IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39. NO. 8, AUGUST 1992

79 1

A Random Dipole Model for Spontaneous Brain Activity Jan C. de Munck, Peter C. M. Vijn, and Fernando H. Lopes da Silva

Abstract-The statistical properties of the EEG and the MEG are described mathematically as the result of randomly distributed dipoles. These dipoles represent the interactions of cortical neurons. For certain dipole distributions, the first- and second-order moments of the electric and magnetic fields are derived analytically. If the dipoles are in a spherical volume conductor and have no preference for any direction, the variance of a differentially measured EEG-signal is only a function of the electrode distance. In this paper, the theoretically derived variance function will be compared with EEG- and MEGmeasurements. It is shown that a dipole with a fixed position and a randomly fluctuating amplitude is an adequate model for the a-rhythm. An expression for the covariance between the magnetic field and a differentially measured EEG-signal is derived. This covariance is considered as a function of the magnetometer position, and is compared with the measurements of Chapman et al. [23]. The theory can be used to obtain a (spatial) covariance matrix of the background noise, which occurs in evoked potential measurements. Such a covariance matrix can be used to obtain a maximum likelihood estimator of the dipole parameters in evoked potential studies, to evaluate the merits of the so-called “Laplacian derivation,” and for the interpolation of electromagnetic data.

I. INTRODUCTION serious difficulty in the interpreting evoked electromagnetic fields is the presence of noise in the data. The background EEG is considered as the major origin of noise for evoked potentials (e.g., [1]-[3]). Also for evoked magnetic fields the spontaneous brain activity is an important factor [4]. When the statistical properties of the noise are known, confidence limits of estimated dipole parameters can be calculated [5] and noise filters can be designed to improve the signal-to-noise ratio. However, little research has been reported on the statistical properties of the ongoing EEG, in spite of the importance of this problem, with the exceptions of the studies of [6]-[9].

A

Manuscript received August 6, 1990; revised July 3 I , 1991. This work was supported by a grant from S.T.W., the Dutch Foundation for Applied Research. J . C. de Munck is with Low Temperature Department, Technical University of Enschede, 7500 AE Enschede, The Netherlands. P. C. M. Vijn is with The Netherlands Ophthalmic Research Institute, 1100 AC Amsterdam ZO, The Netherlands. F. H. Lopes da Silva is with the Department of Zoology, University of Amsterdam, Kruislaan 320, 1090 SM Amsterdam, The Netherlands. IEEE Log Number 9201482.

However, these studies were limited to the temporal characteristics of a single-channel EEG. For the application to multichannel EP’s, we need not only a temporal description of the noise, but also its spatial characteristics. We have started the study of the ongoing EEG by making a mathematical model, which describes the EEG in terms of its underlying generators. The random component of the scalp potentials is explained by the action of randomly varying sources. The elecfric potential ~ ( 2t ), and the random magnetic induction E(?, t ) depend on the spatiotemporal distribution of the sources. When it is assumed that the generators are dipoles, the spatial characteristics of the EEG and MEG can be determined from the spatial distribution of these dipoles. The temporal characteristics of the EEG and MEG are expressed in terms of the temporal characteristics of the sources. No attempts are made to explain the latter on the basis of neuronal interactions, because a lot of research on the temporal aspects has already been performed, (for reviews see [lo], [111>. When the theoretical predictions of the random dipole model are compared with EEG-measurements, the effect of the reference electrode has to be taken into account. In Section I11 it is shown that this can be done by considering the variance of each pair of electrodes, instead of the covariance. For a spherically symmetric distributions of dipoles, the variance of the potential is calculated analytically and plotted as a function of the electrode distance. The concept of random dipoles has been put forward previously [12]-[14], but apart from the study of Katznelson, this idea has not been worked out as a model for the spontaneous EEG. For MEG, a similar model has been developed independently [15], [16] to describe the correlation of temperature noise. In this case the random dipoles represent the movement of free charges in a conducting medium. The first objective of our study was to get a statistical description of the background noise of EP-data. The model we developed for this purpose is so general that other applications of the theory are possible. In Section IV some applications of the analysis will be discussed: Spatial Wiener filtering, improvement of parameter estimation on the basis of EP-data, localization of random sources (such as the a-rhythm) and improvement of interpolation techniques for the construction of brain maps.

0018-9294/92$03.00 0 1992 IEEE

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

11. A MATHEMATICAL MODELFOR EEG A N D MEG ACTIVITY In this section a mathematical model that describes the EEG in terms of its underlying generators is presented. For MEG, the model is identical to that for the EEG, and therefore MEG will not always be mentioned separately. Since there are many innumerable neurons (more than 10’’ in the cortex) and because every neuron has many connections to other neurons, the mechanism underlying the ongoing EEG is highly complex. Although the gross mechanism of interaction between neurons is well understood, it is unfeasible to model the interactions between all neurons of the human brain precisely. Therefore, the question whether the nature of the ongoing EEG is either deterministic or stochastic, is an academic one. Analogous to the theory of statistical physics, a description of the EEG in statistical terms is the best we can do. Evoked potentials are usually+described as a time varying potential distribution $(?, P ( t ) ) , generally by a limited number of neural populations, which Leact deterministically to the stimulus. The parameters P ( t ) denote the positions, orientations and strengths of the neural sources. We assume here that the physical mechanism, responsible for the background EEG, is similar to the mechanism of the sources of EP’s. Contrary to EP’s, the source paramgers of the EEG are described by stochastic processes P ( t ) . The stochastic character of the sources is responsible for the random behavior of the surface potentials. The potential at position x’ is represented by the stochastic variable ~ ( x ’t,) , with

X(7, t ) = $(?, &). (1) The horizontal bars below the symbols denote random variables. Equation (1) expresses the essence of our theory. It implies that when ,an EEG is recorded, a finite number of sources (since P contains a finite number of parameters) is active at randomly distributed positions and orientations. The source parameters result from stochastic processes. The stochastic signals caused by the instantaneous source strengths are conducted through the volume conductor following the laws of Maxwell. This conductance is determined by the volum_econductor model used, i.e., by the dependence of $ on P . At the scalp the result of the superposition of all elementary processes can be measured as EEG-signals. Equation (1) shows that the statistics of the sources completely determine the statistics of the potentials. However, the dependence of the statistics of the potentials on the statistics of the sources, is very complex in its general form. To make the theory applicable, simplifying assumptions are needed. Another need for making such assumptions occurs when means and covariances are estimated from measurements. In that case it has to be assumed, for instance, that samples which are separated by large intervals, are statistically independent [7] so that the average of these samples have a “narrow” distribution. These kinds of assumptions, however, fall outside the scope of this paper. Only simplifying assumptions are

considered, which are necessary to obtain manageable formulas. They are introduced systematically in the remainder of this section. With each assumption the mathematical convenience and the physiological relevance are discussed. A . Moments The density function f.&;2, t ) of the potentials t ) can be cal_ulated from the joint density function of the sources f p ( P ; t ) , by expressing all moments of into terms offp. Although occasionally the third and the fourth moment of the potential have to be considered [8], for Gaussian distributions the first two moments of are sufficient. Therefore, we will only consider the mean and covariance of the potential. The mean of the can be expressed as follows:

x(?,

x

x x

=

\

d$fp($; t)$(?,

F).

