Synchronization of Elliptic Bursters

Periodic bursting behavior in neurons is a recurrent transition between a quiescent state and repetitive spiking. When the transition to repetitive spiking occurs via a subcritical Andronov--Hopf bifurcation and the transition to the quiescent state occurs via fold limit cycle bifurcation, the burster is said to be of elliptic type (also known as a "subHopf/fold cycle" burster). Here we study the synchronization dynamics of weakly connected networks of such bursters. We find that the behavior of such networks is quite different from the behavior of weakly connected phase oscillators and resembles that of strongly connected relaxation oscillators. As a result, such weakly connected bursters need few (usually one) bursts to synchronize, and synchronization is possible for bursters having quite different quantitative features. We also find that interactions between bursters depend crucially on the spiking frequencies. Namely, the interactions are most effective when the presynaptic interspike frequency matches the frequency of postsynaptic oscillations. Finally, we use the FitzHugh--Rinzel, Morris--Lecar, and Hodgkin--Huxley models to illustrate our major results.