Dynamical Systems in Neuroscience:

490 Synchronization (see www.izhikevich.com)to get the new phase equation˙ϑ = 1 + εp(t)/f(x(ϑ)) , (10.24)which is equivalent to (10.23) for any, not necessarily small, ε.We can also get (10.24) using any of the three methods of reduction of oscillatorsto phase models:• Malkin’s method is the easiest one: We do not even have to solve the onedimensionaladjoint equation (10.10) having the form ˙Q = −f ′ (x(t)) Q, becausewe can get the solution Q(t) = 1/f(x(t)) directly from the normalization conditionQ(t)f(x(t)) = 1.• Kuramoto’s method relies on the function ϑ(x), which we can find implicitly.Since the phase at a point x(t) on the limit cycle is just t, x(ϑ) is the inverse ofϑ(x). Using the rule for differentiating of inverse functions, ϑ ′ (x) = 1/x ′ (ϑ), wefind grad ϑ = 1/f(x(ϑ)).• Winfree’s method relies on PRC (ϑ), which we find using the following procedure:A pulsed perturbation at phase ϑ moves the solution from x(ϑ) to x(ϑ) + A,which is x(ϑ + PRC (ϑ, A)) ≈ x(ϑ) + x ′ (ϑ)PRC (ϑ, A) when A is small. Hence,PRC (ϑ, A) ≈ A/x ′ (ϑ) = A/f(x(ϑ)), and the linear response is Z(ϑ) = 1/f(x(ϑ))when A → 0.Two coupled identical oscillatorsẋ 1 = f(x 1 ) + εg(x 2 )ẋ 2 = f(x 2 ) + εg(x 1 )can be reduced to the phase model (10.17) with G(χ) = H(−χ) − H(χ), whereH(χ) = 1 T∫ T0Q(t) g(x(t + χ)) dt = 1 T∫ T0g(x(t + χ))f(x(t))dtThe condition for exponential stability of the in-phase synchronized state, χ = 0, canbe expressed in the following three equivalent forms∫ T0g ′ (x(t)) dt > 0as we prove in Ex. 24.or10.4.2 SNIC oscillators∫S 1 g ′ (x)f(x) dx > 0 or ∫S 1f ′ (x)g(x) dx > 0 , (10.25)f 2 (x)Let us go through all the steps of derivation of the phase equation using a neuronmodel exhibiting low-frequency periodic spiking. Such a model is near the saddle-node