Dynamics of Structured Networks of Winfree Oscillators

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.

Volcano transition in populations of phase oscillators with random nonreciprocal interactions

“…However, nonreciprocal interactions are found almost everywhere, from aggregates of neurons to self-motile active particles, see, e.g., [13]. Populations of phase oscillators with asymmetric couplings are found in models inspired in neuroscience [14,15], society [16], hydrodynamically coupled flagella [17], etc. The effect of nonreciprocity on glassy phases in the context of synchronization is attracting attention [18] but remains scarcely explored; particularly in comparison to random neural networks, see, e.g., the discussion in [19,20] and references therein.…”

The effects of degree distributions in random networks of Type-I neurons

“…Increasing the width of the degree distribution decreases the value of g at which collective oscillations start. This bifurcation is reminiscent of that which occurs in all-to-all connected networks of Winfree oscillators [51]: increasing the coupling strength causes the onset of oscillations through a Hopf bifurcation [30,52,53]. As is also seen in networks of Winfree oscillators, decreasing ∆ (the level of heterogeneity) has the same effect as increasing g, producing oscillations via a Hopf bifurcation (not shown).…”

“…Note that under the transformation V = tan (θ/2) a network of theta neurons is exactly equivalent to a network of quadratic integrateand-fire neurons with infinite threshold and reset values [29]. We now proceed to analyse the network dynamics, using ideas similar to those in [9,13,21,22,30].…”

Global and Local Reduced Models for Interacting, Heterogeneous Agents

“…Secondly, in the example considered the oscillators synchronized so that the state of an oscillator is a smooth function of the model heterogeneity. However, for many networks such synchrony does not occur, but the distribution of the state of an oscillator is a smooth function of the model heterogeneity [104][105][106] . In such cases it would be interesting to learn the dynamics governing the lowest few moments of these distributions.…”