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.

中文翻译：

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神经场方程行波解的有限尺寸效应。

神经场方程用来描述连续极限中神经元的突触耦合群体网络中活动的时空演化。他们的启发式推导涉及两个近似步骤。在网络中每个人口都很大的假设下，以人口平均数来描述活动。然后用一个连续体来近似离散网络。在本文中，我们将两个近似步骤明确化。通过扩展Bressloff和Newby的模型，我们描述了由马尔可夫链在有限人口离散网络中活动的演变。为了确定有限大小的效应（由于网络中种群数量的有限而导致偏离均值场极限），我们分析了此马尔可夫链的波动并建立了扩散过程的近似系统。我们表明，获得了一个具有噪声项的适定随机神经场方程，该噪声项解释了行波解的有限大小效应，并将其作为强连续极限。