EEG-DISTRIBUTED INVERSE SOLUTIONS FOR A SPHERICAL HEAD MODEL

The theoretical study of the minimum norm solution to the MEG inverse problem has been carried out in previous papers for the particular cas e of spherical symmetry. However, a similar study for the EEG is remar kably more difficult due to the very complicated nature of the express ion relating the voltage differences on the scalp to the primary curre nt density (PCD) even for this simple symmetry. This paper introduces the use of the electric lead field (ELF) on the dyadic formalism in th e spherical coordinate system to overcome such a drawback using an exp ansion of the ELF in terms of longitudinal and orthogonal vector field s. This approach allows us to represent EEG Fourier coefficients on a 2-sphere in terms of a current multipole expansion. The choice of a su itable basis for the Hilbert space of the PCDs on the brain region all ows the current multipole moments to be related by spatial transfer fu nctions to the PCD spectral coefficients; Properties df the most used distributed inverse solutions are explored on the basis of these resul ts. Also, a part of the ELF null space is completely characterized and those spherical components of the PCD which are possible silent candi dates are discussed.