So Hodgkin Huxley were not satisfied by just drawing a circuit. They want to even further, pinpoint exactly what is the mechanism, what could be the mechanism in the membrane that enables these two currents and this difference in kinetics of the sodium versus the potassium. So what they did, really, can be shown here. First of all, we have to note that they recorded currents, potassium and sodium currents, as I showed you before, they recorded the currents. But they could also estimate the conductances, the membrane conductances underlying this currents. Why? Because you know remember that we already spoke that any ionic current, is actually according to Ohm's law is the multiplication of the conducatance and what we call the driving force for this current. So it's the membrane voltage minus the battery of this particular ion multiplied by the conductance of this ion that gives rise to this current. This is true for sodium, potassium. This is true for sodium. The only difference, of course, is the conductance of the sodium channels. And this is the battery, or the driving force, for the sodium. So this is, this holds true for any current. And remember that in the case of voltage clamp we fix Vm, we don't allow Vm to change. So if I know Vm, if I fix the voltage, the, the, the voltage clamp to a given value, and if I know the battery, which, which depends on the concentration difference between the outside and the inside of sodium. Or between the outside and the inside of the potassium. If I know the voltage clamp value. And if I know the battery of this particular ion. And I know the current that I measure. They, I can then, then extract the conductance. So this is exactly what Hodgkin Huxley did. The voltage clamp, the membrane to different values. So here is small depolarization, but still supra-threshold, a stronger depolarization, a stronger depolarization, voltage clamping the membrane step by step. So, jumping the voltage from here to here to here to here and measuring the conductants of both the potassium here, and of the sodium. So, the, the different votages. So let's look, for example, at the potassium. The conductance of the potassium. You can see that the conductance of the potassium, becomes stronger and stronger, as you depolarize more and more, the membrane. So the larger the voltage clamp the more conductance opening of a conductance, voltage-dependent conductance. So I can call it voltage-dependent conductance because when I change the voltage clamp, I get more and more conductance. You can see, the, the, the potassium conductance responds slowly to the abrupt change in the, in the voltage. So, I change the voltage fast. Step, it's a step voltage clamp. And it takes time, as we saw before, for the opening of the potassium conductants. It takes time. And by the way this times becomes faster as your voltage clamp more stronger. So you can see that it grows faster, and also to a more higher conductance as you depolarize more and more with voltage clamp. And when you stop the voltage clamp you can see the continuation of the. Potassium conductance, the continuation of potassium conductance to it's resting venue. So I can say again, this is a voltage dependant conductance. This ion channels that are sensitive to voltage in this case these channels open slowly, it takes them time to respond. When they respond they are open to a given value. And this given value the amount of conductance the degree of conductance change depends on how big the depolarization you have. You have big depolarization, you have big conductance change for the potassium. You have small depolarization, you have smaller conductance. And all this conductance take time to develop, so this is the later current. This is the later outward current that we saw which does not inactivate during the voltage clamp. But if we look now at the sodium conductance, and you see now the conductance, you are used, you are used to look at the current, which is an inward current, I showed you before. But now we look at conductance. So this is the, this is the reverse direction of the current. So the current is inward, but the conductance of course is not inward, conductance is always positive. So this is the conductance of the sodium. As a function of this voltage clamp, so again, like here, we voltage clamp to this value, to this value, to this value, to this value And Hodgkin Huxley recorded that conductance of the sodium for this value, for this value, for this value, of depolarization. And here again, you see something interesting. First again, you see that this sodium conductance responds fast. So as soon as you have voltage clamp, very early on, you get a conductance change. Very early on. This is why we call it fast conductance change. This is a slow conductance change for the potassium, this is fast conductance change for the sodium, fast conductance change. And again, like here, when, when you, when the depolarization is more and more extreme you get more and more conductance. You get more and more conductance. So the conductance becomes stronger or larger with depolarization. From minus 40 millivolts to minus 20 to zero to even plus 20. You get more and more sodium conductance. This is in the beginning. But still, you continue to voltage climb, and the conductance fades away. Somehow it inactivates, and it disappears during the voltage climb. So this is again this conductance that is early which enables very early on to carry sodium channels, sodium current inside the cell, early on. But after some times, this conductance turns off, and there is no sodium current anymore, although you continue the voltage clamp. This is not the case for the potassium that continues to be open as long as the voltage clamp descends to a given depolarization. So these are the different two conductances that underlying, the voltage sensitivity of the membrane of the axon. So, I'm, I'm summarizing here. The slow, the slow potassium current, or potassium conductance, does not inactivate during voltage clamp. Know that the k conductants rises slower. Then it attenuates, and this was important for Hodgkin Huxley later on to write down the appropriate equation to describe these curves. So it's a slowing growing curve and faster attenuating curve at the end of the voltage clamp. And you see this sodium current this early sodium current, or conductance, inactivates during