Thus, the expected value of the potential can be written %s a weight sum of the potentials caused by the sources P . Since (2) can be alternatively interpreted as the potential M , caused by a distribution of point sourcesfp with potentials $, we may conclude that the mean of the EEG, caused by dipoles which are randomly and homogeneously distributed over a closed surface, with an orientation normal to this surface, is zero. Such a dipole distribution can therefore never be detected by simply estimating the mean of the EEG-signals. Since EEG-signals are always high pass-filtered by the EEG-amplifiers, there is another reason why the first moment of x gives no information about the underlying sources. Suppose that the applied EEG-filter L [ ] is linear. Then, the mean of the measured signal is E { L [ X ( x ’ ,01) = U E { X ( x ’ , 011 r e

=

s

d$L[fp(?;

1

t)]$(n’,

F).

(3)

If the EEG is a stationary stochastjc process, then fp is independent of t , and hence L [ f p ( P ;t ) ] = 0. Therefore, higher order moments are necessary to obtain theoretical predictions, which can be tested experimentally. For this purpose, the covariance of two potentials will be expressed in terms of the density function of the sources: cov (XI,

x2)

= E(x(x’1, 4)x(?2,

I-

I-

t2)I

de

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er al.: DIPOLE

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MODEL FOR BRAIN ACTIVITY

Note that in this definition of covariance, it is assumed time lagged versions of the first neurons. According to that the mean vanishes. Higher order moments of the po- this model, the existence of randomly moving sources is tential can be derived similarly. However, it is clear from excluded. A relationship between the (time) power spec(4) that these expressions will _be rather complicated, be- tra of the sources and the density of their positions can be cause more integrations over P will be needed, and the put into the model by combining the appropriate time coresulting expression will depend on more position and variance functions Sl and probability density functions time parameters. Even the second order moment is no: & ( p ’ ) . In this way a model can be built in which, for easy to handle in its general form, because the vector example, the a-rhythm is generated in a different part of contains the information of all neurons. Even if the neu- the brain than the 6-rhythm. Finally, we will make assumptions about the volume rons belonging to the same neu_ral mass are considered as one source, the dimension of is enormous. Therefore, conductor and about the character and distribution of the some assumptions will be made in the following section sources. These will be discussed in the following secto obtain simpler expressions that are general enough to tions. have physiological relevance. C. Electric Covariance for a Spherically Symmetric Model B. Basic Assumptions Apart from the distribution of the sources, the statistics Similar to the fixed dipole model for evoked potentials [ 171, it is assumed that the potential measured at position of the electromagnetic field depends on the geometry of x’ is the result of the contribution of L sources, weight by the volume conductor. Contrary to Katznelson [13], who modeled the head as a semi-infinite medium, we will use their instantaneous source strengths: a multilayered sphere to describe the head. The sources L will be dipoles, which are randomly distributed without x(x’, t) = I = I s1(t)J/r(Z9 (5) preference for any 6- or cp-direction. Under these condiHere the source strengths s l ( t ) are random processes and tions, the covariance of two potential measurements is the source parameters f i r are Fndom variables. Thus the only a function of the distance between the electrodes and we could say that the EEG is “spatially stationary.” collection of all ammeters P is partitioned into (p’ -rr -.T This covariance will be referred to as the “electric coP2, P3, * * PL). The second assumption is that each source strength is variance, opposed to the “magnetic covariance,” which statistically independent of the other source parameters, is the covariance between two magnetic measurements. and that the source parameters of different sources are mu- First, the distribution of the dipole positions is considered tually independent. The source strengths, however, may to be statistically independent of the orientations. This debe correlated. In the Appendix it is shown that under these viates slightly from reality, because very superficial diassumptions the mean and the covariance are simply re- poles, which are more frequently located in a gyrus, will lated to the means and covariances of the individual more likely be radially oriented than tangentially. Howsources. Consequently, in the following we will only con- ever, one can include this effect in a further analysis by taking for& the superposition of appropriately chosen insider the single source covariances: dependent states. Thus we have

e

c

E/).

r,

- 2 9

y’

&(@)= @(ro)3(6)

(8)

with 6 the orientation of the dipole in the local coordinate frame. Since the distribution has no preference for any direction, it is independent of the other orientation angle Sl(t1, t2) = ds, dS2fSS,(Sl,s2; tl, t2). (7) cp, and also independent of the position angles bo and cpo. In this section the source index 1 will be omitted. The Here the function Sl defines the time covariance function dipole function is written as follows: of source 1. These functions will not be modeled in this thesis, but it will be noted that any model can be substituted here [lo]. The assumptions which we have made until now have a clear physiological meaning. They imply that the generators of the EEG occur at random positions, and with random orientations. The source strengths are time de(9) pendent and result from stochastic processes. These sources excite other neurons , the position and orientation where is the monopole potential at zl due to a source of which is unrelated to the first neurons. The second at To. In this formula ro, Go and p0are the spherical cogroup of neurons generate fields which are, for example, ordinates of the position vector Zo. With (6), (8), and (9) with

s

j

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

we find Cov

( x l , x2)

=

1 1

1

s12 d z o a(ro) d o

sin

zr)

dcp3(,j)

the orthonormality of the spherical harmonics in these variables. For the e g e n t i a l part only the integral over cpo is simple. The following integral can be used to find the integral over a0 [191:

m2 +-sin2 P,, tJ0 '

(cos 6

a + ar,

COS

a + sinro cpsinsin6,6 -) acpo

sin 6 a ro a60

cp

$mon(?29

20)

=

(lo)

where SI2is the time covariance (7). Although (10) might seem a little unfriendly, we will demonstrate that it takes a more pleasant form, when the monopole potentials are expanded in spherical harmonics. First, however, the integration over the orientation angles will be performed. Then with

I

Pa.

