Abstract
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 (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.
Highlights
A network of oscillators can form sets of sub-networks oscillating with phase lags among them [1]; these are often referred to as phase clusters
In this paper we study a specific type of clustering dynamics observed in synaptically all-to-all coupled networks of identical HH neurons, but for which certain synaptic coupling parameters slightly vary across the population, making the whole network heterogeneous
We demonstrate a coarse-graining approach enabling the analysis of the lowdimensional dynamics of single- and double-cluster states, which provides an efficient way of studying the coarse-grained clustering dynamics of an arbitrarily large network
Summary
A network of oscillators can form sets of sub-networks oscillating with phase lags among them [1]; these are often referred to as phase clusters. The dynamical nature of the transitions between different cluster states in phase-reduced oscillator models and slow switching along the heteroclinic orbits involved have been discussed [3, 14,15,16,17,18,19] In these studies of cluster dynamics, it is often assumed that the constituting entities are identical and/or the coupling strength is weak, allowing dimensional reduction via complete synchronization within each cluster and/or through the phase reduction procedure. The paper is organized as follows: The model and the parameter values used in it are described, and some observations on the clustering dynamics in networks of heterogeneously coupled neurons are presented in Sect.
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