🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001304s)
  1. Topological Descriptors for Coral Reef Resilience Using a Stochastic Spatial Model (2022)

    Robert A. McDonald, Rosanna Neuhausler, Martin Robinson, Laurel G. Larsen, Heather A. Harrington, Maria Bruna
    Abstract A complex interplay between species governs the evolution of spatial patterns in ecology. An open problem in the biological sciences is characterizing spatio-temporal data and understanding how changes at the local scale affect global dynamics/behavior. We present a toolkit of multiscale methods and use them to analyze coral reef resilience and dynamics.Here, we extend a well-studied temporal mathematical model of coral reef dynamics to include stochastic and spatial interactions and then generate data to study different ecological scenarios. We present descriptors to characterize patterns in heterogeneous spatio-temporal data surpassing spatially averaged measures. We apply these descriptors to simulated coral data and demonstrate the utility of two topological data analysis techniques--persistent homology and zigzag persistence--for characterizing the spatiotemporal evolution of reefs and generating insight into mechanisms of reef resilience. We show that the introduction of local competition between species leads to the appearance of coral clusters in the reef. Furthermore, we use our analyses to distinguish the temporal dynamics that stem from different initial configurations of coral, showing that the neighborhood composition of coral sites determines their long-term survival. Finally, we use zigzag persistence to quantify spatial behavior in the metastable regime as the level of fish grazing on algae varies and determine which spatial configurations protect coral from extinction in different environments.
  2. The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics From Neural Data (2018)

    Samir Chowdhury, Bowen Dai, Facundo Mémoli
    Abstract We develop of a line of work initiated by Curto and Itskov towards understanding the amount of information contained in the spike trains of hippocampal place cells via topology considerations. Previously, it was established that simply knowing which groups of place cells fire together in an animal’s hippocampus is sufficient to extract the global topology of the animal’s physical environment. We model a system where collections of place cells group and ungroup according to short-term plasticity rules. In particular, we obtain the surprising result that in experiments with spurious firing, the accuracy of the extracted topological information decreases with the persistence (beyond a certain regime) of the cell groups. This suggests that synaptic transience, or forgetting, is a mechanism by which the brain counteracts the effects of spurious place cell activity.
  3. Modelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes (2017)

    P. Corcoran, C. B. Jones
    Abstract This paper presents a model of swarm behavior that encodes the spatial-temporal characteristics of topological features, such as holes and connected components. Specifically, the persistence of topological features with respect to time is computed using zig-zag persistent homology. This information is in turn modelled as a persistence landscape, which forms a normed vector space and facilitates the application of statistical and data mining techniques. Validation of the proposed model is performed using a real data set corresponding to a swarm of fish. It is demonstrated that the proposed model may be used to perform retrieval and clustering of swarm behavior in terms of topological features. In fact, it is discovered that clustering returns clusters corresponding to the swarm behaviors of flock, torus, and disordered. These are the most frequently occurring types of behavior exhibited by swarms in general.