On the Weak and Numerical Solutions of the Forward Problem in EEG
DIANA RUBIO1 and MARÍA INÉS TROPAREVSKY2
1
Instituto de Ciencias, Universidad Nacional de General Sarmiento
José M. Gutiérrez entre José L. Suárez y Verdi 1613 - Los Polvorines - Prov. de Bs. As.
1
Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín
Calle 91 Nº 3391, V. Ballester – Prov. de Bs. As.
2
Departamento de Matemática, Facultad de Ingeniería
Universidad de Buenos Aires
ARGENTINA
Abstract: - The Forward Problem in EEG consists on finding the scalp potential due to current sources within the
brain. Electrical activity in the brain is governed by the laws of electromagnetics. Thus, to solve the problem,
Maxwell Equations must be solved in a volume representing the head. In this work we consider a simplified
model of the human head and a static approximation of the dynamical model. We use BEM techniques to solve
the resulting equations. We illustrate some properties of the solution by typical numerical examples.
Key-Words: - Bioelectric Potentials, Boundary Element Method, Forward Model, Spherical Head Model
1 Introduction
The potential distribution on the scalp produced by
current sources within the brain can be measured by
an Electroencephalograph (EEG). For several
decades, neurologists have been interested in solving
the problem of determining the location of these
current sources from the measured potential on the
scalp. In the case of epilepsy there are a few dipoles
that give major contribution in the generation of the
electric field and we are interested in finding their
location based on the superficially measured potential
values. In order to solve this problem, called the
INVERSE PROBLEM in EEG, we must be able to
calculate the superficial potential for any possible
dipole configuration which is called the FORWARD
PROBLEM (FP) in EEG. In this work we present
some results concerning Numerical Solutions of the
Forward Problem in EEG.
Electrical activity in the brain is governed by the
laws of electromagnetics. Thus, to solve the Forward
Problem, Maxwell's equations must be solved in a
volume representing the head. In order to solve
numerically the problem of finding the location of the
source of an epileptic seizure we need to propose a
mathematical model, i.e., we need to choose the
equations that will represent the process and the
domain where these equations will be solved. At this
stage of our work we simplified the model in two
ways. The first one is about the differential system of
equations arising from Maxwell's equations: we
consider the static approximation to the equations.
The static approximation of the dynamical model is
standard in neurology and it is justified by the high
speed of propagation of waves in the head (see [6],
[9]). This approximation uncouples the magnetic and
electric equations, in consequence the model for the
electrical activity consists of the Poisson equation
with boundary conditions (see [12]). The second
simplification of the model is about the domain
where we solve the equations: the human head is a
complicated anisotropic media with tissues of
different conductivity values. We simplify the
geometry of the head representing it as three volumes
with different conductivity values. Each volume is
surrounded by the next one. They represent the brain,
the skull and the scalp. In this case where the domain
is described by concentric spheres it is possible to
solve the differential boundary value problem
exactly by a series of functions (see [3], [4], [11]
among others). For general domains it is not possible
to obtain the solution in this way. Therefore it is
necessary to use numerical methods. As we are
interested in finding a solution on a realistic head
model, at a first step we propose an approximated
solution although we are working with spherical
models. We intend to adapt this scheme to the
realistic head model. To solve numerically the
differential problem we use the boundary element
technique which allows us to reduce a problem in a
volume to a problem over surfaces. In this way the
numerical burden is alleviated. Other approaches with
FEM techniques [15], the reciprocity theorem [13]
and the cubic Hermite boundary element procedure
[1] were also developed. The paper is organized as
follows: in Section 2 we present the differential
system and establish some properties of its weak
