Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 481(a) (b) identify (c)2identify2211 1Figure 10.27: Torus knot of type (2,3) (a) and its representation on the square (b).The knot produces frequency locking and phase locking. (c) Torus knot that does notproduce phase locking.that H(χ) = Q(χ) · A/T in the case of pulse-coupling (10.1), so that H(χ) is justre-scaled PRC.A special case of (10.15) is when H is replaced by its first Fourier term, sin. Theresulting system written in the slow time τ = εtϕ ′ i = ω i +n∑c ij sin(ϕ j − ϕ i + ψ ij )j=1is called the Kuramoto phase model (Kuramoto 1975). Here, the frequency deviationsω i are interpreted as intrinsic frequencies of oscillators. The strengths of connectionsc ij are often assumed to be equal to K/n for some constant K, so that the model canbe studied in the limit n → ∞. The phase deviations ψ ij are often neglected for thesake of simplicity.To summarize, we transformed the weakly coupled system (10.11) into the phasemodel (10.15) with H given by (10.16) and each Q being the solution to the adjointproblem (10.10). This constitutes the Malkin theorem for weakly coupled oscillators(Hoppensteadt and Izhikevich 1997, Theorem 9.2).10.3 SynchronizationConsider two coupled phase variables (10.12) in a general form˙ϑ 1 = h 1 (ϑ 1 , ϑ 2 ) ,˙ϑ 2 = h 2 (ϑ 1 , ϑ 2 ) ,with some positive functions h 1 and h 2 . Since each phase variable is defined on thecircle S 1 , the state space of this system is the 2-torus T 2 = S 1 ×S 1 depicted in Fig. 10.27,with ϑ 1 and ϑ 2 being the longitude and the latitude, respectively. The torus can berepresented as a square with vertical and horizontal sides identified, so that a solutiondisappearing at the right side of the square appears at the left side.