Mathematical Methods for the Neuroelectromagnetic Inverse Problem for Focal and Unbiased Brain Imaging

This dissertation advances Bayesian methods for estimating and localizing focal neuroelectromagnetic activity in the human brain using measurements obtained non-invasively by electroencephalography (EEG) and magnetoencephalography (MEG). These methods are designed to localize focal brain activity while enduring modeling errors and significant measurement noise, which are unavoidable in neuroelectromagnetic problems.

In this research, we introduce a family of hierarchical Bayesian methods (HBM) distinguished by the parametrization of the exponential power prior distribution. First, we show that a particular choice of the prior parameters yields a focal estimation of neuroelectromagnetic activity. Second, we improve the HBM source estimates by introducing localization bias-reduction techniques. In particular, we study two techniques: randomized multiresolution scanning (RAMUS) and standardization, known from the context of standardized low-resolution brain electromagnetic tomography (sLORETA). Specifically, we develop the standardization technique within the Bayesian modeling framework and derive conditions for perfect source localization estimates. Moreover, we analyze the robustness of this technique against measurement noise. The proposed standardization technique is then applied to the Bayesian recursive filter for Gaussian multivariate distributions, commonly known as the Kalman filter. The standardized Kalman filter enables us to account for the temporal (time-varying) nature of brain signals, allowing for precise and focal tracking and localization of neuroelectromagnetic activity.

Through a series of individual articles, we compare these novel methods with classical and well-established neuroelectromagnetic estimation techniques, using both simulated and real data from somatosensory evoked potentials and epilepsy studies. The results demonstrate that striving for unbiasedness leads to more accurate and robust source estimations.