Dynamical Systems in Neuroscience:

480 Synchronization (see www.izhikevich.com)Andronov-Hopf oscillatorvan der Pol oscillator0.5H ij ( )0.5H ij ( )00-0.5-0.5G( )G( )0 2 4 T0 2 4 6T20I Na +I K -model (Class 1) I Na +I K -model (Class 2)H ij ( )4H ij ( )0-2G( )-4G( )0 2 4 6phase difference,T0 1 2 3phase difference,TFigure 10.26: Solid curves: Functions H ij (χ) defined by (10.16) with the inputg(x i , x j ) = (x j1 − x i1 , 0) corresponding to electrical synapse via gap-junction. Dashedcurves: Functions G(χ) = H ji (−χ) − H ij (χ). Parameters as in Fig. 10.3.with the first term capturing fast free-running natural oscillation ˙ϑ i = 1, and thesecond term capturing slow network-induced build-up of phase deviation from thenatural oscillation. The relationship between x i (t), ϑ i (t) and ϕ i (t) is illustrated inFig. 10.25.Substituting (10.13) into (10.12) results in˙ϕ i = ε Q i (t + ϕ i ) ·n∑g ij (x i (t + ϕ i ), x j (t + ϕ j )) . (10.14)j=1Notice that the right hand-side is of order ε, reflecting the slow dynamics of phasedeviations ϕ i seen in Fig. 10.25. Thus, it contains two time scales: fast oscillations(variable t) and slow phase modulation of phase (variables ϕ). The classical methodof averaging, reviewed by Hoppensteadt and Izhikevich (1997, Chap. 9) consists in anear-identity change of variables that transforms the system into the formwhereH ij (ϕ j − ϕ i ) = 1 T˙ϕ i = εω i + ε∫ T0n∑H ij (ϕ j − ϕ i ) , (10.15)j≠iQ i (t) · g ij (x i (t), x j (t + ϕ j − ϕ i )) dt , (10.16)and each ω i = H ii (ϕ i − ϕ i ) = H ii (0) describes a constant frequency deviation fromthe free-running oscillation. Figure 10.26 depicts the functions H ij corresponding togap-junction (i.e., electrical; see Sect. 2.3.4) coupling of oscillators in Fig. 10.3. Prove