M? =

0

d 6 3(6) sin 6 cos2 6

2n(n 2n

tim*

(cos 60)PnPm(cos

60)

+ 1) (n + m)! + 1 (n - m ) ! '

Here the apostrophe denotes a differentiation with respect to cos O0. The factor m 2 originates from the differentiation with respect to cpo in (13). In this way we find the following simple expression for the covariance:

n2 x2) =

2n = O U3 (rh)rh2"drh Pd(cos

P12)

(1 1)

and

1, d 6 3(6)sin3 6 P*

M:

=

(12)

we find cov (XI, x 2 )

Note that this expression is valid for any dipole distribution, which is independent of cp. Next, is expanded as follows [18, (lo), (33), and (36)]:

Ynma(61,

cpl)ynma(60, PO)

(14)

and similarly $2. Here, the functions gn and Rk') depend on the conductivity and radii of the concentric sphere model. These functions are described in detail in [18]. For example, if rl = 1 is the boundary of the volume conductor then for the infinite medium these functions equal 1 / E and for the homogeneous sphere they equal (2n + l)/(m). When the dipoles are in the innermost sphere, Ri')(ro)reduces to r:. Calculating the integrals over d o 0 sin 6,dcpo in (13) is rather simple for the radial part, since use can be made of

Here is the angle between and Z2, viewed from the center of the sphere. In the equation it is assumed that the dipole is in the innermost compartment of the set of concentric spheres. Therefore, it is assumed that the dipole density function (R (rh) vanishes for rh larger than the radius of the innermost sphere. If the dipoles are all located on a shell with radius ro and if both measurement points 2,and J2are on the same radius r , then (16) becomes

- P d ( C 0 S P12).

(17)

The distribution of sources here described, do not contribute to the mean of the electric potential nor to the mean of the magnetic induction, no matter whether the statistical processes are stationary or not. Any tangential dipole is compensated by another tangential dipole with an opposite sign. If we take M8 = 0, then the sphere (0,ro) is homogeneously covered with dipoles, that are normally oriented to this sphere. Consequently, also the radial dipoles compensate each other. The covariance, on the other hand, will deviate from zero, as follows from (17).

D. Magnetic Covariance for a Spherically Symmetric Model The generators of the spontaneous EEG also induce a random magnetic induction. The covariance between the magnetic induction measured at two different points can be calculated in a similar manner as the electriczovariance. We have to deal with the random variable Er(?, t )

de MUNCK et al.: DIPOLE

=

195

MODEL FOR BRAIN ACTIVITY

&(2, al(t)) where

Gl is the magnetic field, caused by

m

- (A, a,)] nc= O u,PL(a, 3,) + [A, a, - ($1 - $2)(A] a,)] - [A, - a, - (21 a,)(& - a,)]

the lth source. It appears to be convenient to c_onsiderfirst the (random) magnetic potential U,(?,t ) = _Ul(?, al(t)). When the covariance between two magnetic potentials has been found, the covariance between the magnetic induction can be obtained by direct differentiation with respect to both measurement points. If the source is a random dipole in a spherically symmetric conductor and if its distribution function is given by (8), we can find an expression for Cov (U1, U,). with The following expression for the magnetic potential at 2,caused by a unit dipole will be used [20]:

*

*

*

*

*

m

c anP&(3, - a,)

n=O

*

and ii = Z i / r , . The prime and the double prime in the Legendre polynomials denote the first and the second deg2,respectively. rivative with respect to cos &, = The first term in (22) denotes the covariance between two radially oriented magnetometers. This term simplifies to

-

with Am=--

sin

PO

2n+ln+ r' 1

(

sin d ro

Q

a as,

With (8), (18) and (19) we find for the covariance of the magnetic potential:

c

S12M2gPi 1 n cov (U,, U,)= - -4 n=O2n I n 1

+

+

- Pm(g1 - i 2 ) .

(24)

In practice, the magnetic induction is almost never measured directly, but a gradiometer is used, which measures the difference between the fields of a pick up coil and a similarly directed compensation coil. An expression for the covariance of two gradiometers is therefore given by

- BT, B: - B;) = cov (I?;, B;) - cov (B;, B;) - COV(B;, E;) + COV(E;, Bz).

COV(GI,Gz) = COV(BY

(20) When the dipoles are homogeneously distributed over a spherical surface of radius ro, (20) reduces to

(25)

Here, the tilded symbols refer to the compensation coils.

* PnO(c0s P I , ) . (21) To find the covariance between signals measured with magnetometers at x', and 2,, in the directions A, and A,, (21) has to be differentiated with respect to the magnetometer positions: cov (BY, B;) = (A, * V , ) ( f i , * V,) cov (U], U,)

E. Electromagnetic Covariance for a Spherically Symmetric Model The above theory can similarly be applied to calculate the covariance between a potential measurement at 3, and a magnetic measurement at Z2. We find

- sin 8 d q 3 ( 0 ) a

+ sin e

+ [A1 - (A,

A, - (fi, *

i,)(A]

*

- 9,)(fi,

(

sin 9

a

sin d ro

-a-

Q

sin

Q

1 *

3,)

a,) + (21 * Q(fi1 - a,)

F Z

=

0.

I I

-

~ 0 s p s i n . r 9a

201

as,

cos Q sin t9 ro sin so

a

G)

-

-

196

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

For the electromagnetic covariance we find

+ P,,

d (COS8,) -Pn*,,,(COS80) d 80

which follows from straight-forward partial integration. Note that the covariance of a magnetic and an electric signal is only zero for a spherically symmetric distribution of sources. For other distributions the covariance does not vanish in general.

F. Localized Random Activity Although simple expressions can be obtained for a completely spherically symmetric geometry, this situation is in general not realistic. For instance, EEG activity caused by an epileptic focus will be highly localized. To describe these kinds of processes it will be attempted to generalize the results derived in the previous section. In the most general case, the covariance function is dependent on two time parameters, and on six position parameters. Therefore, if this function is expanded in spherical harmonics, a double series results:

cov

(XI,

C Anma(r1, r2)Ynma(81,

~ 2 = ) ~ 1 2 nma

-

klb

klb

’ yk/b(827

(28)

p2).

