Topological data analysis (TDA) is an emerging subfield of applied mathematics that can bring new insights to neuroscientific data. Persistent homology, a popular TDA algorithm, is traditionally applied to point cloud data. Yet persistent homology can also be a powerful technique for analyzing matrix structure. We will start from the basics, building up to the definition of persistent homology and Betti curves. Importantly, different null models produce distinctive sets of Betti curves. This lets us use Betti curves to compare data to null models. As an application, we will analyze neural correlation matrices from calcium imaging data for spontaneous activity in the optic tectum of zebrafish larvae. These techniques may also be useful for analyzing connectivity matrices arising from connectome data. If time permits, we will provide a quick demo using open source software for TDA algorithms.
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