🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000972s)
  1. Extracting Insights From the Shape of Complex Data Using Topology (2013)

    P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, G. Carlsson
    Abstract This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.
  2. 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.