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The language of graph theory, or network science, has proven to be anexceptional tool for addressing myriad problems in neuroscience. Yet, the useof networks is predicated on a critical simplifying assumption: that thequintessential unit of interest in a brain is a dyad -- two nodes (neurons orbrain regions) connected by an edge. While rarely mentioned, this fundamentalassumption inherently limits the types of neural structure and function thatgraphs can be used to model. Here, we describe a generalization of graphs thatovercomes these limitations, thereby offering a broad range of newpossibilities in terms of modeling and measuring neural phenomena.Specifically, we explore the use of \emph{simplicial complexes}, a theoreticalnotion developed in the field of mathematics known as algebraic topology, whichis now becoming applicable to real data due to a rapidly growing computationaltoolset. We review the underlying mathematical formalism as well as the buddingliterature applying simplicial complexes to neural data, fromelectrophysiological recordings in animal models to hemodynamic fluctuations inhumans. Based on the exceptional flexibility of the tools and recentground-breaking insights into neural function, we posit that this framework hasthe potential to eclipse graph theory in unraveling the fundamental mysteriesof cognition.

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