Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators

Abstract

Abstract. Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network by a network of excitable three-cluster states. The resulting "excitable network" has the same structure as the heteroclinic network and navigation around the excitable network is also possible by applying large impulsive inputs. We also discuss features that have implications for related models of neural activity.

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

“…Nonetheless, the bifurcations described are for the most part generic in the context of the symmetries present and hence are both robust and independent of exact choice of coupling function. Consideration of more general coupling functions can certainly give rise to dynamics that is not visible in (2); for example [6] consider a similar system of identical oscillators with an extra parameter β, g(x) = sin(x + α) + r sin(2x + β), and find other nontrivial types of clustering appear for larger N compared to the case β = 0. They also find chaotic attractors in the case N = 5 that we do not find for N ≤ 4.…”

“…In the ξ-plane a symmetry of the system system (6) corresponds to a symmetry of an equilateral triangle. The in-phase solution (origin) and the manifold of antiphase solutions M (3) are particularly significant in organizing the bifurcation behaviour of (6). Note that the latter consists of a point in each invariant triangle…”

Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions

“…[11,12] and references therein). It appears that randomly chosen initial states in the course of evolution can eventually become identical, and the final configuration consists of clusters of identically equal states.…”

Neuronal Assembly Dynamics in Supervised and Unsupervised Learning Scenarios

“…In particular, Ashwin et al (2007) and Wordsworth & Ashwin (2008) showed, when adopting a specific g, that the model is able to display heteroclinic cycles, a fundamental mechanism of cognition according to some authors (Ashwin & Timme, 2005;Rabinovich et al, 2012). Additionally, Orosz and collaborators (Orosz et al, 2009) demonstrated how to design g so that the network organizes itself in an arbitrary number of stable clusters with a given phase relationship between clusters.…”