse calculations were performed at each time point. Some high-frequency noise can be observed in the inversely recovered EEG, resulting from the ill-conditioned inverse estimation procedure. Good recovery is achieved when the amplitude of EEG alpha waves is large or when the signal-to-noise ratio is large

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331

showed that usir/g the SVP or the global field power is a good way to compress a large amount of data. To check the feasibility of employing the SVP, the TMD-generated potentials (Fig. 12) at 21 electrode positions on the scalp were calculated from the data shown in Fig. 3 at each time point. F r o m comparison of the TMD-generated potentials with the recorded alpha waveforms, it is clear that the equivalent dipole model can recover the basic rhythm and pattern of the scalp-recorded L E G alpha activity. Fig. 12 also contains some high-frequency noise components which cannot be seen in the original data (Fig. 3), especially existing in areas where the amplitudes of the alpha wave are small. This probably comes from the inverse estimation procedure which is well known as an illconditioned process. Careful comparison between Figs. 3 and 12 also indicates that the inverse recovery is good when the alpha activity is strong. This result suggests the usefulness of the SVP for not only data compression but also data selection. The estimated equivalent dipoles should give information on the neural sources of the alpha activity. However, one should note that the neural sources will not be located at a few points as dipoles, but actually distributed over extended spatial volume. The estimated inverse dipole solutions equivalently represent the neural sources, in the sense that the field pattern observed on the scalp is almost the same. When the neural activity is limited to small areas of .the brain, the estimated equivalent dipoles' locations would represent directly the locations of neural activity. When neural activity such as alpha activity is spread in the brain, the equivalent dipole estimation procedure will not behave directly as functional localisation of the small activated areas in the brain. Rather, it will behave as functional classification and provide useful information about h u m a n high-order neural activity. Further investigations are necessary to explore the relationship between the estimated equivalent dipoles and the realistic generators for the alpha activity and to understand the underlying physiology as well as psychology. Acknowledgment--The authors appreciated discussions with Drs S. Homma, Y. Nakajima, K. J. Eriksen and H. Ozaki as well as comments from Drs B. E. H. Saxberg and K. Yana on this manuscript. We are also grateful to W. Ye and H. Sasaki for their assistance in LEG data collection. This work was partially supported by the Casio Science Promotion Foundation.

References AOKI, M., OKAMOTO, Y., MUSHA, T. and HARUMI, K. (1987) Three-dimensional simulation of the ventricular depolarization and repolarization processes and body surface potentials: normal heart and bundle branch block. IEEE Trans., BME-34, 454-462. BREBBIA, C. A. and WALKER, S. (1980) Boundary element techniques in engineering, Butterworth & Co. Ltd., London. CHAPMAN, R. M., ILMONIEMI,R. J., BARBANERA,S. and ROMANI, G. L. (1984) Selective localization of alpha brain activity with neuromagnetic measurements. Electroenceph. Clin. Neurophysiol., 58, 569-572. COHEN, D. (1968) Magnetoencephalography: evidence of magnetic fields produced by alpha rhythm currents. Science, 161, 784-786. DE MUNCK, J. C., DIJK, B. W. and SPEKREIJSE,H. (1988) Mathematical dipoles are adequate to describe realistic generators of human brain activity. IEEE Trans., BME-35, 960-966. GEDDES, L. A. and BAKER, L. E. (1967) The specific resistance of biological material--a compendium of data for the biomedical engineer and physiologist. Med. & Biol. Eng., 5, 271-293. GRANDORI, F. (1982) Potential fields evoked by the peripheral auditory pathway: inverse solution. Int. J. Biomed. Comput., 13, 512528. 332

