Pattern formation in next generation neural field models

神经场模型通常用于描述大脑组织水平上的波传播和碰撞吸引子。虽然这些模型的动机是生物学，但本质上是现象学的。它们是建立在神经组织以近乎同步的方式运作的假设之上的，因此，不能解释潜在模式同步的变化。在研究群体内部同步时，习惯上使用尖峰神经网络模型。不幸的是，众所周知，这些高维模型很难从中获得洞察力。这篇论文将考虑一个二次积分和火神经元网络，它最近被证明在没有空间成分的情况下承认一个精确的平均场描述。此外，它将考虑扩展到一个连续体模型，可以解释非局部空间相互作用。模型中的模式将使用图灵不稳定性分析，弱非线性分析（通过多尺度方法）和直接数值模拟的混合分析。对于振荡状态，将进行相位振幅分析。本文将着重于超越标准（现象学）神经场模型的时空模式。

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.