  A neural field models the large scale behaviour of large groups of neurons.
We extend results of van Gils et al. [2013] and Dijkstra et al. [2015] by
including a diffusion term into the neural field, which models direct,
electrical connections. We extend known and prove new sun-star calculus results
for delay equations to be able to include diffusion and explicitly characterise
the essential spectrum. For a certain class of connectivity functions in the
neural field model, we are able to compute its spectral properties and the
first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical
example, we find that the addition of diffusion suppresses non-synchronised
steady-states, while favouring synchronised oscillatory modes.
