0:20

And the first two lectures focused on the

biological background that's necessary to understand this model.

Now in part three of these series of lectures,

we're going to actually get into the Hodgkin-Huxley model itself.

And in particular, we're going to focus on how Hodgkin and

Huxley derived the model equations directly from the experimental records.

And that's going to require us to go through a few steps.

First we want to convert from currents to conductances.

And in part two we talked about how Hodkin

and Huxely recorded ionic currents using the voltage clamp technique.

And how they separated sodium current and potassium current.

0:55

Here step one is going to be to convert from currents to conductances.

Then we're going to analyze the two conductances,

the potassium conductance and the sodium conductance.

And we're going to notice something very important about

how these conductances change as a function of time.

Potassium conductance is going to increase monotonically with a delay.

And sodium conductance is going to show dramatically different behavior.

It's going to show an increase, and then a decrease.

And that decrease is what, we're going to, process we're going to call inactivation.

And then we're going to go through to see how we can describe these,

these different processes mathematically, with the equations

that were developed by Hodgkins and Huxley.

This slide here shows where we left off at

at the end of lecture two on action potential models.

Where we have these voltage clamps steps, that were applied to

the squid giant axon preparation, holding for minus, holding at minus 60.

Depolarizing the, the membrane to a series of

membrane potentials from minus 30 up to plus 60.

As color coded here in the different colors.

And we finished the last lecture by talking about how Hodgkin and Huxely

were able to separate sodium current over here and potassium current over here.

2:05

Now it's important to remember, something we talked in previous lectures.

Which is that current can be calculated as the conductance times the driving force.

Where the driving force is voltage you're at,

minus the reversal potential for the particular ion.

So when we talk about x here, x is a generic

term which would be a sodium current or, or potassium current.

And the reason I, I showed this equation again, is that if you

have an increase in the current, for instance in the potassium current here.

You can see the one that's black is greater than the one that's green.

And greater than the one that's magenta.

This can either reflect a change in the driving force.

At a higher voltage here you're

farther away from the potassium reversal potential.

Or it can reflect a change in the conductance.

It can reflect the difference in how permeable the membrane

is to allow potassium to, to cross the cell membrane.

2:54

And so when you're looking at currents, you're looking

at the combination of the conductance and the driving force.

So what we want to do, is we want to eliminate one of those variables.

And we want to convert from currents into conductances.

And this slide here shows what happens when

we convert from ionic currents to ionic conductances.

These are the sodium, this is the so, so, the

sodium and potassium currents that we saw in the last slide.

And as we discussed, sodium current is

sodium conductance times the driving force for sodium.

And then an analogous equation for a potassium current.

Therefore, we can compute the conductances just by

taking the currents as a function of time.

Dividing by the, driving force for sodium and then do the same thing for potassium.

And when we do that, we get these, traces here

for sodium conductance and these traces here for potassium conductance.

And when we do this, we see

something that's fundamentally different about these two.

That was already eluded to, in the, in the introduction.

Which is that potassium conductance increases monotonically, and

then reaches a plateau, and stays at that plateau.

Whereas sodium conductance increases and then it decreases rapidly.

And much of the rest of this lecture is going to be focused on

how we understand the difference between what's

going on with the potassium conductance here.

Versus what's going on with the sodium conductances, over here.

Let's start by discussing how we develop equations for the

one that's simpler to explain, which is the potassium conductance.

4:21

This again is our, our set of potassium

conductance traces for a series of memory potentials.

Depolarizations ranging from minus 30, which is

the lowest one, the black point here.

Minus 15, zero all the way up to plus sixty milli-volts.

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First, we notice two things when we look at these traces.

These time courses for potassium conductance.

One is that when you change the voltage, it changes both the steady

state, gK, the, the plateau level, and it changes the rate of rise.

So as we move from the black to the blue,

all the way up to the, to the second black line.

We see that the plateau level, the steady state level it reaches, gets bigger.

5:14

And the, the second insight that, that Hodgkin and Huxley saw when they

looked at these is that, they said the time course of potassium conductance.

The time course of gK, has an increase that's

similar to an exponential function raised to a power.

