THE SLOW PASSAGE THROUGH A HOPF-BIFURCATION - DELAY, MEMORY EFFECTS, AND RESONANCE
THE SLOW PASSAGE THROUGH A HOPF-BIFURCATION - DELAY, MEMORY EFFECTS, AND RESONANCE
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DOI：
10.1137/0149003
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发表时间：
1989-02-01
影响因子：
1.9
通讯作者：
RINZEL, J
中科院分区：
文献类型：
--
作者：
BAER, SM;ERNEUX, T;RINZEL, J
本文探讨分析和数值，在神经膜兴奋性的FitzHugh-Nagumo模型的背景下，一个有趣的现象，已被描述为延迟或记忆效应。当一个参数缓慢地通过一个Hopf分叉点，系统的响应从一个缓慢变化的稳态变为缓慢变化的振荡时，就会发生这种情况。定量观察发现，实现的过渡时，参数是远远超出预测的值从一个简单的分叉分析，忽略了动态方面的参数变化。描述了该时滞及其对参数变化速度的依赖性，研究了多参数模型的奇异极限。一个特别的是缓慢通过一个低频率的霍普夫分岔，系统的响应从一个缓慢变化的稳定状态，以缓慢变化的松弛振荡。我们发现，在这种情况下，振荡的开始表现出一个提前，而不是延迟。可以预期振荡的开始，但是小幅度噪声和接近谐振频率的周期性环境扰动可以减小延迟并破坏记忆效应。本文认为，确定性和随机方法将是重要的比较系统中的理论和实验结果的缓慢通过通过一个霍普夫分岔的基本机制的发病“振荡。
This paper explores analytically and numerically, in the context of the FitzHugh–Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described.The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset of oscillations exhibits an advance rather than a delay.This paper shows that in general delays in. the onset of oscillations may be expected but that small amplitude noise and periodic environmental perturbations of near resonant frequency may decrease the delay and destroy the memory effect. This paper suggests that both deterministic and stochastic approaches will be important for comparing theoretical and experimental results in systems where slow passage through a Hopf bifurcation is the underlying mechanism for the onset of 'oscillations.