Electroencephalographic Source Reconstruction by the Finite-Element Approximation of the Elliptic Cauchy Problem

Objective: This paper develops a novel approach for fast and reliable reconstruction of EEG sources in MRI-based head models. Methods: The inverse EEG problem is reduced to the Cauchy problem for an elliptic partial-derivative equation. The problem is transformed into a regularized minimax problem, which is directly approximated in a finite-element space. The resulting numerical method is efficient and easy to program. It eliminates the need to solve forward problems, which can be a tedious task. The method applies to complex anatomical head models, possibly containing holes in surfaces, anisotropic conductivity, and conductivity variations inside each tissue. The method has been verified on a spherical shell model and an MRI-based head. Results: Numerical experiments indicate high accuracy of localization of brain activations (both cortical potential and current) and rapid execution time. Conclusion: This study demonstrates that the proposed approach is feasible for EEG source analysis and can serve as a rapid and reliable tool for EEG source analysis. Significance: The significance of this study is that it develops a fast, accurate, and simple numerical method of EEG source analysis, applicable to almost arbitrary complex head models.