Bayesian Approaches to Inverse Problems in Functional Neuroimaging

In the last decades, the availability of novel functional neuroimaging methods has revolutionized the field of neuroscience and has given us a deeper understanding of the function of the human brain. Recently, functional magnetic resonance imaging (fMRI) has become a prominent method, enabling localization of neuronal activity within millimeters. The temporal resolution of fMRI on the other hand is limited due to the indirect mechanism the measurements are related to neuronal activity and due to the low sampling rate. In contrast, electroencephalography (EEG) and magnetoencephalography (MEG) can attain very high temporal resolutions as they provide direct measures of electrical activity in the brain via voltage sensors, or magnetic field sensors, respectively. However, the spatial resolution of EEG and MEG is considered low as localizing the sources of activity in the brain is an ill-posed inverse problem.;This work is concerned with Bayesian approaches to inverse problems in functional neuroimaging. We address the problem of limited spatio-temporal resolution of current neuroimaging methods by developing a method for fusing EEG and fMRI, i.e., information from fMRI and signal priors are used to find a solution to the ill-posed source localization problem. Recently, the estimation of connectivity between brain regions has become increasingly popular in neuroscience. However, effective (directed) connectivity estimation for fMRI is a difficult problem due to the non-linear relationship between time series and due to the indirect fMRI observation process. The second method developed in this work addresses this problem by employing a hierarchical model consisting of a vector autoregressive (VAR) connectivity model in combination with a linear fMRI observation model. Last, we propose a method for simultaneous sparse approximation with smooth signals, a problem which shares similarities with the M/EEG source localization problem.;We employ Bayesian formulations for all methods presented in this work. The use of sophisticated signal priors and Bayesian inference techniques enable the proposed methods to estimate latent variables with high accuracy. Novel methods, such as the ones presented in this work, play a central role in advancing the field of neuroscience and our understanding of the human brain.