Abstract
In this article, we analyze a nonlocal ring network of adaptively coupled phase oscillators. We observe a variety of frequency-synchronized states such as phase-locked, multicluster and solitary states. For an important subclass of the phase-locked solutions, the rotating waves, we provide a rigorous stability analysis. This analysis shows a strong dependence of their stability on the coupling structure and the wavenumber which is a remarkable difference to an all-to-all coupled network. Despite the fact that solitary states have been observed in a plethora of dynamical systems, the mechanisms behind their emergence were largely unaddressed in the literature. Here, we show how solitary states emerge due to the adaptive feature of the network and classify several bifurcation scenarios in which these states are created and stabilized.
Highlights
Adaptive networks appear in many real-world applications
The coupling weights can change in response to the relative timings of neuronal spiking as it is the case for spiketiming dependent plasticity [3,4,5,6,7]
We restrict our analysis to one specific coupling structure, a nonlocally coupled ring, on which the coupling weights are adapted depending on the dynamics of the network
Summary
Adaptive networks appear in many real-world applications. One of the main motivations for studying such networks comes from the field of neuroscience where the weights of the synaptic coupling can adapt depending on the activity of the neurons that are involved in the coupling [1,2]. We restrict our analysis to one specific coupling structure, a nonlocally coupled ring, on which the coupling weights are adapted depending on the dynamics of the network. Adaptive Kuramoto-Sakaguchi type models have been shown to exhibit diverse complex dynamical behavior. Such a structure leads to significantly different frequencies of the clusters and, as a result, to their uncoupling This phenomenon is reported for adaptive networks of Morris-Lecar, HindmarshRose, and Hodgkin-Huxley neurons with either spike-timing-dependent plasticity or Hebbian learning rule [39,40,41]. Several types of synchronization patterns are found in the numerical simulations, depending on the values of α and β
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