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 far, we've noticed, that Hodgkin and Huxley replaced the composite resistance,

composite membrane resistance RM, with separate pathways for each of the

different ions. In this lecture, we consider what is the

mathematical equation for each of these three pathways.

Let's take them in turn. The equation for the potassium pathway, IK

= gK (Vm - EK). There are a couple of important things to

notice here. First, as we were saying last time around,

pathway resistance has been replaced by the pathway conductance.

That means that in this equation, we have the conductance that mesh, that multiplies

the voltage, the voltage term, rather than having the current divided by the

resistance, as is normally the case for equations written as Ohm's law.

But we see, in a different way, this is another form of Ohm's law, in the sense

that it has the current equals the conductance times the voltage.

However, however, this voltage is not just the voltage.

It's not just Vm. It's Vm - EK.

This whole entire composite effect, that is to say, the current that flows, IK, is

depending on the Vm, Vm will be from here to there.

Take away EK - EK. So the current that flows is Vm - EK,

current that flows on the K pathway is proportional to Vm - EK.

Now there's some things that are worth thinking about there.

First of all. Gk seems to be a variable rather than a

constant. We have an equation here.

Let's talk about that equation. That equation says the value of gK, at any

particular moment, that value, is equal to the maximum value of gK, the maximum it

could ever have, that's signified by this bar over the gK, the maximum value, times

this other quantity, n to the fourth. How does one interpret that?

Well very loosely, right here at the beginning, we would say [couch] very

loosely right here at the beginning, we would say what that means is that the

potassium conductance, gK, is the conductance that would be present if the

channel was completely open g bar K times the probability that the channel is open.

Which is given as n to the fourth, that's a little mysterious but we'll get to it in

a moment. So we really are thinking not for just one

channel, but for a whole aggregate of channels, the thousands of channels that

are present in a patch. So for that aggregate of channels, the

conductivity is the maximum conductivity, the conductivity of all the channels were

open, times the probability of a channel being open.

Seems like a strange way to do it, but it works, so we like it.

Let's make a sketch of what's this n to the fourth idea is.

So, the idea is not too hard, it says, suppose we have a channel.

Channel is going from the inside to the outside of this membrane.

How is the internal channel structured conceptually?

We're not talking about how is it really built, but conceptually how is it.

And what we're saying is, we're saying there are four gates the channel has to

pass through. One, an ion has to pass to, through to get

through the channel. Probability that one particle, it's

Hodgkin and Huxley call it, probability that one of these gates, one particle, is

open, is n. Probability, four are open, is n to the

fourth. That is to say, the product of the

probabilities of each one separately. So you think of the whole pathway from

inside to outside, the whole channel, for potassium, as being a channel that has

four units within it, four particles, each particle has to be in the open position,

and when all four are open, then the channel as a whole is open.

Thats where the n forth comes from. Now what is n?

Well we haven't gotten there yet but the probability is but once we know it we know

how to find out what the probability is that the whole channel is open.

For the sodium pathway we have a similar structure.

Once again, voltage term is the difference between trans-membrane potential and the

sodium equilibrium potential. As you know, this is a hugely different

combination than Vm - EK. If we look at gNa, gNa, we have a similar

but not equivalent form. Gna conductivity of sodium channels b,

that is, exist at a particular moment, is the maximum value, the value if all were

open, timed m cubed h. So m cubed h is again a probability and

let's see if we can make a sketch. We would say m cubed h that means, we have

three particles of the m type. Also we have one particle of the h type.

Here's the inside, here's the outside. For the channel as a whole to be open, all

four of these particles have to be open. So that says the probability that all four

are open. What's the product of these probabilities?

H for that channel times m cubed and cube being because there are three m channels.

So in other words it's equal to h times m times m times m.

Why do you suppose there are two different kinds of gates that are built into this

model? Two different kinds of particles.

And the answer to that is it has to do with timing.

This is a mechanism by which the right timing can be introduced.

And, as you'll see later on, what happens is that, the timing is such that m, the m,

m gates control the opening, control the timing of the opening and the h gates

control the timing of the closing. We haven't shown that here, but you'll see

that later on. That leaves us only with the L pathway.

Well, good news. The L pathway, the leakage pathway, is

like an ordinary hole in the membrane. There are not very many of these, so that

means that gL is not a very big number. But it's an important number, even though

small. And gL goes according to Vm - EL.

El is usually taken as the equilibrium potential for chlorine ions.

So, we know, we have a number for that value once we know this EL ion

concentration in the interior and the exterior.

So that concludes our description of the equations for each of the pathways, but we

haven't yet talked about how the probabilities are found.

When we get to the next segment we'll talk about that.

Thank you.

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