From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data

Abstract

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research.

A topological data analysis-based method for gait signals with an application to the study of multiple sclerosis

“…TDA has been applied to time series in medicine and biology since the 2010s. Applications include the study of cardiac arrythmia with electrocardiograms [25], motor learning with fMRI data [26,27], gene expression time series [28], wheeze in breathing signals [29], epileptic seizures with electroencephalograms [30], the spread of COVID-19 [31] and autism spectrum disorder [32].…”

Statistical inference for persistent homology applied to simulated fMRI time series data

“…Because we focus on time series data, we describe a method for determining whether topological characteristics within a set of time series intervals are significantly different from those in another set of time series intervals, where these intervals are related to different experimental conditions. This investigation builds on our previous work [25] where we demonstrated the use of persistent homology to characterize structure in fMRI data, though without a framework for statistical inference.…”

“…The fMRI signal is high dimensional, and characterized by hidden properties, the nature of which are not always known a priori. In this vein, topological data analysis (TDA) is in fact a viable option for exploring associations related to the topological or geometric characteristics of fMRI [25]. Within TDA, persistent homology is one of the best known tools for characterizing topological features of a set of points in a relatively high-dimensional space, such as the four-dimensional space (three spatial dimensions, and one signal amplitude dimension) in which fMRI data naturally sit.…”

“…With the rise of the use of topological methods in data analysis in the past ten years (see [28] for an introduction and brief survey) and in fMRI in particular (see [25] for an introduction and brief survey), here, we motivate a combination of statistical inference with topological methods. The idea is to calculate the persistence diagram (see [20] for an introduction) of the time-series data at each time index separately, and then to ask whether the temporal organization of the data into epochs can be recovered from the persistence diagrams in some statistically significant way.…”

“…For an extended discussion of the relevance of topological summaries to fMRI, and a detailed comparison of the kinds of insights about fMRI data obtainable via persistent homology, but not by classical statistical methods (e.g. regression analysis), we refer the reader to the paper [25], which is devoted to that topic. There are a variety of papers on TDA applied to time-series data, including [26], [18], and [22]; see [21] and [23] for nice surveys of some current ideas.…”