Vector-valued spline method for the spherical multiple-shell electro-magnetoencephalography problem

Abstract

Human brain activity is based on electrochemical processes, which can only be measured invasively. Thus, 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 the visualization of the cerebral activity is one of the main research tools in cognitive neuroscience. Here, using an isotropic multiple-shell model for the human head and a quasi-static approach for the electro-magnetic processes, we derive a novel vector-valued spline method based on reproducing kernel Hilbert spaces in order to reconstruct the current from the measurements. The presented method follows the path of former spline approaches and provides classical minimum norm properties. Besides, it minimizes the (infinite-dimensional) Tikhonov-Philips functional which handles the instability of the inverse problem. This optimization problem reduces to solving a finite-dimensional system of linear equations without loss of information, due to its construction. It results in a unique solution which takes into account that only the harmonic and solenoidal component of the neuronal current affects the measurements. Furthermore, we prove a convergence result: the solution achieved by the novel method converges to the generator of the data as the number of measurements increases. The vector splines are applied to the inversion of three synthetic test cases, where the irregularly distributed data situation could be handled very well. Combined with five parameter choice methods, numerical results are shown for synthetic test cases with and without additional Gaussian white noise. Former approaches based on scalar splines are outperformed by the novel vector splines results with respect to the normalized root mean square error. Finally, results for real data acquired during a visual stimulation task are demonstrated. They can be computed quickly and are reasonable with respect to physiological expectations.