Human brain activity is based on electrochemical processes, which can only be measured invasively. For this reason, induced quantities such as magnetic flux density (via MEG) or electric potential differences (via 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 tools in brain science and diagnosis.
Using an isotropic multiple-shell model for the geometry of the human head
and a quasi-static approach for modelling the electro-magnetic processes, a singular-value decomposition of the continuous forward operator between infinite-dimensional Hilbert spaces is derived. Due to a full characterization of the operator null space, it is revealed that only the harmonic and solenoidal component of the neuronal current affects the measurements. Uniqueness of the problem can be achieved by a minimum-norm condition. The instability of the inverse problem caused by exponentially decreasing singular values requires a stable and robust regularization method.
The few available measurements per time step ($\approx 100$) are irregularly distributed with larger gaps in the facial area. On these grounds, a vector spline method for regularized functional inverse problems based on reproducing kernel Hilbert spaces is derived for dealing with these difficulties. Combined with several parameter choice methods, numerical results are shown for synthetic test cases with and without additional Gaussian white noise. The relative normalized root mean square error of the approximation as well as the relative residual do not exceed the noise level. Finally, also results for real data are demonstrated. They can be computed with only a short delay time and are reasonable with respect to physiological expectations.