  Neural field models are commonly used to describe wave propagation and bump
attractors at a tissue level in the brain. Although motivated by biology, these
models are phenomenological in nature. They are built on the assumption that
the neural tissue operates in a near synchronous regime, and hence, cannot
account for changes in the underlying synchrony of patterns. It is customary to
use spiking neural network models when examining within population
synchronisation. Unfortunately, these high dimensional models are notoriously
hard to obtain insight from. In this paper, we consider a network of
$\theta$-neurons, which has recently been shown to admit an exact mean-field
description in the absence of a spatial component. We show that the inclusion
of space and a realistic synapse model leads to a reduced model that has many
of the features of a standard neural field model coupled to a further dynamical
equation that describes the evolution of network synchrony. Both Turing
instability analysis and numerical continuation software are used to explore
the existence and stability of spatio-temporal patterns in the system. In
particular, we show that this new model can support states above and beyond
those seen in a standard neural field model. These states are typified by
structures within bumps and waves showing the dynamic evolution of population
synchrony.
