Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 489Let us assume that the frequencies ω i are distributed randomly around 0 with asymmetric probability density function g(ω), e.g., Gaussian. Kuramoto (1975) hasshown that in the limit n → ∞, the cluster size r obeys the self-consistency equationr = rK∫ +π/2−π/2g(rK sin ϕ) cos 2 ϕ dϕ (10.22)derived in Ex. 21. Notice that r = 0, corresponding to the incoherent state, is alwaysa solution of this equation. When the coupling strength K is greater than a certaincritical value,K c = 2πg(0) ,an additional, nontrivial solution r > 0 appears, which corresponds to a partiallysynchronized state. It scales as r = √ 16(K − K c )/(−g ′′ (0)πK 4 c ), as the reader canprove himself by expanding g in a Taylor series. Thus, the stronger the coupling Krelative to the random distribution of frequencies, the more oscillators synchronize intoa coherent cluster. The issue of stability of incoherent and partially synchronized statesis discussed by Strogatz (2000).10.4 ExamplesBelow we consider simple examples of oscillators to illustrate the theory developed inthis chapter. Our goal is to understand which details of oscillators are important inshaping the PRC, the form of the function H in the phase deviation model, and hencethe existence and stability of synchronized states.10.4.1 Phase oscillatorsLet us consider the simplest possible kind of a non-linear oscillator, known as the phaseoscillator:ẋ = f(x) + εp(t) , x ∈ S 1 , (10.23)where f(x) > 0 is a periodic function, for example, f(x) = a + sin x with a > 1.Notice that this kind of oscillator is quite different from the two- or high-dimensionalconductance-based models with limit cycle attractors that we considered in the previouschapters. Here, the state variable x is one-dimensional defined on a circle S 1 , that is,it may be interpreted as a measure of distance along a limit cycle attractor of a multidimensionalsystem.Consider the unperturbed (ε = 0) phase oscillator ẋ = f(x), and let x(t) be itssolution with some period T > 0. Following Kuramoto’s idea, we substitute x(ϑ) into(10.23) and use the chain rule,f(x(ϑ)) + εp(t) = {x(ϑ)} ′ = x ′ (ϑ) ϑ ′ = f(x(ϑ))ϑ ′ ,