Dynamical Systems in Neuroscience:

474 Synchronization (see www.izhikevich.com)Figure 10.20: Yoshiki Kuramoto in1988, while he was visiting Jim Murray’sinstitute at University of Oxford.rewrite the corresponding Poincare phase map (10.3)PRC{ }} {ϑ(t n+1 ) = {ϑ(t n ) + Z(ϑ(t n )) εp(t n )h +h} mod T} {{ }Ain the formϑ(t n + h) − ϑ(t n )hwhich is a discrete version of= Z(ϑ(t n ))εp(t n ) + 1 ,˙ϑ = 1 + εZ(ϑ) · p(t) , (10.6)in the limit h → 0.To be consistent with all the examples in the previous section, we implicitly assumehere that p(t) perturbs only the first, voltage-like variable x 1 of the state vector x =(x 1 , . . . , x m ) ∈ R m and Z(ϑ) is the corresponding sensitivity function. However, thephase model (10.6) is also valid for an arbitrary input p(t) = (p 1 (t), . . . , p m (t)). Indeed,let Z i describe the linear response to perturbations of the ith state variable x i , andZ(ϑ) = (Z 1 (ϑ), . . . , Z m (ϑ)) denote the corresponding linear response vector-function.Then, the combined phase shift Z 1 p 1 + · · · + Z m p m is just the dot product Z · p in(10.6).10.2.2 Kuramoto’s approachConsider the unperturbed (ε = 0) oscillator (10.5), and let the function ϑ(x) denotethe phases of points near its limit cycle attractor. Obviously, isochrons are the levelcontours of ϑ(x) since the function is constant on each isochron. Differentiating thefunction using the chain rule yieldsdϑ(x)dt= grad ϑ · dxdt= grad ϑ · f(x) ,