  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.
