A theory of synchrony by coupling through a diffusive chemical signal

Highlights: • A PDE–ODE model that couples two active compartments is formulated and analyzed. • Phase diagrams where anti-phase and in-phase oscillations occur are found. • Analysis predicts a wild parameter range where stable in-phase oscillations occur. • Our theory of bulk-mediated oscillations is extended to a periodic chain of units. We formulate and analyze oscillatory dynamics associated with a model of dynamically active, but spatially segregated, compartments that are coupled through a chemical signal that diffuses in the bulk medium between the compartments. The coupling between each compartment and the bulk is due to both feedback terms to the compartmental dynamics and flux boundary conditions at the interface between the compartment and the bulk. Our coupled model consists of dynamically active compartments located at the two ends of a 1-D bulk region of spatial extent $2L$. The dynamics in the two compartments is modeled by Sel'kov kinetics, and the signaling molecule between the two-compartments is assumed to undergo both diffusion, with diffusivity $D$, and first-order, linear, bulk degradation. For the resulting PDE–ODE system, we construct a symmetric steady-state solution and analyze the stability of this solution to either in-phase synchronous or anti-phase synchronous perturbations about the midline $x=L$. The conditions for the onset of oscillatory dynamics, as obtained from a linearization of the steady-state solution, are studied using a winding number approach. Global branches of either in-phase or anti-phase periodic solutions, and their associated stability properties, are determined numerically. For the case of a linear coupling between the compartments and the bulk, with coupling strength $\beta$, a phase diagram, in the parameter space $D$ versus $\beta$ is constructed that shows the existence of a rather wide parameter regime where stable in-phase synchronized oscillations can occur between the two compartments. By using a Floquet-based approach, this analysis with linear coupling is then extended to determine Hopf bifurcation thresholds for a periodic chain of evenly-spaced dynamically active units. Finally, we consider one particular case of a nonlinear coupling between two active compartments and the bulk. It is shown that stable in-phase and anti-phase synchronous oscillations also occur in certain parameter regimes, but as isolated solution branches that are disconnected from the steady-state solution branch.