🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.00117s)

  1. Statistical Inference for Persistent Homology Applied to Simulated fMRI Time Series Data (2023)

    Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew Salch
    Abstract Time-series data are amongst the most widely-used in biomedical sciences, including domains such as functional Magnetic Resonance Imaging (fMRI). Structure within time series data can be captured by the tools of topological data analysis (TDA). Persistent homology is the mostly commonly used data-analytic tool in TDA, and can effectively summarize complex high-dimensional data into an interpretable 2-dimensional representation called a persistence diagram. Existing methods for statistical inference for persistent homology of data depend on an independence assumption being satisfied. While persistent homology can be computed for each time index in a time-series, time-series data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multi-level block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting task-dependent topological organization is then demonstrated on simulated fMRI data. This new statistical test is therefore suitable for analyzing persistent homology of fMRI data, and of non-independent data in general.

  2. Feasibility of Topological Data Analysis for Event-Related fMRI (2019)

    Cameron T. Ellis, Michael Lesnick, Gregory Henselman-Petrusek, Bryn Keller, Jonathan D. Cohen
    Abstract Recent fMRI research shows that perceptual and cognitive representations are instantiated in high-dimensional multivoxel patterns in the brain. However, the methods for detecting these representations are limited. Topological data analysis (TDA) is a new approach, based on the mathematical field of topology, that can detect unique types of geometric features in patterns of data. Several recent studies have successfully applied TDA to study various forms of neural data; however, to our knowledge, TDA has not been successfully applied to data from event-related fMRI designs. Event-related fMRI is very common but limited in terms of the number of events that can be run within a practical time frame and the effect size that can be expected. Here, we investigate whether persistent homology—a popular TDA tool that identifies topological features in data and quantifies their robustness—can identify known signals given these constraints. We use fmrisim, a Python-based simulator of realistic fMRI data, to assess the plausibility of recovering a simple topological representation under a variety of conditions. Our results suggest that persistent homology can be used under certain circumstances to recover topological structure embedded in realistic fMRI data simulations.How do we represent the world? In cognitive neuroscience it is typical to think representations are points in high-dimensional space. In order to study these kinds of spaces it is necessary to have tools that capture the organization of high-dimensional data. Topological data analysis (TDA) holds promise for detecting unique types of geometric features in patterns of data. Although potentially useful, TDA has not been applied to event-related fMRI data. Here we utilized a popular tool from TDA, persistent homology, to recover topological signals from event-related fMRI data. We simulated realistic fMRI data and explored the parameters under which persistent homology can successfully extract signal. We also provided extensive code and recommendations for how to make the most out of TDA for fMRI analysis.