The definite non-uniqueness results for deterministic EEG and MEG data

Abstract

The solvability of the inverse problems of electroencephalography and magnetoencephalography has been studied extensively in the literature using a variety of models, including spherical and non-spherical geometries, homogeneous and inhomogeneous head models, and neuronal excitations involving the discrete and continuous distribution of dipoles. Among the important methods used are the methods based on spectral decompositions, physical arguments and integral representation techniques. Regarding the uniqueness of these inverse problems, a general result, independent of the geometry and the homogeneity of the conducting medium, has been obtained recently by the second author. This paper summarizes this result which appears to be mathematically definitive, in the sense that the geometry is arbitrary, the brain is surrounded by shells of varying conductivities, the neuronal current is arbitrary, the data are complete and the proofs are analytical. Furthermore, this paper includes a summary of the main steps of the proofs leading to the above result. In addition, it demonstrates the consistency of this general uniqueness result with all earlier results known in the literature.