A center manifold result for delayed neural fields equations

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 etal. together with a theory of linear functional differential equations in a history space larger than the commonly used set of time-continuous functions

A general center manifold theorem on fields of Banach spaces

A general local center manifold theorem around stationary trajectories is proved for nonlinear cocycles acting on measurable fields of Banach spaces

Kernel reconstruction for delayed neural field equations

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues. In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for the inverse problem. We employ spectral regularization techniques for its stable solution. A sensitivity analysis of the regularized kernel reconstruction with respect to the input signal u is carried out, investigating the Frechet differentiability of the kernel with respect to the signal. Finally, we use numerical examples to show the feasibility of the approach for kernel reconstruction, including numerical sensitivity tests, which show that the integral equation approach is a very stable and promising approach for practical computational neuroscience

On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Exponential stability of the stationary distribution of a mean field of spiking neural network

International audienceIn this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in \cite{de_masi_hydrodynamic_2015,fournier_toy_2016}. We complement the convergence result proved in \cite{fournier_toy_2016} using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation

Smooth centre manifolds for impulsive delay differential equations

Preprint of an article published in Journal of Differential Equations, available at: https://doi.org/10.1016/j.jde.2018.04.021The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.National Sciences and Engineering Research Council of Canada, Alexander Graham Bell Canada Graduate Scholarships-Doctoral Progra

ERRATUM: A Center Manifold Result for Delayed Neural Fields Equations

International audienceLemma C.1 in [R. Veltz and O. Faugeras, SIAM J. Math. Anal., 45(3) (2013), pp. 1527-1562] is wrong. This lemma is used in the proof of the existence of a smooth center manifold, Theorem 4.4 in [5]. An additional assumption is required to prove this existence. We spell out this assumption, correct the proofs and show that the assumption is satisfied for a large class of delay functions τ . We also weaken the general assumptions on τ