Here I'll show you, what I, what I mean by that.

If you just take the equation 1 minus E raised to minus t over Tau.

Where, where little t here is time and Tau represents the time constant.

You see a function that looks like this.

This is what an exponential function looks like.

If you were to take this exponential function

and square it, you get the red term.

If you raise it to the third power you get the, the, the green trace here.

And then if you raise it to the fourth power you see this magenta trace here.

So what happens if you take this exponential and you raise

it to a power, is you see this lag at the beginning.

And that's one of the things that Hodgkin and Huxley

noticed when they, when they looked at their potassium conductance traces.

Is that goes up somewhat like an

exponential but there's a lag at the beginning.

And the lag at the beginning is what you see if

you have an exponential in its race to, to some power.

6:21

And that left you a fundamental insight in,

in terms of how they developed their model.

They said that these facts suggest a following

model for how potassium conductance changes in the membrane.

They imagined some variable, little n, that represented the

fraction of particles that are in a permissive state.

6:41

And conductance would be proportional to, to n to the 4th.

So, the idea is that you have these particles that are in the membrane.

And the membrane can become permeable to potassium.

Potassium can go through the membrane.

When four of your, all four of your particles are in a permissive state.

But then when one of them is, is in a non-permissive

state, it won't be proportion, that you won't be permeable to potassium.

So the idea is that you c-,c-, at any time

you can calculate the potassium

conductance, times the max, maximum conductance.

because maximum conductance can be reached if your fraction n is equal to one.

7:38

Over here on the left you have your, you have 1 minus n.

These are the, this is the fraction that's in the non-permissive state.

And over here you have n, which is a

fraction of particles that are in the permissive state.

And then you can say that the, these, these

transitions from non-permissive to permissive with some reconstant alpha.

And then they go back the other direction with some reconstant beta.

And therefore, this is just like a law of mass action type equation here.

Like we encountered previously in our lectures of dynamical systems.

You can write down your differential equation dn/dt = a(1-n) - Bn.

That's just taking the amount that's over here times the rate constant.

The amount that's over here times the rate constant.

8:19

So this is a very simple differential equation.

And what makes this a little more complicated is that

alpha and beta in general are, are functions of voltage.

In other words, the, these particles that

Hodgkin and Huxley hypothesized in the membrane.

You know, they, they don't transition from non-permissive to

permissive use at the same rate, at all voltages.

When you change the voltage, you're going to change the relative

proportion that are in permissive state or, or non-permissive state.

And remember that this variable n which we're going to refer to

later as a gating variable, is always between 0 and 1.

Because n represents a fraction.

So you're never going to get a case where n

is equal to 17 or, or anything like that.

As we just mentioned in the last slide, alpha and beta are the

rate constants that determine transitions for

the non-permissive state to the permissive state.

And as we said, alpha and beta can depend on voltage.

Now the question is, how we can determine, alpha is a function of voltage?

And, and beta is a function of voltage?

9:19

This was our differential equation, dn/dt.

That shows how alpha and beta

determine these transitions from non-permissive to permissive.

And when alpha and beta are constant, this

equation is going to have a steady state solution.

If you set this differential equation equal to 0, you can

say that it's a steady state as time goes to infinity.

Then the value you're going to get for n

is a value that we're going to call n infinity.

Which is Alpha over Alpha plus Beta.

10:20

But what this tells us, is that if we know an infinity is a function of voltage.

And if we know tau is a function of voltage.

Then we can determine alpha and beta.

And if we know alpha and beta, then we know our differential equation for, for n.

And now we have a differential equation telling

us how potassium conductance varies as a function.

Now what I want to show you next is how n infinity Voltage

and tao of voltage can be extracted directly from the experimental data.

And that's what we want to address next.

How can you get n infinity and tao directly from the experimental data?

Lets see how we can get n infinity and tao

from the data by first remembering what happens to potassium conductance.

These are our, our traces for gK as

a function of, of time at different voltages.

And what we want to argue is that gK as a function of time, will tell us, as,

as a function of time and as a function of voltage will tell us an infinity and tao.

11:16

Well, what we have in all these different

at, as we extend this voltage clamp depolarization.

