Magnetoencephalography associates a neuronal current within the functional brain with the magnetic flux den- sity and it creates outside the head. In particular, the direct problem consists of the calculation of the exterior magnetic field when the primary neuronal current is given, and the inverse problem seeks to identify the current that generates a given exterior magnetic field. The main difficulty with the inverse problem of Magnetoencephalography is due to the fact that, besides the excitation of the neuronal current within the conductive brain tis- sue, a secondary induction current is generated which, in a sense, makes the primary neuronal current less “visible” by the exterior magnetic field. The induction current is supported on the conductive brain tissue and therefore it depends on the geometry of the brain-head system. Even for relatively simple geometrical models such as a triaxial ellipsoid the calculations are very complicated [1] . An excellent review of the electromagnetic activity of the human brain can be found in [2] , while the standard book for an introduction of the brain imaging modalities of Electroencephalography and Magnetoencephalography is the book by Malmivuo and Plonsey [3] .

The most important question for the inverse problem is the question of uniqueness, that is, whether it is possi- ble to recover the complete information about the current from the exterior magnetic data. The answer to this question is negative. The fact that a current within a conductor cannot be identified, from measurements of the exterior magnetic field it generates, was known to Helmholtz 160 years ago [4] . However, the problem of deter- mining exactly what part of the current can be recovered was a topic of intense investigation during the last de- cade, and the final answer was given in [5] . A complete discussion of all the existing results on the topic can be found in [6] . Today we know that, no matter what the geometry of the brain-head system is, complete knowledge of the electric potential on the surface of the head can give us no more than one third of the neuronal current, and complete knowledge of the exterior magnetic field can give us no more than two thirds of the neuronal current, one of which is the one obtained from the electroencephalographic measurements. Hence, even in the very difficult situation where we have complete and synchronous data from both electroencephalographic and Magnetoencephalographic data, still one third of the current cannot be identified.

The next question one can ask, in connection to the inverse problem of Magnetoencephalography, has to do with the possibility to identify the position and the extent of a localized current. Albanese and Monk [7] , have shown that this is not possible if the support of the current is a three-dimensional set. However, this is possible if the support of the current is a two-, one-, or zero-dimensional set. That is we can identify a localized current if it is supported on a set of isolated points, on a segment of a curve, or on a surface patch.

The purpose of this work is to provide a concrete example of the Albanese-Monk result. In particular, we con- sider a neuronal current that is supported on a small circular disk, orthogonal to the radius that passes through its center, and we construct an algebraic algorithm that solves the inverse Magnetoencephalographic problem. In fact, the identification of the location and the size of the supporting disk is reduced to the solution of a nonlinear algebraic system which can be solved numerically.

The paper is organized as follows. The solution to the problem of calculating the exterior magnetic potential for a single dipolar current within a conductive sphere is briefly described in Section 2. This solution plays the role of the fundamental solution for the problem of Magnetoencephalography in spherical geometry. Section 3 contains the calculations that lead to the magnetic potential in the case where the source current is distributed on a small disk. Finally, Section 4 develops the algorithm for the inversion of the measured data in the case des- cribed in the previous section. An Appendix at the end of the paper provides some explanations about the long and tedious calculations that are used in Section 3.

Quasi-Static theory of Electromagnetism Magnetoencephalography [8] -[10] assumes that the time derivatives of the electric and magnetic fields are very small and therefore the corresponding terms in Maxwell’s equations can be neglected. Then, it can be shown that the magnetic field, generated by a dipolar current at the point hav- ing moment, is given by the Geselowitz formula [11]

where is the electric potential on the boundary of the conducting medium representing the brain- head system. In Formula (1), denotes the exterior domain, is the constant conductivity of the brain tissue, is the common magnetic permeability both inside and outside and stands for the outward unit normal on the boundary.

If is a sphere of radius, then we know from the solution of the corresponding Electroencephalography problem [12] , that the electric potential on the boundary of the sphere is given by

where stands for the normalized complex spherical harmonics

and denotes the Legendre functions of the first kind.

Inserting expression (2) in the Formula (1) and performing the indicated integration we obtain the magnetic field outside the sphere. However, since the magnetic field in the exterior to the sphere is both solenoidal and irrotational it follows that there exists a scalar magnetic potential, which is also harmonic, such that [10]

Then, utilizing the orthogonality properties of the spherical harmonics we can calculate the magnetic potential and arrive at the following expression [12] [13]

The above expression provides the magnetic potential in the exterior of the sphere due to a single current dipole {, }. Therefore, it can be considered as the fundamental solution of the MEG problem for the sphe- rical geometry [14] . Consequently, any discrete, or continuous, current distribution can be obtained through sum- mation, or integration, respectively, of the above fundamental solution [15] . Then, Formula (3) gives the exterior magnetic field.

