0:20

And the first two lectures focused on the

Â biological background that's necessary to understand this model.

Â Now in part three of these series of lectures,

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

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

Â Huxley derived the model equations directly from the experimental records.

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

Â First we want to convert from currents to conductances.

Â And in part two we talked about how Hodkin

Â and Huxely recorded ionic currents using the voltage clamp technique.

Â And how they separated sodium current and potassium current.

Â 0:55

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

Â Then we're going to analyze the two conductances,

Â the potassium conductance and the sodium conductance.

Â And we're going to notice something very important about

Â how these conductances change as a function of time.

Â Potassium conductance is going to increase monotonically with a delay.

Â And sodium conductance is going to show dramatically different behavior.

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

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

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

Â these different processes mathematically, with the equations

Â that were developed by Hodgkins and Huxley.

Â This slide here shows where we left off at

Â at the end of lecture two on action potential models.

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

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

Â Depolarizing the, the membrane to a series of

Â membrane potentials from minus 30 up to plus 60.

Â As color coded here in the different colors.

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

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

Â 2:05

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

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

Â Where the driving force is voltage you're at,

Â minus the reversal potential for the particular ion.

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

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

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

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

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

Â And greater than the one that's magenta.

Â This can either reflect a change in the driving force.

Â At a higher voltage here you're

Â farther away from the potassium reversal potential.

Â Or it can reflect a change in the conductance.

Â It can reflect the difference in how permeable the membrane

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

Â 2:54

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

Â at the combination of the conductance and the driving force.

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

Â And we want to convert from currents into conductances.

Â And this slide here shows what happens when

Â we convert from ionic currents to ionic conductances.

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

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

Â And as we discussed, sodium current is

Â sodium conductance times the driving force for sodium.

Â And then an analogous equation for a potassium current.

Â Therefore, we can compute the conductances just by

Â taking the currents as a function of time.

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

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

Â for sodium conductance and these traces here for potassium conductance.

Â And when we do this, we see

Â something that's fundamentally different about these two.

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

Â Which is that potassium conductance increases monotonically, and

Â then reaches a plateau, and stays at that plateau.

Â Whereas sodium conductance increases and then it decreases rapidly.

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

Â how we understand the difference between what's

Â going on with the potassium conductance here.

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

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

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

Â 4:21

This again is our, our set of potassium

Â conductance traces for a series of memory potentials.

Â Depolarizations ranging from minus 30, which is

Â the lowest one, the black point here.

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

Â [BLANK_AUDIO]

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

Â These time courses for potassium conductance.

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

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

Â So as we move from the black to the blue,

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

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

Â 5:14

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

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

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

Â similar to an exponential function raised to a power.

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

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

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

Â You see a function that looks like this.

Â This is what an exponential function looks like.

Â If you were to take this exponential function

Â and square it, you get the red term.

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

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

Â So what happens if you take this exponential and you raise

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

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

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

Â Is that goes up somewhat like an

Â exponential but there's a lag at the beginning.

Â And the lag at the beginning is what you see if

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

Â 6:21

And that left you a fundamental insight in,

Â in terms of how they developed their model.

Â They said that these facts suggest a following

Â model for how potassium conductance changes in the membrane.

Â They imagined some variable, little n, that represented the

Â fraction of particles that are in a permissive state.

Â 6:41

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

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

Â And the membrane can become permeable to potassium.

Â Potassium can go through the membrane.

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

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

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

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

Â you can calculate the potassium

Â conductance, times the max, maximum conductance.

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

Â 7:38

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

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

Â And over here you have n, which is a

Â fraction of particles that are in the permissive state.

Â And then you can say that the, these, these

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

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

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

Â Like we encountered previously in our lectures of dynamical systems.

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

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

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

Â 8:19

So this is a very simple differential equation.

Â And what makes this a little more complicated is that

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

Â In other words, the, these particles that

Â Hodgkin and Huxley hypothesized in the membrane.

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

Â permissive use at the same rate, at all voltages.

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

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

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

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

Â Because n represents a fraction.

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

Â is equal to 17 or, or anything like that.

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

Â rate constants that determine transitions for

Â the non-permissive state to the permissive state.

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

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

Â And, and beta is a function of voltage?

Â 9:19

This was our differential equation, dn/dt.

Â That shows how alpha and beta

Â determine these transitions from non-permissive to permissive.

Â And when alpha and beta are constant, this

Â equation is going to have a steady state solution.

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

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

Â Then the value you're going to get for n

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

Â Which is Alpha over Alpha plus Beta.

Â 10:20

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

Â And if we know tau is a function of voltage.

Â Then we can determine alpha and beta.

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

Â And now we have a differential equation telling

Â us how potassium conductance varies as a function.

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

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

Â And that's what we want to address next.

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

Â Lets see how we can get n infinity and tao

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

Â These are our, our traces for gK as

Â a function of, of time at different voltages.

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

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

Â 11:16

Well, what we have in all these different

Â at, as we extend this voltage clamp depolarization.

Â We see that gK reaches a plateau level, right?

Â So this, these different plateau levels tell us

Â our steady state values of, of gK time infinity.

Â 11:33

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

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

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

Â that's going to be proportional to the fourth root

Â of whatever our infinity value of potassium conductance is.

Â In other words you take all

Â these potassium conductances at different voltages.

Â Measure it at 1 times.

Â 11:58

So because this is, this conductance is flat we know

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

Â Now that's the infinity value.

Â And you take these potassium conductances.

Â Take the fourth root of them.

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

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

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

Â 12:59

an exponential race to some power.

Â And in particular they said it resembled

Â an exponential increase raised to the fourth power.

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

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

Â And then they chose the best fit.

Â You know, just qualitatively, we can see

Â that this one is going up relatively slowly.

