It is important to realize that the solution behavior (bifurcation diagrams) we
have described depend on other parameters in the model. The temperature
parameter 
is particularly convenient, with useful interpretative value,
for additional parametric tuning. This parameter, since it plays no role in
,
does not affect the values along the S-shaped curve of steady states in
Figure
6 , or the corresponding curve in Figure 2. The
stability of a steady state does
however depend on 
. As is seen from equation (19), when 
is
large, oscillatory destablization is precluded; Hopf bifurcation from a steady
state only occurs when the time scale of w is slow compared to that of
V.
Thus for
large 
both the upper and lower branches of the S-curve are stable; the
middle branch is of course unstable. This system is bistable. In this
large-
limit, the kinetics of the 
-system are so fast (essentially
instantaneous with 
) that the model reduces to one dynamic
variable, V. Then stability is determined only by the slope of 
so the
two ``outer'' states are stable and the ``middle'' is unstable. This simple
example also shows that sometimes a model can be conveniently reduced to a lower
dimension when there are significant time scale differences between variables.
For intermediate values of 
, the dynamics of both V and w influence
stability and the upper branch is unstable for a certain range of I. Figure 7
shows a bifurcation diagram analogous to that in Figure 6A.
Compute the bifurcation diagram for this by running AUTO. Set up the AUTO (Axes) with xmin=-20, xmax=80, ymin=-70, ymax=25 and set up the AUTO (Numerics) with DSMAX=5, Par Min=-20, Par Max=70. Run from the steady state, then grab the Hopf bifurcation and track the periodic orbits.
As in Figure 6A,
the branch of steady states is S-shaped and the stable rest state
disappears at a turning point (point A). The high voltage
equilibrium is stable for large currents but as the current is reduced
loses stability at a subcritical Hopf bifurcation (point B). An unstable
branch of periodic solutions emanates from the Hopf bifurcation point
and then becomes stable at a turning point (label C). Unlike
Figure 6A, however, this branch of
stable periodic orbits (solid circles) does not terminate on the knee
( A) but instead
on the unstable middle branch (point
D on the diagram) as the current decreases to a critical value,
.
Again the
frequency of the limit cycle tends to zero for this branch, but not as
the square root. Rather, the frequency is proportional to
At the critical value of current, 
,
the closed orbit has infinite period; it is called a
saddle-loop homoclinic orbit.
Recall that the middle branch of solutions is a saddle
point. One branch of the unstable manifold of this saddle-point exits
the singular point and returns via a branch of the stable manifold
(cf Figure 8a.)
(Contrast this with the saddle-node loop homoclinic
in Figures 4-6.) For certain values of the current, this system is
tristable. That is, there are three stable ststes. If I is
chosen to lie between the I-values for
points B and C then, the lower
branch still exists and is stable, the upper branch of equilibria is
stable, and there is a stable periodic orbit. Figure 8A shows the
phaseplane for this case. The stable manifold for the saddle point (bold
dashed trajectory) acts to separate the stable periodic orbit (SPO)
from the
lower rest state. The small unstable periodic orbit (UPO) separates
the upper rest state from the stable periodic solution. As in Figure
3B , we can use brief current pulses to switch between states. Figure
8B shows the effect of three 5 msec current pulses switching from the
periodic orbit to the lower rest state, back to the periodic orbit,
and then to the upper rest state. (Note that perturbations from the
upper rest state decay very slowly.)
(A.) Try several different initial conditions; find three stable solutions. Find the saddle point and track the invariant sets. (It is about V=-23.9 and w=0.) (B.) Set the total amount of time to 600 msec, set V=5 and w=0.4, and set the stimulus parameters, s1=30, s2=50, s3=40 and integrate. Plot V versus t to see the switching between the three behaviors.
The HH model, adjusted for higher than normal external potassium, exhibits similar multistable behavior [29].
This example of coexistence between a depolarized limit cycle and a lower resting state is important as it also forms the basis for a general class of bursting phenomena.