Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 493(a)(b)20normalized PRC (Q1( ))10.80.60.40.20sin 2I=4.52I=4.55I=4.6I=4.7I=5I=10membrane potential, (mV)0-20-40-60inflectionpoints0phase of oscillation,T-800phase of oscillation,TFigure 10.36: (a) Numerically found PRCs of the I Na + I K -oscillator in Class 1 regime(parameters as in Fig. 4.1a) and various I using the MATLAB program in Ex. 12.(b) Corresponding voltage traces show that the inflection point (slowest increase) of Vmoves right as I increases.Gap junctionsNow consider two oscillators coupled via gap junctions, discussed in Sect. 2.3.4,x ′ 1 = 1 + x 2 1 + ε(x 2 − x 1 ) ,x ′ 2 = 1 + x 2 2 + ε(x 1 − x 2 ) .Let us determine the stability of the in-phase synchronized state x 1 = x 2 . The correspondingphase model (10.12) has the formϑ ′ 1 = 1 + ε(sin 2 ϑ 1 )(cot ϑ 1 − cot ϑ 2 ) ,ϑ ′ 2 = 1 + ε(sin 2 ϑ 2 )(cot ϑ 2 − cot ϑ 1 ) .The function (10.16) can be found analytically:H(χ) = 1 π∫ π0sin 2 t (cot t − cot(t + χ)) dt = 1 sin 2χ ,2so that the model in the phase deviation coordinates, ϑ(t) = t + ϕ, has the formϕ ′ 1 = (ε/2) sin{2(ϕ 2 − ϕ 1 )} ,ϕ ′ 2 = (ε/2) sin{2(ϕ 1 − ϕ 2 )} .The phase difference, χ = ϕ 2 − ϕ 1 , satisfies the equation (compare with Fig. 10.26)χ ′ = −ε sin 2χ ,and, apparently, the in-phase synchronized state, χ = 0, is always stable while theanti-phase state χ = π/2 is not.