Dynamical Systems in Neuroscience:

462 Synchronization (see www.izhikevich.com)K + activation gate, n0.60.50.40.30.20.10pulsepulsepulsex pulsepulsepulsePRC newypulsenew= +PRC( )pulse-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane potential, V (mV)Figure 10.5: The geometrical relationship between isochrons and the phase responsecurve (PRC) of the I Na + I K -oscillator in Fig. 10.1.phases, we can measure the phase response curve (also called phase resetting curvePRC, or spike time response curve STRC)PRC (ϑ) = {ϑ new − ϑ} mod T (shift = new phase – old phase) ,depicted in Fig. 10.4, right. Positive (negative) values of the function correspond tophase advances (delays) in the sense that they advance (delay) the timing of the nextspike.In contrast to the common folklore, the function PRC (ϑ) can be measured for anarbitrary stimulus, not necessarily weak or brief. The only caveat is that to measurethe new phase of oscillation we need to wait long enough for transients to subside. Thisbecomes a limiting factor when PRCs are used to study synchronization of oscillatorsto periodic pulses, as we do in Sect. 10.1.5.There is a simple geometrical relationship between the structure of isochrons of anoscillator and its PRC, illustrated in Fig. 10.5, see also Ex. 6. Let us stimulate theoscillator at phase ϑ by a pulse, which moves the trajectory from point x lying onthe intersection of isochron ϑ and the limit cycle attractor to a point y lying on someisochron ϑ new . From the definition of PRC, it follows that ϑ new = ϑ+PRC (ϑ).In general, one uses simulations to determine PRCs, as we do in Fig. 10.4. Using theMATLAB program presented in Ex. 5, we can determine PRCs of all four oscillators inFig. 10.3 and plot them in Fig. 10.6. It is a good exercise to explain the shape of eachPRC in the figure, or at least its sign, using the geometry of isochrons of correspondingoscillators. In Sect. 10.2.4 we discuss pitfalls of using the straightforward method inFig. 10.4 to measure PRCs in biological neurons, and we present a better technique.Notice that PRC of the I Na + I K -model in Fig. 10.6 is mainly positive in the Class 1regime, i.e., when the oscillations appear via saddle-node on invariant circle bifurcation,