In case of spherical symmetry all terms for which n and k , m and 1, or a and b are different vanish. If, on the other hand, there is only axial symmetry the situation is more complicated. This occurs when the dipole generators are randomly distributed on a flat circular disk in the x-yplane. The only simplification we get is that Cov (xl, x2) depends on the difference of p1and (p2, but not on p1and p2 separately. However, the function will be separately dependent on 8,and O z . Although it is not difficult to find the coefficients A,, for this configuration, the result is klb

only of theoretical interest, since it is not easy to handle. This holds true even more if subsequentially ridged body transformations are applied on the spherical harmonics to move the disk to a realistic position. It is more practical to describe the EEG as the sum of “clustered states. ” For this purpose the density function &@’)is expressed as follows: K

f,(F) = k = 1 pk where

(29)

- ?k)

is the chance that the source parameters equal of the j i k are chosen close together, a localized state is described. With (29) the covariance becomes pk

Fk.If the position parts cov (XI,

x2)

=

s12

pk+(?l, k

$k)$(221

Fk).

(30)

111. COMPARING THEORY AND EXPERIMENT Before we compare the theoretical predictions with experimental measurements we have to pay some attention to the fact that the recordings are time-filtered and to the effect of the reference electrode.

A. The Reference Electrode Since the signals from the EEG-amplifiers are the potential differencesbetween two electrodes, the covariance between two EEG-signals measured with a common reference electrode, depends on the positions of three electrodes. This covariance can be expressed in terms of the covariances of potentials: cov

- xo, x 2 - xo)

(Xl

=

cov (XI,

x2)

- cov

(Xl,

xo) - cov ( x 2 , xo)

+ cov (xo, xo)

(32)

where xo is the potential at the reference electrode. Since each term in (32) depends, moreover, on two time parameters, the estimation of covariances does not afford insight in the comparison of theory and experiment. For the same reason it is inconvenient to study correlations or coherences of electric derivations. Instead, we suggest to estimate the variance of each pair of measured potential differences: Var

(xl - x 2 ) = Cov (XI - x 2 , X I - x 2 ) = Var (xd + Var ( x 2 ) - 2 Cov (XI, x 2 ) (33)

because these statistics depend only on two electrode positions (and two time parameters). Therefore, variances of differently measured channels provide more insight than covariances do. On the other hand, the covariance between any pair of electrodes can be expressed in terms of variances of differentially measured channels: cov

(Xl

=

- x3,

x 2 - x4)

;w a r

(Xl

- Var

(XI -

-

x4)

+ Var ( x 2

x2) -

Var

(x3

- x3)

-

x4)).

(34)

Hence no information is lost if only variances are considered. In particular, the covariance between two channels measured with a common reference (32) can be expressed in terms of these variances:

191

de MUNCK el al.: DIPOLE MODEL FOR BRAIN ACTIVITY

Theoretical var lance function

When the source model and the electrode grid are given, (35) can be used to calculate the covariance matrix.

B. Temporal Filtering From now on, we will assume that the statistical processes acting in-the brain are wide-sense stationary. This implies thatfp(P) is independent of time, and thatfpp(P P2; tl, t2) only depends on the time difference 7 = tl - t2 [21]. As a result M ( x ) will be independent of time, and Cov (xl, x2) depends only on 7. It has been demonstrated that these consequences are not contradicted by experimentally obtained results [9], provided that the time segments are not too long. As shown earlier (3) the mean of the potential vanishes everywhere because the EEG is high-pass filtered; for the same reason the magnetic induction also vanishes. Under these conditions (A.8) reduces to L

COV( x l , x 2 ) =

c S'"(7)X")(x'l,

I= 1

22)

(36)

Electrode dlstance (&weed

Fig. 1 . Variance as a function of electrode distance. The variance of the potential, measured between two electrodes, is plotted against the electrode distance in degrees. The functions are normalized such that the variance of the potential with respect to the center of the sphere, is unity. The generating dipole sources are located at various depths (0.2 5 ro 5 0.8),in a three-shell volume conductor. This conductor consists of spherical compartment with radius 0.87 and conductivity 1, surrounded by a spherical shell with outer radius 0.92 and conductivity 0.0125, which in turn is surrounded by a spherical shell of radius 1 and conductivity 1.

where S ( ' ) ( 7 )is the autocovariance of the lth process, and X'')(21, Z2) is its spatial covariance. For the covariance of the magnetic induction an equivalent expression applies. When the complex valued filter characteristic of the EEG-amplifier is given by H ( w ) , the measured covariance in the frequency domain is L

cov (21, 22) =

c H(w)H*("S'"(w)X"'(x'l,

I= 1

22) (37)

where S'"(w) is the Fourier transform of S " ) ( T ) . If the filters are adjusted as band-pass filters, it is possible to investigate whether in some frequency bands the sources are spherically symmetric distributed. C. Simulations If the sources are distributed spherically symmetric, the variance of the potential difference will only be a function of the distance between the electrodes [(17) and (33)]. In order to study this function we plotted in Fig. 1 the variance against Pl2 for several depths of the source layer. The standard three-sphere model [22] was used for the volume conductor; M$ and M , were taken equal. The curves were normalized with Var (x). It appears that the function is very insensitive to depth. If ro is small the function approaches 2 - 2 cos (Pl2) which is the first term of the expansion in Legendre polynomials. This implies that higher order terms have a relatively small contribution if ro is small. The invariance to depth shows that also for more superficial layers the convergence of the series is fast. One important reason is that the smoothing-effect of the skull acts quadratically in (17). In Fig. 2 we plotted the variance function for the same dipole distribution in an infinite medium. It shows that the sensitivity to ro is now much larger than in the case of the three-sphere model. Other simulations demonstrate that it is almost impossible to discriminate between purely ra-

0

50

100

150

200

electrode &stance

Fig. 2. As in Fig. 1 , but then for random dipoles in an infinite medium.

dial, and purely tangential dipoles on the basis of the covariance function for either volume conductor model. This also follows from (17), since M8 and M, appear nearly symmetrical. In Section 11-D we found an expression for the magnetic covariance function. In Fig. 3 we plotted the magnetic covariance function (24), for radially oriented magnetometers. It is assumed that the (radial) magnetometers were located on a spherical shell with a radius of 1 . 1 times the head radius. It is clear that magnetic measurements are much more sensitive to depths variations than electric measurements. If the dipoles are at a spherical shell of radius 0.7 (solid line), the MEG-recordings which are separated more than about 60" are almost completely uncorrelated. For the electric potential such a correlation distance cannot be given in general, since it depends on the position of the reference electrode. As expected, the covariance function of a gradiometer is more sensitive to depth variations. This is illustrated in Fig. 4, where we

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

.....................

