Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 405Review of Important Concepts• Oscillations are described by their phase variables ϑ rotating on a circleS 1 . We define ϑ as the time since the last spike.• The phase response curve, PRC (ϑ), describes the magnitude of the phaseshift of an oscillator caused by a strong pulsed input arriving at phase ϑ.• PRC depends on the bifurcations of spiking limit cycle, and it definessynchronization properties of an oscillator.• Two oscillators are synchronized in-phase, anti-phase, or out-of-phase,when their phase difference, ϑ 2 − ϑ 1 , equals 0, half-period, or some othervalue, respectively; see Fig. 10.1.• Synchronized states of pulse-coupled oscillators are fixed points of thecorresponding Poincare phase map.• Weakly coupled oscillatorscan be reduced to phase modelsẋ i = f(x i ) + ε ∑ g ij (x j )˙ϑ i = 1 + ε Q(ϑ i ) ∑ g ij (x j (ϑ j )) ,where Q(ϑ) is the infinitesimal PRC defined by Malkin’s equation.• Weak coupling induces a slow phase deviation of the natural oscillation,ϑ i (t) = t + ϕ i , described by the averaged model(˙ϕ i = ε ω i + ∑ )H ij (ϕ j − ϕ i ) ,where the ω i denote the frequency deviations, andH ij (ϕ j − ϕ i ) = 1 T∫ Tdescribe the interactions between the phases.0Q(t) g ij (x j (t + ϕ j − ϕ i )) dt• Synchronization of two coupled oscillators correspond to equilibria of theone-dimensional system˙χ = ε(ω + G(χ)) , χ = ϕ 2 − ϕ 1 ,where G(χ) = H 21 (−χ) − H 12 (χ) describes how the phase difference χcompensates for the frequency mismatch ω = ω 2 − ω 1 .