Figure 1.
Schematic phase plane of a two-dimensional bistable model.
The system has two stable steady-states (green dots) and an unstable one (red dot). These steady states lie at the intersections of the nullclines (black dashed curves), and
The stable manifold (green curve) of the saddle point divides the phase plane in the two basins of attraction of the stable equilibria and is therefore called a separatrix. The saddle point’s unstable manifold (red curve) connects the three equilibrium points. Any perturbation pushing the trajectory across the separatrix induces a switch in the final decision.
Figure 2.
Time-scale separation at saddle point (A) and saddle node bifurcation (B-C).
(A) A time-scale separation between the stable and unstable manifolds of the saddle point enhances its temporary attractivity. Trajectories (grey curves) converge to the unstable manifold (red dashed line) in the fast time-scale before sliding to a stable equilibrium point (green dot) in the slow time-scale. (B) For the nullclines (black curves) intersect a three equilibrium points, the system is bistable. For
the nullclines (grey dashed curves) intersect at a simple equilibrium point, the system is monostable. (C) Corresponding diagram of bifurcation. At
the “on” stable equilibrium branch (solid green curve) merges with the saddle branch (red dashed curve) in a so-called saddle node bifurcation.
Figure 3.
Switches with input-strength dependent delays.
In Figure (A), the system is bistable. The switch occurs when the signal strength is greater than a particular threshold
The switching time depends on
Figure (B) shows the corresponding trajectories in the phase plane. For inputs close to
trajectories start in the vicinity of the stable manifold (green dashed curve) and converge rapidly to a close neighborhood of the saddle point (red dot). The escape from the saddle point is slow, causing the time-delay. Figures (C-D) and (E-F) show how the switch is modified by adding a production term
to equation (1). (C-D)
only the “off” state remains while the value of
is close to the bifurcation point
The saddle point has disappeared but its ghost creates a similar delay. One can still observe switches with input-strength dependent delays. (E-F)
both the bottleneck and the switches with delays disappear.
Figure 4.
Results of the analysis of the model of apoptosis proposed by Eißing et al. [3].
(A) Model description. In response to a pro-apoptotic input signal, initiator caspases C8 become activated and activate the effector caspase C3. Activated C3, C3*, activate C8 in return through a positive feedback loop. Inhibitors CARP and IAP bind to C8* and C3* to prevent apoptosis. (B) Time-scale separation at saddle point. Trajectories rapidly converge to the unstable manifold (red dashed line) of the saddle point (red dot) and then slowly escape to reach either the life (green square) or the death steady-states. For the grey trajectory, equally distributed time markers are depicted () showing how trajectories are delayed in the vicinity of the saddle point. (C) Sum of relative sensitivities at saddle point. The saddle point is insensitive to parameters
and
(see Supplementary table 1). These parameters have a high degree of robustness (D). (E) Output trajectories for increasing input. For input above the threshold, the system switches to the unexcited state, see the corresponding trajectories in the phase plane (B). Trajectories have been normalized such that the output equals zero in the unexcited state and equals one in the exited state. Depending on the input strength, the switch is more or less delayed. By observing trajectories in the phase plane (D), one can see that trajectories starting close to the stable manifold of the saddle point fast converge in the neighborhood of the saddle point where there are delayed before converging to the excited state creating a mechanism of delayed decision making.
Table 1.
Model of Eißing: Eigenvalues and ratio .
Figure 5.
Results of the analysis of the model of Schliemann et al. [9].
(A) Magnitude of the real part of the eigenvalues of the Jacobian matrix at the saddle point, the stable ones are depicted in black while the unstable one is depicted in red. The zooms in on the three slowest eigenvalues. (B) Output trajectories for impulse inputs, slightly below (light blue solid curve), slightly above (dark grey dashed curve), above (dark grey solid curve) and significantly above (black solid curve) the decision making threshold, (C) Corresponding trajectories in the phase plane. Trajectories passing close to the saddle point are delayed. Trajectories follow the unstable manifold of the saddle point (red dashed curve) before reaching the survival or death state.
Table 2.
Model of Schliemann: Eigenvalues and ratio .
Figure 6.
Sensitivity analysis at saddle point for the model of apoptosis proposed by Schliemann et al. [9].
The parameters have been divided in three sets. The first one include the parameters controlling the reactions involving the binding of TNF to receptor, the second one the parameters controlling the activity of NF-B and the last one the parameters linked to the reactions governing caspases and their inhibitors.
Figure 7.
Model of Aslam et al. [6].
(A) The model describes the positive feedback loop between the protein -CaMKII and the translation factor CPEB1. The protein
-CaMKII can be in one of three states: inactive (X), active (X
) and phosphorylated (X
). When active and phosphorylated,
-CaMKII phosphorylates CPBE1 which in turn can initiate the translation of a new
-CaMKII protein [6]. (B) Trajectories for increasing inputs showing the delay close to the threshold
(C) Magnitude of the real part of the eigenvalues of the Jacobian matrix at the saddle point, the stable ones are depicted in black while the unstable one is depicted in red. The inlet zooms in on the three slowest eigenvalues. (D) Sensitivity at saddle point and (E) degrees of robustness (DOR). Parameters with a high sensitivity (red) have a low degree of robustness. Conversely, parameters with a low sensitivity (dark blue) have a high DOR.
Table 3.
Model of Aslam: Eigenvalues and ratio .
Figure 8.
Parameter perturbation of the model of Aslam et al. [6].
(A)-(D) The switching is depicted for nominal values of the parameters (black curve), 10 of parameter perturbation (blue dashed curve), 20
(green dashed-doted curve) and 30
of variation (red curve). The system is simulated for an input slightly above the threshold, i.e
where the threshold
is recomputed for each parameter perturbation. (A) (Ca
)
-CaM (parameter 22), (B) k
(parameter 18), (C) k
(parameter 1) and (D) k
(parameter 20). Both the switching threshold and time are affected by perturbation of parameters (Ca
)
-CaM and k
. In contrast, the switching threshold and time are insensitive to a perturbation of parameter kSYN2.
Figure 9.
Influence of the distance to a bifurcation on the switching for the model of Aslam et al. [6].
(A)-(D) Switching responses for the Aslam model for different input intensities above the threshold
:
(doted line),
(dashed-doted line),
(dashed line),
(solid line). (A) Nominal model. (B)-(D) Perturbed models with single parameter set to 0.99 of its upper bifurcation value. (E)-(G) Bifurcation diagrams for the parameters
-CaM, k
and k
. (B) and (E)
-CaM, (C) and (F)
and (D) and (G) k
.