Topological Data Analysis for Neural Time Series: Persistent Homology and Brain Dynamics

Johnathan Bush ()
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Johnathan Bush: James Madison University

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 703-736 from Springer

Abstract: Abstract Understanding complex, high-dimensional temporal patterns in brain activity represents a central challenge in neuroscience. This chapter presents topological data analysis as a framework complementary to standard analytical approaches for characterizing brain dynamics. The computational pipeline is developed from time series to topological signatures and statistical summaries that enable hypothesis testing and comparison between brain states. The approach applies across recording modalities and spatial scales, from single neurons to whole-brain networks. Through examples drawn from oscillatory dynamics, network organization, and population-level behavioral states, the chapter shows how neural recordings can be transformed into geometric objects whose shapes, summarized through persistence diagrams and barcodes, yield topological signatures that support statistically rigorous comparisons across conditions. The chapter assumes minimal background and is intended to equip readers from neuroscience, applied mathematics, and data science with the foundation needed to apply these methods to their own data.

Keywords: Topological data analysis; Persistent homology; Brain dynamics; Time series analysis; Sliding window embedding; Persistence diagrams; Neural oscillations; State space reconstruction; Network topology; Statistical inference (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_90

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