Departamento de Matemática Aplicada and IUMA, University of Zaragoza, E-50009 Zaragoza, Spain.
Departamento de Matemáticas, University of Oviedo, E-33007 Oviedo, Spain.
Chaos. 2020 May;30(5):053132. doi: 10.1063/1.5138919.
Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme.
突发现象存在于各种各样的快-慢系统中。在本文中，我们考虑 Hindmarsh-Rose 神经元模型，在该模型中，正如文献中所指出的，爆发动力学涉及同宿分叉。然而，全局同宿结构远未被完全理解。在一个三参数空间中工作，我们的数值分析结果显示了一个复杂的分岔图集，它从奇异极限扩展到快速-缓慢视角不再适用的区域。基于这些信息，我们提出了一个全局理论描述。在快速-缓慢状态下，同宿分岔面的余维一表面彼此之间的距离呈指数级接近。值得注意的是，根据这些表面的特定性质，我们展示了 Hindmarsh-Rose 模型如何在适当的二维切片中考虑三参数空间时表现出同宿分岔的孤岛。另一方面，这些同宿分岔面包含对应于显示附加退化的参数值的曲线。这些余维二分叉曲线组织了与尖峰添加过程相关的分叉，它们的行为类似于“书本的脊骨”，收集了周期性轨道分叉的“页面”。根据参数空间的探索方式，同宿现象可能不存在或相距甚远，但它们在爆发动力学中的组织作用是毋庸置疑的，因为所涉及的分叉是在其中产生的。这在全局分析和提出的理论方案中得到了展示。
