  We study coarse pattern formation in a cellular automaton modelling a
spatially-extended stochastic neural network. The model, originally proposed by
Gong and Robinson [36], is known to support stationary and travelling bumps of
localised activity. We pose the model on a ring and study the existence and
stability of these patterns in various limits using a combination of analytical
and numerical techniques. In a purely deterministic version of the model, posed
on a continuum, we construct bumps and travelling waves analytically using
standard interface methods from neural fields theory. In a stochastic version
with Heaviside firing rate, we construct approximate analytical probability
mass functions associated with bumps and travelling waves. In the full
stochastic model posed on a discrete lattice, where a coarse analytic
description is unavailable, we compute patterns and their linear stability
using equation-free methods. The lifting procedure used in the coarse
time-stepper is informed by the analysis in the deterministic and stochastic
limits. In all settings, we identify the synaptic profile as a mesoscopic
variable, and the width of the corresponding activity set as a macroscopic
variable. Stationary and travelling bumps have similar meso- and macroscopic
profiles, but different microscopic structure, hence we propose lifting
operators which use microscopic motifs to disambiguate between them. We provide
numerical evidence that waves are supported by a combination of high synaptic
gain and long refractory times, while meandering bumps are elicited by short
refractory times.
