Abstract
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".
Highlights
Neural Fields are a useful mathematical formalism for representing the dynamics of cortical areas at a macroscopic level, see the reviews [9,19,24]
We study a simplified model of the representation of colors in the primate primary cortical visual area V1
We conjecture that we can rely on the Equivariant Branching Lemma (EBL) to predict the existence of patterns that survive in large parameter domains but not to predict their stability
Summary
Neural Fields are a useful mathematical formalism for representing the dynamics of cortical areas at a macroscopic level, see the reviews [9,19,24]. The well-posedness of these equations has been studied in depth by various authors [3, 27, 49, 65] with a special attention to the stationary solutions, i.e. those which do not depend upon time These solutions, called persistent states, are interesting because they appear to provide good models of memory holding tasks on the time scale of the second [18, 28, 42].
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call