# American Institute of Mathematical Sciences

May  2009, 3(2): 243-257. doi: 10.3934/ipi.2009.3.243

## Vector ellipsoidal harmonics and neuronal current decomposition in the brain

 1 Department of Chemical Engineering, University of Patras, Greece, Greece

Received  November 2008 Revised  March 2009 Published  May 2009

Vector ellipsoidal harmonics are introduced here for the first time and their analytic peculiarities, as well as their limitations, are analyzed. A novelty of these vectorial base functions is that we need to introduce two different inner products in order to obtain orthogonality on the surface of any ellipsoid. Furthermore, in contrast to the vector spherical harmonics which are independent of the radial variable, the vector ellipsoidal harmonics can not be defined uniformly over a family of confocal ellipsoids. An expansion theorem is proved which secures completeness of the vectorial harmonics as well as a non-trivial algorithm that determines the coefficients of the expansion. Then, these new functions are used to prove that, for the realistic ellipsoidal model of the human head, there exists a component of the neuronal current that is invisible by the electroencephalographic measurements while it is detectable through magnetoencephalographic measurements in the exterior of the head. Furthermore, in contrast to the case of the sphere, where no part of the current contributes both to the electric potential and to the magnetic field, we prove here that, in the case of the ellipsoid, there is a part of the current that influences the electroencephalographic as well as the magnetoencephalographic recordings.
Citation: George Dassios, Michalis N. Tsampas. Vector ellipsoidal harmonics and neuronal current decomposition in the brain. Inverse Problems and Imaging, 2009, 3 (2) : 243-257. doi: 10.3934/ipi.2009.3.243
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