06028602

j

06028602 Fokas, A.S. Kurylev, Y. Electro-magneto-encephalography for the three-shell model: minimal $L^2$-norm in spherical geometry. Inverse Probl. 28, No. 3, Article ID 035010, 11 p. (2012). 2012 IOP Publishing Ltd., Bristol EN electro-encephalography and magneto-encephalography continuously distributed neuronal current

doi:10.1088/0266-5611/28/3/035010

The problem of determining a continuously distributed neuronal current inside the brain within the framework of the three-shell model was analysed by the first author in [``Electro-magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries ", J. R. Soc. Interface 6, No. 34, 479--488 (2009; \url{doi:10.1098/rsif.2008.0309})], where it was shown that the simultaneous use of electro-encephalography and magneto-encephalography yields information about two of the three scalar functions specifying the interior current. In particular, for the spherical and ellipsoidal geometries, it is possible to determine the angular parts of these two functions, as well as to obtain certain explicit constraints satisfied by their radial parts. The complete determination of the radial parts of these two functions, as well as the determination of the third unknown function, requires some a priori assumption about the current. One such possible assumption is that the current minimizes the $L^{2}$-norm. In this investigation it is shown that this assumption yields a unique and explicit formula for the current in the case of spherical geometry. \"Omer Kavaklioglu (Washington)