Dynamical Systems in Neuroscience:

472 Synchronization (see www.izhikevich.com)Figure 10.17: Arthur Winfree, 2001.the form “there is an ε 0 such that for all ε < ε 0 , the following holds...” (Hoppensteadtand Izhikevich 1997), with ε 0 depending on the function f(x) in (10.5) and sometimestaking not so small values, e.g., ε 0 = 1.Notice that if ε = 0 in (10.5), then we can transform ẋ = f(x) to ˙ϑ = 1 usingthe theory presented in the previous section. What happens when we apply the sametransformation to (10.5) with ε ≠ 0? In this section we present three different butequivalent approaches that transform (10.5) into the phase model˙ϑ = 1 + ε PRC (ϑ)p(t) + o(ε) .Here, the Landau’s “little oh” function o(ε) denotes the error terms smaller than ε sothat o(ε)/ε → 0 if ε → 0. For the sake of clarity of notation, we omit o(ε) throughoutthe book, and implicitly assume that all equalities are valid up to the terms of ordero(ε).Since we do not impose restrictions on the form of p(t), the three methods arereadily applicable to the casep(t) = ∑ sg s (x(t), x s (t)) ,where the set {x s (t)} denotes oscillators in the network connected to x, and p(t) is thepostsynaptic current.10.2.1 Winfree’s approachA sufficiently small neighborhood of the limit cycle attractor of the unperturbed(ε = 0) oscillator (10.5), magnified in Fig. 10.18, has nearly collinear uniformly spacedisochrons. Collinearity implies that a point x on the limit cycle in Fig. 10.18 has thesame phase-resetting as any other point y on the isochron of x near the cycle. Uniformdensity of isochrons implies that the phase resetting scales linearly with the strengthof the pulse, i.e., a half-pulse at the point z in Fig. 10.18 produces a half-resetting ofthe phase.