Â Going up relatively slowly represents a large tau.

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

Â Much more quickly represents a, a smaller tau.

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

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

Â And they said, okay, if we set tau

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

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

Â Maybe that's too fast.

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

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

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

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

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

Â would be an appropriate time constant to describe this?

Â And when you do that sort of procedure, either

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

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

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

Â And voltage, on the x-axis.

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

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

Â Now that you have tau for all these different values of

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

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

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

Â So this is how we can get our functions for alpha

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

Â Now let's look a little more carefully

Â at the time course of, of conductance changes.

Â In particular the potassium conductance.

Â 15:04

This is the current that results.

Â You have this increase in current with the delay here.

Â When we re-polarize back to minus

Â 60 millivolts, we have this instantaneous change.

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

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

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

Â 15:30

Then when we drop down to minus 60,

Â we're much closer to the potassium reversal potential.

Â So that's why the current instantaneously gets smaller.

Â But we're not interested in this instantaneous change.

Â We're in this instantaneous change.

Â We're interested in how this declines as a function

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

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

Â So this is the rising phase of the conductance

Â and this is the falling phase of the conductance.

Â And one of the things that Hodgkin and Huxley

Â noticed, is that if you plot these on a

Â normalized scale, you can see that the rising phase,

Â this black curve here, goes up with the delay.

Â Nut the falling phase doesn't have a delay.

Â The falling phase goes to its new steady state value.

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

Â 16:14

Basically like a, like an exponential.

Â So how do you explain this?

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

Â Is a consequence of the fact that potassium conductance

Â is proportional to n raised to the fourth power.

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

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

Â And they said that the membrane will be permeable to potassium

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

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

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

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

Â Then all four charged particles must move.

Â But what happens when you're decreasing the conductance?

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

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

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

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

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

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

Â And all four of them must be in a

Â particular permissive state for the membrane to be permeable.

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

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

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

Â As soon as one of them flips to

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

Â So that's what we're seeing here.

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

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

Â And this was something that Hodgkin and Huxley put into

Â their model just to explain this aspect of the data.

Â And they and they speculated about what it might be.

Â 17:59

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

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

Â Namely that most an, ion channels are tetramers.

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

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

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

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

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

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

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

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

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

Â They, they, built this model.

Â But now we have a well established physical basis for

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

Â Now let's look at sodium conductance.

Â Sodium conductance is, is clearly

Â more complicated than, than potassium conductance.

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

Â Sodium conductance goes up and then it goes down.

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

Â 19:31

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

Â And the second gate which is the inactivation gate.

Â So for potassium conductance we just have

Â one variable and raised to the fourth power.

Â But for sodium we have two different processes.

Â One which is going to allow the channel,

Â allow the memory to become permeable to sodium.

Â In the second process, it's just going to

Â make the membrane impermeable to, to sodium.

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

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

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

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

Â And if we plot the time course of what happens

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

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

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

Â H is some value around 0.7 or something like that.

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

Â And so if you have one process that's

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

Â And then you multiply them together.

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

Â Something that rises and then, and then it falls.

Â So if you have this particular time course of m.

Â And this particular time course of h.

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

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

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

Â like these sodium conductances that we have up here.

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

Â 21:08

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

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

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

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

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

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

Â So, this turning, this increase in m is

Â what we call the activation of sodium current.

Â The activation of the sodium conductance.

Â And then this decrease in h is what we call

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

Â 21:47

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

Â And we said that the fact that you had to

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

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

Â current also has a well established physical basis now.

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

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

Â But there's been a lot of work on looking

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

Â And correlating changes of the structure of the

Â sodium channels with, with changes in the function.

Â To be able to look at the molecular level at

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

Â 22:26

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

Â this level, is remember that m, when m goes

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

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

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

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

Â This is the way that goes back to zero.

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

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

Â We're not going to through, go through this

Â step by step, in the interest of time.

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

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

Â 23:07

So, the idea here is that they made

Â their voltage clamp protocol a little bit more complicated.

Â Everything we've looked at so far is holding

Â at some potential around minus 60 milli volts.

Â Depolarizing to some level.

Â And then going back down to minus 60 milli volts.

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

Â And then they would hyper polarize to say minus 100 or

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

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

Â You see that you get different sodium currents

Â activated depending on, on your initial voltage here.

Â But they didn't care about that.

Â What they cared about more was what happens to what kind

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

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

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

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

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

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

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

Â We get less and less and less current.

Â 24:50

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

Â blue one, is because all of these

Â channels have been activated before the second pulse.

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

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

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

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

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

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

Â This membrane has undergone an inactivation process.

Â 25:21

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

Â going to give you values of

Â the steady-state inactivation variable, h infinity.

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

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

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

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

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

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

Â a plateau level as the membrane gets more negative.

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

Â And then as I get more and more and more

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

Â 26:01

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

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

Â And then gets smaller as the voltage increases.

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

Â Large hyper-polarized potentials and then getting

Â smaller as we get more depolarized.

Â 26:20

This is going to be proportional to h infinity.

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

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

Â And so, this is the way they were able to

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

Â Okay.

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

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

Â 27:17

But then probably the most important point to

Â get across for this lecture is that, terms describing

Â how the gating variables depend on voltage can

Â be extracted directly from the experimental voltage clamp data.

Â We went through that in detail with the potassium conductance.

Â How we get n infinity.

Â And, and tau n is a function of voltage.

Â And then how we can alpha and beta for n

Â directly from the n infinity and tau of n plots.

Â And you can use similar logic for m and for h

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

Â But in either case, the key thing is that

Â because Hodgkin and Huxley had these good voltage clamp data.

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

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

Â So this gives us a good sense of how the

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

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

Â sorts of predictions they were able to make with this.

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

Â [BLANK_AUDIO]

Â