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So, hello again. This is Roger Coke Barr for the
Bioelectricity course, Week six, Lecture segment number ten.
In this segment, I have the fun of showing you, something that is done, with a
mathematical derivation that is really, really smart.
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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.
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Any combination of parameters that doesn't change the values of these constants is
going to have the same action potential as their solution.
So, if we want to, we can say, let's define a constant, K equal to a over two
Ri theta squared. That leads us to the following very
important result. We turn this equation around.
This one. Take that one and turn it around.
We say, the very same action potential will remain the solution so long as this
relationship applies. Putting that a little bit differently, if
the relationship between a and theta follows this equation, the same action
potential will be the solution. Well, think about that.
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The tissue wants to produce an action potential.
So, what the tissue will do, is it'll keep the relationship between the velocity and
the diameter, What is shown right here.
The velocity will go as the square root of the diameter under most circumstances, cuz
under most circumstances, Ri will not change.
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This is if you will, an imperative to the construction of, nerve fibers or other
tissue where the velocity of propagation is significant.
That is to say, a significant aspect of function.
If you wanted to go faster, if the goal is to have a higher propagation velocity,
Then you've got to have a bigger diameter. And in particular, you have to have a
diameter that is four times the size in order to get twice the velocity, since the
relationship is as a square root. There are many dis, many demonstrations of
this fact in different tissue systems, both of animals and of humans, where you
see that a larger diameter axon for example, is associated with a need for
high conduction velocity, a very functional and useful, practical rule
which we've arrived at through a rather elegant mathematical argument.
Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics
