Alternating pattern formation in oscillatory media with spatially uniform and nonuniform global feedback

Alternating (temporally period-2) stripes, clusters, and labyrinths have been observed in chemical and biochemical reactions, in Min-protein dynamics in E. coli and in vitro systems, and in calcium alternans in cardiac myocytes. However, the mechanisms underlying the formation of these spatiotemporal patterns and the transitions among them remain incompletely understood. In this article, we use an amplitude-equation (AE) approach to perform theoretical analysis and numerical simulations to elucidate the mechanisms of the alternating pattern formation and selection, and we then use reaction-diffusion models to validate the predictions of the AE model. It was shown previously that in chemical reactions and cardiac myocytes, a global feedback loop plays a key role in the formation of alternating clusters. Based on previous AE models developed to describe cardiac alternans dynamics, we first develop an AE model that incorporates either a spatially uniform global feedback (UGF) loop or a spatially nonuniform global feedback (NUGF) loop. The NUGF is represented by a spatially Gaussian-weighted function. We then use the AE model to conduct nonlinear stability analysis to reveal the mechanisms underlying the emergence of alternating patterns under different feedback conditions. Under positive UGF, no patterns can form. When the UGF is negative, the stable patterns are only multiple stripes and circular clusters. For stripe patterns, they may be spatially periodic or random as long as they satisfy the requirements for pattern selection. For circular clusters, a medium can support one or multiple circular alternating clusters of the same radius depending on the size of the medium. Under NUGF, in addition to stripe and circular-cluster patterns, lateral instability arises when the width of the Gaussian function falls within an intermediate range, leading to transitions from stripes or circular clusters to labyrinths. Other initial-condition-dependent patterns are also observed under NUGF. Under UGF, the width of the stripe can be periodic or random, but under NUGF, the patterns become less random in space as the feedback strength increases. Finally, we incorporate the UGF and NUGF loops into several reaction-diffusion models, including the Belousov-Zhabotinsky reaction model, the carbon monoxide oxidation model, and the periodically paced FitzHugh-Nagumo model. These models successfully reproduce the alternating pattern dynamics predicted by the AE model, validating the theoretical predictions.