**Subcritical Elliptic Bursting of Bautin Type
**

*SIAM Journal on Applied Mathematics *(2000),
60:503-535

Eugene M. Izhikevich

*Systems Science Center, Box 7606,
Arizona State University,
Tempe, AZ 85287-7606. *

**Abstract.** 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 non-invertible) 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 limit cycle (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-of-phase synchronization is 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.

**Keywords:** Subcritical elliptic
burster, subcritical Andronov-Hopf bifurcation, double limit cycle bifurcation,
Bautin bifurcation, normal form, canonical model, slow passage effect,
weakly connected networks, fast threshold modulation, FM interactions,
FitzHugh-Rinzel model

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