But if your not mathematically inclined, the result is something, that is really,
really useful. First, let's do the mathematics.
And if your mathematically inclined, you'll really enjoy it.
If your not, well, you can take a nap for a few minutes and we'll, we'll get to
something that very useful, very quickly. Let's do a very quick derivation.
We began with the equation that said the membrane current was the capacity current
plus the ionic current. We've done that many times.
In an earlier segment, the one about the track, we found an expression for the
membrane current. The membrane current goes as the second
derivative of the transmembrane potential. With, that is to say, second derivative
with respect to space, d square Vmdx square of the transmembrane potential.
There also are some important constants that precede the second derivative within
the equation. Suppose now, we think of uniform
propagation. By uniform propagation, I mean, we have a
cylindrical fiber or other uniform structure and the action potential is
just, so to speak, going down the track. It's propagating from one site to another
to another to another. And it's doing it uniformly.
In that case, there's a relation between the spatial second derivative and the
temporal second derivative. This is a well known relationship and I
have written it here as the third equation.
The only thing that's unusual as compared to the way you see it written eslewhere,
is the symbol theta, is the symbol used for velocity.
So, it's not angle, it's velocity. If you take equations one. two, and three,
combine them all together, you get to equation four and you can say the
following. Equation four is the differential equation
that has to be satisfied by the action potential.
That is, each and every point along the action potential has to obey this
equation. So, let's suppose that we have an action
potential. So, obviously it is the solution to this
equation. What other circumstances for, what other
circumstances will that very same action potential also be a solution?
And the answer has to do with these constants.