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Numerical mathematics for the modeling of a current dipole in EEG source reconstruction using finite element head models
Carsten H. Wolters, Harald Köstler, Christian Möller, Jochen Härdtlein, Lars Grasedyck and Wolfgang Hackbusch
In electroencephalography (EEG) inverse source analysis, a mathematical dipole is widely used as the model of the primary current source. The inverse methods are based on solutions to the corresponding forward problem, i.e., the simulation of the electric potential in the head volume conductor for a dipole in the cortex sheet of the human brain. The current dipole introduces a strong singularity on the right-hand side of the governing Poisson-type differential equation that has to be treated specifically when solving the equation towards the electric potential.
In this paper, we give a proof for existence and uniqueness of the weak solution in the function space of zero-mean potential functions, using a subtraction approach. The method divides the total potential into a singularity potential and a correction potential. The singularity potential is due to a mathematical dipole in an infinite region of homogeneous conductivity (the one at the source position). We then state convergence properties of the Finite Element (FE) method for the numerical solution to the correction potential. We validate our approach using high-resolution tetrahedra and regular and geometry-conforming node-shifted hexahedra elements in a three-layer sphere model. Validation has been carried out using sophisticated visualization techniques and statistical metrics for a comparison of the numerical results with analytical series expansion formulas at the surface and within the volume conductor. Finally, we validate the computed potentials of the subtraction method with the results of a direct approach in realistically-shaped FE head volume conductor models.