We see that gK reaches a plateau level, right?

So this, these different plateau levels tell us

our steady state values of, of gK time infinity.

11:33

And remember what we said a couple of slides ago, that

potassium conductance gK is proportional to n raised to the fourth power.

So if we want to get n infinity, all we have to do to get n infinity

that's going to be proportional to the fourth root

of whatever our infinity value of potassium conductance is.

In other words you take all

these potassium conductances at different voltages.

Measure it at 1 times.

11:58

So because this is, this conductance is flat we know

that, you know, approximately what happens when we reach steady state.

Now that's the infinity value.

And you take these potassium conductances.

Take the fourth root of them.

And then you get something that's proportional to, to n infinity.

So directly from these values here at the, long time points, we

can get a plot of n infinity as a function of voltage.

12:59

an exponential race to some power.

And in particular they said it resembled

an exponential increase raised to the fourth power.

And so, what they did is they plotted 1 minus e raised to

the minus t over tao to the fourth power for different values of tao.

And then they chose the best fit.

You know, just qualitatively, we can see

that this one is going up relatively slowly.

Going up relatively slowly represents a large tau.

And this one here, this cyan one goes up much more quickly.

Much more quickly represents a, a smaller tau.

So what Hodgkin and Huxley did, is they said, well what kind of

plot are we going to get if we set tau equal to ten milliseconds.

And they said, okay, if we set tau

equal to ten milliseconds, maybe that's too slow.

Now let's see what happens if we set tau equal to one millisecond.

Maybe that's too fast.

And then iteratively, they just tried different values of tau.

And then they figured out the one that chose the best fit.

These days with, with computers, you can just take these, your, your data traces,

and you can use a, a curve fitting function in a program such as MATLAB.

And say, well, what would be the best, what

would be an appropriate time constant to describe this?

And when you do that sort of procedure, either

manually the way Hodg, Hodgkin and Huxley did or automatically.

The way that we can do it today is you'll get a plot that looks like this.

Which will be tau on the, on the y-axis here.

And voltage, on the x-axis.

And so you can see that for different values of, of

tau, I mean for different values of voltage, your tau can vary.

Now that you have tau for all these different values of

voltage and n infinity for all these different values of voltage.

At any given voltage you can solve alpha is equal to n infinity divided

by tau, and beta is equal to 1 minus n infinity divided by tau.

So this is how we can get our functions for alpha

as a function of voltage, and beta as a function of voltage.

Now let's look a little more carefully

at the time course of, of conductance changes.

In particular the potassium conductance.

15:04

This is the current that results.

You have this increase in current with the delay here.

When we re-polarize back to minus

60 millivolts, we have this instantaneous change.

And this instantaneous change occurs just because we go from plus 20 to minus 60.

Remember our plus 20 is far away from the potassium reversal potential, the

potassium reversal potential is somewhere around minus 70 in the squid giant axon.

15:30

Then when we drop down to minus 60,

we're much closer to the potassium reversal potential.

So that's why the current instantaneously gets smaller.

But we're not interested in this instantaneous change.

We're in this instantaneous change.

We're interested in how this declines as a function

of time after we've switched back to minus 60.

And we can get that if we convert from current again to conductance.

So this is the rising phase of the conductance

and this is the falling phase of the conductance.

And one of the things that Hodgkin and Huxley

noticed, is that if you plot these on a

normalized scale, you can see that the rising phase,

this black curve here, goes up with the delay.

Nut the falling phase doesn't have a delay.

The falling phase goes to its new steady state value.

The new steady state value in this case is going to be zero.

16:14

Basically like a, like an exponential.

So how do you explain this?

Well, the fact that the rising phase has a delay and the falling phase does not.

Is a consequence of the fact that potassium conductance

is proportional to n raised to the fourth power.

Remember, this is our, our simple model, our Hodgkin and

Huxley simple model for the non-permissive state and the permissive state.

And they said that the membrane will be permeable to potassium

when you have four particles that are all in the permissive state.

And that's how you get conductance that's proportional to n to the 4th power.

The idea is that when your conductance is increasing, when you're going

from here at essentially zero conductance, to some higher level of conductance here.

