We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.

中文翻译：

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尖刺神经元网络中的颠簸和震荡

我们在二次积分和发射神经元的空间扩展网络的精确平均场描述中研究局部模式。我们研究了局部解决方案（所谓的颠簸）的存在和稳定性的条件，并给出参数范围的解析估计，其中当一个或多个微观网络参数发生变化时，这些解存在于参数空间中。我们为模型方程式开发了Galerkin方法，从而可以对固定和时间周期的空间扩展解进行数值分叉分析。我们研究的多个凸块，其被布置成蛇和梯子分叉结构如果均相或非均相突触内核适当地选择组成图案的出现。此外，我们在存在外部强迫的情况下，以及在自主的，循环耦合的兴奋性和抑制性网络中检查时间周期的，空间局部化的解决方案（oscillons）。在这两种情况下，我们都观察到周期加倍级联导致混沌振荡。