Synchronous oscillations for a coupled cell-bulk ODE–PDE model with localized cells on $${\mathbb {R}}^2$$

Abstract

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.