Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 487From ϕ ′ n = ω we find that ω 0 = ω, i.e., the common frequency is the frequency ofthe free oscillation of the last, uncoupled oscillator. The phase lag along the chain,χ = ϕ i+1 − ϕ i , satisfies n − 1 identical conditions 0 = H + (χ). Thus, the traveling wavewith a constant phase shift exists when H + has a zero crossing with positive slope, incontrast to Fig. 10.26. The sign of χ, and not the direction of coupling, determines thedirection of wave propagation.10.3.3 NetworksNow let us consider weakly connected networks (10.11) with arbitrary, possibly allto all coupling. To study synchronized states of the network, we need to determinewhether the corresponding phase model (10.15) has equilibria and examine their stabilityproperties. A vector φ = (φ 1 , . . . , φ n ) is an equilibrium of (10.15) whenn∑0 = ω i + H ij (φ j − φ i ) for all i . (10.19)j≠1It is stable when all eigenvalues of the linearization matrix (Jacobian) at φ have negativereal parts, except one zero eigenvalue corresponding to the eigenvector along thecircular family of equilibria (φ plus a phase shift is a solution of (10.19) too since thephase differences φ j − φ i are not affected).In general, determining the stability of equilibria is a difficult problem. Ermentrout(1992) found a simple sufficient condition. Namely, if• a ij = H ′ ij(φ j − φ i ) ≥ 0, and• the directed graph defined by the matrix a = (a ij ) is connected, (i.e., each oscillatoris influenced, possibly indirectly, by every other oscillator),then the equilibrium φ is neutrally stable, and the corresponding limit cycle x(t + φ)of (10.11) is asymptotically stable.Another sufficient condition was found by Hoppensteadt and Izhikevich (1997). Itstates that if system (10.15) satisfies• ω 1 = · · · = ω n = ω(identical frequencies), and• H ij (−χ) = −H ji (χ) (pair-wise odd coupling)for all i and j, then the network dynamics converge to a limit cycle. On the cycle, alloscillators have equal frequencies 1 + εω and constant phase deviations.The proof follows from the observation that (10.15) is a gradient system in therotating coordinates ϕ = ωτ + φ, with the energy functionE(φ) = 1 n∑ n∑∫ χR ij (φ j − φ i ) , where R ij (χ) = H ij (s) ds .2i=1j=1One can check that dE(φ)/dτ = − ∑ (φ ′ i) 2 ≤ 0 along the trajectories of (10.15), withequality only at equilibria.0