Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 469(a)synchronizationin-phase anti-phase out-of-phase(b)phase-locking1:2 2:13:2Figure 10.12: Examples of fundamental types of synchronization of spiking activity toperiodic pulsed inputs (synchronization is 1:1 phase-locking).T-T sunstablestabilityregionPRC00stablephaseshiftstimulus phase,stabilityregionTFigure 10.13: Fixed points of the Poincarephase map (10.1.5).marked by the bold curves in Fig. 10.13.Now consider the Class 1 and Class 2 I Na + I K -oscillators shown in Fig. 10.6. ThePRC in Class 1 regime is mostly positive, implying that such an oscillator can easilysynchronize with faster inputs (T − T s > 0) but cannot synchronize with slower inputs.Indeed, the oscillator can only advance its phase to catch up faster pulse trains, but itcannot delay the phase to wait for the slower input. Synchronization with the inputhaving T s ≈ T is only marginal. In contrast, Class 2 I Na + I K -oscillator does not havethis problem because its PRC has well-defined positive and negative regions.10.1.8 Phase lockingp:q-phase-locking occurs when the oscillator fires p spikes for every q input pulses,such as the 3:2-phase-locking in Fig. 10.12b or 2:2 phase-locking in Fig. 10.14, whichtypically occurs when pT ≈ qT s . The integers p and q need not be relatively prime inthe case of pulsed-coupled oscillators. Synchronization, i.e., 1:1 phase-locking, as wellas p:1 phase-locking corresponds to a fixed point of the Poincare phase map (10.3) withp fired spikes per one input pulse. Indeed, the map tells only the phase of the oscillatorat each pulse, but does not tell the number of oscillations made between the pulses.Each p:q-locked solution corresponds to a stable periodic orbit of the Poincarephase map with the period q (so that ϑ n = ϑ n+q for any n). Such orbits in maps (10.4)