Okay, so let me try to graphically show you how we think about the potassium channel, or the potassium conductance. So this will now signify a pore through the membrane. This will be the inside of the membrane. This will be the outside of the membrane. So the membrane is here and this is a potassium channel. And we just said that you can think about this n to the power of four as if you have four gates in this channel. These gates, when they are closed, this n gate, when they are closed, potassium current cannot flow through this channel because this channel is closed. And the only way to open this channel completely is to move these gates to the open state. So in this case we shall call the n gate to be zero. Zero means that it's closed. And we know that with depolarization, when you depolarize the membrane, make the inside more positive, these gates are starting to open. Physically, they start to open when you have depolarization, [COUGH] and eventually, schematically, you will get something like this from this pore. So if the depolarization was strong enough, you will move the gates into the open state and this will enable potassium to flow from the inside to the outside. Okay, you can think about n as the probability of a gate to be open. Because as I said, n lies between zero to one. It's like probability. So you need both this one and this one and this one and this one to be open to be at the state that enables the potassium to flow. There is no half open channel. There is either open channel or closed channel. This is a closed channel, this is an open channel. But there is a probability of the channel to be open due to the probability of n to get the value of one. So this is the transition between a closed channel and an open channel. And when you have an open channel, potassium goes from the inside to the outside. So that's the equation that Hodgkin-Huxley wrote. We already saw that. You need 4 n to open the channel, and this n parameter, this activation parameter, according to the Hodgkin-Huxley equation, relies on another parameter, this alpha n and this beta n. These are the rate functions. So Hodgkin-Huxley thought about the channel as if it is either in a closed state or in an open state. And there is a parameter, a rate parameter, that shifts depending on time and depending on voltage, because the alpha and beta are voltage dependent. Depending on voltage and depending on time, you move from the closed state to the open state as a function of voltage and time. So if alpha is big, it shifts from the closed state to the open state, and beta shifts the open state to the closed state. And this is the differential equation that describe the dependence of n, of this activation n variable, on voltage, because alpha and beta depend only on voltage. Alpha becomes larger with voltage, and as this becomes larger, it moves the closed state to the open state. Beta becomes smaller with voltage, so less is moved from the open state to the closed state. And eventually if the voltage is big enough, the depolarization is big enough, most of the channels will be in their open state. So n will be close to 1. So this is the idea of Hodgkin-Huxley how to mathematically describe fully the n variable that controls the opening of this potassium channel. But remember we need four of them to be open in order to get the flow of potassium through the channel. I'm not going to repeat the equation, but it's a little bit more complicated because in the case of sodium you have activation and inactivation for the same voltage, for the constant voltage. This means that here you have to have another variable, not only an activating variable, which they call m, but they also have an inactivating variable, h. So this is the equation of Hodgkin-Huxley. They said that if the sodium conductance is m3, three activating variables that tend to open the channels of u depolarization, but there is also this h variable that closes the channel during depolarization. So this opens very fast, but this slowly, slowly closes the channel, and these are the two differential equations that describe the m variable, like the n 1 and the h variable, like the n 1. So these are two additional equations, and this is what Hodgkin-Huxley showed when they clamped the cell again and again. I showed you before for the potassium, this is a fit for the sodium. You can see with depolarization you get more sodium and less and less and less with less depolarization, but always the sodium is open fast, and then, inactivates slowlier. And this is now a schematic representation that I drew for you to explain for you how does the sodium channel looks like. So we have now three gates that behaves like the n gates of the potassium. This is the m gate for the sodium. This m gate opens with the polarization. It's a fast gate, so it opens early on. It is activated early on, but there is this other h gate that is inactivated. It goes to zero. It closes the channel, but slowly. So, lucky for us, lucky for the action potential, you have a fast gate that opens early on, enables the flow of sodium from the inside to the outside early on. And this is the beginning of the spike, because sodium goes in, and when sodium goes in there is depolarization due to the sodium. Sodium goes in, but with time, there is this closing of this h gate, this red gate. And if it is closed there is no sodium in anymore, and that's what you saw here. So m, m gates open. There is fast inward sodium, and then h gates gets in and inactivates this channel. Today, with modern times, we can really reconstruct the membrane channels. I'm not going to go into it, but beautifully we now really can see the disprediction of Hodgkin–Huxley. Which is really just based on recordings and a very, very good mathematical capability to capture the behavior of the channels. We know today that indeed ion channels through the membrane. This is the outside. This is the inside. There are these parts of the channels that