Statistical and Generative Methods for Brain Functional Connectivity Matrices

Abstract

Functional connectivity (FC), derived from resting-state functional Magnetic Resonance Imaging (fMRI), serves as a powerful tool for revealing brain co-activation pattern, identifying functional networks, and understanding of brain organization. FC is commonly represented by covariance matrices that characterize the temporal dependencies among predefined brain regions. Despite the richness of information covariance matrices provide, statistical modeling of such covariance matrices remains challenging due to dimensionality, limited interpretability, and the non-Euclidean geometry of symmetric positive definite (SPD) matrices. To address these challenges, we introduce a suite of novel statistical methods that enable interpretable and flexible modeling of FC-derived covariance matrices to advance the understanding of brain mechanism. We first propose a causal mediation framework with covariance matrix as graph mediator. We define causal estimands under a structural equation modeling framework, introduce a low-rank representation of covariance matrices, and develop likelihood-based estimators for identifying both mediation effects and low-dimensional structures. Simulation studies demonstrate that the proposed method achieves comparable performance to existing approaches under various scenarios. The framework is applied to resting-state fMRI data to evaluate the mediation effect of FC in explaining differences in motor task performance by sex. Next, we propose a parsimonious clustering model that integrates a Mixture-of-Experts structure with a covariance regression framework. This approach clusters subjects based on their brain connectivity while allowing cluster membership to vary with subject-level covariates. Simulation results demonstrate the superior performance of proposed method relative to existing methods. Application in fMRI data reveals clinically relevant subgroups along with associated cognitive and demographic characteristics. Lastly, we address the inherent challenge of fMRI data scarcity by developing a geometry-aware generative modeling framework for functional connectivity data. Recognizing the non-Euclidean geometry of SPD matrices, we adopt a Log-Euclidean representation for generative modeling of FC matrices. We further introduce a novel integration of diffusion transformers (DiT) and a rectified-flow strategy to achieve scalable and efficient synthesis of realistic functional connectivity matrices.