solution. In Section 3 we present the integral
formulation of the problem. Section 4 introduces the
Discrete System. Numerical examples are shown in
Section 5. Finally we present the conclusions in
Section 6.
2 The Differential Equations
2.1 Maxwell Equations
The electrical activity of the brain consists of currents
generated by biochemical sources at the cellular
level. The electric and magnetic fields that they
produce can be measured and predicted because they
obey physical laws: Maxwell's Equations [7] and the
equation
of
conservation
of
the
charge .J   / t , where E is the electric field,
B the magnetic field,  the density of charge and J is
the total current density. The density J can be
decomposed in the current density due to the
macroscopic electric field, and the current density
caused by synaptic activity as follows
J  E  J i
(1)
where  is the conductivity of the media. If we accept
the properties of the tissues involved (conductivity ,
magnetic permeability values and dielectric values to
be constant, see [9]), the velocity of propagation of
the electromagnetic waves caused by potential
changes within the brain is about 10 5 m / s . This
means that the effect of the potential changes may be
detected simultaneously at any point in the brain or in
the surrounding tissues. In other words, the currents
caused by sources in the brain behave in a stationary
way (see [6],[9]). The use of the static approximation
of Maxwell's Equations to describe the process is
then justified. We can simplify (1) to obtain:
.( E )  4
xE  0
4
J.
c
These equations together with
the law of
conservation of charge, that in this case can be
written as
(2)
.J  0
give rise to
(3)
E  u.
Substituting (3) in (1), and taking the divergence of
both sides we arrive to the Second Order Partial
Differential Equation that describes the relationship
between the measured potential u and the current
density J i (the Poisson equation):
.B  0
x( B ) 
.(  ( x ).u( x ))  .J i
(4)
with boundary condition
u( x )
 0 , x  G ,
n
(5)
where G denotes the volume representing the head
and n represents the outward normal. This condition
reflects the fact that the normal derivative to the head
at the boundary must be zero since air is an insulating
material that does not support current flow. As
mentioned in the Introduction, we assume that G can
be described as three concentric spheres where the
radii and conductivity values are given (see [8]). We
denote them from the inner one to the outer: G1 the
brain, G2 the skull and G3 the scalp. The surfaces
between them are denoted by: S1 , S 2 and S 3 (the
scalp) respectively (Fig.1).
Fig.1. The Three Sphere Model
In neuromagetics a current dipole is widely used
to approximate a primary current. In the case of
epilepsy a dipole can be thought as a concentration
of J i to a single point rq :
J i1  M q  ( r  rq ),
(6)
where  (.) is the Dirac delta distribution and M q is
the dipole strength. In this work we considered that
there is an unique dipole responsible for the seizure
process. A current dipole is completely described by
six parameters: three to determine its location and
three to define its strength. There are physical
considerations that must be taken into account. They
can be described by the equations:
u 
Si
0
 u 
 n 
Si
0
where 
. represents the difference between the values
of the functions inside the brackets through the
indicated surface. These equations represent the
continuity of the potential and of its normal
derivative across the different regions.
The scalp potential u(x) is measured as a
difference between the potential value at each point
x  S 3 and its value at a reference point x 0  S 3 ,
thus u( x0 )  0 .
Therefore, the differential equation to be solved is
.(  ( x ).u( x ))  .J i
with boundary condition
u
( x )  0,
n
x  G
and continuity condition for the normal derivatives at
the transition surfaces
 u 
 n 
and it must verify u( x0 )  0 .
u 
Si
0
Si
 0,
2.2 Existence and Uniqueness of Solutions
Equation (4) relates the impressed current J i with
the electric potential u. This equation is a second
order elliptic one and it can be written in the
divergence form as
Lu  .(  ( x ).u( x ))  .J i .
The function (x) that contains the conductivity of
the different tissues at each point x, may be
considered discontinuous and piecewise constant
 1 ,

