Desynchronization Transitions in Adaptive Networks

Abstract

Adaptive networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability approach for a large class of adaptive networks. This approach allows for reducing the synchronization problem for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. We show how the interplay between adaptivity and network structure gives rise to the formation of stability islands. Moreover, we report a desynchronization transition and the emergence of complex partial synchronization patterns induced by an increasing overall coupling strength. We illustrate our findings using adaptive networks of coupled phase oscillators and FitzHugh-Nagumo neurons with synaptic plasticity.

Synchronization in Networks With Heterogeneous Adaptation Rules and Applications to Distance-Dependent Synaptic Plasticity

“…In this work, we study the synchronization on networks with adaptive coupling weights, where the adaptation (plasticity) rule depends on the distance between oscillators (neurons). We consider the model of adaptively coupled phase oscillators, which has proven to be useful for understanding dynamics in neuronal systems with spike timing-dependent plasticity [77,79,48]. The model reads as follows:…”

“…For illustrative purposes, the coupling function is set throughout the paper to g(ϕ) −sin(ϕ + α)/N with the phase lag parameter α [80]. Such a phase lag can account for a small synaptic propagation delay [81,48]. For formal derivations, however, a generic coupling function is used.…”

“…In this section, we derive a framework for the local stability analysis of the synchronous states. We note that the master stability approach for homogeneous adaptations h ij h was introduced in [48,83]. Here we extend the methodology to heterogeneous adaptation rules.…”

“…Since its introduction, the master stability approach has been extended and refined for various complex systems [34][35][36][37][38][39][40][41][42], and methods beyond the local stability analysis have been developed [43][44][45][46][47]. More recently, the master stability approach has been extended to another class of oscillator networks with high application potential, namely adaptive networks [48].…”

“…We introduce the model in Section 2. Building on findings from [48], we develop a master stability approach in Section 3 that takes a heterogeneous adaptation rule in account. In Section 4.1, we provide an approximation of the structural eigenvalues that determine the stability of the synchronous state.…”