Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry

We consider the inverse problem of identifying the current inside the brain, within the framework of the three-shell spherical model, from either electroencephalographic or magnetoencephalographic measurements, under some a priori assumptions about the nature of the current. In particular, we show that under the assumption that the current is localized within a small sphere of radius ε, it is possible to determine explicitly the center of the sphere as well as the characteristics of the current by solving a certain system of linear algebraic equations. In addition, we derive simple algorithms for identifying one or more dipoles. Finally, we present a set of compatibility conditions which can be used to verify whether the given measurements can be described by n dipoles.