are voltage sensors and they feel the voltage across the membrane and the change the configuration of this channel enable to open the channel if you have depolarization, it really enables the flow of sodium or potassium, depending on which channel. And there is also other aspects that a voltage sensor, they tend to turn off the channel. And so we really now connect the physical structure of channels to the model of Hodgkin–Huxley and really Hodgkin–Huxley, in some sense, predicted the structure of ion channels in membrane that are voltage dependent without knowing about channels at all. They just recorded currents and measured conductances. So to summarize this part, this is now the circuit that I mentioned before, capacitance, conductance for sodium, conductance for potassium, and the reconductance, and these are the four equation that I mention in the beginning that summarize everything that you want to know about the spike. So this is the, grand equation, because it says that the current, the actual potential current, depends on capacitance cividity plus sodium conductance multiplied by Hm to the power of 3, multiplied by the driving force for the sodium current, so this is the sodium current. The voltage dependant sodium current and the voltage dependants come to the h and to the m. This is the potassium current. This is the driving force for the potassium. This is the conductance of the potassium which is voltage dependant. This is the n to the power of 4, and this is the leak passive conductance, G leak. This is the passive current. This is the grand equation, and there are these three additional equations that describe the dependence of each of these parameters, m with voltage and time, N for the potassium, voltage and time, voltage and time and also h. So when you solve these four equations, numerically Hodgkin–Huxley use a very simple computer that they used to have then, really simple hand computer so to speak. They really succeeded to replicate mathematically, the action potential. This is now a mathematical replica, mathematical drawing from these four equations. This is the action potential. When you solve this equation for V, for V, which appears in many places. When you solve this equation for V, you get this spike. You see the spike starts from resting potential. If you depolarize enough, it fires, overshoots, and then it goes down below the resting potential ands ends. So this is V. But because you have the mathematics, you can also see what is the sodium conductance and what is the potassium conductance underlying action potential. And that's the beauty of this theory. Because now you can really see, because you had voltage claim and you know for each voltage what is the expected conductance of sodium and potassium, you can now reconstruct the conductances, and you can see something that I mention many times, but now it's very important to re-emphasize, that early on when you have depolarization, in this case, you inject volt current. But in the regular case, the synapses are those that change the voltage towards depolarization. If it is exacatory synapse, it wants to shift the membrane from rest towards depolarization. So if you reach to a certain threshold, there is an opening of sodium conductance. GNA. And this sodium conductance enables extra flow of sodium ions from outside to inside. When sodium flows from outside to inside, it's on it's own depolarized the cell. Why? Because sodium is a positive ion, goes from outside to inside, make the inside more positive. When it becomes more positive inside due to extra sodium ions, there is an extra depolarization here. The spike starts to grow. When the spike starts to grow there is even more sodium channels going in. More sodium channel going in, more depolarization, more depolarization more sodium channel open, more sodium channel open, more depolarization and so on. So this is the positive feedback between voltage, growth and more conductance for sodium, more sodium enter the cell, more depolarization, more depolarization, more depolarization until a point where first of all, there is the sodium battery. We all ready mentioned that we cannot go beyond the battery of the particular conductance involved. Also the sodium inactivates. So even when it grows, the conductance of the sodium starts to inactivate because of the H value. Luckily it is slower than the M. So we first have an activation. First we have a spike. Then, the spike starts to suffer, so to speak from the closing of the sodium channels. The spike starts to go down. Okay so this is the inactivation part of the sodium channels, but know that at the same time, when you start with the inactivation of the h variable, there is the beginning of the activation of the potassium conductance. This activation of the potassium conductance due to the voltage of the spike, due to the depolarization due to the spike, the activation of the potassium starts to enable the flow of potassium ions from inside to the outside which means hyper-polarizing the membrane because you lose positive ions from the inside to the outside. So this opening of the potassium is an extra help, so to speak, to turn off the spike, to hyperpolarize the membrane. So both the potassium that grows, and the inactivation of the sodium, both are responsible for the hyper-polarization, for the inactivation phase of the spike going down, and here you can see this undershoot below the resting potential is due to still an extra current of potassium. So here the potassium conductance is larger than the rest. This extra potassium conductance enables the flow of potassium channel ions from inside to outside. This makes the inside even more negative than the resting potential. With time, the voltage of the spike goes back slowly, slowly to rest. The conductance of the potassium goes also slowly, slowly to its resting value. And eventually, the spike ends. So the spike starts after he polarized undershoot, then about five