Nullclines

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Figure C3.6.7(a) shows tire u= 0 and i )= 0 nullclines of tliis system along witli trajectories corresponding to sub-and super-tlireshold excitations. The trajectory arising from tire sub-tlireshold perturbation quickly relaxes back to tire stable fixed point. Three stages can be identified in tire trajectory resulting from tire super-tlireshold perturbation an excited stage where tire phase point quickly evolves far from tire fixed point, a refractory stage where tire system relaxes back to tire stable state and is not susceptible to additional perturbation and tire resting state where tire system again resides at tire stable fixed point.

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

Figure 21. The nullclines of the minimal model of glycolysis (schematic). The graphic analysis allows to deduce the qualitative dynamics of the system. Each area in the phasespace is characterized by the signs of the local derivatives, corresponding to increasing or decreasing concentration of the respective variable. The gray arrows indicate the direction a trajectory will go. Note that the trajectories may only intersect vertically or horizontally with the nullclines. For simplicity, the nullclines are depicted schematically only, for the actual nullclines corresponding to the rate equations see Fig. 22C.

To obtain a qualitative impression of the dynamics, Fig. 21 shows a schematic depiction of the nullclines of the system. The nullclines are defined as curves in phasespace on which the derivative of a variable vanishes. Specifically, the nullcline for TP is defined by... 

Intersections of both nullclines correspond to the steady states of the system. A graphical analysis of the nullclines allows to deduce qualitative dynamic features, as exemplified in Fig. 21. 

Figure 22. The nullclines corresponding to the minimal model of glycolysis. Depending on the value of the maximal ATP utilization Vm3, the pathway either exhibits a unique steady state or allows for a bistable solution. Note that the nullcline for TP does not depend on VThe corresponding steady states are shown in Fig. 23. Parameters are Vm 3.1, K 0.57, ki 4.0, K i 0.06, and n 4 (the values do not correspond to a specific biological situation).

For the choice of parameters used here, the simple pathway gives rise to bistability and hysteresis. In particular, Fig. 22 depicts the nullclines for different values of the maximal ATP consumption rate V ,. The corresponding steady states, given as the solution of the equation... 

Figure 24. The nullclines (upper panels, gray lines) and time courses (lower panels) for oscillatory solutions of the minimal model of glycolysis. Left panels Damped oscillations. The...

Fig. 5.6. Typical tc-g parameter plane for the thermokinetic model with the full Arrhenius temperature depedence a region of stationary-state instability lies within the locus of Hopf bifurcation points (solid curve). Also shown, as broken lines, are the loci corresponding to the maximum and minimum in the g(a, 0) = 0 nullcline (see text).

This is the fast motion of the system. We can draw this nullcline , so called because one of the rates of change is zero along this curve, in the a-d plane in Fig. 5.7. It has a maximum and a minimum whose coordinates are relatively easily located as a function of y, as we will see below (they also correspond to the maximum and minimum in the stationary-state locus). 

As a varies according to its own rate eqn (5.61), the slow motion of the system, we can expect the trajectory in the phase plane to try to stay as close as possible to the g(a, d) = 0 nullcline. We can also draw the /(a, d) = 0 nullcline on to the phase plane. This is a curve on which da/d Tis zero, so the concentration of A passes through a maximum or a minimum in time whenever a trajectory crosses it. There is one point where the two nullclines intersect. Here both time derivatives vanish simultaneously this is the sta-... 

Fig. 5.7. The nullclines f(a, 0) = 0 and g(a, 6) = 0 in the phase plane for the thermokinetic oscillator, specifically with y = 0.175 and M/K = 2.0, although the qualitative form of these loci is general for all parameter values (with y < I).

Fig. 5.8. Variation in stationary-state intersection relative to the maximum and minimum in the g(ot, 6) = 0 nullcline with the quotient M/K for a system with y = 0.2. (a) Intersection below maximum, M/K = 1.6 a given trajectory moves quickly to the g(a, 0) = 0 nullcline which it then moves along to the stationary-state solution, (b) Intersection above the minimum, M/K = 20 again a given trajectory will approach the stationary state along the g(a, 0) = 0 nullcline. (c) Intersection lying between the extrema, M/K = 5 now the stationary state is not approached and the time-dependent solutions cycle around the phase plane on the g(x, 6) = 0 nullcline (slow motion) with rapid jumps from one branch to the other (fast motion) at the turning points, (d), (e) Schematic representation of the relaxation oscillations for the conditions in (c).

In order to make quantitative predictions, we must locate the four corners of the oscillation A, B, C, and D. Two of these, A and C, correspond to the turning points in the g(a, 0) = 0 nullcline. For the present model these are also the turning points in the ass(p) locus, whose coordinates have been determined in eqns (4.66)-(4.68). We can content ourselves here with the leading-order forms with small y ... 

We require that the stationary state should lie on the section of the g(a, 0) nullcline along which a is decreasing. Thus 0SS, which is simply the quotient... 

In general terms we wish to calculate the time for motion along the g(oc, 0) = 0 nullcline. Along this curve we have... 

