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

Hello again.

This is Roger Coke Barr for the Bioelectricity course.

We're in week four.

This is segment seven.

And this is a problem session.

It's an interesting problem session.

I think you will enjoy it.

Here's our problem.

We have active tissue.

We want to use the Hodgkin Huxley standard parameter values that

were given a few slides back.

We want to use the state variable set number one that

was given a few slides back.

There's no stimulus current at this time.

And we're asking for this tissue, this membrane.

One of the values of IK, Ina, IL,

the leakage current and then the most interesting part.

Part d, if these currents were maintained without change for 50 microseconds.

These currents, it means IK, Ina, and IL, if these currents were maintained without

change for 50 microseconds, what would be the change in Vm?

So let's talk about how that can be computed.

First let's talk about the situation with IK.

Now IK is easy to find because we have an expression for IK.

We know that IK equals g bar k n to the fourth Vm minus EK,

that is to say, we know that IK goes as the maximum

sodium current that could ever exist if all the channels

were open, the probability to the fourth power.

If one of the particles on a channel in open.

And then the potential difference between the Vm which is the present

trans membrane potential and EK, what Vm would be if everything was in equilibrium.

We have a number value for each one of these from our list above.

So therefore we can just plug in numbers here, here, here, here.

Do the computation.

And we get everything.

A similar situation exists for the other two currents.

With the question ask

us to findb, INa equals

g bar m cubed h Vm minus ENa.

Every one of these numbers is given, so we just read the numbers

out of the standard parameter values and the values of the state variables.

We read the numbers, we plug them in, we get the result.

And similarly, we do the very same thing for IL.

IL has an even simpler expression.

Put that expression in, do the computation, and then you have it.

I'll leave it up to you to actually compute the numbers.

Now here we come to the interesting part of this calculation.

It says if these ionic currents were maintained without change

for 50 microseconds what would be the change in Vm?

Now, 50 microseconds is a pretty big time shift to,

to assume that everything remains constant.

That is to say, to assume that all the membrane currents,

ionic currents are remaining constant.

If it was a computer calculation, which nowadays is what people usually have,

you would not use a step that was 50 microseconds.

You would use a much smaller step.

Maybe five microseconds.

Maybe one microsecond.

Here in this case we're saying 50 microseconds.

Which is about the outer limit, near the outer limit, for hand calculations.

Sometimes people use a 100 microseconds in hand calculations and this larger number

is used because it's more interesting when you're doing it by hand to see the result.

It's a bigger result and so you, you get more out of it,

it's not hidden, and the, in the, digits of lower precision.

In any event, what would the change in Vm be?

Now at this point we don't actually have

an expression that's the right expression to use.

We'll have to make one for ourselves.

What we do have is the expression that was used for passive tissue.

And I've written it right here.

This equation number one.

The problem with equation number one

is that we have this term, Vm over Rm.

And we can't make use of that computation because we have no value for Rm.

We know that the membrane resistance has changed most likely

since the membrane was at rest and we don't have any other value.

However, we remember that Vm over Rm

is simply the ionic current, and we can get that another way.

So we can say Vm over Rm is equal to I ion.

And that's equal to IK plus INa plus IL.

We have just gotten finished computing.

IK plus INa plus IL we'll list and compute them individually.

So we have the I's from the earlier parts for this, this, and this.

Therefore what we can do, we can find the sum so now we will have I ion.

And when we have I ion, we can use it in place of Vm over Rm.

And, and proceed with our computation.

So just rewriting the dVm part of the equation,

we can say, dVm is equal to I stim

minus I ion times dt divided by Cm.

If we look at this part by part thinking now in terms of numbers,

we have I stim which is given as zero.

We have I ion which we just computed.

We have dt which is 50 microseconds and

we have Cm which is one micro farad per centimeter squared.

So we have all the number values that we need.

It's now simply a matter of doing the computation.

It's worth noting as we do the computation, but at one level,

we're just putting numbers in an equation and doing a calculation.

At another level,

when you think about it, we're doing something that is really special.

We're taking a set of values that exist at one moment.

I stim, I ion, Cm, these all exist at one moment, and

now we're extrapolating them, so we have an estimate of Vm.

Finding the change between the present moment and Vm at the next moment.

Or in other words, you'll find the change in Vm as it shifts with time.

And that is a new thing and worth reflecting on.

Here are the standard parameters that we were mentioning in the previous slide.

We could take these values, plug them into the equations on the previous slide, and

then find numerical results.

Well, with this beautiful shot of the woods along Science Drive at

Duke University.

Thank you for watching this segment.

I look forward to talking to you again in the next one.

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