SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 826-852, January 2021.
In this paper, we investigate a family of coupled excitable cells, using a Kuramoto-type model with either fixed excitability and Gaussian (dynamic) noise or a random distribution of the excitability. In both cases, we reduce the coupled system to a low-dimensional system using mean field approaches such as the Ott--Antonsen ansatz. In the case of a Cauchy distribution of excitability, we prove that with pure sinusoidal coupling, there can be no oscillations. However, if the excitability distribution has faster decay or the noise is Gaussian, then we show that there are oscillations and that they occur in a very specific manner organized around a Takens--Bogdanov bifurcation and a degenerate homoclinic bifurcation. We show that if the coupling is slightly more general, then even a Cauchy distribution is able to generate oscillations. Finally, we rescale the reduced equations in the small heterogeneity limit and show the common dynamics in these different models.

中文翻译：

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耦合激励系统中的噪声驱动振荡

SIAM应用动力系统杂志，第20卷，第2期，第826-852页，2021年1月。
在本文中，我们使用具有固定兴奋性和高斯（动态）噪声或兴奋性随机分布的Kuramoto型模型，研究了耦合的兴奋性细胞家族。在这两种情况下，我们都使用Ott-Antonsen ansatz等均场方法将耦合系统简化为低维系统。在柯西分布的可激发性的情况下，我们证明了在纯正弦耦合的情况下，不会有振荡。但是，如果兴奋性分布具有更快的衰减或噪声是高斯分布，则表明存在振荡，并且振荡以围绕Takens-Bogdanov分叉和简并同斜分叉的非常特定的方式发生。我们表明，如果耦合更为普遍，那么即使是柯西分布也能够产生振荡。最后，