2020/03/28 by Avitabile, Daniele, Coombes, Stephen, Lima, Pedro M. · 1 citation
#Biological Physics (physics.bio-ph) #Disordered Systems and Neural Networks (cond-mat.dis-nn) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Pattern Formation and Solitons (nlin.PS)
paper · doi:10.48550/arxiv.2003.12761
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 finite-difference 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.