Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data

TL;DR

This paper introduces simplicial complexes, an algebraic-topological framework that extends traditional graph models to better capture complex, higher-order neural interactions, offering new insights into brain function.

Contribution

It generalizes graph theory using simplicial complexes, enabling modeling of higher-order neural structures and functions beyond dyadic connections.

Findings

Simplicial complexes provide a flexible framework for neural data analysis.

Application to electrophysiological and hemodynamic data demonstrates potential.

The approach may surpass traditional graph methods in understanding cognition.

Abstract

The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit of interest in a brain is a dyad -- two nodes (neurons or brain regions) connected by an edge. While rarely mentioned, this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model. Here, we describe a generalization of graphs that overcomes these limitations, thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena. Specifically, we explore the use of \emph{simplicial complexes}, a theoretical notion developed in the field of mathematics known as algebraic topology, which is now becoming applicable to real data due to a rapidly growing computational toolset. We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data, from electrophysiological recordings in animal models to hemodynamic fluctuations in humans. Based on the exceptional flexibility of the tools and recent ground-breaking insights into neural function, we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition.