Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 463Andronov-Hopf oscillatorvan der Pol oscillator0.2Re z(t)0.2x(t)PRC PRC0-0.2 PRC 1PRC 2PRC 2-0.2 PRC1PRC 1 PRC 20 2 4 T0 2 4 6stimulus phase,stimulus phase,I Na +I K -model (Class 1) I Na +I K -model (Class 2)0.5V(t)V(t)-0.5 PRC 2000.2-0.2PRC 1PRCPRC0T0 2 4 6stimulus phase,T0 1 2 3stimulus phase,TFigure 10.6: Examples of phase response curves (PRC) of the oscillators in Fig. 10.3.PRC 1 (ϑ): Horizontal pulses (along the first variable) with amplitudes 0.2, 0.2, 2, 0.2 forAndnronov-Hopf, van der Pol, Class 1 and Class 2 oscillators, respectively. PRC 2 (ϑ):Vertical pulses (along the second variable) with amplitudes 0.2, 0.2, 0.02, 0.002, respectively.An example of oscillation is plotted as a dotted curve in each subplot (not toscale).but it changes sign in the Class 2 regime, corresponding in this case to the supercriticalAndronov-Hopf bifurcation. In Sect. 10.4 we find PRCs analytically in the case of weakcoupling, and show that PRC of a Class 1 oscillator has the shape sin 2 ϑ (period T = π)or 1 − cos ϑ (period T = 2π), whereas that of Class 2 oscillator has the shape sin ϑ(period T = 2π). We show in Sect. 10.1.7 how the synchronization properties of anoscillator depend on the shape of its PRC.10.1.4 Type 0 and 1 phase responseInstead of phase resetting curves, many researchers in the field of circadian rhythmsconsider phase transition curves (Winfree 1980)Sinceϑ new = PTC (ϑ old ) .PTC (ϑ) = {ϑ + PRC (ϑ)} mod T ,the two approaches are equivalent. PRCs are convenient when the phase shifts aresmall, so that they can be magnified and clearly seen. PTCs are convenient when the