Some theoretical and numerical results for delayed neural field equations

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Abstract

In this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a fairly detailed study of the numerical computation of these solutions. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience.

Introduction

Delays arise naturally when we model neurobiological systems. For example, the finite-velocity propagation of action potentials, or the dendritic and synaptic processing can generate delays on the order of milliseconds. Effective delays can also be induced by the spike-generation dynamics. First, the delay due to propagation of action potentials along the axon depends on the travelled distance as well as on the type of neurons. Indeed, conduction velocities in the axon can range from about 1 m s−1 along unmyelinated axons to more than 100 m s−1 along myelinated ones [1], [2]. This is one of the reasons why significant time delays can emerge in certain brain structures. Second, some cells may have synapses or gap junctions on dendrites far from the cell body. In this case, there can also be a delay associated with the propagation of the action potential along the dendrite. Another delay can occur at a synaptic contact point in the transduction of an electrical signal into a biochemical signal and back again to a post-synaptic potential. Hence, the growing interest in understanding network models with space-dependent delays [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].

In this paper, we focus on space-dependent delays. In particular we incorporate delays in the well-known Wilson and Cowan [13], [14] and Amari [15] models for neural fields. In order to cover both axonal and dendritic delays and contrary to most of the papers on the subject, we deal with very general delay integro-differential equations without specifying the form of the delays.

We present a general mathematical framework for the modeling of neural fields which is based on tools of delay differential equation analysis and an original presentation and analysis of numerical schemes. We illustrate our results with numerical experiments. In Section 2 we briefly introduce the equations; in Section 3 we analyse the problem of the existence and uniqueness of their solutions. In Section 4 we study the problem of their asymptotic stability. In the penultimate section, we present some numerical schemes for the actual computation of the solutions. Each numerical scheme is illustrated by numerical experiments. We conclude in Section 6.

Section snippets

The models

Neural field models first appeared in the 50s, but the theory really took off in the 70s with the works of Wilson and Cowan [13], [14] and Amari [15]. Neural fields are continuous networks of interacting neural masses, describing the dynamics of the cortical tissue at the population level. These neural field models of population firing rate activity can be described, when delays are not taken into account by the following integro-differential equations: ∂tV(r,t)=−LV(r,t)+∫ΩW(r,r′,t)S(V(r′,t))dr′

Existence and uniqueness of a solution

In this section we deal with the problem of the existence and uniqueness of a solution to (2) for a given initial condition. We first introduce the framework in which this equation makes sense.

We start with the assumption that the state vector V is a differentiable (resp., square integrable) function of the time (resp. the space) variable. Let Ω be an open subset of Rq where q=1,2,3 and F be the set L2(Ω,Rn) of the square integrable functions from Ω to Rn. The Fischer–Riesz theorem ensures that

Linear stability analysis in the autonomous case

The goal of this section is to work at a fixed point V0 of (2) and study a linear retarded functional differential equation. The theory introduced by Hale in [16] is based on autonomous systems, this is why we need to impose that the connectivity does not depend on the time t, we note W(r,r′). For the moment, we impose that W∈L2(Ω2,Mn(R)). This section is divided into three parts. First we perform the linearization of (2) at a fixed point, then we study the stability by the method of Lyapunov

Numerical schemes

The aim of this section is to numerically solve Eq. (2) for different n and q. We remind the reader that n is the number of populations of neurons and q is the spatial dimension. This implies developing a numerical scheme that approaches the solution of our equation, and to prove that this scheme effectively converges to the solution.

To our knowledge, only in the paper of Hutt et al. [10] and in the thesis of Venkov [19], has a numerical scheme been explicitly developed, but without really

Conclusion

We have studied the existence, uniqueness, and asymptotic stability of a solution of nonlinear delay integro-differential equations that describe the spatio-temporal activity of sets of neural masses. We have also developed approximation and numerical schemes.

Using methods of functional analysis, we have found sufficient conditions for the existence and uniqueness of these solutions for general inputs. We have developed a Lyapunov functional which provides sufficient conditions for the

Acknowledgement

This work was partially funded by the ERC advanced grant NerVi number 227747.

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