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.

Andrew Salch,Michael J Catanzaro,Adam Regalski,Raviteja Suryadevara,Hassan Abdallah,Vaibhav A Diwadkar,Federico Giove

doi:10.1371/journal.pone.0255859

Andrew Salch, Michael J Catanzaro + Show 5 more

Open Access

https://doi.org/10.1371/journal.pone.0255859

Copy DOI

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.

Highlights

Functional magnetic resonance imaging is the preeminent method for a) reaching inferences regarding brain function, b) charactering the organization of macroscopic brain networks and c) understanding the brain’s structure-function relationships [1,2,3]. fMRI data are high dimensional and complex, and inference is fraught with uncertainty regarding signal origins and their relationship to underlying neurophysiology
As far as we are aware, the only paper in the extant literature which applies Topological data analysis (TDA) to understand the geometry of the fMRI signal, along the lines we propose, is Ellis et al [50]
We show an example of the kind of static structure in fMRI data that non-dynamic methods of TDA are designed to reveal

Summary

Introduction

Functional magnetic resonance imaging (fMRI) is the preeminent method for a) reaching inferences regarding brain function, b) charactering the organization of macroscopic brain networks and c) understanding the brain’s structure-function relationships [1,2,3]. fMRI data are high dimensional and complex, and inference is fraught with uncertainty regarding signal origins and their relationship to underlying neurophysiology. Functional magnetic resonance imaging (fMRI) is the preeminent method for a) reaching inferences regarding brain function, b) charactering the organization of macroscopic brain networks and c) understanding the brain’s structure-function relationships [1,2,3]. FMRI data are high dimensional and complex, and inference is fraught with uncertainty regarding signal origins and their relationship to underlying neurophysiology. Discovery is almost always motivated by the desire to understand structure-function relationships, or to “. FMRI data (like other classes of high dimensional data, and neurobiological data) has structure within the acquired signal data itself, present in the form of meaningful organization, and symmetric patterns. Topological data analysis (TDA) is optimized to search for specific classes of structure within data, and its application to fMRI may provide researchers with concepts and methods that complement existing approaches

Objectives

Methods

Findings

Discussion

Conclusion