The formation of oscillating phase clusters in a network of identical Hodgkin-Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition-through [Formula: see text] (possibly perturbed) period-doubling and subsequent bifurcations-to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neurons can lead to a combinatorially large number of different, yet similar "fine" states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron's "identity" (its own value of the heterogeneity parameter) and its dynamical state. For single- and double-cluster states we demonstrate an effective coarse-graining approach that uses the Polynomial Chaos expansion to succinctly describe the dynamics by these quickly established "identity-state" correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient computations of the neuron ensemble dynamics.

中文翻译：

---

异构耦合神经元的粗粒聚类动力学。

研究了相同的霍奇金-赫克斯利（Hodgkin-Huxley）神经元网络中振荡相簇的形成及其动态行为。神经元以所有方式突触耦合，但是突触耦合特征时间在连接之间是异质的。在N个神经元的网络中，这种异质性以规定的随机变量为特征，振荡的单簇状态可以通过以下方式转换：集群状态。在详细和粗粒度级别上都对聚类动力学行为进行了计算研究，并提出了一种数值方法，该方法可以研究任意大小的网络中的粗粒度动力学。在形成的许多簇状态中，由几乎相等的子网大小组成的双簇被认为是稳定的；有趣的是，每个双聚类组件中的异质性参数都倾向于与整个网络中的随机变量一致：给定一个双聚类状态，排列神经元的动态变量可能导致大量不同的组合，相似的“精细”状态在粗粒度级别上看起来几乎是相同的。对于弱的异质性，我们发现在每个簇中，神经元的“身份”（异质性参数自己的值）与其动态状态之间的相关性迅速发展。对于单集群状态和双集群状态，我们演示了一种有效的粗粒度方法，该方法使用多项式混沌展开来通过这些快速建立的“身份状态”相关性简洁地描述动力学。在无方程式框架内利用这种粗粒度方法来执行神经元集合动力学的有效计算。