Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

Abstract

The problem of determining a continuously distributed neuronal current inside the brain under the assumption of a three-shell model is analysed. It is shown that for an arbitrary geometry, electroencephalography (EEG) provides information about one of the three functions specifying the three components of the current, whereas magnetoencephalography (MEG) provides information about a combination of this function and of one of the remaining two functions. Hence, the simultaneous use of EEG and MEG yields information about two of the three functions needed for the reconstruction of the current. In particular, for spherical and ellipsoidal geometries, it is possible to determine the angular parts of these two functions as well as to obtain an explicit constraint satisfied by their radial parts. The complete determination of the radial parts, as well as the determination of the third function, requires some additional a priori assumption about the current. One such assumption involving harmonicity is briefly discussed.

Mathematical Foundation of Electroencephalography

“…One of the authors has showed [39] that the conductivity profile of the layered structure enters the potential formula by normalizing each term by a constant which incorporates the conductivity jumps across the interfaces and the geometrical characteristics of the layers. Analogous results show a similar effect on the inhomogeneous conductor [31]. Hence, the formula of the electric potential that will serve as the stepping stone for the inverse calculations is the one that corresponds to the most realistic inhomogeneous models that acknowledge the layered conductivity profile of the head-brain system.…”

“…An independent view to the particular problem has been provided by Fokas [31]. After formulating the surface potential with explicit Q dependence (26), computing the surface potential for a continuously distributed current is straightforward.…”

“…In principle, the procedure described for the sphere is generally applicable since it is geometry independent. Note again that every detail regarding the moment Q and position r 0 of the dipole is encoded into the coefficients of expansion (31). which incorporates characteristics, such as the geometrical and physical properties of the conductor, as well as the EEG measurements, the algebraic manipulations leading to the solution are somewhat a little more painless.…”

EEG identification of a localized 1-D neuronal excitation

“…As far as the question of uniqueness of the solutions of these two inverse problem is concerned, that is the characterization of the class of currents that provide identical eclectic potentials on the head, and identical magnetic fluxes outside the head, the ultimate results have been obtain recently [3], [5]. The definitive result states that neither the EEG nor the MEG measurements can recover completely the primary neuronal current, and therefore no uniqueness for the inverse problems exists.…”

On Two Useful Identities in the Theory of Ellipsoidal Harmonics

“…In some of these cases, it is possible, using Gauss theorem, to map the interior of the ellipsoid to its boundary and then it is enough to compute the left-hand side of Equation (11). An example where such a computation is needed, appears in electroencephalography and magnetoencephalography [2][3][4].…”