... .6

. ... .... ... . .

. .. . . ... . . . . . . . . ......... ....... .3 ..... ............ . 2

Fig. 3. Covariance functions of the radial component of a magnetometer at a distance of 1.1 from the center of the conductor, for several shells of random dipoles. The solid line represents the cortex region (ro = 0.7).

. .. .. . .. . . . .. .. .. . . . . . . . . .

................ -. . 3 .2

Fig. 4. Covariance functions for a first-order gradiometer.

plotted the covariance function for radially oriented gradiometers with the pickup coils and the compensation coils at respectively 1.1 and 1.3 times the head radius. It can also be seen that the correlation distance is shorter (about 5J0, forro = 0.7).

fined state. The (32) electrode positions were determined by fitting a sphere to the electrode grid, as described in [24]. For all pairs of electrodes the distance and the variance of the EEG-signal were determined. The variance in various frequency bands was plotted as a function of the electrode distance. Since there are 32 electrodes, each plot contains (1/2) (31 x 32) = 496 points. Three *examples IV . EXPERIMENTS are given in Figs. 5-7. The frequency bands were respecWe did some experiments to test whether in some fre- tively 5-8 Hz, 9-12 Hz, and 13-30 Hz. The variances quency bands the electric variance function is spherically were normalized using the highest variance occurring in symmetric, as assumed in (17). The experimental results each band. of others [4], [23] were used to test the magnetic part of Fig. 5 demonstrates that for the lower frequency band the theory. We recorded segments of EEG of a subject the variance is mainly a function of the electrode distance. with closed eyes to maintain a physiologically well de- With the random dipole model we can explain this result

de MUNCK et al.: DIPOLE MODEL FOR BRAIN ACTIVITY

band fmm 5 U, 8 iiz

h-cy

.

.. ..

. .. . - .

.

.

..

.

,. ..... . .... .. . . . '.

.

..

,.,

.

. .

i

.

.

. .. 1

'

,

.

..+., I

I

I

I

I

I

I

I

Elccmde dismce

I 180

Fig. 5. Experimentally determined variance in the frequency band of 5-8 Hz, as a function of the electrode distance.

Frcquc.cy band h m 9

U,

12

.

.

.

I

HZ

.

I

. . .... .

.

.

.. : . . . . .. . ..

.

I

I

.. ., . . ..

.

.

..

I

I

I F,kcu&

dimmce

I

I 180

Fig. 6. As Fig. 5, but then in the a-band, i . e . , 9-12 Hz.

Elwmde di-

180

Fig. 7. As Fig. 6, for the frequency band of 13-40 Hz.

by assuming a global distribution of independent dipoles. In Fig. 7,which shows the results for the higher frequencies, the points are also scattered in a narrowband. The upper most points appear to correspond to one electrode, which is located at the border of the electrode grid. If Figs. 5 and 7 are compared with the theoretical curve of Fig. 1, the agreement of theory and experiment is striking. For the a-band, however, it clear from Fig. 6 that a spherically symmetric distribution of dipoles is not adequate to describe the resulting variance function. Further experiments, with different subjects in different states [20], confirmed that in the a-band the points are scattered over a large area, and that in the lower frequency band the points have a narrow distribution.

Recently, Bullock and McClune [25] performed a similar analysis on EEG-data obtained with microelectrodes in animal experiments. They plotted the correlation as a function of electrode distance. However, they averaged the spatial correlations for each pair of electrodes having the same distance, and therefore it is impossible to test from their measurements whether the generators are homogeneously distributed or not. Moreover, they did not take the effect of the reference electrode into account adequately. Knuutila and Hamalainen [4] determined the covariance matrix of spontaneous MEG-activity with a seven channel first order gradiometer. Fortunately it was possible to reconstruct the distances between the gradiome-

800

it * r

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

“I

I

B lot if

Fig. 8. Experimentally determined covariance function, plotted as a function of magnetometer distances. Data from [4].

ters from their figures. When their covariance matrix is plotted as function of distance Fig. 8 is obtained. The correspondence between the theoretical curve of Fig. 3 and the experimental result of Fig. 8 is clear. It should be noted, however, that with the seven-channel system the largest distance between two gradiometers is only 4 0 ” . To verify that the correlation tends to zero for larger distances, as predicted by Figs. 3 and 4 , an experimental study with more measurement points is necessary. Chapman et a l . [23] determined the (relative) covariance between an EEG-signal, which was measured between two fixed electrodes (located at Zl and Z 2 ) and a MEG-signal which was measured at various locations x’ above the scalp. These measurements resulted in a “COvariance map,” with a dipolar pattern. If it is assumed that the a-rhythm is generated by one or more clustered random dipoles, then the random dipole model gives a clear interpretation of Chapman’s results. If we assume that there are two dipoles, with parameters jil and ji2,we can find, similar to ( 3 1 ) , the following expression for the electromagnetic covariance: cov

( E , x1

A. Spatial Wiener Filtering Wiener filtering is a method for noise filtering in the time domain. The method is based on knowledge of the power spectra of the signal and the noise. Those frequencies, for which the signal dominates the noise, are amplified. If the two spectra overlap the application of Wiener filtering is of limited value. The idea behind Wiener filtering can also be applied in the spatial domain. Since the signal ($dip) as well as the covariance of the noise ( 1 7 ) are known in spherical harmonics, it is convenient to express the spatial frequencies in these functions. A difficulty is, that the coefficients of the series expansion of a function $(x’) depend on the orientation of the coordinate frame. The coefficients transform if the coordinate frame is rotated. The concept of spatial frequency, however, is independent of the actual orientation of the frame. Therefore, if $ is =

r,,,

nl

a: =

C

m=O,a=O

A:,

which is independent of the orientation of the coordinate frame. For a dipole we find

a 2, =

2~ ~

2n

2 2n-2

+ 1 gnro

[2n2M,2 + n(n

+ l)Mi].

