The Human Brain as a Combinatorial Complex

TL;DR

The paper addresses the limitation of graph-based brain networks in capturing higher-order neural interactions by introducing combinatorial complexes (CCs) built directly from multivariate time series using information-theoretic measures. It defines CCs as $(S, \mathcal{X}, \mathrm{rk})$ to encode both pairwise and higher-order dependencies, and uses $\Sigma$ and $\Omega$ to select synergistic triplets under a dual-threshold scheme, enabling a unified representation that preserves higher-order structure. The authors demonstrate the construction pipeline on NetSim data, providing a concrete example of rank-0 to rank-2 cells and outlining computational considerations, scalability challenges, and future integration with topological deep learning (TDL) architectures. This approach preserves higher-order informational structure invisible to traditional graphs and offers a path toward leveraging TD architectures for brain state classification and cognitive task decoding, with a clear framework for extension to real fMRI data and larger-scale studies.

Abstract

We propose a framework for constructing combinatorial complexes (CCs) from fMRI time series data that captures both pairwise and higher-order neural interactions through information-theoretic measures, bridging topological deep learning and network neuroscience. Current graph-based representations of brain networks systematically miss the higher-order dependencies that characterize neural complexity, where information processing often involves synergistic interactions that cannot be decomposed into pairwise relationships. Unlike topological lifting approaches that map relational structures into higher-order domains, our method directly constructs CCs from statistical dependencies in the data. Our CCs generalize graphs by incorporating higher-order cells that represent collective dependencies among brain regions, naturally accommodating the multi-scale, hierarchical nature of neural processing. The framework constructs data-driven combinatorial complexes using O-information and S-information measures computed from fMRI signals, preserving both pairwise connections and higher-order cells (e.g., triplets, quadruplets) based on synergistic dependencies. Using NetSim simulations as a controlled proof-of-concept dataset, we demonstrate our CC construction pipeline and show how both pairwise and higher-order dependencies in neural time series can be quantified and represented within a unified structure. This work provides a framework for brain network representation that preserves fundamental higher-order structure invisible to traditional graph methods, and enables the application of topological deep learning (TDL) architectures to neural data.

Figures (2)

• Figure 1: Toy combinatorial complex from NetSim neural time series. Rank-0 cells are brain regions (nodes); Rank-1 edges capture pairwise interactions (mutual information $\geq 0.02$); Rank-2 hyperedges denote synergistic triplets (S-information $\geq 0.45$, O-information $\lesssim 0$). Top triplets: (2,3,4) with $S=0.51$, $\Omega=0.06$, and (1,2,3) with $S=0.48$, $\Omega=0.04$. Construction pipeline and implementation details are provided in \ref{['apd:pipeline']}.
• Figure 2: Pipeline for constructing combinatorial complexes (CCs) from fMRI data.