Abstract
Neural Field Equations (NFE) are intended to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind of integro-differential equations proved to be a useful tool to describe the spatiotemporal neuronal activity from a macroscopic point of view, allowing the study of a wide variety of neurobiological phenomena, such as the sensory stimuli processing. The present article aims to study the effects of additive noise in one- and two-dimensional neural fields, while taking into account finite axonal velocity and an external stimulus. A Galerkin-type method is presented, which applies Fast Fourier Transforms to optimise the computational effort required to solve these equations. The explicit Euler-Maruyama scheme is implemented to obtain the stochastic numerical solution. An open-source numerical solver written in Julia was developed to simulate the neural fields in study.
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Code availability
A numerical solver written in Julia, whose purpose is to solve the types of NFEs discussed in this paper, is published in Julia's library and available for installation, more details at https://github.com/tiagoseq/NeuralFieldEq.jl. Technical details of code usage are described in Sequeira (2021). Julia proved to be a great tool to simulate the addressed NF, and allied to the \(\mathcal {RFFT}\) routines it was possible to work in strongly delayed scenarios, when the computational effort increases significantly. The solver is efficient enough to not run out of memory, even in the stochastic case, where we had to compute a large number of trajectories for each noise level. In all simulations carried out, the computing time never exceeded 40 minutes. Note that the computations were performed in a Laptop with a 1.30GHz Intel(R) Core(TM) i7 CPU processor and 16GB memory RAM.
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Acknowledgements
The second author acknowledges the financial support of the portuguese FCT (Fundação para a Ciência e Tecnologia), through projects UIDB/04621/2020, UIDP/04621/2020 and PTDC/MAT-APL/31393/2017. Both authors are grateful to the anonymous reviewers, whose comments and suggestions helped to improve the paper.
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Sequeira, T.F., Lima, P.M. Numerical simulations of one- and two-dimensional stochastic neural field equations with delay. J Comput Neurosci 50, 299–311 (2022). https://doi.org/10.1007/s10827-022-00816-w
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DOI: https://doi.org/10.1007/s10827-022-00816-w