In twodimensional models, the temporal evolution of the variables (u, w)^{T} can be visualized in the socalled phase plane. From a starting point (u(t), w(t))^{T} the system will move in a time t to a new state (u(t + t), w(t + t))^{T} which has to be determined by integration of the differential equations (3.2) and (3.3). For t sufficiently small, the displacement (u,w)^{T} is in the direction of the flow (,)^{T}, i.e.,
Let us consider the set of points with = 0, called the unullcline. The direction of flow on the unullcline is in direction of (0,)^{T}, since = 0. Hence arrows in the phase portrait are vertical on the unullcline. Similarly, the wnullcline is defined by the condition = 0 and arrows are horizontal. The fixed points of the dynamics, defined by = = 0 are given by the intersection of the unullcline with the wnullcline. In Fig. 3.2 we have three fixed points.
So far we have argued that arrows on the unullcline 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 wnullcline. By definition, it separates the region with > 0 from the area with < 0. Suppose we evaluate G(u, w) on the righthand side of Eq. (3.3) at a single point, e.g, at (0, 1). If G(0, 1) > 0, then the whole area on that side of the wnullcline has > 0. Hence, all arrows along the unullcline that lie on the same side of the wnullcline as the point (0, 1) point upwards. The direction of arrows normally^{3.1} changes where the nullclines intersect; cf. Fig. 3.2B.
In Fig. 3.2 there are three fixed points, but which of these are stable? The local stability of a fixed point (u_{FP}, w_{FP}) is determined by linearization of the dynamics at the intersection. With = (u  u_{FP}, w  w_{FP})^{T}, we have after the linearization
Eq. (3.23) is obtained by Taylor expansion of Eqs. (3.2) and (3.3) to first order in . If the real part of one or both eigenvalues of the matrix in Eq. (3.23) vanishes, the complete characterization of the stability properties of the fixed point requires an extension of the Taylor expansion to higher order.
Let us consider the linear dynamics
Because of F_{u} + G_{w} = a  < 0 for a < 0 and F_{u}G_{w}  F_{w}G_{u} = (b  a) > 0, it follows from (3.23) that the fixed point is stable. Note that the phase portrait around the left fixed point in Fig. 3.2 has locally the same structure as the portrait in Fig. 3.3A. We conclude that the left fixed point in Fig. 3.2 is stable.
Let us now keep the wnullcline fixed and turn the unullcline by increasing a to positive values; cf. Fig. 3.3B and C. Stability is lost if a > min{, b}. Stability of the fixed point in Fig. 3.3B can therefore not be decided without knowing the value of . On the other hand, in Fig. 3.3C we have a > b and hence F_{u}G_{w}  F_{w}G_{u} = (b  a) < 0. In this case one of the eigenvalues is positive ( > 0) and the other one negative ( < 0), hence we have a saddle point. The imaginary part of the eigenvalues vanishes. The eigenvectors and are therefore real and can be visualized in the phase space. A trajectory through the fixed point in direction of is attracted towards the fixed point. This is, however, the only direction by which a trajectory may reach the fixed point. Any small perturbation around the fixed point, which is not strictly in direction of grows exponentially. A saddle point as in Fig. 3.3C plays an important role in socalled type I neuron models that will be introduced in Section 3.2.4.
For the sake of completeness we also study the linear system
Since F_{u}G_{w}  F_{w}G_{u} = (a  b) < 0, the fixed point is unstable if a < b. In this case, the imaginary part of the eigenvalues vanishes and one of the eigenvalues is positive ( > 0) while the other one is negative ( < 0). This is the definition of a saddle point.
One of the attractive features of phase plane analysis is that there is a direct method to show the existence of limit cycles. The theorem of PoincaréBendixson (Verhulst, 1996; Hale and Koçak, 1991) tells us that, if (i) we can construct a bounding surface around a fixed point so that all flux arrows on the surface are pointing towards the interior, and (ii) the fixed point in the interior is repulsive (real part of both eigenvalues positive), then there must exist a stable limit cycle around that fixed point.
The proof follows from the uniqueness of solutions of differential equations which implies that trajectories cannot cross each other. If all trajectories are pushed away from the fixed point, but cannot leave the bounded surface, then they must finally settle on a limit cycle; cf. Fig. 3.4. Note that this argument holds only in two dimensions.
In dimensionless variables the FitzHughNagumo model is
A comparison of Fig. 3.5A with the phase portrait of Fig. 3.3A, shows that the fixed point is stable for I = 0. If we increase I the intersection of the nullclines moves to the right; cf. Fig. 3.5C. According to the calculation associated with Fig. 3.3B, the fixed point looses stability as soon as the slope of the unullcline becomes larger than . It is possible to construct a bounding surface around the unstable fixed point so that we know from the PoincaréBendixson theorem that a limit cycle must exist. Figures 3.5A and C show two trajectories, one for I = 0 converging to the fixed point and another one for I = 2 converging towards 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. 3.5D.
