In the phase plane treatment, the rest state of the model is realized as the intersection of the two nullclines; such steady state solutions are also referred to as singular or equilbrium points. From the geometrical viewpoint, one sees how different parameter values could easily lead to multiple singular points - by changing the shapes and positions of the nullclines. In Figure 1, the unique singular point is attracting.
Try running this simulation with different initial values of the voltage. Find the threshold voltage for excitation. The ODE file allows you to simulate a pulse. The parameter s1 is the amplitude of the pulse. Imitate a real current pulse experiment by setting the voltage to rest, V=-60.8. Find the minimum value of s1 to elicit a spike. (It should be about 220.)
Technically, we say it is
asympotically stable, i.e. for any nearby initial point the solution tends to
the singular point as 
. In general, the local stability of
a
singular point can be determined by a simple algebraic criterion [6,34]. The procedure
is to linearize the differential equations and evaluate the partial derivatives
at the singular point (this matrix of partial derivatives is called the
Jacobian). Then one asks whether the exponential solutions to this
constant coefficient system have any growing modes. If so, then the singular
point is unstable; if all modes decay, then it is stable. For equations
(4)-(6), the linearized equations which describe the behavior of
small disturbances, 
, from the singular
point are
where,
Solutions are of the form 
, 
where
are the eigenvalues of the Jacobian matrix in equations
(11)-(12); they are roots of the quadratic:
For the parameters of Figure 1, the two eigenvalues are both real and negative.
As parameters are varied, the singular point may lose stability. In our
example, the rest state could then no longer be maintained and the behavior of
the system would change - it may fire repetitively or tend to a
different steady
state (if a stable one exists). Let us consider the effect of a steady applied
current, and ask how repetitive firing arises in this model. We will apply
linear stability theory to find values of I for which the steady state is
unstable. First, we note that for equations (4)-(6), and for
nerve membrane models of the general form (1)-(2), a
steady state solution 
for a given I must satisfy 
,
where 
is the steady state I-V relation of the model which is given
by:
If 
is N-shaped, there will be three steady states for some range of
I. However if 
is monotonic increasing with V, as for the case
of Figure 1, then there is a unique 
for each I, and moreover,
cannot lose stability by having a single real eigenvalue pass
through zero. Destabilization can only occur by a complex conjugate pair of
eigenvalues crossing the axis 
as I is varied through a
critical value 
. At such a transition, a periodic solution to equations
(4)-(6) is born - and we have the onset of repetitive
activity. This solution, for I close to 
, is of small amplitude and
frequency proportional to 
. Emergence of a periodic solution
in this way is called a Hopf bifurcation [6,34].
From equations (11)-(12), or (17),
we know that 
. Thus, loss of stability occurs
for the I whose corresponding satisfies
The first term here is the slope of the instantaneous I-V relation
and the second is the rate of the recovery process; this condition also applies
approximately to the HH model [28]. From (19) we conclude
that loss of stability occurs: (1)
only if the instantaneous I-V relation has negative slope at 
; (2)
when the destabilizing growth rate of V from this negative resistance
just
balances the recovery rate; and (3) only if recovery is sufficiently slow,
i.e. if 
is small (low ``temperature'').
Run AUTO for this picture (File Auto). Set the AUTO plotting axes (Axes HiLo) as in the picture A (so that xmin=0,xmax=300,ymin=-80,ymax=50 ) and the AUTO numerics (Numerics) so that DSMAX=5 and so that the parameter ranges between 0 and 300 ( Par Min=0, Par Max=300 ). Choose (Run) (Steady State) to get the steady state curve. Then choose (Grab) and move the arrows or (Tab) until the first Hopf bifurcation point is reached (labeled HB at the bottom) and Pt 2 in the diagram. Choose (Enter) to accept the point. Choose (Run) (Periodic) to trace out the periodic orbit. Choose (ABORT) if it looks like it is retracing. Choose (Axes) (Frequency) and clik on (Ok) and then (Axes) (Fit) and finally (rEdraw) to see the frequency as a function of the parameter. NOTE If you run AUTO, don't forget to use the (File) (Reset Diagram) command to clean up all the junk produced by AUTO.
In Figure 2A, 
is
plotted
versus I (this is the steady state I-V relation, but shown as V
against I) and the region of instability is shown dashed.
Figure 2A also shows the maximum and minimum values of V for the oscillatory
response. Just as a singular point can be unstable, so too can a periodic
solution [34]; unstable periodics are indicated by open circles.
Here
we
see that the small amplitude periodic solution born at 
A/cm
from the loss in stability
of 
is itself unstable; it would not be directly observable. (In
the phase plane, but not generally for higher order systems, an unstable
periodic orbit can be determined by integrating backwards in time.) Note, that
solutions along this branch depend continuously upon parameters and they gain
stability at
the turning point or knee at 
A/cm
.
A stable periodic solution is called a limit
cycle. The upper branch (solid) corresponds
to the limit cycle of observed repetitive firing. The frequency
increases with I over most of this branch (Figure 2B).
At sufficiently large I repetitive firing ceases as
regains stability at 
A/cm
.
This figure is referred to as a bifurcation
diagram; it depicts steady state and periodic solutions, and their stability,
as functions of a parameter and it shows where one branch bifurcates\
(from
the Greek word for branch) from
another. Bifurcation theory allows one to characterize solution behavior
analytically in the neighborhood of bifurcation points, e.g., the frequency of
the emergent oscillation at the Hopf point is proportional to 
. When the Hopf
bifurcation is to unstable periodic solutions, back into the parameter region
where the steady state is stable, then it is called subcritical (i.e., a
hard oscillation); if the opposite occurs, the bifurcation is
supercritical.
For a range of I values (between the knee, 
and the Hopf
bifurcation, 
) this model
exhibits bistability: a stable steady state and a stable oscillation
coexist.
Find the fixed point and its stability. Try different initial conditions and find the stable limit cycle as well. Integrate backwards to get the unstable periodic. Set the voltage and recovery to the rest state, (-26.59,0.129), and set the total amount of time to 800. Set s1=30,s2=30 to turn on the stimulus and simulate Fig 3b. Plot V versus t.
Figure 3A illustrates the phase plane profile in such a case; a periodic
response here appears as a closed orbit. There is a stable fixed
point shown as the intersection of the two nullclines and a stable
periodic orbit (labeled SPO).
The two attractors are separated by
an unstable periodic orbit (UPO). Initial values inside
the unstable orbit tend to the attracting steady state and
initial conditions outside of
it will lead to the limit cycle of repetitive firing. A brief current
pulse, whose phase
and amplitude are in an appropriate range, can switch the system out of the
oscillatory response back to the rest state. Such behavior has been seen for
many models and observed, for example, in squid axon membrane
[15]. In Figure 3B, two 30 
current pulses 5
msec in duration are given at t=100 msec and then at t=470
msec. The first switches the membrane from rest to repetitive
firing while the
second pushes the membrane back to rest.
This bistable behavior is critical for the occurence of bursting
oscillations when a very slow conductance is added to the model.
