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We analyze oscillatory instabilities for a coupled PDE-ODE system modelingthe communication between localized spatially segregated dynamically activesignaling compartments that are coupled through a passive extracellular bulkdiffusion field in a bounded 2-D domain. Each signaling compartment is assumedto secrete a chemical into the extracellular medium (bulk region) and it canalso sense the concentration of this chemical in the region around itsboundary. This feedback from the bulk region, resulting from the entirecollection of cells, in turn modifies the intracellular dynamics within eachcell. In the limit where the signaling compartments are circular disks with asmall common radius $\epsilon \ll 1$ and where the bulk diffusivity isasymptotically large, a matched asymptotic analysis is used to reduce thedimensionless PDE-ODE system into a nonlinear ODE system with global coupling.For Sel'kov reaction kinetics, this ODE system for the intracellular dynamicsand the spatial average of the bulk diffusion field is then used to investigateoscillatory instabilities in the dynamics of the cells that are triggered dueto the global coupling. In particular, numerical bifurcation software on theODEs is used to study the overall effect of coupling defective cells (cellsthat behave differently from the remaining cells) to a group of identicalcells. Moreover, when the number of cells is large, the Kuramoto orderparameter is computed to predict the degree of phase synchronization of theintracellular dynamics. Quorum sensing behavior, characterized by the onset ofcollective behavior in the intracellular dynamics as the number of cellsincreases above a threshold, is also studied. Our analysis shows that the cellpopulation density plays a dual role of triggering and then quenchingsynchronous oscillations in the intracellular dynamics.

Localized Signaling Compartments in 2-D Coupled by a Bulk Diffusion Field: Quorum Sensing and Synchronous Oscillations in the Well-Mixed Limit