Matrix Kernels for the Forward Problem in EEG and MEG

This technical report has been submitted for review and possible publication in a journal. Because changes may be made before publication, this document is made available with the understanding that any journal version supersedes this document. Until such journal publication occurs, please cite this work using the above technical report number. Magnetoencephalography (MEG) and electroencephalography (EEG) (collectively E/MEG) are noninvasive methods for monitoring brain function. To estimate the location of the current sources that produce E/MEG signals, we must first solve the quasi-static forward problem relating a putative source to the E/MEG fields that it would produce. The head models that determine these solutions generally assume that the head is a piecewise homogeneous conductor. E/MEG models contain an incremental field element, commonly known as the " lead field, " that linearly relates an incremental source element (the current dipole) to the magnetic field or voltage potential at a distant point. The explicit form of the lead field is dependent on the head modeling assumptions and sensor configuration. The lead field can be partitioned into the product of a vector dependent on sensor characteristics and a matrix kernel dependent only on head modeling assumptions. Here we review analytic solutions for the spherical head model and boundary element methods (BEMs) for arbitrary head geometries. These results are presented in a unified form in terms of their matrix kernels. Using this formulation and a recently developed approximation formula for EEG, based on the " Berg parameters, " we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different BEM methods and present evidence that improvements over currently published E/MEG BEM methods can be realized using alternative error weighting methods.