Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 485non-relaxation oscillatorG( )G( )relaxation oscillator000 phase difference, T0 phase difference, TFigure 10.31: Functions G(χ) for weakly coupled oscillators of non-relaxation (smooth)and relaxation type. The frequency mismatch ω creates a phase difference in thesmooth case, but not in the relaxation case.discontinuity at χ = 0; see Sect. 10.4.4 below. An immediate consequence is thatthe in-phase synchronization is rapid and persistent in the presence of the frequencymismatch ω. Indeed, if G is smooth, then χ slows down while it approaches theequilibrium χ = 0. As a result, complete synchronization is an asymptotic processthat requires an infinite period of time to attain. In contrast, when G is discontinuousat 0, the variable χ does not slow down and it takes a finite period of time to lock.Changing the frequency mismatch ω shifts the root of −ω = G(χ) in the continuouscase, but not in the discontinuous case. Hence, the in-phase synchronized state χ = 0of coupled relaxation oscillators exists and it is stable in a wide range of ω.10.3.2 ChainsUnderstanding synchronization properties of two coupled oscillators helps one in studyingthe dynamics of chains of n > 2 oscillatorsϕ ′ i = ω i + H + (ϕ i+1 − ϕ i ) + H − (ϕ i−1 − ϕ i ) , (10.18)where the functions H + and H − describe the coupling in the ascending and descendingdirections of the chain, as in Fig. 10.32. Any phase-locked solution of (10.18) has theform ϕ i (τ) = ω 0 τ + φ i , where ω 0 is the common frequency of oscillation and φ i areconstants. These satisfy n conditionsω 0 = ω 1 + H + (φ 2 − φ 1 ) ,ω 0 = ω i + H + (φ i+1 − φ i ) + H − (φ i−1 − φ i ) , i = 2, . . . , n − 1 ,ω 0 = ω n + H − (φ n−1 − φ n ) .A solution with φ 1 < φ 2 < · · · < φ n or with φ 1 > φ 2 > · · · > φ n (as in Fig. 10.32)is called a traveling wave. Indeed, the oscillators oscillate with a common frequencyω 0 but with different phases that increase or decrease monotonically along the chain.Such a behavior is believed to correspond to central pattern generation (CPG) incrayfish, undulatory locomotion in lamprey and dogfish, and peristalsis in vascular andintestinal smooth muscles. Below we consider two fundamentally different mechanismsof generation of traveling waves.