Then all four charged particles must move.

But what happens when you're decreasing the conductance?

What happens when you're going back to minus 60?

Well when the conductance decreases, only one out of four moving is sufficient.

And that's how we can get delay on the rising phase.

And we don't have a delay on the falling phase here.

Another way to think of it is when, if you have four charged particles

that have to move in order to, to change your, to increase your conductance.

And all four of them must be in a

particular permissive state for the membrane to be permeable.

Then when you're when you're rising, if there's four of them, it's whichever one

is the slowest one out of the four that's going to, to control it.

When it's falling, it's whichever one is the fastest one.

As soon as one of them flips to

the non permissive state, that's going to be enough.

So that's what we're seeing here.

When the conductance increases, all four charged particles must move.

When it decreases, one out of four moving is sufficient.

And this was something that Hodgkin and Huxley put into

their model just to explain this aspect of the data.

And they and they speculated about what it might be.

17:59

But what was really remarkable, remarkable about that model and their, the

way they put this together was this now has a well-established physical basis.

Namely that most an, ion channels are tetramers.

And so, it's well-established that the potassium channels,

voltage gated potassium channels at least, have four sub-units.

And there are charged alpha-helices in these in these channels.

And all four of them have to move into, into all four

of them have to change position in order for the channels to open.

But then if one of, out of four changes position that's enough to close a channel.

And so it was, it was very, really remarkable that Hodgkin and Huxley

developed their model like this, just to be able to fit these data.

They didn't really know what an ion channel was at the time.

They, they, built this model.

But now we have a well established physical basis for

this, for these equations that they developed many, many years ago.

Now let's look at sodium conductance.

Sodium conductance is, is clearly

more complicated than, than potassium conductance.

Because potassium conductance goes up and then it stays up at a plateau level.

Sodium conductance goes up and then it goes down.

So it increases and then it decreases even when the voltage is constant.

19:31

Is proportional to some gate here m, which is the activation gate.

And the second gate which is the inactivation gate.

So for potassium conductance we just have

one variable and raised to the fourth power.

But for sodium we have two different processes.

One which is going to allow the channel,

allow the memory to become permeable to sodium.

In the second process, it's just going to

make the membrane impermeable to, to sodium.

And because they're, we're multiplying them together.

Both of them must be greater than zero for appreciable sodium conductance.

So when you have a product in this case, if either m is equal to 0

or h is equal 0, then your sodium conductance is going to be equal to 0.

And if we plot the time course of what happens

during a voltage clamp stop, we can see how this works.

So the idea is that when you, when you're at the resting potential,

when you're holding at minus 60, m is essentially 0 And h is appreciable.

H is some value around 0.7 or something like that.

When you depolarize, m goes up and then h goes down.

And so if you have one process that's

going up and a second process that's going down.

And then you multiply them together.

Then you can get a time course that looks like this.

Something that rises and then, and then it falls.

So if you have this particular time course of m.

And this particular time course of h.

And you multiply m raised to the third power times h.

Then you're going to get a time course that looks like this.

Something that rises, and then falls, like these pota-,

like these sodium conductances that we have up here.

Of course, this only works if m is faster than h.

21:08

If m were to go up slowly and h were to decrease quickly.

Then, m cubed times h would always be equal to 0.

But the way that Hodgkin and Huxley built the model, is they

had, the, the m gate changing more rapidly than the h gate.

And that allowed them to have a, a transient peak in sodium conductance.

When m had moved to a high value and h had not yet declined.

So, this turning, this increase in m is

what we call the activation of sodium current.

The activation of the sodium conductance.

And then this decrease in h is what we call

the inactivation of the sodium current, or the sodium conductance.

21:47

And just like we've, we've looked at the potassium conductance.

And we said that the fact that you had to

have four particles moving, now has a well established physical basis.

It's also true that this inactivation of, of sodium

current also has a well established physical basis now.

Which is what's known as a ball-and-chain inactivation.

We're not going to go into that in, in detail.

But there's been a lot of work on looking

at the structure of, of the channels that pass sodium.

And correlating changes of the structure of the

sodium channels with, with changes in the function.

