A Variational Method for Analyzing Stochastic Limit Cycle Oscillators
A Variational Method for Analyzing Stochastic Limit Cycle Oscillators
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DOI：
10.1137/17m1155235
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发表时间：
2018-01-01
影响因子：
2.1
通讯作者：
MacLaurin, James N.
中科院分区：
文献类型：
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作者：
Bressloff, Paul C.;MacLaurin, James N.
本文介绍了一种分析高斯噪声驱动下R-d中极限环振子的变分方法。这使我们能够推导出解的振幅和相位的精确随机微分方程，这些方程在数阶（Cbc(-1)）上是精确的，其中c是噪声的振幅，b是横向波动的衰减幅度。在变分框架内，幅相分解的不同选择对应于内积空间R-d的不同选择。对于具体问题，我们采用加权欧几里得范数，使得最小化方案通过使用Floquet向量将完整解投影到极限环上来确定相位。由于振幅和相位方程之间存在耦合，即使在弱噪声极限下，随机轨迹偏离极限环邻域的概率很小，但非零。我们使用振幅和相位方程来限定它这样做的概率：发现系统离开振荡器的一个邻域所需的典型时间为exp（Cb(-1)）。我们还展示了变分方法如何为计算随机相位提供一个数值上易于处理的框架，我们使用神经元的Morris-Lecar模型的修改版本来说明这一点。
We introduce a variational method for analyzing limit cycle oscillators in R-d driven by Gaussian noise. This allows us to derive exact stochastic differential equations for the amplitude and phase of the solution, which are accurate over times of order (Cbc(-1)), where c is the amplitude of the noise and b the magnitude of decay of transverse fluctuations. Within the variational framework, different choices of the amplitude-phase decomposition correspond to different choices of the inner product space R-d. For concreteness, we take a weighted Euclidean norm, so that the minimization scheme determines the phase by projecting the full solution onto the limit cycle using Floquet vectors. Since there is coupling between the amplitude and phase equations, even in the weak noise limit, there is a small but nonzero probability of a rare event in which the stochastic trajectory makes a large excursion away from a neighborhood of the limit cycle. We use the amplitude and phase equations to bound the probability of it doing this: finding that the typical time the system takes to leave a neighborhood of the oscillator scales as exp(Cb epsilon(-1)). We also show how the variational method provides a numerically tractable framework for calculating a stochastic phase, which we illustrate using a modified version of the Morris-Lecar model of a neuron.