Phase Equations for Relaxation Oscillators

We use the Malkin theorem to derive phase equations for 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 μ → 0, which provides a simple alternative illustration to the major conclusion of the fast threshold modulation (FTM) theory by Somers and Kopell [Biological Cybernetics, 68 (1993), pp. 393--407] that synchronization of relaxation oscillators has properties that are quite different from those of smooth (nonrelaxation) oscillators. We use Bonhoeffer--Van Der Pol relaxation oscillators to illustrate the theory numerically.