Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 499(a)p(x)=1(c)y 1 (t)y 2 (t)p(x)=0(b)(d)f(x,f(x,f(x,y)=0y)+=0f(x,y)=0y)+=0dd'ab'ab'Figure 10.40: Fast threshold modulation of relaxation oscillation. (a) The Heavisideor sigmoidal coupling function p(x) is constant while x is on the left or right branch ofthe x-nullcline. (b) In the relaxation limit µ = 0, the synchronized limit cycle consistsof the left branch of the nullcline f(x, y) = 0 and the right branch of the nullclinef(x, y) + ε = 0. When oscillator 1 is ahead of oscillator 2 (c), the phase differencebetween them decreases after the jump (d).Kopell (1993, 1995) referred to such a coupling in the relaxation limit µ → 0 as fastthreshold modulation (FTM), and found a simple criterion of stability of synchronizedstate that works even for strong coupling.Since the oscillators are identical, the in-phase synchronized state exists, duringwhich the variables x 1 and x 2 follow the left branch of the x-nullcline defined byf(x, y) = 0, see Fig. 10.40b, until they reach the jumping point a. During the instantaneousjump, they turn on the mutual coupling ε, and land at some point b ′ onthe perturbed x-nullcline defined by f(x, y) + ε = 0. They follow the new nullclineuntil the right (upper) knee, and then jump back to the left branch.To determine the stability of the in-phase synchronization, we consider the casewhen oscillator 1 is slightly ahead of oscillator 2, as in Fig. 10.40c. We assume thatthe phase difference between the oscillators is so small, or alternatively, the strengthof coupling is so large, that when oscillator 1 jumps and turns on its input to oscillator2, the latter, being at point d in Fig. 10.40d, is below the left knee of the perturbedx-nullcline f(x, y) + ε = 0 and therefore jumps too. As a result, both oscillators jumpto the perturbed x-nullcline and reverse their order. Although the apparent distancebetween the oscillators, measured by the difference of their y-coordinates, is preservedduring such a jump, the phase difference between them is usually not.The phase difference between two points on a limit cycle is the time needed totravel from one point to the other. Let τ 0 (d) be the time needed to slide from pointd to point a along the x-nullcline in Fig. 10.40d, i.e., the phase difference just before