Integrating simplicial complex structures into statistical models for brain health

This proposal aims to develop statistical models that associate brain connectivity with human health outcomes. It uses a mathematical framework that quantifies not only pairwise co-activation of brain regions (nodes), but also encodes three-way and higher-order interactions, and their densities, using the mathematical framework of simplicial complexes (SCx). The methods developed here will enable the statistical analysis of cognitive function in large neuroimaging studies by modeling connectivity patterns in ways that are more extensive than those currently used. These methods will provide new insights into the complexities of brain-related health conditions because they quantify neuro-activation patterns in new and interpretable ways. Aim 1, extends the investigators’ previous scalar-on-matrix regression to include generalized linear and mixed models, then moves beyond adjacency matrix predictors to upper-adjacency edge (UAE) matrices, defined via three-way co- activation. These higher-order analogues of connectivity matrices involve edge relationships and have a low- rank structure not captured by standard approaches. They also lead to a new concept of edge communities (1- simplexes) that share a triangle (2-simplexes), or maximal edge communities (MEC). In Aim 2, estimated health-associated connectivity patterns in penalized regression models also incorporate higher-order simplicial structures—as predictors and regularizers. These model path structure by viewing node-pairs as boundaries of paths, and modeling their effective resistance (ER), which quantifies network-wide robustness of communication among nodes. Aim 2 leverages the UAE matrix to define a “lifted graph”, and the corresponding lifted-graph Laplacian is used for penalized regression on edges. These models encompass kernel-based methods that involve subject similarities based on simplicial structures. Aim 3 considers matrix- on-scalar regression models to estimate community-level associations between scalar predictors and adjacency-matrix responses. Rather than regressing based on prescribed mesoscale structure associations this form of model is extended to higher-order adjacencies structures, including MECs and other SCx structures. Aim 4 explores the recent concept of persistent Laplacians. This new operator relates the properties of two simplicial complexes when one is embedded in another. This allows analysis of a population of networks/simplices, which do not necessarily share all edges or triangles, by relating them to a common “core” SCx. Participant-wise discrepancies from this core, using the SCx algebra framework, leads to a new type of analysis. Successful completion of the proposed research will provide urgently needed extensions to current analytical methods with new models and software tools aimed at understanding common neurobiological disorders. Project Number: 1R01MH141931-01 | Fiscal Year: 2026 | NIH Institute/Center: National Institute of Mental Health (NIMH) | Principal Investigator: Jaroslaw Harezlak (+1 co-PI) | Institution: TRUSTEES OF INDIANA UNIVERSITY, BLOOMINGTON, IN | Award Amount: $831,656 | Activity Code: R01 | Study Section: Analytics and Statistics for Population Research Panel A Study Section[ASPA] View on NIH RePORTER: https://reporter.nih.gov/project-details/11245888