  Neural field equations are used to describe the spatiotemporal 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.
