Neural field models with transmission delays and diffusion

Abstract

A neural field models the large scale behaviour of large groups of neurons. We extend previous results for these models by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states while favouring synchronised oscillatory modes.

Dynamics of delayed neural field models in two-dimensional spatial domains

“…First we deal with the essential spectrum, σ ess (A), the part of the spectrum which is invariant under compact perturbations. We can leverage the fact that DG(0) is compact with Theorem 27 of [15] to find σ ess (A) = {−α}.…”

“…Delayed neural field models take the form of an integro-differential equation with space dependent delays. By choosing the proper state space, they can be reformulated as an abstract delay differential equation [16,15], where many available functional analytic tools can be applied. When the neuronal populations are distributed over a one-dimensional domain and a special class of connectivity functions is considered, a quite complete description of the spectrum and resolvent problem of the linearized equation is known [5,16].…”

“…When the neuronal populations are distributed over a one-dimensional domain and a special class of connectivity functions is considered, a quite complete description of the spectrum and resolvent problem of the linearized equation is known [5,16]. Recently, this model has been extended by including a diffusion term into the neural field, which models direct, electrical connections, [15]. We analyze the evolution of a delayed neural field equation corresponding to a single population of neurons on a two-dimensional spatial domain.…”

“…The paper is organized as follows. In Section 2, we summarize the functional analytic setting from [16,15] that casts the integro-differential equation into an abstract DDE. It is shown in Section 3 how to obtain an explicit representation of some eigenvectors of the linearized problem for a particular choice of connectivity function, expressed as a finite linear combination of exponential functions.…”

Sun dual theory for bi-continuous semigroups

“…At a peak of these developments van Neerven [51] finally provided a general comprehensive treatment of the theory, together with many new results clarifying especially on the topological aspects, see also [52][53][54]. Since then, the interest in dual semigroups which fail to be strongly continuous remained, and we name particularly applications in mathematical neuroscience [48,50].…”