We have seen in the previous example that, while I is increased, the behavior of the system changes qualitatively from a stable fixed point to a limit cycle. The point where the transition occurs is called a bifurcation point, and I is the bifurcation parameter. Note that the fixed point (u(t), w(t)) = (u_{FP}, w_{FP}) remains a solution of the dynamics whatever the value of I. At some point, however, the fixed point looses its stability, which implies that the real part of at least one of the eigenvalues changes from negative to positive. In other words, the real part passes through zero. From the solution of the stability problem (3.23) we find that at this point, the eigenvalues are
Unfortunately, the discussion so far does not tell us anything about the stability of the oscillatory solution. If the new oscillatory solution, which appears at the Hopf bifurcation, is itself unstable (which is more difficult to show), the scenario is called a subcritical Hopfbifurcation. This is the case in the FitzHughNagumo model where due to the instability of the oscillatory solution in the neighborhood of the Hopf bifurcation the dynamics blows up and approaches another limit cycle of large amplitude; cf. Fig. 3.5. The stable largeamplitude limit cycle solution exists in fact already slightly before I reaches the critical value of the Hopf bifurcation. Thus there is a small regime of bistability between the fixed point and the limit cycle.
In a supercritical Hopf bifurcation, on the other hand, the new periodic solution is stable. In this case, the limit cycle would have a small amplitude if I is just above the bifurcation point. The amplitude of the oscillation grows with the stimulation I.
Whenever we have a Hopf bifurcation, be it subcritical or supercritical, the limit cycle starts with finite frequency. Thus if we plot the frequency of the oscillation in the limit cycle as a function of the (constant) input I, we find a discontinuity at the bifurcation point. Models where the onset of oscillations occurs with nonzero frequency are called type II excitable membrane models. Type I models have an onset of oscillations with zero frequency as will be discussed in the next subsection.
In the previous example, there was exactly one fixed point whatever I. If I is slowly increased, the neuronal dynamics changes from stationary to oscillatory at a critical value of I where the fixed point changes from stable to unstable via a (subcritical) Hopf bifurcation. In this case, the onset occurs with nonzero frequency and the model is classified as type II.

A different situation is shown in Fig. 3.6. For zero input, there are three fixed points: A stable fixed point to the left, a saddle point in the middle, and an unstable fixed point to the right. If I is increased, the unullcline moves upwards and the stable fixed point merges with the saddle and disappears. We are left with the unstable fixed point around which there must be a limit cycle provided the flux is bounded. At the transition point the limit cycle has zero frequency because it passes through the two merging fixed points where the velocity of the trajectory is zero. If I is increased a little, the limit cycle still `feels' the disappeared fixed points in the sense that the velocity of the trajectory in that region is very low. Thus the onset of oscillation is continuous and occurs with zero frequency. Models which fall into this class are called type I; cf. Fig. 3.7.
From the above discussion it should be clear that, if we increase I, we encounter a transition point where two fixed points disappear, viz., the saddle and the stable fixed point (node). At the same time a limit cycle appears. If we come from the other side, we have first a limit cycle which disappears at the moment when the saddlenode pair shows up. The transition is therefore called a saddlenode bifurcation on a limit cycle.
The appearance of oscillations in the FitzHughNagumo Model discussed above is of type II. If the slope of the wnullcline is larger than one, there is only one fixed point, whatever I. This fixed point looses stability via a Hopf bifurcation.
On the other hand, if the slope of the wnullcline is smaller than one, it is possible to have three fixed points, one of them unstable the other two stable; cf. Fig. 3.2. The system is then bistable and no oscillation occurs.
Depending on the choice of parameters, the MorrisLecar model is of either type I or type II. In contrast to the FitzHughNagumo model the wnullcline is not a straight line but has positive curvature. It is therefore possible to have three fixed points so that two of them lie in the unstable region where u has large positive slope as indicated schematically in Fig. 3.6. Comparison of the phase portrait of Fig. 3.6 with that of Fig. 3.3 shows that the left fixed point is stable as in Fig. 3.3A, the middle one is a saddle point as in Fig. 3.3C, and the right one is unstable as in Fig. 3.3B provided that the slope of the unullcline is sufficiently positive. Thus we have the sequence of three fixed points necessary for a type I model.
Consider the onedimensional model
Let us now reduce the amplitude of the applied current I. For I 0, the velocity along the trajectory around = 0 tends to zero. The period of one cycle T(I) therefore tends to infinity. In other words, for I 0, the frequency of the oscillation = 1/T(I) decreases (continuously) to zero. For I < 0, Eq. (3.30) has a stable fixed point at = 0; see Fig. 3.8.
The model (3.30) is a canonical model in the sense that all type I neuron models close to the bifurcation point can be mapped onto (3.30) (Ermentrout, 1996).
A B

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