Section: New Results

Neural fields theory

Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk

Participants : Grégory Faye [correspondent] , James Rankin, David, J.B. Lloyd [Department of Mathematics at the University of Surrey] .

We analyze radially symmetric localized bump solutions of an integro-differential neural field equation posed in Euclidean and hyperbolic geometry. The connectivity function and the nonlinear firing rate function are chosen such that radial spatial dynamics can be considered. Using integral transforms, we derive a PDE for the neural field equation in both geometries and then prove the existence of small amplitude radially symmetric spots bifurcating from the trivial state. Numerical continuation is then used to path-follow the spots and their bifurcations away from onset in parameter space. It is found that the radial bumps in Euclidean geometry are linearly stable in a larger parameter region than bumps in the hyperbolic geometry. We also find and path-follow localized structures that bifurcate from branches of radially symmetric solutions with D6-symmetry and D8-symmetry in the Euclidean and hyperbolic cases, respectively. Finally, we discuss the applications of our results in the context of neural field models of short term memory and edges and textures selectivity in a hypercolumn of the visual cortex. This work has been accepted for publication in Nonlinearity [57] .

Center manifold for delayed neural fields equations

Participants : Olivier Faugeras [correspondent] , Romain Veltz [Salk Institute,San Diego,USA] .

We develop a framework for the study of delayed neural fields equations and prove a center manifold theorem for these equations. Specific properties of delayed neural fields equations make it impossible to apply existing methods from the literature concerning center manifold results for functional differential equations. Our approach for the proof of the center manifold theorem uses the original combination of results from Vanderbauwhede and colleagues together with a theory of linear functional differential equations in a history space larger than the commonly used set of time-continuous functions. This work has been submitted to the SIAM Journal on Applied Mathematics and is under revision [27] .

Reduction method for localized solutions

Participant : Grégory Faye [correspondent] .

We present a reduction method to study localized solutions of an integrodifferential equation defined on the Poincaré disk. This equation arises in a problem of texture perception modeling in the visual cortex. We first derive a partial differential equation which is equivalent to the initial integrodifferential equation and then deduce that localized solutions which are radially symmetric satisfy a fourth order ordinary differential equation. This work has appeared in the “Comptes Rendus Mathematique” [15] .

Spatially localized solutions

Participants : Pascal Chossat, Grégory Faye [School of Mathematics,University of Minnesota, correspondent] , James Rankin.

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist. This work has appeared in the Journal of Mathematical Biology [16] .

Bumps in the Poincaré disk

Participants : Grégory Faye [School of Mathematics,University of Minnesota, correspondent] , David, J.B. Loyd, James Rankin.

We analyze radially symmetric localized bump solutions of an integro-differential neural field equation posed in Euclidean and hyperbolic geometry. The connectivity function and the nonlinear firing rate function are chosen such that radial spatial dynamics can be applied. Using integral transforms, we derive a PDE of the neural field equation in both geometries and then prove the existence of small amplitude radially symmetric spots bifurcating from the trivial state. Numerical continuation is then used to path-follow the spots and their bifurcations away from onset in parameter space. It is found that the radial bumps in Euclidean geometry are linearly stable in a larger parameter region than bumps in the hyperbolic geometry. We also find and path follow localized structures that bifurcate from branches of radially symmetric solutions with D6-symmetry and D8-symmetry in the Euclidean and hyperbolic cases, respectively. Finally, we discuss the applications of our results in the context of neural field models of short term memory and edges and textures selectivity in a hypercolumn of the visual cortex. This work has been submitted to Nonlinearity.