Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 475xgrad (x)isochronf(x)limit cycleFigure 10.21: Geometrical interpretationof the vector grad ϑ.where grad ϑ = (ϑ x1 (x), . . . , ϑ xm (x)) is the gradient of ϑ(x) with respect to the statevector x = (x 1 , . . . , x m ) ∈ R m . However,dϑ(x)dtnear the limit cycle, because isochrons are mapped to isochrons by the flow of thevector-field f(x). Therefore, we get a useful equality= 1grad ϑ · f(x) = 1 . (10.7)Figure 10.21 shows a geometrical interpretation of grad ϑ(x): it is the vector based atpoint x, normal to the isochron of x and with the length equal to the number densityof isochrons at x. Its length can also be found from (10.7).Kuramoto (1984) applied the chain rule to the perturbed system (10.5)dϑ(x)dt= grad ϑ · dxdtand, using (10.7), obtained the phase model= grad ϑ · {f(x) + εp(t)} = grad ϑ · f(x) + ε grad ϑ · p(t) ,˙ϑ = 1 + ε grad ϑ · p(t) , (10.8)which has the same form as (10.6). Subtracting (10.8) from (10.6) yields (Z(ϑ) −grad ϑ) · p(t) = 0. Since this is valid for any p(t), we conclude that Z(ϑ) = grad ϑ;see also Ex. 6. Thus, Kuramoto’s phase model (10.8) is indeed equivalent to Winfree’smodel (10.8).10.2.3 Malkin’s approachYet another equivalent method of reduction of weakly perturbed oscillators to theirphase models follows from Malkin theorem (1949,1956), which we state in the simplestform below. The most abstract form and its proof is provided by Hoppensteadt andIzhikevich (1997).Malkin’s theorem. Suppose the unperturbed (ε = 0) oscillator in (10.5) has anexponentially stable limit cycle of period T . Then its phase is described by the equation˙ϑ = 1 + εQ(ϑ) · p(t) , (10.9)