🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000842s)

  1. On the Local Behavior of Spaces of Natural Images (2008)

    Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian
    Abstract In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches ℳ. We develop a theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of ℳ. We find the best-fitting embedding of the Klein bottle into the ambient space of ℳ. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.

  2. Topological Analysis of Population Activity in Visual Cortex (2008)

    Gurjeet Singh, Facundo Memoli, Tigran Ishkhanov, Guillermo Sapiro, Gunnar Carlsson, Dario L. Ringach
    Abstract Information in the cortex is thought to be represented by the joint activity of neurons. Here we describe how fundamental questions about neural representation can be cast in terms of the topological structure of population activity. A new method, based on the concept of persistent homology, is introduced and applied to the study of population activity in primary visual cortex (V1). We found that the topological structure of activity patterns when the cortex is spontaneously active is similar to those evoked by natural image stimulation and consistent with the topology of a two sphere. We discuss how this structure could emerge from the functional organization of orientation and spatial frequency maps and their mutual relationship. Our findings extend prior results on the relationship between spontaneous and evoked activity in V1 and illustrates how computational topology can help tackle elementary questions about the representation of information in the nervous system.