ON THE WEAK SOLUTIONS OF THE FORWARD PROBLEM IN EEG

The process underlying the generation of the EEG signals can be described as a set of current sources within the brain. The potential distribution produced by these sources can be measured on the scalp and inside the brain by means of an EEG recorder. There is a well-known mathematical model that relates the electric potential in the head with the intracerebral sources. In this paper, we study and prove some properties of the solutions of the model for known sources. In particular, we study the error in the potential, introduced by considering an approximated shape of the head.


Introduction
The electric process underlying the generation of the EEG can be described as a set of current sources within the brain. In the case of epilepsy, there are epileptogenic zones that give major contribution in the generation of the electric field and, for several decades, neurologists have been interested in solving the problem of determining the location and orientation of these current sources from the measured potential on the scalp. This problem is known as the inverse problem in EEG. A first step towards its solution is to solve the forward problem (FP) in EEG that consists in calculating the superficial potential for any possible configuration of the sources. A typical mathematical model that describes this process is a differential boundary value problem of second order, based on the static approximation of the Maxwell equations (see [5]). In order to calculate the solution, a simplified head model is adopted.

The mathematical model
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 estimated by means of Maxwell's equations (see [5,6]). Based on the properties of the tissues involved (see [6]), the velocity of propagation of the electromagnetic waves caused by potential changes within the brain is such that the effect of the potential changes may be detected simultaneously at any point in the brain or in the surrounding tissues. In consequence, the use of a static approximation of Maxwell's equations is justified. This approximation uncouples the equations for the magnetic and electric fields. Consequently, the second-order partial differential equation relates the measured electric potential u and the impressed current J i , usually modeled as a dipole (associated with the microscopic currents). The function σ(x) contains the value of the conductivity of the different tissues.

M. I. Troparevsky and D. Rubio 649
Air is an insulating material that does not support current flow, therefore, the normal derivative to the head at the boundary must be zero: where G is the volume representing the head, ∂G is its external surface, and ν represents the outward normal. We assume that G can be described as three homogeneous sets, each one surrounded by the next one, where the radii and conductivity values are given. We denote them from the inner one to the outer one: G 1 the brain, G 2 the skull, and G 3 the scalp. The surface between them are denoted by S 1 , S 2 , and S 3 , respectively. Note that S 3 = ∂G.
The function σ(x) that contains the conductivity of the different tissues at each point is positive, usually assumed to be discontinuous and piecewise constant There are physical considerations that must be taken into account: (i) the potential is continuous across the different regions; (ii) the normal derivative of the potential is continuous across the different regions; (iii) 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 If we denote by [·] the difference between the values of the functions inside the brackets through the indicated surface, they can be written as respectively.
Therefore, the resulting boundary value problem is

Existence and uniqueness of solutions for the FP
In order to assure the existence of solutions of (2.7) with boundary condition (2.8), we need to introduce some definitions and notations that will lead us to the definition of weak solution of (2.7) with boundary condition (2.8).
We denote by the inner product in G. Let · L n (G) be the norm in L n (G): and · L ∞ (G) the norm in L ∞ (G): We We denote by µ(G) the Lebesgue measure of the set G. Suppose that u is a solution of (2.1) and multiply this equation by a function v. Assuming that both u and v are regular enough to apply the integral theorems to the resulting equation, the solution u must verify the following identity: (3.4) or, equivalently,

M. I. Troparevsky and D. Rubio 651
A weak solution of (2.7) with boundary condition (2.8) is a function u that verifies (3.5) for all functions v with weak derivative of first order, that is, v ∈ H 1 , where In this case, we say that w is the first-order weak derivative of v.
From now on, we work with weak solutions since any classical solution to the problem is also a weak solution. Proof. It can be proved (see [3,7]) that if σ(x) is positive and piecewise C 1 , there exist weak solutions, identical up to a constant, of the secondorder equation (2.7) subject to G ∇ · J i = 0, (3.8) or, equivalently,

9)
that is automatically fulfilled because J i has finite support inside G 1 (dipole). The uniqueness of solution of FP is justified since the potential verifies u(x 0 ) = 0 at the reference point x 0 on the scalp.