 ( x )   2 ,
 ,
 3
x  G1
x  G2 .
x  G3
In order to assure the existence of a solutions of
(4) with boundary condition (5) we need to work with
a generalized solution [5]. It can be proved that there
exist generalized solutions u of (4) if Ji verifies
 .J
i
values chosen for solving the equations are not
accurate [17].
We state that the weak solutions of the equations
are continuous with respect to . That is, if the values
of two different  are close, the corresponding scalp
potential solutions will be close too. Suppose that u is
the generalized solution obtained for a given , and

that u is the generalized solution for  . Considering
the weak form of the equation (4) it can be proved


that u  u 2  C     where C is a constant that
depends on G, the electric field and the
conductivities.
We will illustrate this relationship with several
examples in Section 5.
In order to calculate a solution of (4), in the next
Subsection we present Integral Equations for u.
3 Integral Equations for the Potential
Integral equations can be derived from the differential
boundary value problem using Green's integral
theorems (see [6], [14]). The following formulas are
obtained
 ( r )u( r )   1v( r ) 
3  
r  r'
j
j 1

u( r' )
dS'
3

4
j 1
r

r
'
S
j
for r  S j , where
v( r )  
 0 and that these solutions differs only on a
G
constant. In the case we are considering, the above
condition can be written as
J
i
 0 and it is
G
automatically fulfilled because, under our hypothesis,
J i has finite support inside G1 (dipole).
The uniqueness of solution is also justified
because at the reference point, x 0 , on the scalp the
potential u( x0 )  0 . Finally if u is a solution of the
problem, it is also a generalized solution.
2.2.1 Continuity of the Solution with respect to
the Conductivities
When solving the equation (4) we need to know the
conductivity of the brain, skull and scalp. There exist
typical values for these constant [9]. However, it may
differ from the real ones, thus it is important to
estimate the error that appears if the conductivity
1
.J i
 r  r'
41 G
dr' .
(7)
Let r  r j  S j to obtain another equation for
points on the surfaces of conductivity transition S j :
 k   k 1
2
u( r )   1v( r ) 
 j   j 1
r  r'
u( r' )n( r' )
dS'
3

4
j 1
r

r
'
S
3

(8)
j
for r  S k .
The above formulas provide the solution u at
points on the surfaces of conductivity transition Sj. In
the next Section we present a technique for the
discretization of the integral equations (8) and
describe the way we solved them.
4 The Discrete Problem
In this section we propose a discretized version
of the integral equations for the potential.
4.1 The BEM Technique
The idea behind the Boundary Element Method
(BEM) is to transform a differential equation given
on a domain into an integral equation on the
boundary of that domain. Thus, the dimension of the
problem is reduced yielding to a reduced order
numerical system (see [2]).
In order to apply the BEM technique to our
problem we must discretize the surface integrals (8),
and approximate the volume integral over G for v(r),
given by equation (7).
at the nodal points, that is, at the vertices of the
elements E j ,k of the grid. More precisely, if r is a
nodal
point,
U(r,r')  u(r')n(r')
r  r'
H ( r , r' )  n( r' )
r  r'
3
4.2.1 The Grid
If we divide the surfaces into elements Ej,k such that
verifies S j   E j ,k then the grid coincides exactly
k
with each surface Sj j=1,2,3, and there is no error due
to the surface discretization. We divide the surface
into a set of spherical elements and choose the nodal
points to be the vertices of these elements. We
construct spherical rectangles as follows. We
consider the spherical coordinates >0,   0,2 )
and   0,  . On each sphere is fixed then the
parameters are and . We divide [0,2) into n
intervals and [0,] into n=n/2 intervals. This choice
give rise to a grid of N=n(n nodal points and
M=nn spherical elements. These elements are
rectangles except for   0, h and    - h , ,

v H1



where they become triangles as shown
in Fig. 2.
and
, we approximate each

u j ,k ,i
E j ,k i 1
The discretization error depends on the grid and on
the function and integral approximations.
3
Sj
Nk
The Discretization of the Equations
r  r'
surface integral I S j ( r )  U ( r , r' )dS' by
I S j ( r )  
4.2
r  r'
Nk
 H ( r ,r' )dS' ,
(9)
E j ,k
where uj,k,i are the value of the potential at vertex i of
the element Ej,k and Nk is the number of vertices of
the element Ej,k.
As we are working on spheres, we can
calculate the actual normal at each nodal point
exactly. For simplicity we take the same amount of
points, say N, on each surface that give rise to M
elements per surface (see Section 4.2.1 for a
description of the chosen grid).
There exist other ways to discretize the potential.
In [16] there are several comparisons of the results
obtain for different approximations.
4.2.3 The Integral Approximation
Finally, to calculate the integrals involved in (7) and
(9) we need to select a numerical rule.
The integral on (7) is easy to solve since we
model the primary current Ji as a dipole (see equation
(6)). The integral on (9), that is,
r - r'
H(r,r' )dr' 
dS ,
r - r'


