Uncovering Brain Dynamics Using Directional Statistics and Functional Neuroimaging Data

Unraveling the complexities of human brain function requires advanced computational models that capture the time-varying nature of functional brain activity and connectivity. This thesis develops novel mathematical and statistical methods for analyzing dynamic brain connectivity, focusing on two key research themes: (1) phase coherence in functional magnetic resonance imaging (fMRI) and (2) multimodal and multisubject modeling of electroand magnetoencephalographic (EEG/MEG) data. An overarching theme is the development of sign- and scale-invariant models applicable to brain state modeling in all three functional neuroimaging modalities. In the first research theme, probabilistic models are developed to assess dynamic interregional phase coherence in resting-state and task fMRI as a proxy for instantaneous connectivity. By leveraging statistical distributions defined on Riemannian manifolds that are suitable for various manifold representations of signal phase and phase coherence, these models overcome limitations of existing approaches. Mixture and Hidden Markov modeling (HMM) techniques are introduced to enhance the modeling capacity of dynamic connectivity states. Furthermore, statistical distributions that allow modeling component covariances of a pre-specified rank are introduced, showing that these are better suited for modeling brain connectivity. The second research theme develops methods for multimodal and multisubject modeling, integrating time-locked EEG and MEG data to capture neural dynamics across data sources, allowing intermodal and intersubject variability and ensuring component correspondence. Matrix decomposition techniques are employed and generalized to an entire family of data fusion models with various constraints on extracted spatial source maps and temporal component mixing trajectories. A key contribution is the development of polarity-invariant microstate modeling, enabling a continuous representation of transient brain states across different neuroimaging modalities. Many of the proposed methods are of increased complexity compared to existing models, including probabilistic modeling and HMM, modeling a certain rank of clustering components, and modeling phase coherence as complex numbers. We find that model reliability is generally high, though we only investigated this on few synthetic and real neuroimaging datasets. We suggest for future studies to assess, e.g., the need for explicitly modeling temporal smoothness with an HMM at increased computational cost, as well as to assess the reliability of model estimates over multiple initializations and ensuring model convergence. Future research should focus on refining these methods, improving scalability, and exploring their applicability in broader neuroimaging contexts, thereby potentially providing new knowledge on the function of the human brain.