🍩 Database of Original & Non-Theoretical Uses of Topology

(found 6 matches in 0.001506s)

  2. Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence (2018)

    Woojin Kim, Facundo Memoli
    Abstract When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects. Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik.

  3. Topological Data Analysis of Biological Aggregation Models (2015)

    Chad M. Topaz, Lori Ziegelmeier, Tom Halverson
    Abstract We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

  4. Reconceiving the Hippocampal Map as a Topological Template (2014)

    Yuri Dabaghian, Vicky L. Brandt, Loren M. Frank
    Abstract The role of the hippocampus in spatial cognition is incontrovertible yet controversial. Place cells, initially thought to be location-specifiers, turn out to respond promiscuously to a wide range of stimuli. Here we test the idea, which we have recently demonstrated in a computational model, that the hippocampal place cells may ultimately be interested in a space's topological qualities (its connectivity) more than its geometry (distances and angles); such higher-order functioning would be more consistent with other known hippocampal functions. We recorded place cell activity in rats exploring morphing linear tracks that allowed us to dissociate the geometry of the track from its topology. The resulting place fields preserved the relative sequence of places visited along the track but did not vary with the metrical features of the track or the direction of the rat's movement. These results suggest a reinterpretation of previous studies and new directions for future experiments.

  5. Community Structures in Simplicial Complexes: An Application to Wildlife Corridor Designing in Central India -- Eastern Ghats Landscape Complex, India (2020)

    Saurabh Shanu, Shashankaditya Upadhyay, Arijit Roy, Raghunandan Chundawat, Sudeepto Bhattacharya
    Abstract The concept of simplicial complex from Algebraic Topology is applied to understand and model the flow of genetic information, processes and organisms between the areas of unimpaired habitats to design a network of wildlife corridors for Tigers (Panthera Tigris Tigris) in Central India Eastern Ghats landscape complex. The work extends and improves on a previous work that has made use of the concept of minimum spanning tree obtained from the weighted graph in the focal landscape, which suggested a viable corridor network for the tiger population of the Protected Areas (PAs) in the landscape complex. Centralities of the network identify the habitat patches and the critical parameters that are central to the process of tiger movement across the network. We extend the concept of vertex centrality to that of the simplicial centrality yielding inter-vertices adjacency and connection. As a result, the ecological information propagates expeditiously and even on a local scale in these networks representing a well-integrated and self-explanatory model as a community structure. A simplicial complex network based on the network centralities calculated in the landscape matrix presents a tiger corridor network in the landscape complex that is proposed to correspond better to reality than the previously proposed model. Because of the aforementioned functional and structural properties of the network, the work proposes an ecological network of corridors for the most tenable usage by the tiger populations both in the PAs and outside the PAs in the focal landscape.

  6. A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology (2012)

    Y. Dabaghian, F. Mémoli, L. Frank, G. Carlsson
    Abstract An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a “learning region” that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity., Our ability to navigate our environments relies on the ability of our brains to form an internal representation of the spaces we're in. The hippocampus plays a central role in forming this internal spatial map, and it is thought that the ensemble of active “place cells” (neurons that are sensitive to location) somehow encode metrical information about the environment, akin to a street map. Several considerations suggested to us, however, that the brain might be more interested in topological information—i.e., connectivity, containment, and adjacency, more akin to a subway map— so we employed new methods in computational topology to estimate how basic properties of neuronal firing affect the time required to form a hippocampal spatial map of three test environments. Our analysis suggests that, in order to encode topological information correctly and in a biologically reasonable amount of time, the hippocampal place cells must operate within certain parameters of neuronal activity that vary with both the geometric and topological properties of the environment. The interplay of these parameters forms a “learning region” in which changes in one parameter can successfully compensate for changes in the others; values beyond the limits of this region, however, impair map formation.