0:00
So, hello again. This is Roger Coke Barr for the
bioelectricity course. We're in week four, lecture five.
Let's talk about where we are just for a moment.
We're trying explain action potentials. We saw that Hodgkin and Huxley replaced
the membrane resistance, with three paths, a path for each ion, potassium, sodium,
and leakage. We saw the equation for each path.
So the equation was something like g bar K, n to the fourth.
Now we are want to talk about this n business, the probabilities and how they
change, how these probabilities change as the voltage across the membrane changes.
Hodgkin and Huxley taught us that each of these probabilities changes in one of
these right changing equations. Let's look at what the terms are.
It says, if we want for a moment to think of n as being the number of channels that
are open, it's a probability, but let's say it's the probability out of a million
channels, how many are open? So the number that are open, changing with
the time, dn/dt is equal to this coefficient alpha which tells us what is
the rate at which the closed channels are opening.
So if n is the number open, one minus n is the number closed, n will be the number
open and beta, this coefficient, beta is the coefficient that tells us how rapidly
channels that are open are going closed. So beta is the right, is the right of
closings, alpha is the rate of opening. And if we look at the effect as a whole,
we'd say the rate at which the number of channels, number of n type particles is
opening, is equal to the rate of opening, alpha n times the number of closed
channels, one minus n, subtracting the rate of closing, beta n, that rate, times
the number of channels that are open. When you think about this as a mental
picture, it seems pretty simple. You have a lot of channels that are open.
As time goes by, some of the open channels close.
And as time goes by, some of the closed channels open.
Think back to the pictures that we were seeing of individual channel function.
We saw those last week. You remember that they were up and down
and up and down. Just going across you remember, up down
up, there'd be a wall. Up, up, up.
I'm not drawing it too well, but you remember the randomness.
So channels that were open closed, channels that were closed opened.
And so the number that is open or that is closed depends on the average on the
values of alpha and beta, but for individual channels, they continue to go
to this opening and closing process. Now there's a very important thing here.
All the alphas and betas change when Vm changes.
There is an equation for alphas and betas that all change when Vm changes.
So this equation on the top for dn/dt. That's a pretty simple equation as long as
the alphas and betas are constants. But as you can imagine, things can change
around considerably, if the alphas and betas change.
Now we'll look at another part of the slide.
You remember the sodium and pot, the sodium channel was controlled by two
probabilities, m and h. These probabilities have exactly the same
form as far as the form of the equation as does the equation for n.
So everything corresponds going down vertically, alpha n, alpha m, alpha h,
beta n, beta, beta n, beta m, beta h. All those are pretty much the same.
That does not mean that the number values are the same.
Do not kid yourself. The actual number values exa, for example,
for alpha m are entirely different than the number values for alpha h.
In fact, as Vm increases, Vm is going up. Alpha m is going up.
5:54
Alpha h, is, is going down. So they not only aren't the same value,
they may not go in the same direction. They also may not change in the same
degree. So these values of alpha and beta, alpha m
is not equal to alpha h. For that matter it's not equal to alpha n.
They have similar form but entirely different values.
So as a result you've really got to know what they are, before you can make any use
at all out of these equations. There is one more general result that can
be determined however from looking at these equations, and that is what is the
long term equilibrium value of n, or m, or h, if the voltage does not change.
So if Vm is constant, Vm is a constant, then n goes to alpha n over alpha n plus
beta n. Remember we saw this last week when we
were talking about channels. Remember last week, I asked you to
actually find the solution of an equation that had this form.
I hope you did that, and I hope you can remember it, and use it again this week.
8:13
That is to say they take the dt and put it on the other side of the equation in each
case. They do that because they want to use the
following idea. In the top panel, dn and dt are
infinitesimal in the sense of calculus. They are the limits as the changes are
very, very sh, very, very short in a temporal sense.
However, to make use of these equations and computation, the thought is, you say,
well, delta t, a, a finite time interval is about the same thing as dt, so as a
result, delta n, the actual change in the probability, or the actual change in the
number of open channels if you multiply it by the number of channels, delta n would
be about equal to dn. That's a good thing to do and has been
used successfully in millions and millions of calculations.
So long as, so long as delta t must be short.
9:52
I'll erase and write that again. Delta t about one microsecond.
So sometimes under some circumstances people say oh, I can get away with delta t
about ten microseconds. That is probably the case.
But if somebody else pushes it on out there and says I want my delta t to be
about a hundred microseconds. I can tell you that it will not work.
So don't let your delta t's get anywhere near that big.
So long as you keep them down near one microsecond, and maybe even up to ten
microseconds, things will do okay. Once they get longer than that, you're
going to have trouble on your hands in the form of results that are not just a little
bit more off, but rather, results that are grossly incorrect.
Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics
