Electrical Neuroimaging with Irrotational Sources

Abstract

This paper discusses theoretical aspects of the modeling of the sources of the EEG (i.e., the bioelectromagnetic inverse problem or source localization problem). Using the Helmholtz decomposition (HD) of the current density vector (CDV) of the primary current into an irrotational (I) and a solenoidal (S) part we show that only the irrotational part can contribute to the EEG measurements. In particular we present for the first time the HD of a dipole and of a pure irrotational source. We show that, for both kinds of sources, I extends all over the space independently of whether the source is spatially concentrated (as the dipole) or not. However, the divergence remains confined to a region coinciding with the expected location of the sources, confirming that it is the divergence rather than the CDV that really defines the spatial extension of the generators, from where it follows that an irrotational source model (ELECTRA) is always physiologically meaningful as long as the divergence remains confined to the brain. Finally we show that the irrotational source model remains valid for the most general electrodynamics model of the EEG in inhomogeneous anisotropic dispersive media and thus far beyond the (quasi) static approximation.