This is a continuation of the single cell FHN model .
Since the FHN model consist of only two variables, the qualitative properties of the model can be explored by a phase-plane analysis.
Therefore this applet also includes a phase-space plot of the variables U and v,
in which the computed time series U(t) and v(t) are plotted as ordered pairs (U,v) as a
function of time. The phase-space view shows how the system starts at a fixed point and, if excited above
threshold and allowed to return to rest, follows a closed loop back to the fixed point. The system's nullclines
are plotted as well to further explain the dynamics. The nullclines are found by setting the derivatives of U and v in time equal to
zero
and solving both equations in the form v = f(u), resulting for this system in a straight
line corresponding to the v nullcline (green) and a cubic curve for the U nullcline (red). The system's fixed
point is located at the intersection of the nullclines. Raising the system above threshold corresponds to
increasing U beyond the central portion of the cubic nullcline, so that it is in the area under the local
maximum. As U and v evolve, they roughly follow the course of the nullclines back to the resting state.
Because epsilon is the time scale difference between the U and v processes, decreasing epsilon causes the solution to
follow more closely the cubic nullcline, since that process is faster. Compare, for example, the phase
trajectory produced by using epsilon =0.01 and the one produced by using epsilon =0.1. The evolution
of the phase space trajectory in time can be observed by changing the integration time. Extending the integration
time from 1 to 20 in increments of 1 allows visualization of the different speeds involved in the phase space process.
Changing parameter values changes the nullclines, which in turn change the dynamics.
For instance, the parameter delta can shift the resting potential from stable to unstable and make
the cell auto-oscillatory. Try changing delta from 0 to 0.04 and to 0.15 and observe the oscillatory patterns,
while also noticing how the v nullcline shifts. The oscillations occur because whenever
he fixed point is located along the central region (positive slope) of the U nullcline, the fixed point is
unstable, giving rise to oscillations. If delta is increased further to 0.22, the oscillations cease, as the
fixed point is once again located in a region of negative slope on the U nullcline. However, the fixed point
is now much higher than the initial state, and the system does not exhibit a full cycle. As another example,
try changing gamma from 1 to 3. This reduces the slope of the v nullcline so that it intersects the U nullcline
in three locations, rather than just one. The middle point is unstable, but the other two are both stable.
The initial stimulus moves the system to the upper fixed point, where it remains until the application of
the second stimulus, which moves the system back to the lower fixed point. For further information on
parameters and nullcline analysis we refer to
Winfree, A.T., 1992. The Geometry of Excitability. In: Nadel, L. and Stein. D. (Eds.) 1992 Lectures in Complex Systems. Addison-Wesley, 207-298.