Pattern formation in next generation neural field models

神经场模型通常用于描述脑组织水平上的波传播和凹凸吸引子。尽管这些模型是由生物学驱动的，但它们本质上是现象学的。它们建立在假设神经组织在接近同步的状态下工作，因此无法解释模式潜在同步性的变化。在研究种群同步时，通常使用尖峰神经网络模型。不幸的是，众所周知，这些高维模型很难获得洞察力。这篇论文将考虑一个二次积分点火神经元网络，它最近被证明在没有空间分量的情况下允许精确的平均场描述。此外，它还将考虑对能够解释非局部空间相互作用的连续统模型的扩展。模式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)和直接数值模拟相结合的方法进行分析。对于振荡态，将进行相位-幅度分析。本论文将关注标准(现象学)神经场模型中的时空模式。

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.