The relationship between two fast/slow analysis techniques for bursting oscillations
The relationship between two fast/slow analysis techniques for bursting oscillations
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DOI：
10.1063/1.4766943
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发表时间：
2012-12-01
期刊：
影响因子：
2.9
通讯作者：
Bertram, Richard
中科院分区：
文献类型：
--
作者：
Teka, Wondimu;Tabak, Joel;Bertram, Richard
可兴奋系统中的爆发性振荡反映了多时间尺度的动态。通常通过将方程分成快速子系统和慢速子系统来在数学模型中研究这些振荡。通常，人们将慢变量视为快子系统的参数并研究该子系统的分叉结构。它具有 z 曲线（静止分支）和 Hopf 分岔等关键特征，从而产生周期性尖峰解的分支。在垂体细胞爆发模型中，我们最近使用了一种不同的方法，重点关注慢子系统的动力学。这种方法的特征是折叠节点奇点和临界流形。在本文中，我们研究了两种分析技术的关键结构之间的关系。我们发现，两快/一慢分解的 z 曲线和 Hopf 分岔分别与一快/两慢分解的电压零斜线和折叠节点奇异性密切相关。它们在双奇点极限下变得相同，其中电压无限快而钙无限慢。 (C) 2012 年美国物理研究所。 [http://dx.doi.org/10.1063/1.4766943]
Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4766943]