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The inverse problem of magnetoelectroencephalography is wellposed: it has a unique solution that is stable with respect to perturbations
A. S. Demidov^{ab} ^{a} Lomonosov Moscow State University
^{b} Moscow Institute of Physics and Technology
Abstract:
Contrary to the already prevailing for several decades opinion about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form $\mathbf q=\mathbf q_0+\mathbf p_0\delta_{\partial Y}$, where $\mathbf q_0$ is an ordinary function defined in the domain of the region $Y$ occupied by the brain, and $\mathbf p_0 \delta_{\partial Y}$ is a $\delta$function on the boundary of the domain $Y$ with a certain density $\mathbf p_0$. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell's equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the soughtfor current dipoles are also concentrated in the cerebral cortex.
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A. S. Demidov, “The inverse problem of magnetoelectroencephalography is wellposed: it has a unique solution that is stable with respect to perturbations”, Fundam. Prikl. Mat., 21:4 (2016), 17–22
Citation in format AMSBIB
\Bibitem{Dem16}
\by A.~S.~Demidov
\paper The inverse problem of magnetoelectroencephalography is wellposed: it has a~unique solution that is stable with respect to perturbations
\jour Fundam. Prikl. Mat.
\yr 2016
\vol 21
\issue 4
\pages 1722
\mathnet{http://mi.mathnet.ru/fpm1746}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3783795}
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This publication is cited in the following articles:

A. S. Demidov, “Inverse problems in magnetoelectroscanning (in encephalography, for magnetic microscopes, etc.)”, J. Appl. Anal. Comput., 8:3 (2018), 915–927

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