Volume 8, Number 3, 2018, Pages 915927 DOI：10.11948/2018.915 
Inverse problems in magnetoelectroscanning (in encephalography, for magnetic microscopes, etc.) 
Alexandre Sergeevich Demidov 
Keywords:Inverse problems, integral equations, pseudodifferential operators, magnetoelectroscanning 
Abstract: 
Contrary to the prevailing opinion about the incorrectness of the inverse MEEGproblem, we prove its unique solvability in the framework of the system of Maxwell''s equations [3]. The solution of this problem is the distribution of ${\bf y} \mapsto {\bf q}({\bf y})$ current dipoles of brain neurons that occupies the region $Y \subset \mathbb{R}^3 $. It is uniquely determined by the noninvasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient""s head. The solution can be represented in the form ${\bf q}={\bf q}_0+{\bf p}_0\delta\Big_{\partial Y}$, where ${\bf q}_0$ is the usual function defined in $Y,$ and ${\bf p}_0\delta\Big_{\partial Y} $ is a $\delta$function on the boundary of the domain $Y$ with a certain density ${\bf p}_0$. It is essential that ${\bf p}_0$ and ${\bf q}_0$ are interrelated. This ensures the correctness of the inverse MEEGproblem. However, the components of the required 3dimensional distribution $ {\bf q} $ must turn out to be linearly dependent if only the magnetic field ${\bf B}$ is taken into account. This question is considered in detail in a flat model of the situation. 
PDF Download reader



