In many micro- and macro-scale systems, collective dynamics occurs from the coupling of small spatially segregated, but dynamically active, units through a bulk diffusion field. This bulk diffusion field, which is both produced and sensed by the active units, can trigger and then synchronize oscillatory dynamics associated with the individual units. In this context, we analyze diffusion-induced synchrony for a class of cell-bulk ODE–PDE system in \({\mathbb {R}}^2\) that has two spatially segregated dynamically active circular cells of small radius. By using strong localized perturbation theory in the limit of small cell radius, we calculate the steady-state solution and formulate the linear stability problem. For Sel’kov intracellular reaction kinetics, we analyze how the effect of bulk diffusion can trigger, via a Hopf bifurcation, either in-phase or anti-phase intracellular oscillations that would otherwise not occur for cells that are uncoupled from the bulk medium. Phase diagrams in parameter space where these oscillations occur are presented, and the theoretical results from the linear stability theory are validated from full numerical simulations of the ODE–PDE system. In addition, the two-cell case is extended to study the onset of synchronous oscillatory instabilities associated with an infinite hexagonal arrangement of small identical cells in \({\mathbb {R}}^2\) with Sel’kov intracellular kinetics. This analysis for the hexagonal cell pattern relies on determining a new, computationally efficient, explicit formula for the regular part of a certain periodic reduced-wave Green’s function.

在许多微观和宏观尺度的系统中，集体动力学是通过空间扩散场将小的空间分离但动态活跃的单元耦合而产生的。由主动单元产生和感测的整体扩散场可以触发然后同步与各个单元相关联的振荡动力学。在这种情况下，我们分析了\（{\ mathbb {R}} ^ 2 \）中一类细胞大量ODE-PDE系统的扩散诱导同步性。具有两个在空间上隔离的，半径较小的动态活动圆形单元。通过在小单元半径的极限内使用强局部扰动理论，我们计算出稳态解并提出了线性稳定性问题。对于Sel'kov细胞内反应动力学，我们分析了本体扩散的影响如何通过Hopf分叉触发同相或反相细胞内振荡，否则，与本体介质解耦的细胞将不会发生同相或反相细胞内振荡。给出了出现这些振荡的参数空间中的相位图，并通过ODE–PDE系统的完整数值模拟验证了线性稳定性理论的理论结果。此外，\（{{mathbb {R}} ^ 2 \）具有Sel'kov细胞内动力学。对六角形单元模式的这种分析依赖于为某个周期性减小波格林函数的正则部分确定一个新的，计算效率高的显式公式。