**Synchronization of Elliptic Bursters
**

*SIAM Review * (2001), 43:315-344

*Featured SIGEST paper* in the SIAM Review selected on the basis of its exceptional interest to the entire SIAM community.

Eugene M. Izhikevich

*The Neurosciences Institute,
10640 John Jay Hopkins Drive,
San Diego, CA, 92121. *

**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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An introduction by the Editors:This issue's SIGEST paper, "Synchronization of Elliptic Bursters," by Eugene M. Izhikevich, which first appeared in SIAM Journal of Applied Mathematics, volume 60 (2000), is an outstanding example of sophisticated mathematics with close ties to a scientific application area. The paper's topic is the synchronization dynamics of weakly connected networks of bursters. (Bursting occurs when neuron activity alternates between a quiescent state and repetitive spiking; bursting is usually caused by a slow-voltage or calcium-dependent process that may modulate fast spiking activity.) Identification, classification, and analysis of bursters are key elements in understanding the neuro-computational properties of brain cells.

An especially nice feature of the paper is its mixture of mathematics and pictures that illuminate the sometimes highly subtle differences in synchronization of individual spikes and bursts. (The striking cover of this issue of SIAM Review is based on Figure 1.6.) We thank the author for making the paper more accessible, and we urge interested readers to look at the Web sites mentioned in the paper to see animations of excitability and bursting.