The time AT taken to traverse a particular section of the nullcline can be evaluated by integrating da/dT between the appropriate limits. Using the relationship (5.72) in the rate eqn (5.61) we have... 

To all intents and purposes, the Hopf prediction is the exact result. It is not difficult to construct an intersection between the maximum and minimum which is a stable stationary state. For instance, with y = 0.02, k = 0.12, and /i = 0.2 we have 0X = 1.042, 9C = 2400, and 0SS = i/k = 5/3. The corresponding phase plane nullclines are shown in Fig. 5.10, together with a trajectory spiralling in to the stationary-state intersection. The trace of the Jacobian matrix is negative for this solution (tr(J) = —4.1 x 10 2) indicating its local stability. This is not, however, a particularly fair test of the relaxation analysis because the parameters /i and k are not especially small. In the vicinity of the origin (where is small) both approaches converge. 

Fig. 5.10. The possibility of a stable stationary-state intersection on the middle branch of the g(a, 9) = 0 nullcline the nullclines are shown as broken curves, the solid curve gives the evolution of a typical trajectory towards the stable focal state.

The driving force for this excitable behaviour can be revealed from the model studied above, again assuming e is small and looking at the nullclines in the phase plane. A suitable orientation of the curves /(a, 0) = 0 and g(tx, 0) = 0 is shown in Fig. 5.11. The stationary-state intersection lies outside the range of instability , i.e. just before the maximum in the g(a, 0) = 0 nullcline, reflecting stability. If a small perturbation momentarily decreases a, or induces either a decrease or a very small increase in 0, the system merely jumps back on to that nullcline and then moves along it to the intersection... 

Fig. 5.11. Excitability in a chemical system, (a) The nullclines /(a, 0) = 0 and g(a,0) = 0 intersect just to the left of the maximum. A suitable perturbation must make a full circuit, as shown by a typical trajectory, before returning to the stable stationary state, (b), (c) The corresponding evolution of the concentration of intermediate A and the temperature excess in time showing the large-amplitude excursion.

To sketch the phase portrait, it is helpful to plot the nullclines, defined as the curves where either x=0 or y = 0. The nullclines indicate where the flow is purely horizontal or vertical (Figure 6.1.3). For example, the flow is horizontal where y = 0, and since y = -y, this occurs on the line y = 0. Along this line, the flow is to the right where x = x -i-1 > 0, that is, where x > -1. 

The nullclines also partition the plane into regions where x and y have various signs. Some of the typical vectors are sketched above in Figure 6.1.3. Even with the limited information obtained so far. Figure 6.1.3 gives a good sense of the overall flow pattern. 

For each of the following systems, find the fixed points. Then sketch the nullclines, the vector field, and a plausible phase portrait. 

Nullcline vs. stable manifold) There s a confusing aspect of Example 6.1.1. The nullcline x = 0 in Figure 6.1.3 has a similar shape and location as the stable manifold of the saddle, shown in Figure 6.1.4. But they re not the same curve To clarify the relation between the two curves, sketch both of them on the same phase portrait. 

Consider the following rabbits vs. sheep problems, where x,y>0. Find the fixed points, investigate their stability, draw the nullclines, and sketch plausible phase portraits. Indicate the basins of attraction of any stable fixed points. 

Nondimensionalize the model and analyze it. Show that there are two qualitatively different kinds of phase portrait, depending on the size of K, (Hint Draw the nullclines.) Describe the long-term behavior in each case. 

When polar coordinates are inconvenient, we may still be able to find an appropriate trapping region by examining the system s nullclines, as in the next example. 

Solution First we find the nullclines. The fi rst equation shows that x = 0 on the curve y = x/(a + x ) and the second equation shows that y = 0 on the curve y = b lla + x ). These nullclines are sketched in Figure 7.3.4, along with some representative vectors. 

How did we know how to sketch these vectors By definition, the arrows are vertical on the X = 0 nullcline, and horizontal on the y = 0 nullcline. The direction of flow is determined by the signs of i and y. For instance, in the region above both nullclines, the governing equations imply x > 0 and y < 0, so the arrows point down and to the right, as shown in Figure 7.3.4. 

Now consider the region bounded by the dashed line shown in Figure 7.3.5. We claim that it s a trapping region. To verify this, we have to show that all the vectors on the boundary point into the box. On the horizontal and vertical sides, there s no problem the claim follows from Figure 7.3.4. The tricky part of the construction is the diagonal line of slope —1 extending from the point b, b/a to the nullcline y = x/(a + x ). Where did this come from ... 

Can we conclude that there is a closed orbit inside the trapping region No There is a fixed point in the region (at the intersection of the nullclines), and so the conditions of the Poincard-Bendixson theorem are not satisfied. But if this fi xed point is a repeller, then we can prove the existence of a closed orbit by considering... 

To justify this picture, suppose that the initial condition is not too close to the cubic nullcline, i.e, suppose y-F(x) O(l). Then (4) implies (xj ( (/z) l whereas y 1 hence the velocity is enormous in the horizontal di-... 

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