Partial entropy decomposition reveals higher-order structures in human brain activity

The standard approach to modeling the human brain as a complex system is with a network, where the basic unit of interaction is a pairwise link between two brain regions. While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly. In this work, we present a method for capturing higher-order dependencies in discrete data based on partial entropy decomposition (PED). Our approach decomposes the joint entropy of the whole system into a set of strictly non-negative partial entropy atoms that describe the redundant, unique, and synergistic interactions that compose the system's structure. We begin by showing how the PED can provide insights into the mathematical structure of both the FC network itself, as well as established measures of higher-order dependency such as the O-information. When applied to resting state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses. This synergistic structure distinct from structural features based on redundancy that have previously dominated FC analyses. Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording. We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated, and that the temporal pattern is structured in time. These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models. This synergistic "shadow structures" is dynamic in time and, likely will illuminate new and interesting links between brain and behavior.