Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of interest satisfies a parametrised set of algebraic equations, and branches of solutions are followed as the parameter is varied. An effective way to do this is with pseudo-arclength continuation. We give an introduction to pseudo-arclength continuation and then demonstrate its use in investigating the behaviour of a number of models from the field of computational neuroscience. The models we consider are high dimensional, as they result from the discretisation of neural field models-nonlocal differential equations used to model macroscopic pattern formation in the cortex. We consider both stationary and moving patterns in one spatial dimension, and then translating patterns in two spatial dimensions. A variety of results from the literature are discussed, and a number of extensions of the technique are given.

中文翻译：

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高维神经模型的数值分叉理论。

数值分叉理论涉及随着参数的变化找到并遵循微分方程的某些类型的解，然后确定它们是否经历了任何分叉（行为的质变）。做到这一点的主要技术是数值连续，其中感兴趣的解满足参数化的代数方程组，并且随着参数的变化遵循解的分支。一种有效的方法是使用伪弧长延续。我们对伪弧长延续进行了介绍，然后展示了其在研究计算神经科学领域的许多模型的行为中的用途。我们认为的模型是高维的，因为它们是由神经场模型-用于模拟皮质中宏观图案形成的非局部微分方程离散化而产生的。我们在一个空间维度上同时考虑固定和移动模式，然后在两个空间维度上平移模式。讨论了来自文献的各种结果，并给出了该技术的许多扩展。