voltage climb. It's early, so this is the first current that flows when there is voltage change, it's a sodium current, but this sodium current, early sodium current, inactivates during voltage climb, unlike the potassium current. And here is an example, a direct example of Hodgkin Huxley trying to fit equations, we should talk soon about equations. I'm just showing you now a result of an experiment, so this is an experiment of Hodgkin Huxley, recording the potassium current you see this continuous curve. This is a recoding from Hodgkin actually, and the dots here are numerical, or mathematical fitting, that we shall talk soon about, that, about the mathematics, the fitting of the mathematics to this current. You see, again, that this potassium current grows slowly, and attenuates faster at the end of the voltage clamp. So there is a voltage clamp here somewhere, and this is the response. So you can see, already, now, without talking abut the equation, that they somehow realized how to write an equation that will. The equation, the dots. Will fit very well the actual recorded current. Actually Hodgkin Huxley realized that the fact that it grows slower, the sodium, the potassium current grows slower and attenuates faster, they realized that they can fit. The up stroke of the potassium current or conductance and the down stroke the decay of the potassium current. They can mathematically write the up stroke as a function that looks like one minus exponent to the power of four and this means. Minus t here, this means that the growth as you increase the exponent, the growth here would be slower. So if it would be to the power of one it would be faster, and if it would be to the power of six it will be even slower. Yes, because this is the minus t here. So they realized that they can fit very well this rising phase. By this kind of one minus exponent to the power of four, and it could fit very well the decay with, with the single exponent, not to the power of four. So, the decay and the rise time have a different speed. Of rising and decaying, slow speed and fast decay speed, and this, on its own, Hodgkin Huxley realized that this must mean something about the mechanism of this specific potassium pore, or membrane channel. Something about this channel behaves like it opens like something to the power of four and attenuates like something basically to the power of one. And then they wrote this equation. This was a big conceptual jump. They said that the potassium conductance in the membrane, behaves like a maximal conductance, so you may think about this like the total, absolute total number of channels, potassium channels available in this patch of membrane. This is what they call the maximal conductance, this is gK bar. So this is all their valuable conductance but not always all these conductance is open, because it is multiplied the actual conductance, is multiplied by a factor that they called n. It's an activating parameter, it is a voltage-dependent parameter, this n. This n goes between zero and one. So when n is zero, there is no potassium conductance. When n is equal to one, the conductance of the potassium is maximum. Because n to the power of, one to the power of four is one, so if n is one, you get all the conductance available of this potassium channels. And if n is between zero to one, you get less. The claim was, that for a particular voltage clamp, n gets the value somewhere between zero to one. At zero, at zero voltage without any voltage clamp, n is very close to zero. At the very, very big, at the very, very big voltage clamp, n is becoming close to one. And this to the power of four is exactly what makes things fit this growth, as I said. It looks like, like this, to the power of four. So this is the basic equation for the potassium conductance, but of course we have to discuss what is this parameter, which is really the parameter that makes makes their current both voltage dependent but also time dependent. So n is between zero and one, but n depends on both voltage and time. So n represents the proportion of K-ion channels In the open state. You can say that if n is equal 0.5, 0.5 multiplied to the power of four is the propulsion. It's actually not n, but n to the power of four that represents how much of the conductance is available. If n equal one, all the conductance, all the conductance is available. So n is this special parameter, the activating n parameter for the voltage, for the Hodgkin Huxley equation. So Hodgkin Huxley wrote these equations may be given a physical basis, if we assume that the potassium ions can only cross the membrane when four similar particles, particles occupy a certain region of the membrane. So this is another jump. They are now trying to interpret bio physically, almost anatomically, what is this form means. They want to say that this power form means that this channel, this potassium channel, that this potassium channel is open. Only if four particles, if four gates in the channels are open so that the channel enables the flow of the current through the channel. So they are trying to give now an interpretation to the number four to the, to the exponent four here, to the to this four using a biophysical intuition which we are going to talk about in a second. Okay, may be before we go to talk in a second I just want to complete just showing you another graph on Hodgkin Huxley. About the potassium. This is not really another success of the Hodgkin Huxley. Just to show you this is the potassium conductance for various voltage clamp. You can see what we discussed before, that as you increase the voltage clamp, you get more and more and more potassium current or potassium conductants. And the disconductants does not activate inactivate, it continues it's maximal, it takes time to reach the maximal but when it gets to the maximal it stays there so it does not inactivate. And you can see by the way again that this activation is not only voltage dependent. Because, with voltage, you get more and more activation. So n becomes more and more close to one. It's not only activated more and more. But also, it becomes faster and faster and faster activated. So the activation end, activation viable end, is both increase, increases with voltage towards one, and also, this activation viable end becomes faster. With depolarization faster and faster, or slower and slower for lower voltages.