milliseconds or ten milliseconds, it ends completely. That's the ending of this fantastic phenomena, the spike. Another verification of Hodgkin–Huxley. [COUGH] So this is an experiment, where they injected different strengths of current, you already saw it before. Subthreshold current, you don't get a spike, both in the model and in the experiment. In the larger stimulus, larger and larger and larger stimulus. You've got a full blown spike, a full blown spike and all on non-spike, both in the experiment and in the model. So this is the verification that the model replicates very well. The behavior of increasing stimuli both in the model and the experiment, the spikes moves to the left. Starts earlier, but it is the same spike all over. Either spike or no spike. And let me almost complete this lesson by a phenomena that is extremely important, which is called the refractory period. Hodgkin–Huxley saw and people knew already that this spike, the action potential after you inject the first action potential. You initiate it at the first action potential. Very early on after the first action potential, you cannot get a second action potential. You have to wait enough time for a second stimuli to get the second action potential here. About ten milliseconds until you get a full action potential here, like the first action potential. This is called refractory period. So what is this refractory period? It's a very important thing, because it means that you cannot get one spike after the other very nearby in time. There should be some difference in time some gap, some delay between the first action potential and the second action potential. This means that the frequency of action potential is limited in actions. It can go 1 and then 10 milliseconds another one, which means that maximally in the squid giant axon under this condition, you can get 100 spikes per second. One and then a full one, 10 milliseconds later and then a full one 20 millisecond later and so on. This is called refractory period, this is experimental finding. So there is a period where you cannot get the full spike. And also the experiment, the model also shows this exactly the same. You get the first spike, if you stimuli enough depolarization. Here first spike, but then you cannot get a second spike until you wait enough and here you get the second spike. So what is the source of this refractoriness? Hodgkin–Huxley explained it very, beautifully. You can see just looking at the mathematics and the behavior of both the h variable, which goes down to zero with time after the action potential. So here is the action potential. Here is the h variable, the inactivation variable. You can see that just after the spike, the h becomes very close to zero. And when h is zero, it means that the channel is closed. The sodium channel is closed, no matter what you do, you cannot get a second spike, cuz there is no extra sodium current. So this inactivation variable, which is slow is responsible for what we call the absolute refractory period. Here when the h is very, very small, no matter what you do, you can still rely here very strongly there will be no second spike. Also, the sodium conductance impedes the spike. It doesn't allow the second spike to occur, because it is against the spike. It carries the spike downward. So these two parameters, the h inactivation and the potassium activation, both are responsible for the refractory period. The reason absolute refractory period, whereby you cannot absolutely get a spike. But if you wait a little bit more, you get a partial spike here and then a full blown spike here, so you repeat the zero one. There is really no spike, no healthy spike. The spike is very sick, either does not exist here. Or a very sick spike, a partial spike, which cannot really propagate later on. But then you will get a full spike when everything is recovered, let's say around here. You will get again a second spike, if you stimulate again. A refractory period, which is a very important limitation on the frequency that cells can generate. They cannot generate one spike attached to the other. There always be a difference in time between the first spike and the second spike. So the clock, if you want to look at the axon as a clock, the frequency of the clock is about 100 hertz, 100 spikes per second. Some cells can fire maybe 200 spikes per second, but they cannot fire too fast. So, it's a slow clock. There is a bit of information, another bit of information with a clock difference of about ten milliseconds. So you can see spikes in any nerve cell. Let me show you an example of a cortical cell you already saw. So this is from the squid to the cortical axon. And actually, every excitable cell in every system, it's an invention of the nervous system, the spike. This is a very special signal that carries information, represents information. And you already saw it, but I want to show you another one taken from the David McCormick lab recording from the cortical pyramidal cell. And suddenly, you see the spike that you saw in the squid. Here, you see in cortical pyramidal cell. And here we see. [SOUND] Spike, spike, spike, spike, spike. Again, spike, spike, spike, spike, all are non-spike. So this is a response to a cortical send to an injection of current. And you will see later on that this current, as we already discussed is coming from synapses and it is the synapses from the dendritic tree. All the excited synapses that are activated many, many places, 10,000 of them. When they reach the soma and depolarize the axon enough and not too many inhibitor synapses are active, then you will get enough depolarization at the cell body and the actual initial segment and the output of this will be this, [SOUND] which is the message that the axon sends this ramification axon sends to whoever wants to listen to the spikes. So let's meet next lesson and this time, we should talk about learning in plasticity using spikes, using synopsis. See you next week.