Phase Equations For Relaxation Oscillators

SIAM Journal on Applied Mathematics (2000), 60:1789-1805

Eugene M. Izhikevich

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

Abstract. We use Malkin theorem to derive phase equations for general networks of weakly connected relaxation oscillators. We find an explicit formula for the connection functions when the oscillators have one-dimensional slow variables. The functions are discontinuous in the relaxation limit, which provides a simple alternative illustration to the major conclusion of the FTM (Fast Threshold Modulation) theory by Kopell and Somers that synchronization of relaxation oscillators has properties that are quite different from those of smooth (non-relaxation) oscillators. We use Bonhoeffer -- Van Der Pol relaxation oscillators to illustrate the theory numerically.

Keywords: Weakly connected oscillators, FTM (fast threshold modulation), synchronization, Class 2 excitability, pulse-coupled oscillators

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