# Inverse MEG and EEG

## Inverse Problems of MEG and EEG

MEG (magnetoencephalography) and EEG (electroencephalography) are techniques for the measurement of quantities related to the magnetic and electric fields of the brain. These fields are caused by current distributions and electric signalling inside the brain. The task of the inverse problems of MEG and EEG is to compute the neuronal currents inside the brain out of MEG and EEG data. These inverse problems are, like many inverse problems in practice, (severely) ill-posed. This is, in particular the case because the solution is not unique and not stable, i.e. small errors in the measurements can cause completely different solutions.

In cooperation with AS Fokas at the University of Cambridge and Olaf Hauk at the MRC Cognition and Brain Sciences Unit in Cambridge, UK, the Geomathematics Group in Siegen is investigating these inverse problems with respect to theoretical and numerical aspects.

For a spherical multiple-shell model of the brain structure, we are able to prove that only the solenoidal and harmonic part of the neuronal current affects the MEG and EEG measurements. Besides, we are able to derive for both problems singular value decompositions of the corresponding compact operators with infinite-dimensional range. The exponentially-fast decreasing of the singular values reveals the severe ill-posedness of these problems. By means of the developed approach, we are able to handle the non-uniqueness of these problems via additional non-uniqueness constraints like the minimum-norm constraint. Moreover, the detailed study of the problems reveals similarities to the inverse gravimetric problem from physical geodesy. We were able to use this synergy to derive further theoretical results and develop enhanced numerical methods.

In order to solve the ill-posed problems numerically, we developed tailor-made regularization methods for its solution, such as a scalar- and vector-valued spline method based on reproducing kernels on the ball. In addition, we adapted the regularized (orthogonal) functional matching pursuit algorithm (ROFMP) to these particular problems. Especially, the vector spline method and the ROFMP yield good numerical results in synthetic test cases. Besides, some first numerical experiments with real data yield promising results.

References:

- A.S. Fokas, O. Hauk, V. Michel:
*Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry*, Inverse Problems, 28 (2012), 035009 (28pp) - S. Leweke: The Inverse Magneto-electroencephalography Problem for the Spherical Multiple-shell Model, PhD thesis, submitted 2018, University of Siegen, Department of Mathematics, Geomathematics Group.
- S. Leweke, V. Michel, R. Telschow:
*On the non-uniqueness of gravitational and magnetic field data inversion (survey article)*, in: Handbook of Mathematical Geodesy (W. Freeden and M.Z. Nashed, eds.), Birkhäuser, Basel, 2018, pp. 883-919. - V. Michel, S. Orzlowski:
*On the null space of a class of Fredholm integral equations of the first kind,*Journal of Inverse and Ill-Posed Problems 24 (2016), 687-710, https://doi.org/10.1515/jiip-2015-0026