Efficient Computation of Lead Field Bases and Influence Matrix for the FEM-based EEG and MEG Inverse Problem

The inverse problem in EEG and MEG aims at reconstructing the underlying current distribution in the human brain. The finite element method, used for the forward problem, is able to realistically model tissue conductivity inhomogeneities and anisotropies. So far, the computational complexity is quite large when using the necessary high resolution finite element models. It is already known that the so-called reciprocity can strongly reduce this complexity with regard to the EEG modality. We will derive algorithms for the efficient computation of EEG and MEG lead field bases which exploit the fact that the number of sensors is generally much smaller than the number of reasonable dipolar sources. Each finite element forward solution is then reduced to a simple matrix-vector multiplication instead of an expensive iterative finite element solution process. Our approaches can be applied to inverse reconstruction algorithms in both continuous and discrete source parameter space for EEG and MEG. In combination with modern solver methods, the presented approach leads to a highly efficient solution of FE-based source reconstruction problems.