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
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 . 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 ; 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 . 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.