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Journal Articles Comptes Rendus. Mathématique Year : 2022

Spatial and color hallucinations in a mathematical model of primary visual cortex

Abstract

We study a simplified model of the representation of colors in the primate primary cortical visualarea V1. The model is described by an initial value problem related to a Hammerstein equation. The solutionsto this problem represent the variation of the activity of populations of neurons in V1 as a function of spaceand color. The two space variables describe the spatial extent of the cortex while the two color variablesdescribe the hue and the saturation represented at every location in the cortex. We prove the well-posednessof the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in spacesolutions. We show that the model equation is equivariant with respect to the direct productGof the groupof the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group ofcolor transformations, isomorphic toO(2), and study the equivariant bifurcations of its stationary solutionswhen some parameters in the model vary. Their variations may be caused by the consumption of drugsand the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcatedsolutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axialsubgroups ofG. These define bifurcated solutions which are invariant under the action of the correspondingaxial subgroup. We compute analytically these solutions and illustrate them as color images. Using advancedmethods of numerical bifurcation analysis we then explore the persistence and stability of these solutionswhen varying some parameters in the model. We conjecture that we can rely on the EBL to predict theexistence of patterns that survive in large parameter domains but not to predict their stability. On our waywe discover the existence of spatially localized stable patterns through the phenomenon of "snaking".
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Dates and versions

hal-03388400 , version 1 (20-10-2021)
hal-03388400 , version 2 (25-02-2022)

Identifiers

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Olivier C Faugeras, Anna Song, Romain Veltz. Spatial and color hallucinations in a mathematical model of primary visual cortex. Comptes Rendus. Mathématique, 2022, 360, pp.59-87. ⟨10.5802/crmath.289⟩. ⟨hal-03388400v2⟩
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