On the Inverse EEG Problem for a 1D Current Distribution

Abstract

Albanese and Monk (2006) have shown that, it is impossible to recover the support of a three-dimensional current distribution within a conducting medium from the knowledge of the electric potential outside the conductor. On the other hand, it is possible to obtain the support of a current which lives in a subspace of dimension lower than three. In the present work, we actually demonstrate this possibility by assuming a one-dimensional current distribution supported on a small line segment having arbitrary location and orientation within a uniform spherical conductor. The immediate representation of this problem refers to the inverse problem of electroencephalography (EEG) with a linear current distribution and the spherical model of the brain-head system. It is shown that the support is identified through the solution of a nonlinear algebraic system which is investigated thoroughly. Numerical tests show that this system has exactly one real solution. Exact solutions are analytically obtained for a couple of special cases.

Mathematical Foundation of Electroencephalography

“…In previous sections, we swiftly examined currents of zero dimensionality, namely dipoles. In what follows, we will explore the forward and inverse EEG problems for one-and two-dimensional continuously distributed currents [29,30].…”

“…(11) provides the surface potential in the case of a linearly distributed current [29]. Knowledge of the surface measurements enables us to identify the position, moment, orientation and size of the current.…”

“…In the one-dimensional case, the system consists of a total of 19 equations with four constrains. It is possible to solve the latter at least semi-analytically [29].…”