(41)

If the noise is spherically symmetric, we can express the covariance in normalized spherical harmonics, as follows:

2a n=o(2n *

*

V. APPLICATIONS Some applications of the above theory will be discussed below :

(39)

with ( 9 , p) the normalized spherical harmonics, then the power of the nth spatial frequency component will be

- x2)

(38) where Zl and Z 2 are the locations of the electrodes, x’ is the location of the gradiometer, and S is the autocovariance function of the a-rhythm. In this equation we took the second order derivative of the magnetic induction to approximate the effect of the second-order gradiometer which was used in the experiment. Equation ( 3 8 ) shows that for fixed x’] and Z 2 , the covariance as a function of 2 , must have a dipolar pattern. From Fig. 6 it was already clear that the generators of the a-rhythm do not have a simple spherically symmetric distribution. Further evidence is given by ( 2 6 ) , which shows that for a spherically symmetric distribution, the covariance between the magnetic field and the electric potential, is zero.

C L u z ( r ) X m a ( 8 , CP)

+ 1)

g:rF [ 2 n 2 ~+; n(n

+ ~)M$I

C ynma!(~l,PI)Y n m a ( 9 2 , ~

ma!

2

) (42)

Since ( 4 1 ) and the coefficients of ( 4 2 ) only differ a factor l ) , the noise spectrum and the spectrum of of 1 / ( 2 n the signal will be highly similar. This is confirmed in Fig. 9, in which we plotted the spatial frequency spectra of the covariance of the noise and of the signal. In Fig. 9 we have chosen both the random dipoles and the deterministic source at 70% of the head radius and both to have equal radial and tangential components. The picture shows that the spectra largely overlap, and hence Wiener filtering in the spatial frequency domain does not seem helpful if the EP and the background EEG originate from the same depth. Some authors [26] advocate the use of the so-called ‘‘Laplacian” derivation. In this recording technique, the potential at each electrode is multiplied by four and the potentials of the surrounding four electrodes is subtracted. The resulting ‘‘Laplacian’ ’ is a high-pass filtered signal of the original potential distribution in the spatial

+

80 1

de MUNCK er al.: DIPOLE MODEL FOR BRAIN ACTIVITY .........

-mise

0

8

24

16

is the covariance matrix. Although our random dipole theory does not generally predict Gaussian noise, the noise in the data is the result of many averages, and therefore, it may well be approximately Gaussian. In (44) the inverse of Q has to be taken. Here, ij has to be considered as one index, and i’j‘ as the other one. Therefore, if there are 50 time parameters and 24 electrodes, Q is a 1200 X 1200 matrix, which is not very economic to invert. Assumptions about the covariance matrix will be made to simplify the analysis. Suppose that the noise can be adequately described by one statistical process, so that L = 1 in (36). In that case we have

signal

32

Q... 1JlY’

40

soattal frequency

Fig. 9. Spatial frequency spectra. The power of signal and noise is plotted against the spatial frequency n. The signal consists of the potential generated by a dipole which is half radially, half tangentially oriented. The noise is caused by randomly distributed dipoles, located at the same depth.

= S .J. ,J Xl .l . ,

where S is the temporal part, and X is the spatial part of the covariance. The structure represented by (46) is called a Kronecker product. It can be demonstrated that for Kronecker products we have Q‘.?.” ely’ =

domain. The results of this section demonstrate the “Laplacian” will not be helpful to improve the signal-to-noise ratio because the frequency bands of signal and noise are largely overlapping.

B. Maximum Likelihood Estimation Another possibility of using the noise characteristics for parameter estimation, is offered by the Gauss-Markov theorem [27]. In the following K~ is the recorded_EP-data at the ith electrode at thejth time sample. If Fu(P) are the corresponding model predictions and if no is the noise, then we have +

rij = Fij(P)+ rlij

(43)

where F are now the (deterministic) parameters to be estimated. If E{rzij} = 0, and if E{nurziy.} = Qiiiyr(is the covariance matrix provided by the theory), then the model parameters which minimize

are optimal in the sense that these parameters have minimum variance. This so-called Gauss-Markov theorem is only valid for linear model parameters, and therefore its application to dipole localization is limited. There exists, however, also another interpretation of the model parameters which minimize H. Suppose the probability density function of the noisef, has the following form

f,(n’) = g ( z T p ” n ’ ) , (45) where Q is some positive definite matrix, and g (A) some function, wgch is decreasing for X > 0. Then the model parameters P wQich minimize H give rise to the model errors Fij - s j ( P ) , which are most likely to occur. For this reason, P is called a maximum likelihood estimator. An example of a distribution, for which (45) is satisfied, is Gaussian noise. In that case g (A) = Ce-’, and Q

(46)

S!fl,”x;,”. JJ

(47)

So the inversion of the spatial temporal covariance matrix requires the inversion of a spatial and of a temporal covariance matrix, separately. Moreover, if the stochastic process is stationary, and if the time samples are equidistant, the matrix elements of S only depend on j - j ’ . Hence, S is Toepliz, and relatively easy to invert [28], [29]. With (47) the residual becomes H ( F ) = Tr { ( R -

k)‘Si””(k- R)XinVT). (48)

When it is assumed the background noise has a Gaussian distribution of which the covariance is known, it is rather straight forward to calculate the confidence intervals of the estimated parameters [5]. It was found by Knuutila and Hamalainen [4] that these intervals are rather sensitive for the covariance of the noise. Therefore, the application of (48) instead of ordinary least squares estimation might yield more reliable results in practice. C. Other Applications One of the problems often encountered in electroencephalography is where to place the reference electrode. It is often argued that the reference should be placed far away from the active sources, because otherwise the signals are disturbed. Others argue that, when EEG is used for source localization studies, the position of the reference does not matter at all, because in the model the reference is taken into account (at least it should be). The reference has only an effect on the display of the data, not on the estimated parameters. With the theory and the experiments described in this paper it is possible to consider the effect of the reference electrode from a signal-to-noise point of view. From the theoretical plots (Figs. 1 , 2) and the experimental results (Figs. 5 , 7) it follows that in the largest part of the frequency band the variance of the spontaneous EEG increases with the distance of the electrodes. Therefore, the most favorable position of the reference electrode is in the middle of the other electrodes,

