Subcritical Elliptic Bursting of Bautin Type

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 double limit cycle bifurcation, the burster is said to be of subcritical elliptic type. When the fast subsystem is near a Bautin (generalized Hopf) point, both bifurcations occur for nearby values of the slow variable, and the repetitive spiking has small amplitude. We refer to such an elliptic burster as being of local Bautin type. First, we prove that any such burster can be converted into a canonical model by a suitable continuous (possibly noninvertible) change of variables. We also derive a canonical model for weakly connected networks of such bursters. We find that behavior of such networks is quite different from the behavior of weakly connected phase oscillators, and it resembles that of strongly connected relaxation oscillators. As a result, such weakly connected bursters need few (usually one) bursts to synchronize. In-phase synchronization is possible for bursters having quite different quantitative features, whereas out-ofphase synchronization may be difficult to achieve. 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 model to evaluate how studying local Bautin bursters can contribute to our understanding of the phenomena of subcritical elliptic bursting.