More About the Math:
The Traveling Spike (Part 4)
The next step is to notice that, for our spiking solution and any x, V
(x; t) =
Vrest when t becomes very small or very large. This
is just saying that the membrane starts at rest, the pulse passes any point
by, and then the membrane returns to rest.
Can our equations produce such a solution? Let's think about what this
means. There must be a trajectory that
- Starts at resting values of V, m, n, and h.
- Moves away from this resting state, tracing out the pulse over time.
- Returns exactly to the resting state at large times.
Thinking about it, this seems like hitting a needle in a haystack. Departing
a single point, obeying a highly complex and nonlinear system of equations,
and then returning exactly to the single point where the trajectory
started. So it's no surprise that this does not work for most values of the
assumed pulse speed c: there is no such solution to our equations!
However, if one chooses this wavespeed perfectly—at the precise value
of 18.8 millimeters/second—then a solution with the properties we need
appears. The mathematical method used to find this value is called "shooting";
it's an algorithm for honing in on the value that gives the desired
solution. This technique is used both numerically in computer simulations
and in "pencil and paper" mathematical proofs of the existence
of traveling pulse solutions to Hodgkin and Huxley type equations. (For
more on this, see the references by Hastings and Carpenter below.)
What does the biology tell us? Hodgkin and Huxley found experimentally that
the travelling pulse moves at 21.2 millimeters/second. So, the
mathematical theory got it right, within about 11 percent!
Surprised? That's quite a low margin of error, considering the complexity
of the mathematical model and the experiments done to derive it. The
scientific community was certainly surprised by the success of the theory--
it was a breakthrough that led to the Nobel prize.
References:
Keener, J. and Sneeyd, J. (1998) Mathematical Physiology, IAM.
Carpenter, G. (1977) A geometric approach to singular perturbation
problems with applications to nerve impulse equations, J. Diff. Eqns.
23: 335-367.
Hastings, S.P. (1975) The existence of progressive wave solutions
to the
Hodgkin-Huxley equations, Arch. Rat. Mech. Anal. 60:
229-257.
