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.

Abstract. 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 ``subHopf/fold cycle'' burster). Here we study synchronization dynamics of 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, 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.

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

Full text in PDF file (1.5M), Postscript file (9M)

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.

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