Vasiliki Liontou ; Matilde Marcolli - Gabor frames from contact geometry in models of the primary visual cortex

mna:9766 - Mathematical Neuroscience and Applications, June 6, 2023, Volume 3 - https://doi.org/10.46298/mna.9766
Gabor frames from contact geometry in models of the primary visual cortexArticle

Authors: Vasiliki Liontou ; Matilde Marcolli

    We analyze the interplay between contact geometry and Gabor filters signal analysis in geometric models of the primary visual cortex. We show in particular that a specific framed lattice and an associated Gabor system is determined by the Legendrian circle bundle structure of the $3$-manifold of contact elements on a surface (which models the V1-cortex), together with the presence of an almost-complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a scaling of the lattice, also dictated by the manifold geometry, that ensures the frame condition is satisfied. We then consider a $5$-dimensional model where receptor profiles also involve a dependence on frequency and scale variables, in addition to the dependence on position and orientation. In this case we show that a proposed profile window function does not give rise to frames (even in a distributional sense), while a natural modification of the same generates Gabor frames with respect to the appropriate lattice determined by the contact geometry.


    Volume: Volume 3
    Published on: June 6, 2023
    Accepted on: April 12, 2023
    Submitted on: July 6, 2022
    Keywords: Quantitative Biology - Neurons and Cognition,92C20, 42C15, 57K33, 53D10
    Funding:
      Source : OpenAIRE Graph
    • Funder: Natural Sciences and Engineering Research Council of Canada
    • Geometry and Arithmetic in Theoretical Physics; Funder: National Science Foundation; Code: 1707882
    • Arithmetic and Topological Structures in Physics; Funder: National Science Foundation; Code: 2104330

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