分层激发网络的设计
Design of Hierarchical Excitable Networks
AI总结：
本文提出了一种分层激发网络的设计方法，通过构造具有零阈值的激发性连接和异宿性连接的分层结构，扩展了单纯形实现方法。
摘要：
我们提供了一种系统构造向量场的方法，使得动力学表现出对应于期望的分层连接结构的过渡。该结构由有限个有向图G₁,…,G_N（下层）以及另一个有N个顶点的有向图Γ（上层）组成。G₁,…,G_N的动力实现是异宿网络，可以看作是在给定状态集上的个体连接模式。Γ中的边对应于这些不同模式之间的过渡。在我们的构造中，通过Γ给出的连接不是异宿的，而是具有零阈值的激发性连接。这描述了两个不变集之间的动态过渡，其中第一个集的每个δ邻域都包含一个初始条件，其ω极限在第二个集内。因此，我们证明了一个定理，允许系统地创建上层为激发性、下层为异宿性的分层网络。我们的结果修改并扩展了Ashwin & Postlethwaite提出的单纯形实现方法。
英文摘要：
We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