Solutions on different domains
In this section, we consider that the domain where we solve (2.7) is composed by only one set. The conductivity function σ(x) need not be constant in the domain, actually σ(x) ∈ C 1 and is positive if required. Let G and H be two sets representing the head (see Figure 4.1). We consider that (i) u G is a weak solution of (2.1) on G: (ii) u H is a weak solution of (2.1) on H: where σ G (x) and σ H (x) are the conductivity functions in G and H, and supp J i ⊂ G ∩ H. We prove that if the difference between the sets G and H is small, so it is the L 2 -norm of the difference between the solutions u G and u H . To do so, we calculate a bound for the L 2 -norm of the difference of the solutions for the two different domains G and H. We consider that the conductivity functions σ G and σ H are positive, coincide on G ∩ H, and verify We denote by G H the symmetric difference between the domains G and H, that is, G H = (G − H) ∪ (H − G). We consider that G, H are bounded subsets of R 3 , ∂G ∈ C 1 , ∂H ∈ C 1 . We assume that ∇u G and ∇u H are bounded in G and H, respectively. In the case of the solutions of the FP in EEG, this assumption is reasonable since u G and u H represent the electric potential on the head, and consequently, ∇u G and ∇u H are the electric fields.
In order to establish a bound for u G − u H L 2 (G∩H) , we need some lemmas. Proof. Let G ∪ H ⊂ V , u G , and u H be the extensions of u G and u H , respectively, to R 3 that verify (see [4]) If we define u = u G − u H , it has compact support and u| R 3 −V = 0. From the Poincaré inequality (see [4]), we have In addition, u L 2 (G∪H) ≤ u L 2 (V ) . Combining these inequalities, the first statement of the lemma follows.
Since we assume that ∇u G L ∞ (G) and ∇u H L ∞ (H) are finite, then ∇u G , ∇u H ∈ L ∞ (V ), and ∇u ∈ L ∞ (V ), hence the lemma follows. for some constant K.
Since supp(∇ · J i ) ⊂ (G ∪ H) and u G , u H are weak solutions of (2.7) in G and H, respectively, from (3.5) we obtain Finally, combining equations (4.7) and (4.11), we obtain Now, for the chosen V , the thesis follows.
The following result is a consequence of Lemmas 4.1 and 4.2. Now we apply the result of Lemma 4.3 to the case of the EEG signals.
The following proposition shows that the difference of two solutions of the FP calculated on different domains approaches zero when the measure of the symmetric difference of their domains tends to zero.
Proposition 4.4. Let G and H be two different domains in R 3 that models the head. Let u G and u H be the solutions of the FP in EEG described by (4.1) and (4.2), respectively, and u G and u H the extensions of these solutions to R 3 presented in Lemma 4.1. Then (4.14) Proof.
Inequality (4.15) means that the error produced by considering weak solutions of (2.7) in two different domains, with conductivity function verifying (4.3), is proportional to the Lebesgue measure of the symmetric difference of those domains.
Remark 4.5. The result of Proposition 4.4 can be extended to the case where G is a multicompartment set and σ(x) is a C 1 piecewise positive function that verifies (2.5).

Conclusions
In this paper, we state and prove some theoretical properties of the weak solution of the equations that model the FP in EEG.
Proposition 4.4 states that if (2.7) is solved in two different domains G and H, the L 2 -norm of the difference between the solutions tends to zero when µ(G H) tends to zero. This result is concerned with a real situation: the dimensions and shape of the head are approximated. It is important from a theoretical and qualitative point of view since it gives us some confidence on the solutions obtained in the case of approximated shape and dimensions of the head.