Analysis of a hyperbolic geometric model for visual texture perception

We study the neural field equations introduced by Chossat and Faugeras to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, that is, time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.Mathematics Subject Classification:30F45, 33C05, 34A12, 34D20, 34D23, 34G20, 37M05, 43A85, 44A35, 45G10, 51M10, 92B20, 92C20.

[1]  C. Laing,et al.  Two-bump solutions of Amari-type models of neuronal pattern formation , 2003 .

[2]  H. Sompolinsky,et al.  13 Modeling Feature Selectivity in Local Cortical Circuits , 2022 .

[3]  Oren Shriki,et al.  Rate Models for Conductance-Based Cortical Neuronal Networks , 2003, Neural Computation.

[4]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[5]  Martin Golubitsky,et al.  What Geometric Visual Hallucinations Tell Us about the Visual Cortex , 2002, Neural Computation.

[6]  Olivier Faugeras,et al.  Some theoretical and numerical results for delayed neural field equations , 2010 .

[7]  S. Amari.  Dynamics of pattern formation in lateral-inhibition type neural fields , 1977, Biological Cybernetics.

[8]  Olivier D. Faugeras,et al.  Local/Global Analysis of the Stationary Solutions of Some Neural Field Equations , 2009, SIAM J. Appl. Dyn. Syst..

[9]  S. Helgason.  Groups and geometric analysis , 1984 .

[10]  Bard Ermentrout,et al.  Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses , 2001, SIAM J. Appl. Math..

[11]  Olivier P. Faugeras,et al.  Bifurcation of Hyperbolic Planforms , 2010, J. Nonlinear Sci..

[12]  Paul C. Bressloff,et al.  Breathing Pulses in an Excitatory Neural Network , 2004, SIAM J. Appl. Dyn. Syst..

[13]  Silvere Bonnabel,et al.  Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank , 2008, SIAM J. Matrix Anal. Appl..

[14]  Stephen Coombes,et al.  Waves, bumps, and patterns in neural field theories , 2005, Biological Cybernetics.

[15]  J. Cowan,et al.  A spherical model for orientation and spatial-frequency tuning in a cortical hypercolumn. , 2003, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[16]  H. Sompolinsky,et al.  Theory of orientation tuning in visual cortex. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[17]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[18]  Xiao-Jing Wang,et al.  A Model of Visuospatial Working Memory in Prefrontal Cortex: Recurrent Network and Cellular Bistability , 1998, Journal of Computational Neuroscience.

[19]  M. Golubitsky,et al.  Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. , 2001, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[20]  J. Cowan,et al.  SO3 symmetry breaking mechanism for orientation and spatial frequency tuning in the visual cortex. , 2002, Physical review letters.

[21]  J. Cowan,et al.  Correction for Bressloff and Cowan, A spherical model for orientation and spatial-frequency tuning in a cortical hypercolumn , 2003 .

[22]  D. Hubel,et al.  Receptive fields and functional architecture of monkey striate cortex , 1968, The Journal of physiology.

[23]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[24]  Paul C. Bressloff,et al.  Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn , 2000, Neural Computation.

[25]  J. Cowan,et al.  The visual cortex as a crystal , 2002 .

[26]  Olivier P. Faugeras,et al.  Hyperbolic Planforms in Relation to Visual Edges and Textures Perception , 2009, PLoS Comput. Biol..

[27]  H. Knutsson.  Representing Local Structure Using Tensors , 1989 .

[28]  Roland Potthast,et al.  Existence and properties of solutions for neural field equations , 2009 .

[29]  Carl-Fredrik Westin,et al.  Representing Local Structure Using Tensors II , 2011, SCIA.

[30]  S. Coombes,et al.  Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities , 2007 .

[31]  H. Iwaniec.  Spectral methods of automorphic forms , 2002 .

[32]  J. Bigun,et al.  Optimal Orientation Detection of Linear Symmetry , 1987, ICCV 1987.

[33]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[34]  Olivier D. Faugeras,et al.  Persistent Neural States: Stationary Localized Activity Patterns in Nonlinear Continuous n-Population, q-Dimensional Neural Networks , 2009, Neural Computation.

[35]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[36]  Boris S. Gutkin,et al.  Multiple Bumps in a Neuronal Model of Working Memory , 2002, SIAM J. Appl. Math..

[37]  HighWire Press.  Philosophical Transactions of the Royal Society of London , 1781, The London Medical Journal.

[38]  G. Orban,et al.  Velocity sensitivity and direction selectivity of neurons in areas V1 and V2 of the monkey: influence of eccentricity. , 1986, Journal of neurophysiology.

[39]  Carlo R. Laing,et al.  PDE Methods for Nonlocal Models , 2003, SIAM J. Appl. Dyn. Syst..