Bifurcation mechanism for emergence of spontaneous oscillations in coupled heterogeneous excitable units

Biological systems are often composed of various heterogeneous excitable units. It is a fascinating problem to investigate how this heterogeneity affects collective behavior of biological systems. In this paper, to understand the effect of the unit heterogeneity on the dynamical mechanism for the onset of collective oscillatory behavior, we analyze coupled heterogeneous excitable units. We clarify how spontaneous oscillations emerge depending on the degree of heterogeneity of the units. With an increase in the coupling strength, the system undergoes a saddle-node on invariant circle bifurcation and a heteroclinic bifurcation. Based on bifurcation theory, we reveal that the order of the two bifurcations plays key roles in the mechanism of the emergence of spontaneous oscillations. In addition, we analytically show that when the system has a symmetric property, a 5th-order pitchfork bifurcation occurs instead of the two bifurcations. We also find that spontaneous oscillations are more likely to occur when the sizes of subpopulations of excitable units with different parameters are more balanced.