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802

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

because then the background noise is minimal. However, when the a-band is dominating, the optimum position may be different. Another application of the covariance functions here derived is for interpolation techniques, to construct brain maps. Ilmoniemi and Hamiilainen [30] derived a formula which is based on the inner products of the lead fields:

(Ii

-

,

APPENDIX In this Appendix it is shown that if the potential is generated by a set of fixed sources which are statistically independent, the covariance and the mean can be written as the sums of the covariances of the individual sources. Since the sources are independent of time we have L

X G ,0

-

j j ) = [f(Zo)j(Zi, Zo) j ( Z j , Zo) dZo. (49)

Here, 7 (Zi , Zo) is the lead field of the magnetometer at Zi, for a dipole source at Zo and f(Z0)is a weight function. Although this interpolation technique works well in practice, it appears to ,be vefy time consuming to calculate the matrix elements ( J i * J j ) for all combinations of grid points. With the theory of the present paper the matrix elements of (49) can be interpreted as the covariance of two magnetic measurements (at positions Ziand Z j ) , on a system of random dipoles in a volume conductor. In other words, (49) is, apart from a constant factor, equal to (22). This immediately yields a very fast way to calculate the matrix elements. Instead of tedious 3-D numerical integration, we can use the fast converging series of (22).

VI. SUMMARY In this paper we considered the effect of random dipoles in a volume conductor, on the statistics of the electric and magnetic field. The attention is focused on two extremes: a global distribution and a localized random source. The localized model is adequate to describe the a-band, whereas the global distribution describes the other bands adequately. It has been shown that correlations or coherences are inappropriate to compare the theoretical predictions with EEG-measurements, because it depends on the reference electrode. It is better to study the variance between each pair of electrodes. Since the covariance matrix can be calculated when all these variances are known, no information is lost by this procedure. For MEG this difficulty does not exist. It is shown, on the basis of our model, that the “Laplacian derivation” does not improve the signal to noise ratio of EP-data, because signal and background noise have nearly the same spatial frequency spectra. The random dipole model can be used to obtain more realistic maximum likelihood estimators based on EP- and MEFdata by including the spatio-temporal covariance matrix of the background noise. Finally, the analytic expressions for the covariance can be used to improve the numerical performance of minimum norm interpolation.

I= 1

%(t)d&,

(A. 1)

El).

El

Furthermore, we assume that the source parameters are statistically independent of the time functions 8 ( t ) , which are considered as correlated stochastic process. The firstand second-order statistical descriptions of the sFchastic proczsses x(J, t ) , a F given by the functionsfp(P; t ) and f p p ( P ; t l , t2) where P T = , is the set of all source parameters. We have

(a:,a:, a:, - - -

f P ( E

- -

=f&,

t)

*

$3

L 9

SL;

t) 1IT =1

&[GI)

(A.2)

and L

=fSs(s,,,

f P P ( R tl, t 2 )

-

* *

9

l-I &,(FA.

t 2 ) 1= I

sL2;

(A.3) Here, fs is the joint probability density function of the source strength at time t, andfss is joint probability density of the source strengths at different times c1 and t2. For a detailed description of these functions the reader is referred to [21]. Although the sources are independent, the density functions and the character of each source is allowed to be different. They may be monopoles, dipoles, or dipole layers. With these assumptions the mean and the covariance of the total of sources can be expressed in terms of the means and covariances of the individual sources:

M(x)

=

1- - 1

dsl

*

* * *

L

L

dsL dFl

=

c

P

c M(S~(t))M(X‘”)

I=1

*

L

P

m+l

L

- - dp’tfs(s,, - -

L

P

ACKNOWLEDGMENT The authors wish to thank the (anonymous) referee for his helpful suggestions to improve the manuscript.

=

*

,

de MUNCK et al.: DIPOLE MODEL FOR BRAIN ACTIVITY

803

and similarly it is found that Cov

( x l , x2) =

[

J

[ dsll

*

J

*

dsL2d F l

*

*

L

t2)

k= 1

f,,(P’k)

k+m

=

cov

( x p , xirn’),

(A. 6)

and if 1 # m , we have

1

‘

’

j

dP’l

’ ’ ’ dP’L

I?f p a ( P ’ k ) $ l ( ’ I ,

k= 1

5l)$rn(’29

P’m)

= M(XI[’)M(Xp). 04-71 If the summation in ( A S ) is split into terms for which m and 1 are equal, and terms for which they are unequal, we find L

cov ( X I ,

x2)

=

c cov

1=1

[Sl(tl), S l ( t 2 ) I

cov (XP, X P )