E j,k
E j,k
is the solid angle subtended by the element Ej,k at the
point r of the grid. We apply the Simpson’s rule for
r  E j ,k and geometric properties of the solid angle
in the improper case (see [10]), that is when r  E j,k .
4.3 The Linear System
Fig.2. The Spherical Grid
4.2.2 The Potential on the Grid
The next step is to decide the way in which the
function u will be approximated over the elements of
the grid. We consider u to be an average of its values
The implementation of all these approximation
schemes lead to a Linear System of Equations
(D -A) u=C
(10)
where D is the block diagonal matrix arising from the
right hand side of (8), that is, D=diag(D1,D2,D3)
where Di 
 i   i 1
2
for i=1,2,3. A is the matrix
resulting from the discretization of the surface
integrals ISj (9), C is the discretization of v(r) and u is
the vector containing the values of the potential at the
nodes uj,k,i.
Once the grid is chosen, the matrices of the
system, D and A, depends only on the conductivities
of the brain, skull and scalp.
Note that all the values of uj,k,i are involved in the
calculation of the whole vector u. After solving (10)
we can reconstruct the scalp potential interpolating
the values obtained at the nodal points of the external
surface (representing the scalp).
5 Numerical Results
We solve the Forward Problem for a volume
conductor G composed by three concentric spheres of
radii .071, .078 and .085 mm. The conductivity
values
commonly
used
are
(see
[9])
  (  1 , 2 , 3 )  ( 0.33,0.0042,0.33 ) Am/m for
the brain, skull and scalp respectively. We considered
dipoles oriented in different ways. In Fig.3 the scalp
potentials due to a transversal and radial dipoles are
shown.
Fig.4. Surface Potentials for Different Skull
Conductivities.
To quantify the differences between the
potential distribution for different  i i  1,2,3 we
compare the 2-norm of the vector u containing the
values of the potential over the three surfaces with the
2-norm of the vector  . We present some results in
the following table
i  j
ui  u j
4.50000e-003
2.20000e-003
2.30000e-003
4.03150e-003
2.75879e-003
1.27304e-003
2.23663e-004
9.95365e-008
5.86464e-008
4.16362e-008
4.58856e-009
3.09301e-009
1.50926e-009
1.46143e-010

Fig.3. Scalp Potential for a Tangential Dipole (on left)
and Radial Dipole (on right).
5.1 Numerical
Results
Conductivity Values
for
2,3
1, 3
1, 2
4, 5
5, 6
4, 6
4, 1
1
2
Brain Cond.
0.330000
0.330000
Skull Cond.
0.004200
0.006500
Scalp Cond.
0.330000
0.330000
3
0.330000
0.002000
0.330000
4
0.329900
0.004205
0.330200
5
0.327000
0.004150
0.333000
6
0.329000
0.004180
0.331100
Different
As mentioned in Section 2 the solutions of the FP are
continuous with respect to the values of  . To
illustrate this relationship we numerically solve the
FP and found the potential distribution for different
values of  i i  1,2,3 around the references values
mentioned above. We observe that for different
values of  i the patterns of the potential distribution
are very similar, they only differ in their magnitude,
not in their position. This fact can be observed in
Fig.4 where the scalp potential for a dipole located at
rq  (0,0.0062,0.04)
and
with
M q  (1.2 10 10 ,0.6 10 10 ,0.6 10 10 ) , for different
conductivities is shown. In the first column
  ( 0.33,0.0065,0.33 ) , in the second one
  ( 0.33,0.0042,0.33 ) and in the third one
  ( 0.33,0.0020,0.33 ) .
It can be observed that as the order of  i  
j
decreases, the order of u i  u j decreases too.
6 Conclusions
In this work we established the continuity of the weak
solution to the FP of EEG with respect to the vector
parameter  (conductivity) and shown how this fact
is reflected in a numerical solution. From the
examples presented in Section 5, it can be seen that
the changes in the conductivity produces variations
only in the magnitude of the potential as we pointed
out.
We proposed a spherical grid that exactly matches
the surfaces of transition of the head model which
reduces the discretization error.
With respect to the potential, we discretized it at
each grid element and used geometric properties to
calculate the resulting integrals.
These results were illustrated by some typical
numerical examples.
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