Using persistent homology on cubical complexes to study task-modulated brain activity.
Functional MRI (fMRI) measures blood-oxygen-level-dependent (BOLD) signals across the brain over time, producing a 4D dataset: three spatial dimensions of voxels plus time. Standard analyses often reduce this to summary statistics or connectivity matrices, which can miss the fine-grained spatial structure of how regions activate.
This project applies topological data analysis to fMRI data using the natural geometry that the voxel grid provides. Each brain region of interest is modeled as a cubical complex, a topological space built from cubes rather than triangles. Cubical complexes are amenable to efficient homological calculations and are well-adapted to the rectangular grid of voxels. The fMRI BOLD signal then induces a filtration on this complex: at each threshold value, we include all voxels whose signal exceeds that value. Running persistent homology on this filtration yields a persistence diagram encoding how clusters, loops, and voids in the activation pattern appear and disappear across signal intensities.
The main application is the anterior cingulate cortex (ACC), a brain region involved in cognitive control and task switching. The question is whether topological features of ACC activation systematically differ between task conditions and is something connectivity-based methods may not capture. A general mathematical framework for this approach is laid out in (Salch et al., 2021), and results applying it to the ACC are in (Catanzaro et al., 2023). This work is joint with Vaibhav Diwadkar, Sam Rizzo, Peter Bubenik, Andrew Salch, Adam Regalski, Hassan Abdallah, and Raviteja Suryadevara.
References
2023
Topological Data Analysis Captures Task-Driven fMRI Profiles in Individual Participants: A Classification Pipeline Based on Persistence
Michael J. Catanzaro, Sam Rizzo, John Kopchick, Asadur Chowdury, David R Rosenberg, Peter Bubenik, and Vaibhav A Diwadkar
BOLD-based fMRI is the most widely used method for studying brain function. The BOLD signal while valuable, is beset with unique vulnerabilities. The most notable of these is the modest signal to noise ratio, and the relatively low temporal and spatial resolution. However, the high dimensional complexity of the BOLD signal also presents unique opportunities for functional discovery. Topological Data Analyses (TDA), a branch of mathematics optimized to search for specific classes of structure within high dimensional data may provide particularly valuable applications. In this investigation, we acquired fMRI data in the anterior cingulate cortex (ACC) using a basic motor control paradigm. Then, for each participant and each of three task conditions, fMRI signals in the ACC were summarized using two methods: a) TDA based methods of persistent homology and persistence landscapes and b) non-TDA based methods using a standard vectorization scheme. Finally, using machine learning (with support vector classifiers), classification accuracy of TDA and non-TDA vectorized data was tested across participants. In each participant, TDA-based classification out-performed the non-TDA based counterpart, suggesting that our TDA analytic pipeline better characterized task- and condition-induced structure in fMRI data in the ACC. Our results emphasize the value of TDA in characterizing task- and condition-induced structure in regional fMRI signals. In addition to providing our analytical tools for other users to emulate, we also discuss the unique role that TDA-based methods can play in the study of individual differences in the structure of functional brain signals in the healthy and the clinical brain.
2021
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, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, and Vaibhav A. Diwadkar
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.
