  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 \textit{C. 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.
