The importance of the whole: topological data analysis for the network neuroscientist

The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying sets of node pairs that are densely connected and that collectively support brain function. Yet the restriction to pairwise interactions prevents us from realizing intrinsic topological features such as cavities within the interconnection structure that may be just as crucial for proper function. To detect and quantify these topological features we must turn to methods from algebraic topology that encode data as a simplicial complex built of sets of interacting nodes called simplices. On this substrate, we can then use the relations between simplices and higher-order connectivity to expose cavities within the complex, thereby summarizing its topological nature. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the underlying mathematics and perform demonstrative calculations on the mouse structural connectome, electrical and chemical synapses in textitC. elegans, and genomic interaction data. Finally we suggest avenues for future work and highlight new advances in mathematics that appear ready for use in revealing the architecture and function of neural systems.