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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] byincluding a diffusion term into the neural field, which models direct,electrical connections. We extend known and prove new sun-star calculus resultsfor delay equations to be able to include diffusion and explicitly characterisethe essential spectrum. For a certain class of connectivity functions in theneural field model, we are able to compute its spectral properties and thefirst Lyapunov coefficient of a Hopf bifurcation. By examining a numericalexample, we find that the addition of diffusion suppresses non-synchronisedsteady-states, while favouring synchronised oscillatory modes.

Neural Field Models with Transmission Delays and Diffusion