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

Autor: Faugeras, Olivier D., Song, Anna, Veltz, Romain
Přispěvatelé: Mathématiques pour les Neurosciences (MATHNEURO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Imperial College London], Imperial College London, Haematopoietic Stem Cell Laboratory, The Francis Crick Institute, London, UK, The Francis Crick Institute [London]
Rok vydání: 2022
Předmět:
Snaking
General Mathematics
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
FOS: Physical sciences
Neural Fields
Dynamical Systems (math.DS)
Pattern Formation and Solitons (nlin.PS)
[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]
Visual Hallucinations
[NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS]
Julia programming
FOS: Mathematics
Mathematics - Numerical Analysis
Equivariant Bifurcations
Mathematics - Dynamical Systems
Initial Value Problem
[INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS]
Numerical BifurcationAnalysis
[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT]
[SCCO.NEUR]Cognitive science/Neuroscience
[SDV.NEU.SC]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]/Cognitive Sciences
Numerical Analysis (math.NA)
[INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA]
Nonlinear Sciences - Pattern Formation and Solitons
Integro-partial differential equations
37L10
35B06
35B32
35G25
37M20
45B05
45K05
46E35
47G20
47H30
65J15
65R0
9208
9210
92B20
Quantitative Biology - Neurons and Cognition
FOS: Biological sciences
Equivariant Branching Lemma
Neurons and Cognition (q-bio.NC)
Color Perception
OPAL-Meso
Zdroj: Comptes Rendus. Mathématique
Comptes Rendus. Mathématique, 2022, 360, pp.59-87. ⟨10.5802/crmath.289⟩
ISSN: 1778-3569
1631-073X
Popis: We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product G of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to O(2), and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of G . These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of "snaking".
Comment: 30 pages, 12 figures
Databáze: OpenAIRE
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