量化噪声吸引子：从异宿到可激发网络
Quantifying noisy attractors: from heteroclinic to excitable networks
摘要：
动力系统的吸引子可以是相空间中的网络，其可以是异宿的（其中简单不变集之间存在动力学连接）或可激发的（其中扰动阈值需要被越过到“节点”之间的动力学连接）。这样的网络吸引子可以在访问的相空间区域和网络周围的转换序列方面显示出对噪声的高度敏感性。这两种类型的网络是密切相关的-一个可以直接分叉到另一个。在本文中，我们试图量化加性噪声对这种网络吸引子的影响。噪声增加了网络被探索的平均速率，并可能导致网络周围的“宏观”随机运动。我们进行了一个渐近分析的局部行为的逃逸模型附近的异宿/可兴奋节点的限制噪声$η\rightarrow 0^+$作为一个模型的平均停留时间$T$附近的平衡。我们还探讨了各向异性噪声存在下网络节点之间的转移概率。对于低水平的噪声，数值结果表明，一个（异宿或兴奋）网络可以近似实现任何一组的转移概率和任何足够大的平均停留时间在给定的节点。我们表明，这可以很好地模拟在我们的示例网络中的多个独立的逃逸过程，其中第一次逃逸的方向决定了过渡。这表明设计具有任意马尔可夫转移概率和停留时间的噪声网络吸引子是可行的。
英文摘要：
Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between "nodes"). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related---one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored, and can result in "macroscopic" random motion around the network. We perform an asymptotic analysis of local behaviour of an escape model near heteroclinic/excitable nodes in the limit of noise $η\rightarrow 0^+$ as a model for the mean residence time $T$ near equilibria. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realise any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modelled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.
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