Numerical investigation of a neural field model including dendritic processing

Abstract

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finitedifference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

Projection Methods for Neural Field Equations

“…x ) in space. Results of this type are currently presented in the literature for discrete schemes, where quadrature rules are prescribed [39,38,11]. We show here that they are a consequence of the theory presented in the previous chapters.…”

“…Intuitively, a numerical scheme can be derived by picking a spatial grid, approximating the integral with a quadrature scheme, and using a time stepper for the corresponding set of ODEs. Convergence results are limited to these schemes (which we will classify later as discrete collocation schemes) in deterministic neural fields [39], neural fields with anisotropic diffusion [11], stochastic neural fields [38,33], and neural fields with distributed delays [25,44].…”

“…For neural fields, the abstract function space formulation allows us to derive bounds that are insightful with respect to the few existing convergence results available for discrete schemes. For instance, it emerges that the choice of a quadrature scheme (which is made upfront in discrete schemes for neural fields [47,39,38,11]) is in fact dependent on the choice of a projector P n : the latter determines the accuracy that must be matched by the former, when one commits to a discretization of the problem. As we will show here, a discrete scheme formulated without accounting for P n may waste resources or pollute the convergence of the projector scheme from which it is derived.…”