The Inverse Problem of Magneto-Electroencephalography is Well-Posed: it has a Unique Solution that is Stable with Respect to Perturbations

Abstract

Contrary to the opinion that has prevailed for the last several decades about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form q = q0 + p0δ|∂Y, where q0 is an ordinary function defined in the domain of the region Y occupied by the brain, and p0δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p0. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell’s equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the sought-for current dipoles are also concentrated in the cerebral cortex.