The complete ellipsoidal shell-model in EEG imaging

This work provides the solution of the direct Electroencephalography (EEG) problem for the complete ellipsoidal shell-model of the human head. The model involves four confocal ellipsoids that represent the successive interfaces between the brain tissue, the cerebrospinal fluid, the skull, and the skin characterized by different conductivities. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be physically complete, since the effect of the substance surrounding the brain is taken into account. The direct EEG problem consists in finding the electric potential inside each conductive space, as well as at the nonconductive exterior space. The solution of this multitransmission problem is given analytically in terms of elliptic integrals and ellipsoidal harmonics, in such way that makes clear the effect that each shell has on the next one and outside of the head. It is remarkable that the dependence on the observation point is not affected by the presence of the conductive shells. Reduction to simpler ellipsoidal models and to the corresponding spherical models is included.