Suppose now that the neuronal current is supported on a small part of a smooth surface which is located around a central point. Let this little surface be represented by the equation

where is some interior point of this surface. Then the current is described by the function

which, if it is assumed to be small, it can be represented by the linear part of its Taylor expansion

In particular, if the surface that curies the current is a small disk of radius, which is centered at and has the vector perpendicular to its plane, then, this disk is defined by

Utilizing the freedom we have to select a coordinate system we choose the direction to coincide with the -axis. Then a parametric representation of the circle is given by

and. In this case, the third Cartesian coordinate of is always equal to, and the linear Taylor approximation assumes the form

where, denote the constant vectors

and denotes the average moment over the small circle. In order to interpret Formula (11) in an invariant form, we introduce a third vector, which is also constant, as well as the dyadic [16]

Then, the absence of a term involving in Formula (11) can be interpreted as the constrain

which means that, the left contraction of the dyadic with the direction normal to the disk vanishes. Conse- quently, in the general case of an arbitrary oriented system, we should have the condition

which gives the three scalar conditions

Consequently, the approximate current (11) is now written as

We can utilize the coordinate invariance principle to simplify our calculations. We actually want to calculate the total magnetic potential, given in (5), which is generated by the approximate current (28). Since our ultimate goal is to invert the MEG data that will give us the quantities, and we will calculate as many terms of the expansion (5) as we actually need.

For the generic excitation dipole, Formula (5) is written as

and through some direct calculation with the Legend polynomials [17] we obtain the following relations, which are written in dyadic form [16] in order to isolate the factors that are going to be integrated

The symbol denotes the identity dyadic, “:” defines the double contraction [16]

and similarly the triple contraction [16] is defined as

Furthermore, the exterior magnetic potential given in (29) can be written in its Cartesian form [13] , [15] as follows

where the coefficients

are homogeneous harmonic functions [15] .

In order to obtain the magnetic potential, generated by the neuronal current which is distributed over the disk of radius, we need to substitute the expressions (19) in (27), (28) and (29) and integrate the resulting equations over the disk. We perform the calculations in the special case where and express the results in terms of. Then, in the resulting expressions we replace by the arbitrary vector to obtain the corresponding generic results. The final expressions are given in the sequel

where

is the vector invariant [16] of the dyadic.

The actual calculations that led to the above expressions, and especially to the Formula (32), are involved and quiet long. In order to facilitate the reader we provide, in the Appendix, an outline of the basic steps that lead the above expressions.

If we insert the above expressions of the harmonic functions in the expansion (26) we arrive at the Cartesian representation of the exterior magnetic potential up to the terms of order. That solves the forward MEG problem for a neuronal excitation that is supported on a small disk.

In the previous section we defined the functions, and which are homogeneous harmonic polynomials of degree 1, 2 and 3, respectively. These functions assume the forms

with

and

with

In the present work we assume the idealized case where the exterior magnetic potential is known. This means that expansion (26) is known and therefore the coefficients, and are also known. Conse- quently, if we rewrite the polynomials and in terms of the Cartesian monomials that appear in (34), (35) and (37), then we can utilize their linear independence to equate each monomial with the corres- ponding known coefficient or. In doing so for the first degree polynomial, we obtain from (30) and (34)

or

where is the vector invariant of the dyadic, defined by Equation (33). Equating the coefficients of the harmonic polynomials of the second degree, given in the Equations (31) and (35), we arrive at the relations

where the constrain (36) is easily verified.

Next we consider the cubic polynomials (32) and (37). The coefficients of the cubic monomials provide the equations

The coefficients of the monomials involving the cross-product terms, , provide the following equations

where it is straightforward to verify the constraints (38)-(40). Finally, from the equality of the coefficients of the product terms in Equations (32) and (37) we obtain

The solution of the above nonlinear system of algebraic equations determines the position, the orientation and the size of the disk that supports the primary neuronal current. A series of numerical tests to obtain the solution of this system show that there exists a unique real solution.

The present work is part of the project “Functional Brain”, which is implemented within the “ARISTEIA” Ac- tion of the “OPERATIONAL PROGRAMME EDUCATION AND LIFELONG LEARNING” and is cofunded by the European Social Fund (ESF) and National Resources.