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In this chapter, we analyze multicluster states in a network of adaptively coupled phase oscillators. Multicluster states are composed of several one-clusters with distinct frequencies. Starting from random initial conditions, our numerical simulations show two different types of states. These are the splay and the antipodal type multicluster states. A third mixed type multicluster state is found by specially prepared initial conditions. For all these states the collective motion of oscillators, the shape of the coupling structure, and the interaction between the frequency clusters are presented in detail. It turns out that due to the adaptive dynamics of the coupling weights, the oscillators are able to form groups of strongly connected units. The interaction between the groups is weak compared to the interaction within the groups. We provide an analytical description for the existence of multicluster states by using methods from asymptotic analysis. Employing the fact of weak inter-cluster coupling, we find stability conditions for multicluster states.

In this chapter, we analyze the phenomenon of frequency clusters in a population of coupled Hodgkin–Huxley neurons with spike timing-dependent plasticity. We show their persistence against external noise and introduce a phenomenological phase oscillator model to understand their emergence. We observe that the splitting of a neural population to a few clusters that are synchronized at different frequencies may lead to a slow waxing and waning of the mean field amplitude and might relate to low-frequency modulation found in brain signals like local field potentials (LFP) or electroencephalographic (EEG) signals.

In this chapter, 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. A special type of multicluster states are solitary states which have been observed in a plethora of dynamical systems. 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.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit., 16 pages, 8 figures

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in: Berner, R., Yanchuk, S., Maistrenko, Y., & Schöll, E. (2021). Generalized splay states in phase oscillator networks. In Chaos: An Interdisciplinary Journal of Nonlinear Science (Vol. 31, Issue 7, p. 073128). AIP Publishing. https://doi.org/10.1063/5.0056664 and may be found at https://doi.org/10.1063/5.0056664., Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized m-splay states constituting a special subclass of phase-locked states with vanishing mth order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto-Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models. Published under an exclusive license by AIP Publishing.

We report the phenomenon of frequency clustering in a network of Hodgkin-Huxley neurons with spike timing-dependent plasticity. The clustering leads to a splitting of a neural population into a few groups synchronized at different frequencies. In this regime, the amplitude of the mean field undergoes low-frequency modulations, which may contribute to the mechanism of the emergence of slow oscillations of neural activity observed in spectral power of local field potentials or electroencephalographic signals at high frequencies. In addition to numerical simulations of such multi-clusters, we investigate the mechanisms of the observed phenomena using the simplest case of two clusters. In particular, we propose a phenomenological model which describes the dynamics of two clusters taking into account the adaptation of coupling weights. We also determine the set of plasticity functions (update rules), which lead to multi-clustering.

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.