Array response kernel for EEG in four-shell ellipsoidal geometry

We present an analytical forward modeling solution in the form of an array response kernel for electroencephalography (EEG) assuming a four-shell ellipsoidal geometry that approximates the anatomy of the brain, cerebrospinal fluid (CSF), skull, and scalp, while a current dipole models the source. The use of an ellipsoidal geometry is useful for cases when incorporating the anisotropy of the head is important but a better model cannot be defined. The structure of our forward solution facilitates the analysis of the inverse problem by factoring the lead field into a product of the current dipole source and a kernel containing the information corresponding to the head geometry and location of the source and sensors. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search. Furthermore, we introduce an schematic representation to easily derive the conductivity function involved in our forward solution. This schematic representation generalizes the calculation of the conductivity function for any number of shells, and offers a better understanding of the effect of the geometry on the shell conductivities.