Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 483Figure 10.29: A major part of computational neuroscience concerns coupled oscillators.of the oscillators comprising the network. In addition, a network can exhibit partialsynchronization, when only a subset of oscillators is synchronized.Synchronization of oscillators with nearly identical frequencies is described by thephase model (10.15). Existence of one equilibrium of (10.15) implies the existence ofthe entire circular family of equilibria, since translation of all ϕ i by a constant phaseshift does not change the phase differences ϕ j − ϕ i and hence the form of (10.15).This family corresponds to a limit cycle of (10.11), on which all oscillators, x i (t + ϕ i ),have equal frequencies and constant phase shifts, i.e., they are synchronized, possiblyout-of-phase.10.3.1 Two oscillatorsConsider (10.11) with n = 2, describing two coupled oscillators, as in Fig. 10.29. Letus introduce the “slow” time τ = εt and rewrite the corresponding phase model (10.15)in the formϕ ′ 1 = ω 1 + H 1 (ϕ 2 − ϕ 1 ) ,ϕ ′ 2 = ω 2 + H 2 (ϕ 1 − ϕ 2 ) ,where ′ = d/dτ is the derivative with respect to slow time. Let χ = ϕ 2 − ϕ 1 denotethe phase difference between the oscillators. Then the two-dimensional system abovebecomes one-dimensionalχ ′ = ω + G(χ) , (10.17)