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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 715785, 11 pages
http://dx.doi.org/10.1155/2014/715785
Research Article

On the Inverse EEG Problem for a 1D Current Distribution

Department of Chemical Engineering, University of Patras and ICE/HT-FORTH, Patras, Greece

Received 27 December 2013; Revised 16 May 2014; Accepted 22 May 2014; Published 19 June 2014

Academic Editor: Shan Zhao

Copyright © 2014 George Dassios et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.