An Inverse Source Problem for Maxwell's Equations in Magnetoencephalography

Consider an inverse source problem for Maxwell's equations which arises in determining locations of epileptic foci in the living human brain. The human brain is modeled as a heterogeneous medium, where the electric permittivity, magnetic permeability, and conductivity may all be functions. A current dipole is used to model the epilepsy. The inverse source problem in this context is to determine the current dipole from boundary measurements of the fields. In this paper, a new computational method is introduced for solving the inverse problem. The method is based on a low-frequency asymptotic analysis of Maxwell's equations. Our method is constructive and has good convergence properties. A crucial step of the method is to construct special test functions. Uniqueness and stability results for the inverse problem are also established. The method and results are expected to find applications particularly in magnetoencephalography.