A Model For Diffusively Coupled Self-Oscillating Droplets With Delay

The Oregonator system, consisting in a set of three Ordinary Differential Equations (ODEs), is widely used in the literature to model the oscillatory Belousov-Zhabotinsky (BZ) chemical reaction. Recently, D’ambrosio et al. used adapted numerical methods to follow the apriori known qualitative behaviour of the solution and improve the quantitative matching with experiments. Budroni et al. employed a modified version of the Oregonator to model a network of diffusively coupled inorganic oscillators, confined in micro-compartments by means of a flow-focus microfluidic technique. This class of networks is effective for predicting and understanding the global dynamics of those systems where the diffusion of activatory or inhibitory signals regulates the communication among different individuals. By tuning the lamellarity and permeability of the BZ cell membrane, the relative contribution of the inhibitory and activatory coupling can be finely controlled to induce unprecedented synchronisation patterns. In this talk, we want to address the impact of time delays that, because of compartmentalisation constraints such as lamellarity, can affect the chemical communication among successive microoscillators. We therefore introduced a delay in the coupling term of the ODEs model thus obtaining a system of Delay Differential Equations (DDEs). We next explored the effect of varying this time-delayed feedback as a control parameter. Our contribution provides an alternative modelling approach that might improve the quantitative matching between in-silico simulations and experimental results and, more in general, it represents a further step towards understanding, predicting and classifying possible synchronization scenarios driven by communication delays in ensembles of diffusively coupled oscillators