Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

Abstract

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

“…6). The underlying mechanisms are qualitatively similar to these described in [72,73] involving the standard SAOs discussed above, and will not be discussed further in the context of this paper. • Mixed-mode oscillatory (MMO) patterns (e.g., Fig.…”

“…Here we follow previous work [35,36,73] and focus on the dynamics of the one-cluster globally coupled system (7) with σ 1 = 1 (σ 2 = 0). This is not likely to be a realistic situation, but it provides information about how the global feedback affects the oscillatory dynamics and the occurrence of the canard phenomenon in an autonomous system.…”

“…For this to happen, the forcing effects should not interfere with the canard phenomenon for each oscillator. A richer repertoire of intermediate patterns that are not "purely LAO" or "purely SAO" are expected to result from the complex interactions between oscillators as it happens for other systems [72,73].…”

“…In a phase-locked two-cluster pattern the two oscillators exhibit LAOs. In some cases, they may also exhibit MMOs, which typically reflect the effects of the network connectivity (e.g., inhibition transiently pushing the activator down or terminating an oscillation before it reaches high enough values), but may also reflect the interaction between the connectivity and the intrinsic canard structure [76] of the individual oscillators [72,73].…”

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

“…Typical examples are the phase-locked (e.g., antiphase) oscillatory two-cluster patterns where each cluster exhibits LAOs or MMOs. The latter typically reflect the effects of the network connectivity (e.g., inhibition transiently pushing the activator down or terminating an oscillation before it reaches high enough values) and/or the interaction between the connectivity and the intrinsic canard structure [76] of the individual oscillators [72,73].…”