In two-dimensional models, the temporal evolution of the variables can be visualized in the so-called phase plane. From a starting point the system will move in a time to a new state which has to be determined by integration of the differential equations (4.4) and (4.5). For sufficiently small, the displacement is in the direction of the flow , i.e.,
which can be plotted as a vector field in the phase plane. Here is given by (4.4) and by (4.5). The flow field is also called the phase portrait of the system. An important tool in the construction of the phase portrait is the nullcline, which is introduced now.
Let us consider the set of points with , called the -nullcline. The direction of flow on the -nullcline is in direction of , since . Hence arrows in the phase portrait are vertical on the -nullcline. Similarly, the -nullcline is defined by the condition and arrows are horizontal. The fixed points of the system, defined by are given by the intersection of the -nullcline and the -nullcline. In Fig. 4.7 we have three fixed points.
So far we have argued that arrows on the -nullcline are vertical, but we do not know yet whether they point up or down. To get the extra information needed, let us return to the -nullcline. By definition, it separates the region with from the area with . Suppose we evaluate on the right-hand side of Eq. (4.5) at a single point, e.g, at . If , then the whole area on that side of the -nullcline has . Hence, all arrows along the -nullcline that lie on the same side of the -nullcline as the point point upward. The direction of arrows normally55Exceptions are the rare cases where the function or is degenerate; e.g., . changes where the nullclines intersect; cf. Fig. 4.7B.
In Fig. 4.7 there are three fixed points, but which of these are stable? The local stability of a fixed point is determined by linearization of the dynamics at the intersection. With , we have after the linearization
where , , …, are evaluated at the fixed point. To study the stability we set and solve the resulting eigenvalue problem. There are two solutions with eigenvalues and and eigenvectors and , respectively. Stability of the fixed point in Eq. (4.25) requires that the real part of both eigenvalues be negative. The solution of the eigenvalue problem yields and . The necessary and sufficient condition for stability is therefore
If , then the imaginary part of both eigenvalues vanishes. One of the eigenvalues is positive, the other one negative. The fixed point is then called a saddle point.
Eq. (4.25) is obtained by Taylor expansion of Eqs. (4.4) and (4.5) to first order in . If the real part of one or both eigenvalues of the matrix in Eq. (4.25) vanishes, the complete characterization of the stability properties of the fixed point requires an extension of the Taylor expansion to higher order.
In dimensionless variables the FitzHugh-Nagumo model is
Time is measured in units of and is the ratio of the two time scales. The -nullcline is with maxima at . The maximal slope of the -nullcline is at ; for the -nullcline has zeros at 0 and . For the -nullcline is shifted vertically. The -nullcline is a straight line . For , there is always exactly one intersection, whatever . The two nullclines are shown in Fig. 4.10.
A comparison of Fig. 4.10A with the phase portrait of Fig. 4.8A, shows that the fixed point is stable for . If we increase the intersection of the nullclines moves to the right; cf. Fig. 4.10C. According to the calculation associated with Fig. 4.8B, the fixed point loses stability as soon as the slope of the -nullcline becomes larger than . It is possible to construct a bounding surface around the unstable fixed point so that we know from the Poincaré-Bendixson theorem that a limit cycle must exist. Figures 4.10A and C show two trajectories, one for converging to the fixed point and another one for converging toward the limit cycle. The horizontal phases of the limit cycle correspond to a rapid change of the voltage, which results in voltage pulses similar to a train of action potentials; cf. Fig. 4.10D.
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