Optogenetic Stimulation Shifts the Excitability of Cerebral Cortex from Type I to Type II: Oscillation Onset and Wave Propagation
A: Phase portrait of the model with τi = 4 ms and τe = 2 ms. The system is initially at rest (black dot; Ue = 0.017, Ui = 0.020) whereupon a steady injection current (Je = 2) induces a stable limit cycle (heavy black line) via a supercritical Hopf bifurcation. Nullclines are shown in light gray (excitatory is “cubic”-shaped and inhibitory is “S-shaped”). The dashed nullcline is that of Ue when the injection current is applied. B: Corresponding time plots of Ue, Ui, LFP and the injection current Je. C: Frequency of the oscillation as a function of injection current Je. The frequency is always wi1thin the 40–80 Hz gamma band. D: Return trajectories for a range of perturbations applied to the resting state. Stars mark the initial conditions. Only the largest perturbations induced large excursions in phase space. E: Time plots of the same return trajectories. F: Bifurcation diagram showing the envelope of the oscillations in Ue as a function of the injection current. The critical point of the Hopf bifurcation is labelled HB. The dashed line indicates unstable fixed points. G-I: Same as panels D-F except that the time constant of excitation has been halved (τe = 1 ms). This regime is said to be more excitable because small perturbations produce large responses.