L

+ m ,C l=l

filter versus averaging of evoked responses,” Biol. Cybernet., vol. 27, pp. 147-154, 1977. [4] J. Knuutila and M. S. Hamalainen, “Characterization of brain noise using a high sensitivity 7-channel magnetometer,” in Biomagnetism ’87. Tokio Univ. Press, 1988, pp. 186-189. [5] J. Sarvas, “Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,” Phys. Med. Biol., vol. 32, no. 1, pp. 11-22, 1987. [6] M. G. Saunders, “Amplitude probability density studies on alpha and alpha-like patterns,” Electroenceph. clin. Neurophysiol., vol. 15, pp. 716-767, 1963. [7] J. Persson, “Comments on estimations and tests of EEG amplitude distributions,” Electroenceph. clin. Neurophysiol., vol. 37, pp. 309313, 1974. [8] T. Gasser, “Goodness-of-fit tests for correlated data,” Biometrika, vol. 62, no. 3, pp. 563-570, 1975. [9] J. A. McEwen and G. B. Anderson, “Modeling and stationarity and Gaussianity of spontaneous electroencephalographic activity,” IEEE Trans. Biomed. Eng., vol. BME-22, pp. 361-369, May 1975. [lo] F. H. Lopes da Silva, “EEG Analysis: theory and practice,” in Electroencephalography: Basic Principles, Clinical Applications and Related Fields, 2nd. ed. E. Niedermeyer and F. H. Lopes da Silva, Eds., Urban & Schwarzenberg, Baltimore-Munich, 1987, pp. 871897. [ 1I] -, “Dynamics of EEGs as signals of neural populations: Models and theoretical considerations,” in Electroencephalography: Basic Principles, Clinical Applications and Related Fields, E. Niedemeyer and F. H. Lopes da Silva, Eds. Urban & Schwarzenberg, BaltimoreMunich, 1987, pp. 15-28. [12] B. N. Cuffinand D. Cohen, “Magnetic fields produced by models of biological current sources,”J. AppZ. Phys., vol. 48, no. 9, pp. 39713980, 1977. [ 131 R. D. Katznelson, “Deterministic and stochastic field theoretic models in the neurophysics of EEG,” Ph.D. desertation, Univ. California at San Diego, 1982. [I41 R. P. Gaumond, J. H. Lin, and D. B. Geselowitz, “Accuracy of dipole localization with a spherical homogeneous model, ” IEEE Trans. Biomed. Eng., vol. BME-30, pp. 29-34, 1983. [ 151 T. Varpula and T. Poutanen, “Magnetic field fluctuations arising from thermal motion of electric charge in conductors,” J. Appl. Phys., vol. 55, pp. 4015-4021, 1984. [16] I. Nenonen, T. Katila, and J. Montonen, “Thermal noise of a biomagnetic measurement dewar,” in Advances in Biomagnetisrn. New York: Plenum, 1990. [I71 J. C. de Munck, “The estimation of time varying dipoles on the basis of EP’s,” Electroenceph. clin. Neurophysiol., vol. 77, pp. 156-160, 1990. “The potential distribution in a layered anisotropic spheroidal [18] -, volume conductor,” J. Appl. Phys., vol. 64, no. 2, pp. 464-470, 1988. [19] W. P. Smythe, Static and Dynamic Electricity. New York: McGraw-Hill, 1950, p. 150. [20] J. C. de Munck, “A mathematical and physical interpretation of the electromagnetic field of the brain,” Ph.D dissertation, Univ. Amsterdam, 1989. [21] A. Papoulis, Probability, Random variables and Stochastic Processes. New York: McGraw-Hill, 1984. [22] J. P. Ary, S. A. Klein, and D. H. Fender, “Location of sources of evoked scalp potentials: correction for skull and scalp thickness,” IEEE Trans. Biomed. Eng., vol. BME-28, pp. 447-452, 1981. [23] R. M. Chapman, R. J. Ilmoniemi, S. Barbanera, and G. L. Romani, “Selective localization of alpha brain activity with neuromagnetic measurements,” Electroenceph. clin. Neurophysiol., vol. 58, pp. 569-572, 1984. [24] J. C. de Munck, P. C. M. Vijn, and H. Spekreijse, “A practical method to determine electrode positions on the head,” Electroenceph. clin. Neurophysiol., vol. 78, pp. 85-87, 1991. [25] T. H. Bullock and M. C. McClune, “Lateral coherence of the electrocorticogram: A new measure of brain synchrony,” Electroenceph. clin. Neurophysiol., vol. 73, pp. 479-498, 1989. [26] A. S. Gevins, S. L. Bressler, N. H. Morgan, B. A. Cutillo, R. M. White, D. S. Greer, and J. Illes, “Event-related covariances during a bimanual visuomotor task. I. Methods and analysis of stimulus- and response-locked data,” Electroenceph. clin. Neurophysiol., vol. 74, pp. 58-75, 1989. [27] Y. Bard, Nonlinear parameter estimation. New York: Academic, 1974, pp. 83-140.

~~(~~)IM(XY)W(X~))

cov [s/(tl>,

I#m

(A.8) In this way the mean and the covariance are expressed in the means and covariances of the individual sources.

REFERENCES [I] D. S. Ruchkin, “An analysis of average response computations based upon aperiodic stimuli,” IEEE Trans. Biomed. Eng., vol. BME-12, pp. 87-94, 1965. [2] V. Albrecht and T. Radil Weiss, “Some comments on the derivation of the Wiener filter for average evoked potentials,” Biol. Cybernet., vol. 24, pp. 43-46, 1976. [3] V. Albrecht, P. Linsky, M. Indra, and T. Radil Weiss, “Wiener

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[28] J. R. Jain, “An efficient algorithm for a large Toeplitz set of linear

equations,” IEEE Trans. Acoust. Speech Sign. Proc., vol. ASSP-27, pp. 612-615, 1979. [29] S. Zohar, “Fortran subroutines for the solution of Toeplitz sets of linear equations,” IEEE Trans. Acoust. Speech Sign. Proc., vol. ASSP-27, no. 6, pp. 656-658, 1979. [30] M. S. Hamalainen and R. J. Ilmoniemi, “Interpreting magnetic fields of the brain: minimum norm estimates,” IEEE Trans. Biomed. Eng., to be published.

Jan C. de Munck was born in 1962. He graduated with a degree in experimental physics from the University of Amsterdam, Amsterdam, The Netherlands, in 1985. He wrote a theoretical thesis on the mechanism, used by fish to detect sounds from different directions and distances. At the Netherlands Ophthalmic Research Institute (1985) he developed a method to estimate the positions and orientations of current sources in the brain, based on EEG measurements. His main research interest is on theoretical aspects of forward and inverse modeling. In 1989 he received the Ph.D. degree from the University of Amsterdam. Currently, he is working at University of Twente to study MEG.

Peter C. M. Vijn was born in The Netherlands in 1960. He studied electronics from 1978 to 1980. He received the Master’s degree in biology from the University of Amsterdam, Amsterdam, The Netherlands, in 1978. His research interest is in the analysis of biological signals and the modeling of bioelectric phenomena. He is currently a Ph.D. student in electrophysiology at The Netherlands Ophthalmic Research Institute in Amsterdam. He studies synchrony of single units within the visual cortex, both during rest and during processing of different stimuli. Fernando H.Lopes d a Silva received the medical degree from the University of Lisbon, Lisbon, Portugal. He followed with a post-graduate course on Engineering and Physics at the Imperial College of the University of London and worked at the Department of Physiology and Pharmacology of the National Institute of Medical Research. Thereafter, he joined the scientific staff of the Institute of Medical Physics at Utrecht, The Neth- . erlands. He received the Ph.D. degree from the University of Utrecht for his research on a systems Analysis of Visual Evoked Potentials. He became Full professor in General Animal Physiology at the University of Amsterdam in 1980. From 1981 to 1985, he was professor in Neurophysiology at Twente University within the Biomedical Engineering Program. His principal research activity consists of fundamental scientific research concerning the biophysical basis of electrical activities of the central nervous system and the biophysical basis of the electrical activities of the central nervous system and the physiology of limbic structures of the brain, in particular the mechanisms of epileptogenesis.