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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Hodgkin-Huxley Membrane Models

This week we will examine the Hodgkin-Huxley model, the Nobel-prize winning set of ideas describing how membranes generate action potentials by sequentially allowing ions of sodium and potassium to flow. The learning objectives for this week are: (1) Describe the purpose of each of the 4 model levels 1. alpha/beta, 2. probabilities, 3. ionic currents and 4. trans-membrane voltage; (2) Estimate changes in each probability over a small interval $$\Delta t$$; (3) Compute the ionic current of potassium, sodium, and chloride from the state variables; (4) Estimate the change in trans-membrane potential over a short interval $$\Delta t$$; (5) State which ionic current is dominant during different phases of the action potential -- excitation, plateau, recovery.

- 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. 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.

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.

There's one other thing we should say about these equations.

That's very, very helpful when it comes time to do calculations.

In our original equations, they were written this way.

If, when it comes time to do numerical calculations, people usually turn these

equations around, and instead they write them like this.

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.

It'll work. And how short is short?

Well that's always the question. So when you say delta t must be short you

are thinking that delta t is perhaps one microsecond.

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.

The part we've left out here is talking about the alphas and the betas so we'll

have to come back and talk about those in the next lecture.

Thank you for watching. See you next time well, we'll talk about

those alphas and betas and all their twists and turns.

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