Fig. 2.

Bifurcation diagram and respective time courses of an excitatory (E)-inhibitory (I) rate model, for 3 values of the ratio of rate timescales for the E and I populations. A: bifurcation plots of the 4-variable E-I gamma model (Eqs. 1 and 2) showing the rate variable r_E of the excitatory population vs. excitatory input, I_E. Steady-state rate (black) is flanked by maximum and minimum amplitudes of oscillation. The steady state transitions from stable (solid) to unstable for different ratios of time constants, $\text{ρ}={\text{τ}}_{r_{\text{I}}}/{\text{τ}}_{r_{\text{E}}}$; ρ = 0.25 (dotted), ρ = 1 (dot-dashed), ρ = 4 (dashed). Thin vertical black line corresponds to input value (I_E = 0.9) for example time traces. Colors illustrate oscillation amplitude profiles for varying ρ. The time constant values are ${\text{τ}}_{r_{\text{E}}}$ = 8 ms and ${\text{τ}}_{r_{\text{I}}}$ = 2 ms (ρ = 0.25), ${\text{τ}}_{r_{\text{E}}}$ = ${\text{τ}}_{r_{\text{I}}}$ = 5 ms (ρ = 1), and ${\text{τ}}_{r_{\text{E}}}$ = 2 ms and ${\text{τ}}_{r_{\text{I}}}$ = 8 ms (ρ = 4) for the E and I rate variables with the synaptic timescales fixed to 3 ms (E) and 10 ms (I). The oscillation amplitude near onset changes more abruptly as ρ increases, i.e., the inhibitory rate timescale becomes larger than the excitatory rate timescale. B–D: example time courses for I_E = 0.9 (thin vertical black line in A) for both rate and synapse variables. Oscillation becomes more PING-like as the ratio of inhibitory to excitatory rate timescales increases.