A numerical scheme for neural field models including dendritic processing

Authors

Avitabile, D., Lima, P., and Coombes, S., 2019.

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 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.

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BibTeX

@article{avitabile2020numerical,
  title={Numerical investigation of a neural field model including dendritic processing},
  author={Avitabile, Daniele and Coombes, Stephen and Lima, Pedro M.},
  journal={arXiv preprint arXiv:2003.12761},
  year={2020}
}