HE, B., MUSHA, T., OKAMOTO,Y., HOMMA,S., NAKAJIMA,Y. and SATO, T. (1987) Electric dipole tracing in the brain by means of the boundary element method and its accuracy. IEEE Trans., BME-34, 406414. HE, B., MUSHA, T., YE, W., NAKAJIMA,Y. and HOMMA, S. (1988) The dipole tracing method and its application to the human alpha wave. In LEG topography 1987. J. TsuzuI, J. (Ed.), Neuron Publishing Co. Ltd., 9-17. HE, B., YE, W. and MUSHA,T. (1989) Equivalent dipole tracing of human alpha activities. Proc. llth Ann. Int. Conf. 1EEL Eng. in Med. & Biol. Soc., Seattle, 8th-12th Nov., 1217-1218. HOMMA, S., NAKAJIMA,Y., MUSHA, T., OKAMOTO,Y. and HE, B. (1987) Dipole-tracing method applied to human brain potentials. J. Neurosci. Meth., 21, 195-200. KAVANAGH,R. N., DARCEY,T. M. and FENDER, D. H. (1976) The dimensionality of the human visual evoked scalp potential. Electroenceph. Clin. N europhysiol., 40, 633-644. KOSUGI, Y., ANDO, A., IKEBE,J. and TAKAHASHI,H. (1985) Dipole localization of somatosensory evoked potentials in the brain. Jpn J. Med. Electron. & Biol. Eng., 23, 35~41~ KOWALIK, J. and OSBORNE, M. R. (1968) Methods for unconstrained optimization problems. Elsevier, New York. LEHMANN, D., OZAKI, H. and PAL, I. (1987) LEG alpha map series: brain micro-states by space-oriented adaptive segmentation. Electroenceph. Clin. Neurophysiol., 67, 271-288. LEHMANN, D. and MICHEL, C. M. (1989) Intracerebral dipole sources of LEG FFT power maps. Brain Topography, 2, 155-164. MEIJS, J. W. H., TEN VOORDE,B. J., PETER, M. J., STOK, C. J. and LOPES DA SILVA, F. H. (1988) On the influence of various models on the LEGs and MEGs. In Functional brain mapping. Pfurtsceller, G. and LOPESDA SILVA,F. H. (Eds.). OKAMOTO,Y., TERAMACHI,Y. and MUSHA,T. (1983) Limitation of the inverse problem in body surface potential mapping. IEEE Trans., BME-30, 749-754. PLONSEY,R. (1969) Bioelectric phenomena. McGraw-Hill, New York. WITWER, J. G., TREZEK,G. J. and JEWETT,D. L. (1972) The effect of media inhomogeneities upon intracranial electrical fields. IEEE Trans., BME-19, 352-362. WOOD, C. C. (1982) Application of dipole localization methods to source identification of human evoked potentials. In Evoked potentials. BODIS-WOLLNER,I. (Ed.), Ann. N.Y. Acad. Sci., 388, 139-155. WOOD, C. C., COHEN,D., COFFIN, I . N., YARITA,M. and ALLISON, T. (1985) Electrical sources in human somatosensory cortex: identification by combined magnetic and potential recordings. Science, 227, 1051-1053.

Authors" biographies Bin He was born in Zhejiang, China in 1957. He received the Ph.D. degree in Electrical Engineering with biomedical specialisation from the Tokyo Institute of Technology, Tokyo, Japan, in 1988. After conducting postdoctoral research in neural imaging at the Tokyo Institute of Technology from 1988 to 1989, he joined the Massachusetts Institute of Technology as a Postdoctoral Fellow. He is now a Research Scientist in the Harvard--MIT Division of Health Sciences & Technology. His research interests include electrical activity in the brain and heart, biomedical instrumentation and biomedical signal processing and imaging. Toshimitsu Musha was born in Japan in 1931. He received the Ph.D. degree in Physics from the University of Tokyo, Japan, in 1963. From 1954 to 1964 he was with the Electrical Communication Laboratory, NTT; from 1964 to 1965 with the Massachusetts Institute of Technology, Cambridge, USA; and from 1965 to 1966 with the Royal Institute of Technology, Sweden. He is currently a Professor at the Tokyo Institute of Technology. His interests are l/f fluctuations in electron devices and biological systems, computer diagnosis of heart disease using body surface potential mapping, and dipole tracing in the human brain.

Medical & Biological Engineering & Computing

May 1992