The Inverse Magnetoencephalography Problem and Its Flat Approximation

Abstract

Contrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove that its uniqueness solution in the framework of the electrodynamic system of Maxwell equations [1]. The solution of this problem is the distribution of y → q(y) current dipoles of brain neurons that occupies the region Y ⊂ R 3 . It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient's head. The solution can be represented in the form q = q * + ρδ ∂Y , where q * is the usual function defined in Y, and ρδ ∂Y is a δ-function on the boundary of the domain Y with a certain density ρ. However, in cases where the conductivity is assumed to be everywhere the same (in the brain, skull, ambient air) and, in addition, it is not possible (or impossible) to record in time the electric and magnetic inductions, it is impossible to completely find q. Nevertheless, it is still possible to obtain partial information about the distribution of q : Y y → q(y). This question is considered in detail in a flat model situation.