Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 471amplitude of stimulation, A1:61:4synchronization1:2 3:21:11:3 2:32:15:20T/4 T/3 T/2 3T/4 T 5T/4 3T/2 7T/4 2T 9T/4 5T/2period of stimulation, TsFigure 10.15: Arnold tongues are regions of existence of various phase-locked states onthe “period-strength” plane.cycle slippingT/2Poincare phase mapn+1=f( n)0?ghost ofattractor-T/2-T/2 0 n T/2Figure 10.16: Cycle slipping phenomenon at the edge of the Arnold tongue correspondingto a synchronized state.fold fixed point becomes a ghost attractor that traps orbits and keeps them near thesynchronized state for a long period of time. Eventually the orbit escapes, the synchronizedstate is briefly lost, and then the orbit returns to the ghost attractor to betrapped again. Such an intermittently synchronized orbit typically corresponds to ap:q-phase-locked state with high order of locking p + q.10.2 Weak CouplingIn this section we consider dynamical systems of the formẋ = f(x) + εp(t) , (10.5)describing periodic oscillators, ẋ = f(x), forced by a time-depended input εp(t), e.g.,from other oscillators in a network. The positive parameter ε measures the overallstrength of the input, and it is assumed to be sufficiently small, denoted as ε ≪ 1. Wedo not assume ε → 0 here. In fact, most of the results in this section can be cast in