Abstract
Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an “upstream” and a “downstream” oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.
Highlights
The behavior of networks of coupled oscillators is a topic of ongoing interest (Strogatz, 2000; Pikovsky et al, 2001; Arenas et al, 2008)
Since an oscillator’s dynamics are independent of its downstream oscillators, neither the r(out,out) nor the r(out,in) assortativities influence the overall network dynamics as shown in all the traces of Figures 7C,D. This dynamic interplay is quite different for a network with strong preferential attachment between high in-degree and high out-degree oscillators as when r(in,out) is positive (Figure 7B)
The upstream node of an attached pair may only integrate a small number of inputs, whose behavior is strikingly distinct from an oscillator with many inputs
Summary
The behavior of networks of coupled oscillators is a topic of ongoing interest (Strogatz, 2000; Pikovsky et al, 2001; Arenas et al, 2008). Each oscillator is assumed to have a phase response curve, a function of its own phase, which can be measured from individual neurons, for example Schultheiss et al (2011) and Netoff et al (2005) This describes how an oscillator’s phase changes as the result of input from other oscillators. In the limit N → ∞ the network is described by the probability density function f (θ , ω|k, t) where f (θ , ω|k, t)dθ dω is the probability that an oscillator with degree k has phase in [θ , θ + dθ ] and value of ω in [ω, ω + dω] at time t
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