Statistical Modeling of Brain Network Data

There has been an increasing interest in using brain imaging technologies to better understand the relationship between brain structural connectivity networks -- also known as connectomes -- and human traits, ranging from cognitive abilities to neurological disorders. The brain network for an individual corresponds to a set of interconnections among anatomical regions in the brain. Modern neuroimaging technology allows a very fine segmentation of the brain, producing very large structural brain networks. Therefore, dimensionality reduction and feature extraction for such large complex networks become crucial. We first focus on the problem of studying shared- and individual-specific structure in brain networks or replicated graph data. We proposed Multiple GRAph Factorization (M-GRAF) model to estimate a common structure and low-dimensional individual-specific deviations from replicated graphs, which relies on a logistic regression mapping combined with a hierarchical eigenvalue decomposition. We develop an efficient algorithm for estimation and study basic properties of our approach. Application of our method to human brain connectomics data provides better prediction and goodness-of-fit (in terms of topological properties) to brain networks than some popular dimension-reduction methods.

To relate brain connectivity pattern with human cognitive traits, there is a strong need for accurate and efficient statistical methods on learning a set of small outcome-relevant subgraphs in network-predictor regression, which can greatly improve the interpretation of the association between the network predictor and the response. For example in brain connectomics, the extracted signal subgraphs can lead to discovery of key interconnected brain regions related to the trait and important insights on the mechanism of variation in human cognitive traits. We propose a symmetric bilinear model with $L_1$ penalty to search for small clique subgraphs that contain useful information about the response. A coordinate descent algorithm is developed to estimate the model where we derive analytical solutions for a sequence of conditional convex optimizations. Application of this method on human connectome and language comprehension data shows interesting discovery of relevant interconnections among several small sets of brain regions and better predictive performance than competitors.

Another possible formulation to tackle the problem of interest is by taking the connectivity network as the response and learning how human brain networks vary as a function of a continuous trait. We develop a Bayesian semiparametric model, which combines low-rank factorizations and flexible Gaussian process priors to learn changes in the conditional expectation of a network-valued random variable across the values of a continuous predictor, while including subject-specific random effects. The formulation leads to a general framework for inference on changes in brain network structures across human traits, facilitating borrowing of information and coherently characterizing uncertainty. We provide an efficient Gibbs sampler for posterior computation along with simple procedures for inference, prediction and goodness-of-fit assessments. The model is applied to learn how human brain networks vary across individuals with different intelligence scores. Results provide interesting insights on the association between intelligence and brain connectivity, while demonstrating good predictive performance.