Figure 9

Bifurcation diagrams for a single population. (a) Bifurcations of the synaptic current (s) as a function of the stimulus (\(I_{{\text{app}}}\)). Two saddle-node (SN) bifurcations take place at \(I_{\text{SN}_{1}}=-0.4627\) and \(I_{\text{SN}_{2}}=0.3002\). An unstable limit cycle arises from a subcritical Hopf bifurcation (HB) at \(I_{{\text{HB}}}= \{ -0.07069,-0.01817 \} \) for the 3d (red) and 2d (blue) model (see text). The limit cycle merges with the saddle in a saddle-homoclinic orbit (SHO) at \(I_{{\text{SHO}}}= \{ -0.02013,0.04802 \} \) for the 3d and 2d models. The system is bistable between HB and SN₂. (b) Two-parameter bifurcations of \(I_{{\text{app}}}\) versus self-coupling weight w. The bistable region is between the top limit point (LP) line and the HB line. (c) Two-parameter bifurcations of time constants \(\tau _{s}\) versus \(\tau _{d}\) (both measured in \(\tau _{r}\)). The bistable region is between the left LP line and the HB line. A cusp point is at \((\tau _{s},\tau _{d})\approx (8.0,68.1 )\). (d) Bifurcation sequence in the s-d plane between SN₁ and SN₂. The OFF state remains stable (filled circle with small s and large d). Red (black) lines are stable (unstable) invariant manifolds of the saddle (triangle). The ON state is initially unstable (open circle) and becomes stable at the Hopf bifurcation. The central plot depicts how the unstable limit cycle terminates at the saddle when \(I_{\text{app}}=I_{\text{SHO}}\)