🍩 Database of Original & Non-Theoretical Uses of Topology

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  2. Topological Portraits of Multiscale Coordination Dynamics (2020)

    Mengsen Zhang, William D. Kalies, J. A. Scott Kelso, Emmanuelle Tognoli
    Abstract Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a challenge in both theoretical and empirical realms. The present work shows how integrating approaches from computational algebraic topology and dynamical systems may help us meet this challenge. In particular, we focus on the application of multiscale topological analysis to coordinated rhythmic processes. First, theoretical arguments are introduced as to why certain topological features and their scale-dependency are highly relevant to understanding complex collective dynamics. Second, we propose a method to capture such dynamically relevant topological information using persistent homology, which allows us to effectively construct a multiscale topological portrait of rhythmic coordination. Finally, the method is put to test in detecting transitions in real data from an experiment of rhythmic coordination in ensembles of interacting humans. The recurrence plots of topological portraits highlight collective transitions in coordination patterns that were elusive to more traditional methods. This sensitivity to collective transitions would be lost if the behavioral dynamics of individuals were treated as separate degrees of freedom instead of constituents of the topology that they collectively forge. Such multiscale topological portraits highlight collective aspects of coordination patterns that are irreducible to properties of individual parts. The present work demonstrates how the analysis of multiscale coordination dynamics can benefit from topological methods, thereby paving the way for further systematic quantification of complex, high-dimensional dynamics in living systems.

  3. Topological Gene Expression Networks Recapitulate Brain Anatomy and Function (2019)

    Alice Patania, Pierluigi Selvaggi, Mattia Veronese, Ottavia Dipasquale, Paul Expert, Giovanni Petri
    Abstract Understanding how gene expression translates to and affects human behavior is one of the ultimate goals of neuroscience. In this paper, we present a pipeline based on Mapper, a topological simplification tool, to analyze gene co-expression data. We first validate the method by reproducing key results from the literature on the Allen Human Brain Atlas and the correlations between resting-state fMRI and gene co-expression maps. We then analyze a dopamine-related gene set and find that co-expression networks produced by Mapper return a structure that matches the well-known anatomy of the dopaminergic pathway. Our results suggest that network based descriptions can be a powerful tool to explore the relationships between genetic pathways and their association with brain function and its perturbation due to illness and/or pharmacological challenges., In this paper, we described a gene co-expression analysis pipeline that produces networks that we show to be closely related to either brain function and to neurotransmitter pathways. Our results suggest that this pipeline could be developed into a platform enabling the exploration of the effects of physiological and pathological alterations to specific gene sets, including profiling drugs effects.

  4. Toroidal Topology of Population Activity in Grid Cells (2022)

    Richard J. Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A. Baas, Benjamin A. Dunn, May-Britt Moser, Edvard I. Moser
    Abstract The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

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