Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Bioelectricity: A Quantitative Approach

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Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Propagation

this week we will examine how action potentials in one region normally produce action potentials in adjacent regions, so that there is a sequence of action potentials, an excitation wave. the learning objectives for this week are: (1) Identify the differences between the propagation pattern following sub-threshold versus threshold stimuli; (2) Compute the changes in transmembrane potentials and currents from one time to a short time laterIdentify the outcome of stimulating a fiber at both ends; (3) Quantify the interval after propagation following one stimulus to the time when there will be another excitation wave following a 2nd stimulus; (4) Explain why "propagation" is different from "movement".

- Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics

Biomedical Engineering, Pediatrics

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.

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.

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.

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.

Now first, yes, theta really is the velocity, not some kind of an angle.

Second, yes people use this equation all the time, so it's worth memorizing it, or

at least memorize the fact that theta goes as the square root of a.

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.

Thank you for watching this segment. And we'll see you again soon.

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