The Fitzhugh - Nagumo ODE system models the excitation of nerve cells. From
our basic understanding, a nerve cell has some equilibrium electrical potential
x(t) that it is trying to maintain. Because of this, the solutions of the
differential equation that are periodic cannot be interpretted very easily.
We would expect the nerve cell would eventually reach some equilibrium state.
By this reasoning the only solutions that do seem to have a realistic
interpretation for a "standard" nerve cell are those solutions that
decay to some fixed point as t approaches infinity. Determining the hopf
bifurcation we found these solutions satisfy:
a > -1/c*(c2-b)(1/2)*b+1/c*(c2-b)(1/2)+1/3*b/c3*(c2-b)(3/2)
Or more specifically:
a > -(1-b)(1/2)*b+(1-b)(1/2)+1/3*b*(1-b)(3/2) for c=1
a > -(1-b)(1/2)*b+(1-b)(1/2)+1/3*b*(1-b)(3/2) for
For a that satisfies these equations all trajectories do in fact decay,
however, it may be logical to restrict a even more so that there is only a
single fixed point that a decays to. In most cases we would expect that a
nerve cell would only have one steady state. The equations above are satisfied for cases where a
has either 1 or 3 fixed points. Recall, in the case of 1, there is one sink; and in the caes of three,
there are two sinks and a saddle.
We restrict a again by limit point bifurcation, so that there is only one steady state as
a < ((-1+b)/b)(1/2)*b-((-1+b)/b)(1/2)-1/3*b*((-1+b)/b)(3/2)
It is definately possible that limit cycles and other periodic solutions, and multiple fixed points have realistic interpretations in this nerve excitation model. We do not wish to say that the do not. We only wish to suggest where the model can be most
easily interpreted as an actual physical process.
[ Introduction |