Dynamical Systems in Neuroscience:

484 Synchronization (see www.izhikevich.com)(a)max G0(b)max G0G( )min GG( )0 1 2 3phase difference,T0 1 2 3phase difference,TFigure 10.30: Geometrical interpretation of equilibria of the phase model (10.17) forgap-junction-coupled Class 2 I Na + I K -oscillators (see Fig. 10.26).whereω = ω 2 − ω 1 and G(χ) = H 2 (−χ) − H 1 (χ) ,is the frequency mismatch and the anti-symmetric part of the coupling, respectively,illustrated in Fig. 10.26, dashed curves. A stable equilibrium of (10.17) corresponds toa stable limit cycle of the phase model.All equilibria of (10.17) are solutions to G(χ) = −ω, and they are intersections ofthe horizontal line −ω with the graph of G, as illustrated in Fig. 10.30a. They arestable if the slope of the graph is negative at the intersection. If the oscillators areidentical, then G(χ) = H(−χ) − H(χ) is an odd function (i.e., G(−χ) = −G(χ)),and χ = 0 and χ = T/2 are always equilibria, possibly unstable, corresponding to thein-phase and anti-phase synchronized solutions. The stability condition of the in-phasesynchronized state isG ′ (0) = −2H ′ (0) < 0(stability of in-phase synchronization)The in-phase synchronization of electrically (gap-junction) coupled oscillators in Fig. 10.26is stable because the slope of G (dashed curves) is negative at χ = 0. Simulation of twocoupled I Na + I K -oscillators in Fig. 10.25 confirms that. Coupled oscillators in Class 2regime also have a stable anti-phase solution, since G ′ < 0 at χ = T/2 in Fig. 10.30a.The max and min values of the function G determine the tolerance of the networkto the frequency mismatch ω, since there are no equilibria outside this range. Geometrically,as ω increases (the second oscillator speeds up), the horizontal line −ω inFig. 10.30a slides down, and the phase difference χ = ϕ 2 − ϕ 1 increases, compensatingfor the frequency mismatch ω. When ω > − min G, the second oscillator becomestoo fast, and the synchronized state is lost via saddle-node on invariant circle bifurcation;see Fig. 10.30b. This bifurcation corresponds to the annihilation of stable andunstable limit cycles of the weakly coupled network, and the resulting activity is calleddrifting, cycle slipping, or phase walk-through. The variable χ slowly passes the ghostof the saddle-node point, where G(χ) ≈ 0, then increases past T , appears at 0, andapproaches the ghost again, thereby slipping a cycle and walking through all the phasevalues [0, T ]. The frequency of such slipping scales as √ ω + min G; see Sect. 6.1.2.In Fig. 10.31 we contrast synchronization properties of weakly coupled oscillatorsof relaxation and non-relaxation type. The function G(χ) of the former has a negative