Section: New Results

Neural fields theory

We have started to tackle the problem of rigorously proving the existence of localized solutions to the neural fields equations. Existence of such solutions had been assumed or guessed from numerical simulations by other researchers. In a series of articles starting with  [55] we have used ideas from the theory of ordinary differential equations (existence of homoclinic orbits)  [56] , [19] , and the theory of partial differential equations (Swift-Hohenberg equation) [16] to show the existence of localized solutions for an extended variety of neural fields equations. This is important both theoretically and for neuroscience since these solutions are considered to characterize working (short-term) memory.

We have developed a framework for the study of delayed neural fields equations and proved a center manifold theorem for these equations. Specific properties of delayed neural fields equations make it difficult 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 et al. [1992] 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 appeared in the SIAM Journal on Mathematical Analysis [24] .

Neural field equations describe the activity of neural populations at a mesoscopic level. Although the early derivation of these equations introduced space dependent delays coming from the finite speed of signal propagation along axons, there has been few studies concerning their role in shaping the nonlinear dynamics of neural activity. This is mainly due to the lack of analytical tractable models. On the other hand, constant delays have been introduced to model the synaptic transmission and the spike initiation dynamics. By incorporating the two kind of delays in the neural fields equations, we are able to find the Hopf bifurcation curves analytically which produce many Hopf-Hopf interactions. We use normal theory to study two different types of connectivity that reveals a surprisingly rich dynamical portrait. In particular, the shape of the connectivity strongly influences the spatiotemporal dynamics. This work has appeared in SIAM Journal on Applied Dynamical Systems [25] .

We extend the theory of neural fields which has been developed in a deterministic framework by considering the influence spatio-temporal noise. The outstanding problem that we address here is the development of a theory that gives rigorous meaning to stochastic neural field equations, and conditions ensuring that they are well-posed. Previous investigations in the field of computational and mathematical neuroscience have been numerical for the most part. Such questions have been considered for a long time in the theory of stochastic partial differential equations, where at least two different approaches have been developed, each having its advantages and disadvantages. It turns out that both approaches have also been used in computational and mathematical neuroscience, but with much less emphasis on the underlying theory. We present a review of two existing theories and show how they can be used to put the theory of stochastic neural fields on a rigorous footing. We also provide general conditions on the parameters of the stochastic neural field equations under which we guarantee that these equations are well-posed. In so doing, we relate each approach to previous work in computational and mathematical neuroscience. We hope this will provide a reference that will pave the way for future studies (both theoretical and applied) of these equations, where basic questions of existence and uniqueness will no longer be a cause for concern. This work is available on ArXiV and is under review for a Journal.