The stability of fixed points for a Kuramoto model with Hebbian interactions

Abstract

We consider a variation of the Kuramoto model with dynamic coupling, where the coupling strengths are allowed to evolve in response to the phase difference between the oscillators, a model first considered by Ha, Noh, and Park. We demonstrate that the fixed points of this model, as well as their stability, can be completely expressed in terms of the fixed points and stability of the analogous classical Kuramoto problem where the coupling strengths are fixed to a constant (the same for all edges). In particular, for the "all-to-all" network, where the underlying graph is the complete graph, the problem reduces to the problem of understanding the fixed points and stability of the all-to-all Kuramoto model with equal edge weights, a problem that is well understood.

Synchronization and multi-cluster capabilities of oscillatory networks with adaptive coupling

“…In [33], a relation between adaptive and non-adaptive Kuramoto networks is provided and stability properties of fixed points of adaptive Kuramoto networks are studied. Also, numerical simulations with periodically oscillating behavior of the coupling strength and partially synchronized oscillator phases are presented there.…”

Path-dependent Dynamics Induced by Rewiring Networks of Inertial Oscillators

“…A third possible area for future study is to consider how path-dependence arises in a system of inertial oscillators adhering to an adaptive rewiring scheme, where the states of the oscillators themselves inform the network rewiring process [14,19,20,43,[50][51][52]. In particular, investigating path-dependence in systems of inertial oscillators under Hebbian or anti-Hebbian adaptive rewiring [53][54][55] may be insightful for understanding whether network evolution path-dependence plays a significant role in the development of neuronal networks. It is well known that the human brain undergoes a variety of structural changes during development [56][57][58], not only in synaptic density but also in topological characteristics such as degree heterogeneity, clustering, and modularity [59,60].…”

Synchronicity transitions determine connection fluctuations in a model of coupled oscillators with plasticity

“…Numerous variants of the Kuramoto model have been examined (Acebrón et al, 2005). Some variants incorporate features such as plasticity, which allows for changes in the interaction strength between oscillators (Aoki & Aoyagi, 2009;Bronski et al, 2017;Fialkowski et al, 2023;Ha et al, 2018Ha et al, , 2016Niyogi & English, 2009;Park et al, 2021;Picallo & Riecke, 2011;Poyato, 2019;Ren & Zhao, 2007;Seliger et al, 2002). Furthermore, in the Kuramoto model, the EI balance is conceptualized as a balance between two conflicting forces (EI-Kuramoto model), leading to synchronized oscillations induced through feedback mechanisms (Montbrió & Pazó, 2018) and transfer between different dynamic states, synchronized, bistable, and desynchronized, by modulating the EI-balance (Kuroki & Mizuseki, 2024).…”