Theory of the EEG Inverse Problem

Abstract

We deal here with the EEG neuroimaging problem: given measurements of scalp electric potential differences (EEG: electroencephalogram), find the 3D distribution of the generating electric neuronal activity. This problem has no unique solution. Particular solutions with optimal localization properties are of main interest, since neuroimaging is concerned with the correct localization of brain function. A brief historical outline of localization methods is given: from the single dipole, to multiple dipoles, to distributions. Technical details on the formulation and solution of this type of inverse problem are presented. Emphasis is placed on linear, discrete, 3D distributed EEG tomographies, that have a simple mathematical structure allowing a complete evaluation of their localization properties. One particular noteworthy member of this family is eLORETA (exact low resolution brain electromagnetic tomography [46]), which is a genuine inverse solution (not merely a linear imaging method, nor a collection of one-at-a-time single best fitting dipoles) with zero localization bias in the presence of measurement and structured biological noise.