Dynamical Systems in Neuroscience:

466 Synchronization (see www.izhikevich.com)(a)membrane potential, V(c)phase, n101 2 3 4 5 6 7?T stime, tT345 6 71K + gating variable, b(b)0.60.40.2(d)Tmod Tn+10314-7-80 -60 -40 -20 0 20membrane potential, V102Poincare phase map315,6,74stablefixedpoint021 2 3 4 5 6 7pulse number, n020 10nTFigure 10.10: Description of synchronization of I Na + I K -oscillator in Fig. 10.4 usingPoincare phase map.10.1.5 Poincare phase mapThe phase resetting curve (PRC) describes the response of an oscillator to a singlepulse, but it can be used to study its response to a periodic pulse train using thefollowing “stroboscopic” approach. Let ϑ n denote the phase of oscillation at the timethe nth input pulse arrives. Such a pulse resets the phase by PRC (ϑ n ), so that the newphase right after the pulse is ϑ n +PRC (ϑ n ); see Fig. 10.9. Let T s denote the period ofpulsed stimulation. Then the phase of oscillation before the next, (n + 1)th pulse, isϑ n +PRC (ϑ n ) + T s . Thus, we have a stroboscopic mapping of a circle to itselfϑ n+1 = (ϑ n + PRC (ϑ n ) + T s ) mod T (10.3)called Poincare phase map (two pulse-coupled oscillators are considered in Ex. 11).Knowing the initial phase of oscillation ϑ 1 at the first pulse, we can determine ϑ 2 , thenϑ 3 , etc. The sequence {ϑ n } with n = 1, 2, . . . , is called the orbit of the map, and it isquite easy to find numerically.Let us illustrate this concept using the I Na + I K -oscillator with PRC shown inFig. 10.4. Its free-running period is T ≈ 21.37 ms, and the period of stimulationin Fig. 10.10a is T s = 18.37, which results in the Poincare phase map depicted inFig. 10.10d. The cobweb in the figure is the orbit going from ϑ 1 to ϑ 2 to ϑ 3 , etc. Notice