On the non-uniqueness of the inverse problem associated with electroencephalography

Abstract

We present here a quantitative characterization of the non-uniqueness for the inverse problem of electroencephalography (EEG). First, we identify the singular support of the electric potential generated by a dipolar current which is fired inside the spherical model of the brain. Next, we extend this result to a continuously distributed neuronal current and we derive the equivalent Green's integral representation. Then, using the Hansen representation of the current, we show that among the three scalar representation functions, only two are needed to represent the observed electric potential on the surface or outside the head. The scalar function that is missed by the EEG recordings is exactly the one that is recorded by magnetoencephalography (MEG). Finally, the solution of the inverse EEG problem is reduced to a specific moment problem, which is exactly solved under the minimum-current assumption.