Inversion of electroencephalography data for a 2-D current distribution

Abstract

It is known from the fundamental work of Albanese and Monk that, the recovery of the support of a three dimensional current, within a conducting medium, from measurements of the generated exterior electric potential, is not possible. However, it is possible to recover the support of any other current, which is supported on a set of dimension lower than three. Nevertheless, no algorithm for such an inversion is known. Here, we propose such an algorithm for a two dimensional current distribution, and in particular, we apply this algorithm to the inverse problem of electroencephalography in the case where the neuronal current is restricted to a small disk of arbitrary location and orientation within the brain. The solution of this inverse problem is reduced to the solution of a nonlinear algebraic system, and numerical tests show that the there exists a unique real solution to this system.

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].…”

“…Due to the specific orientation of the primary current, only seven parameters have to be determined. Once more, the system of equations from which these parameters will be decided is nonlinear [30].…”

Magnetoencephalography for epileptic focus localization based on Tucker decomposition with ripple window

“…Third, the modern MEG has been developed very quickly and can be well used on source location with a large number of electrodes 11 . In the presurgical evaluation of epileptics, since the neuronal activity of the patients cannot be directly measured from MEG, it is always obtained via the mathematical modeling of the inverse problem 12,13 . The precision of the spatial localization depends on the employed mathematical models 14,15 .…”

EEG for Current With Two-Dimensional Support

“…Here, we extend the above result for EEG for the case that the current is supported on a closed surface. In [9] and [10] the direct and inverse problems for both EEG and MEG are solved when the current is constrained on a small disc.…”