Vector-valued Spline Method for the Spherical Multiple-shell Electro-magnetoencephalography Problem

Human brain activity is based on electrochemical processes, which can only be measured invasively. Therefore, quantities such as magnetic flux density (MEG) or electric potential differences (EEG) are measured non-invasively in medicine and research. The reconstruction of the neuronal current from the measurements is a severely ill-posed problem though its visualization is one of the main research tools in cognitive neuroscience. Here, using an isotropic multiple-shell model for the geometry of the head and a quasi-static approach for modeling the electro-magnetic processes, we derive a novel vector-valued spline method based on reproducing kernel Hilbert spaces. The presented vector spline method follows the path of former spline approaches and provides classical minimum norm properties. In addition, it minimizes the (infinite-dimensional) Tikhonov-Philips functional handling the instability of the inverse problem. This optimization problem reduces to solving a finite-dimensional system of linear equations without loss of information. It results in a unique solution which takes into account that only the harmonic and solenoidal component of the current affects the measurements. Besides, we prove a convergence result: the solution achieved by the vector spline method converges to the generator of the data as the number of measurements increases. The vector splines are applied to the inversion of synthetic test cases, where the irregularly distributed data situation could be handled very well. Combined with parameter choice methods, numerical results are shown with and without additional Gaussian white noise. Former approaches based on scalar splines are outperformed by the vector splines results with respect to the normalized root mean square error. Finally, reasonable results with respect to physiological expectations for real data are shown.