Figure 3
Results for the k3 vs S analysis of the SK model. (a) 2D bifurcation diagram for k3 and S. In black, the supercritical Hopf curve. In red, the subcritical one. In blue, the Saddle-Node (SN) curves, with the Cusp Point (CP) at the joint, also in blue. The GH bifurcations in black. The Bogdanov-Takens (BT) bifurcations are in red. A bistable region develops for low values of k3 and high values of S, separated from the excitable region by a portion of the subcritical Hopf curve. (b) Zoom of the 2D bifurcation diagram with the Saddle-Homoclinic (SHom, in cyan) and Limit Point Cycle (LPC, in green) curves. The stable limit cycles end at the LPC curves, but the unstable ones increase their period as they approach the SHom bifurcations. (c) Amplitude vs. frequency curves for five different values of k3. Inverted “U” near the top of the oscillatory region, and a transition in the shape as the system approaches the bistable and excitable regions. More robust amplitude, lower frequency values and relatively larger ranges are observed for lower k3. (d) 1D bifurcation diagram for k3 = 0.77 and a scan of S. Steady states: stable in blue lines, unstable in red. Same colors for the limit cycles and circles. An unstable limit cycle is born from the right-hand side subcritical Hopf and continues until the LPC. The stable limit cycle is limited by the LPCs. An unstable limit cycle is limited between the left LPC and an SHom (in a very small range of S). The left subcritical Hopf is “disconnected” from the left LPC: starting from said Hopf, an unstable limit cycle runs for a small range of S and ends at another SHom branch (not shown). (e) Bistable curves in the DP cycle while scanning Kact, for each of the five cases studied. SN points in blue. Clear change in the range for the active kinase, allowing changes in the oscillator frequencies, while the amplitude varies slightly for each k3 value. (f) Time series for two pairs of k3 and S, representing the variety of oscillatory outputs.