To be able to look at the molecular level at

what's causing these changes in conductance as a function of time.

22:26

But what's important for understanding the Hodgkin-Huxley model at

this level, is remember that m, when m goes

up, that's activation of a channel, that's turning it,

making the conductance go up, that's turning it on.

And then when h decreases, that's inactivation of the channel.

And that's the way that the, this conductance shuts off.

This is the way that goes back to zero.

Now the question becomes, how did, Hodgkin and Huxley get differential equations for

m and differential equations for h from their experimental voltage clamp data.

We're not going to through, go through this

step by step, in the interest of time.

But, I do want to point out a very clever experiment

they used to measure steady state values of, of h.

23:07

So, the idea here is that they made

their voltage clamp protocol a little bit more complicated.

Everything we've looked at so far is holding

at some potential around minus 60 milli volts.

Depolarizing to some level.

And then going back down to minus 60 milli volts.

Here's a case where the would start it around minus sixty milli volts.

And then they would hyper polarize to say minus 100 or

hyper polarize to minus 80, minus 60 or depolarize to zero.

And they didn't care what happened in response to this first pulse here.

You see that you get different sodium currents

activated depending on, on your initial voltage here.

But they didn't care about that.

What they cared about more was what happens to what kind

of sodium current do you measure in response to the second pulse.

And the second pulse here is always going to be at the

same level, which in this particular case is minus 10 milli volts.

And if we look at the second pulse here, we can see that we don't have any,

we, we have a very large current with the black one in, in minus 100 milli volts.

And then we have essentially an equally large current for the red one.

But then as we go to green one and the magenta one, and the blue one.

We get less and less and less current.

24:50

So the reason why you don't get a current with the second pulse for the

blue one, is because all of these

channels have been activated before the second pulse.

If we look at the blue one in response to the first

pulse, you get a sodium current and then it goes to essentially zero.

And so because it's gone to essentially zero, that these

channels have inactivated, your h gate has gone to, to zero.

And therefore when you give the second pulse, you're not

able to generate a current because these channels have been inactivated.

This membrane has undergone an inactivation process.

25:21

And as long as this first pulse is long, then this is

going to give you values of

the steady-state inactivation variable, h infinity.

And so what you can do, is you can take, you can say,

I'm going to look at the voltage that I had during the first pulse.

That's going to be what I plot on my x-axis here.

And then what I'm going to plot on my y-axis is something that's

proportional to how much current I get in response to the second pulse.

And that's how you're going to get something that has

a plateau level as the membrane gets more negative.

Remember that minus 100 and minus 80 are, are close to one another.

And then as I get more and more and more

positive I'm going to get less and less and less current.

26:01

So what we're plotting over here is, you know, something

that has a plateau level from very, very hyperpolarized membranes.

And then gets smaller as the voltage increases.

That's similar to what we're seeing here on these traces.

Large hyper-polarized potentials and then getting

smaller as we get more depolarized.

26:20

This is going to be proportional to h infinity.

Because this is saying what, how much inactivation I have in the

steady state depending on what I'm holding at during this first pulse here.

And so, this is the way they were able to

measure the steady state value of h, or, or h infinity.

Okay.

Now to summarize this lecture, our, our third one on the

Hodgkin and Huxley model of the squid giant axon action potential.

27:17

But then probably the most important point to

get across for this lecture is that, terms describing

how the gating variables depend on voltage can

be extracted directly from the experimental voltage clamp data.

We went through that in detail with the potassium conductance.

How we get n infinity.

And, and tau n is a function of voltage.

And then how we can alpha and beta for n

directly from the n infinity and tau of n plots.

And you can use similar logic for m and for h

even though we didn't go through that exactly step by step.

But in either case, the key thing is that

because Hodgkin and Huxley had these good voltage clamp data.

They were able to get their, their rate constants, alpha and

beta, for these gating variables, directly from the experimental voltage clamp data.

So this gives us a good sense of how the

Hodgkin and Huxley model was constructed, based on the data.

What we're going to discuss next is how, is what

sorts of predictions they were able to make with this.

And what sort of experimental results they were able recapitulate using this model.

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