Adaptive Oscillator Networks with Conserved Overall Coupling: Sequential Firing and Near-Synchronized States

TL;DR

This paper studies adaptive oscillator networks with conserved total coupling, revealing multiple phase-locked states, including near-synchronized and splay states, with analytical insights into their dynamics and stability.

Contribution

It introduces a model of adaptive oscillators with conserved coupling, identifying new stable states and analyzing their properties analytically.

Findings

Multiple phase-locked states identified.

Near-synchronized states depend weakly on coupling.

Large networks develop loop-dominated topologies.

Abstract

Motivated by recent observations in neuronal systems we investigate all-to-all networks of non-identical oscillators with adaptive coupling. The adaptation models spike-timing-dependent plasticity in which the sum of the weights of all incoming links is conserved. We find multiple phase-locked states that fall into two classes: near-synchronized states and splay states. Among the near-synchronized states are states that oscillate with a frequency that depends only very weakly on the coupling strength and is essentially given by the frequency of one of the oscillators, which is, however, neither the fastest nor the slowest oscillator. In sufficiently large networks the adaptive coupling is found to develop effective network topologies dominated by one or two loops. This results in a multitude of stable splay states, which differ in their firing sequences. With increasing coupling strength their frequency increases linearly and the oscillators become less synchronized. The essential features of the two classes of states are captured analytically in perturbation analyses of the extended Kuramoto model used in the simulations.