EEG for Current With Two-Dimensional Support

The inverse problem of computing the neuronal current density from scalp EEG is highly ill-posed. In part, this is due to the nonuniqueness of the mapping between current sources and scalp potentials. We develop an explicit formula for the scalp EEG for sources constrained to the cortical surface in terms only of the components of the current that affect the EEG signal.

Starting from the quasi-static form of Maxwell's equations, we develop a formula that involves only the "visible" part of the current (i.e., the part of the current that affects the EEG measurements), as well as certain auxiliary functions that depend on the topology and conductivity of the 3-D domains $\Omega _c$ , $\Omega _f$, $\Omega _b$, and $\Omega _s$, which model the spaces occupied by the cerebrum, cerebrospinal fluid, bone, and scalp, respectively.

we derive expressions for the scalp potential for a general nested topology, as well as for the special case of spherical and ellipsoidal surfaces. We verify that the resulting scalp potential, in the case that the current resides in a spherical shell in the cerebrum of thickness $2\delta$, agrees with the potential obtained via the 3-D formulation for $\delta =10^{-8}\text{m}$.

The "visible" part of the current can be explicitly characterized and consists of a combination of its component normal to the surface and of a certain function generating the remaining tangential components of the current.

The resulting ability to restrict the source space greatly reduces the degree of ambiguity in the inverse solutions, offering the potential for more stable inverse solutions, since the auxiliary functions that define the mapping can be computed efficiently using standard numerical methods.