Pattern formation in next generation neural field models

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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. This thesis will consider a network of quadratic integrate-and-fire neurons, which has recently been shown to admit an exact mean-field description in the absence of a spatial component. Moreover, it will consider the extension to a continuum model that can account for non-local spatial interactions. Patterns in the model will be analysed using a mixture of Turing instability analysis, weakly nonlinear analysis (via the method of multiple scales), and direct numerical simulation. For oscillatory states a phase-amplitude analysis will be pursued. The thesis will focus on the spatio-temporal patterns above and beyond those seen in a standard (phenomenological) neural field model.