We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

中文翻译：

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由体扩散场耦合的二维局部信号隔间：混合极限下的群体感应和同步振荡

我们分析了耦合偏常微分方程 (PDE-ODE) 系统的振荡不稳定性，该系统模拟了局部空间分离的动态主动信号隔间之间的通信，这些隔间通过有界二维域中的被动细胞外散装扩散场耦合。假设每个信号隔室都将一种化学物质分泌到细胞外介质（大块区域）中，并且它还可以感知其边界周围区域中这种化学物质的浓度。这种来自大块区域的反馈，由整个细胞集合产生，反过来又改变了每个细胞内的细胞内动力学。在信号隔间是具有小公共半径的圆盘的极限内ε≪ 1 并且在体扩散率渐近较大的情况下，使用匹配渐近分析将无量纲 PDE-ODE 系统简化为具有全局耦合的非线性 ODE 系统。对于 Sel'kov 反应动力学，该 ODE 系统用于细胞内动力学和体扩散场的空间平均值，然后用于研究由于全局耦合而触发的细胞动力学中的振荡不稳定性。特别是，使用 ODE 上的数值分岔软件来研究耦合的整体效果有缺陷的细胞（与其余单元格行为不同的单元格）到一组相同的单元格。此外，当细胞数量较大时，计算仓本阶参数以预测细胞内动力学的相位同步程度。还研究了群体感应行为，其特征是随着细胞数量增加到阈值以上，细胞内动力学中集体行为的开始。我们的分析表明，细胞群密度在细胞内动力学中起着触发和淬灭同步振荡的双重作用。