Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System
Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System
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DOI：
10.1137/110860136
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发表时间：
2012-02
期刊：
影响因子：
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通讯作者：
M. Krupa;A. Vidal;M. Desroches;F. Clément
中科院分区：
文献类型：
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作者：
M. Krupa;A. Vidal;M. Desroches;F. Clément
在这项工作中，我们研究了 GnRH（促性腺激素释放激素）分泌模型中的混合模式振荡。该模型是一个幻影爆发器，由两个前馈耦合的 FitzHugh-Nagumo 系统组成，具有三个时间尺度。强迫系统（调节器）以最慢的规模演化，并通过移动强迫系统（秘书）的缓慢零斜线来起作用。存在三种动力学模式：脉动（瞬态弛豫振荡）、波动（准稳态）和与慢零斜线通过快速零斜线折叠点相关的小振荡。利用系统的上述功能，我们得出了各种减少的结果。我们得到两个结果；一个是关于参数范围折叠附近的局部动力学，对应于小振荡的存在，另一个是关于全局动力学，更具体地说是关于吸引极限环的存在。我们的局部结果是在具有附加时间尺度的折叠节点的情况下对小鸭翼和旋转扇区的严格表征，该特征允许清晰的几何论证。全局结果给出了一个有吸引力的独特极限环的存在，在某些参数范围内，即使在鸭式爆炸过程中，该极限环仍然具有吸引力和独特性。
In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.