Abstract
Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects—deviations from the mean-field limit due to the finite size of the populations in the network—we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit.
Highlights
The analysis of networks of neurons of growing size quickly becomes involved from a computational as well as from an analytic perspective when one tracks the spiking activity of every neuron in the network
By considering a spatially extended network and letting the density of populations go to infinity, neural field equations are obtained as the continuum limit of these models
We take the continuum limit of the network of diffusions (18), that is, we let the size of the domain and the density of populations go to infinity in order to obtain a stochastic neural field equation with a noise term describing the fluctuations around the deterministic traveling wave solution due to finite-size effects
Summary
The analysis of networks of neurons of growing size quickly becomes involved from a computational as well as from an analytic perspective when one tracks the spiking activity of every neuron in the network. 5.1) we obtain a well-posed L2(R)-valued stochastic evolution equation, with a noise term approximating finite-size effects on traveling waves, which we prove to be the strong continuum limit of the associated network. 3 we introduce our Markov chain model for determining finite-size effects in the voltage-based neural field equation and prove a law of large numbers and a central limit theorem for our choice of jump rates. We use it to set up a diffusion approximation with a noise term accounting for finite-size effects on traveling wave solutions in Sect.
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