Stopping waves: geometric analysis of coupled bursters in an asymmetric excitation field

Bursting is a type of electrical activity seen in many neurons and endocrine cells where episodes of action potential firing are interspersed by silent phases. Here, we investigate partial synchrony and wave propagation in a population of square-wave bursters. In particular, by using a prototypical polynomial bursting model and slow/fast bifurcation analysis, we study why electrically coupled model bursters typically synchronize very easily, as reflected in the tendency for simulated excitation waves to propagate far into the region of silent cells when an excitation gradient is imposed. Such simulation is inspired by, but does not reproduce, experimentally observed Ca2+ waves in pancreatic islets exposed to a glucose gradient. Our analyses indicate a possible modification of the model so that the excitation waves stop at the border between active and silent cells. We verify this property by simulations using such a modified model for a chain, and for a cubic cluster, of coupled cells. Furthermore, we show how our one- and two-parameter bifurcation analyses allow us to predict where the simulated waves stop, for both the original model and the modified version.