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In the last section we left off with this picture, a neuron responding to current

Â steps of different, of different amplitudes, and we saw that it's on the

Â threshold here, of excitable behavior. So our goal in the remainder of this

Â section is to uncover the nonlinearity that leads to this vital property.

Â In the last section, we realized that each ion has its own pathway for current

Â to flow, and a battery of potential difference that's associated with each.

Â We can represent that as these different branches.

Â So let's assemble them now, back into, back into our circuit.

Â We have a branch for sodium. A branch for potassium, and we're also

Â going to include one for the non-specific current flow of the passive membrane.

Â going to call that g link. Now, if the ion channels had a fixed

Â conductance, we'd still have a linear circuit, just a bit more complicated than

Â our original passive, our c circuit. What gives this system its interesting

Â behavior, is that these conductances are not fixed, that's what this variable

Â resistor symbol stands for, they depend on the voltage.

Â So let's take a closer look. So now we're zooming in down to the level

Â of a single potassium channel. The channel is an elaborate molecular

Â machine that contains a gate that prevents ions from entering, and a

Â voltage sensor here, that controls the configuration of the gate channel.

Â The open probability increases when the membrane is depolarized.

Â The gate here consists of four sub-units that need to be in the correct

Â configuration in order for ions to flow through.

Â So the open probability of the channel is the product of the open probability of

Â these four sub-units, so the probability goes as, the open probability of a single

Â sub-unit raised to the power 4. So what is the assumption we're making

Â here? Let's try to visualize this a bit.

Â So here's the closed channel. Each sub-unit fluctuates, open and

Â closed, at some rate. So let's call n the probability that this

Â sub-unit, one of these sub-units is in the open state.

Â Then the probability that it's closed, is 1 minus n.

Â So now, one of those other gates can open and close.

Â Independently, another one flicks open. Sometimes we have more than one open.

Â Finally when all four happen to be open together, the ion channel allows a

Â current to flow. A trans, a transition between states

Â occur at volt, let's say this red circle represents the total probability of the

Â state of one of the sub-units. It has probability n of being in the open

Â state, and probability 1 minus n of being closed.

Â Call this open and this closed. So there's a rate of transitions between

Â the open and the closed state. So the closed to open state we're

Â going to call the transition probability between closed and open alpha, and that's

Â voltage dependent. And there's also some rate, and there's

Â also some rate of transitions between open and closed, which your going to call

Â rate beta, that also is voltage dependent.

Â So now the time derivative of n is then given by the following equation.

Â 3:21

So this first term, represents how much is added to the open state, and the

Â second term, how much is lost? The amount that's added, is proportional

Â to amount that's in the closed state, times the rate of going from closed to

Â open alpha. And this is the amount that's lost, the

Â amount that's actually in the open state, n, times the rate of moving from open to

Â closed. So let's do what we did with the RC

Â circuit, and rewrite this equation in terms of tau and n infinity.

Â So when we do this, you can easily show for yourself that the rates determine the

Â quantities tau and infinity in the following way.

Â Tau is 1 over alpha plus beta. N is alpha over alpha plus beta.

Â We'll come back and use this relationship shortly.

Â But before we do, let's take a look at the sodium channel.

Â So here's the sodium channel. It's similar to the potassium channel,

Â but with one important difference. The sodium channel is open to the joint

Â opening now of three sub-units, but, similar to the previous case.

Â But additionally, to pass current, this channel requires that along with the

Â activation or opening of the three sub-units, there's also an additional

Â gating mechanism. A kind of ball in the socket mechanism,

Â and that's required to not be in place. That is that there's an additional gating

Â factor that must be de-inactivated. We can express the probability of one of

Â these three sub-units being open, as m and the probability that the gate is not

Â in place as h. What's interesting is that while voltage

Â increases, the degree of activation or m, it also decreases h, the level of

Â de-inactivation. So there's a kind of voltage window, in

Â which sodium is able to flow. Generally this results in sodium currents

Â being transient or self-limiting. As soon as sodium starts the flow,

Â because these m gates are open, the voltage moves towards the sodium

Â equilibrium potential, and that increases v and in-activates h, thus closing the

Â channel again. This is one of the mechanisms at work in

Â switching off the spike. So now we have these three gating

Â variables, one for k, that was n, and two for sodium m and h.

Â We can also re-express them in terms of tau and their steady state.

Â So as we did for n we can also write these two equations, a tau for the h

Â variable and a h infinity, and also a tau for m, and a m infinity.

Â 6:02

So now what do we do with these activation and activation variables?

Â We want to combine them to give us these voltage dependent conductance's for the

Â channels. So the probably of the potassium channel

Â being open goes like n the 4th. You're going to multiply that by the

Â total conductance of the channel, and that will give us this voltage dependent

Â conductance. Similarly the probability of the sodium

Â channel being open is given by m cubed of h.

Â We multiply that by sub-maximal conductance for sodium, and now we get

Â the voltage dependent and time dependent conductance for sodium.

Â So now let's pull it all together. The voltage across the membrane changes

Â as a result of changes in the, in the external driving current, and also

Â because these opening and closing probabilities cause the conductance's of

Â these, of these branches to change. And the amount of current going through

Â will change will both with changes in voltage, and with changes in overall

Â conductance. So we can write down that equation here.

Â So we have our Capacitative current, that's the current coming through this

Â branch. We have the Ionic currents, which come

Â down through each of the Ionic branches separately with sub-scripted each of the

Â ions with i. And, and that includes our, our leak

Â which includes non-specific movement of ions through the, through the membrane,

Â and then our external applied current. So, what this gives us is Hodgkin and

Â Huxley's equation, here, in it's, in it's full glory.

Â So we have our equation for the voltrage. And we're going to add to that, these

Â three equations for the different activation and inactivation variables,

Â that specify the conductance's for the different ionic types, sodium potassium.

Â Now let's see how we get to use our understanding of the activation dynamics,

Â to understand the spike. So, remember again, n governs the opening

Â of the potassium channel, and both n and h must be large for the sodium channel to

Â be open. Here's how these activation steady states

Â depend on voltage. They all have this kind of sigmoidal

Â form. As we see from the behavior of n

Â infinity, the potassium channel will have a higher probability of opening for

Â larger voltage. While the sodium channel first has an

Â increase of probability of opening with increasing voltage, because the increase

Â in m with voltage. But then because h is going down to 0,

Â as, as, voltage increases the, the sodium channel will close.

Â It's also very useful to look at the time constants, going back to our equation,

Â this time constant governs how quickly n will approach its final steady state.

Â So the time constants dictate how rapidly each variable responds to a change in

Â volt. Remember the exponential solution.

Â Let's say one changes v, so that's going to give us a new value of the steady

Â state, as a function of v. And then we wait for everything to

Â adjust. Each activation variable will tend toward

Â the steady state for that voltage, with a rate given by this time constant.

Â So, which of these variables here reacts fastest?

Â The variable with the shortest time constant, that is m.

Â So that means that the fastest response to a voltage change, is a change in

Â sodium activation. The dynamics of h and n are slower, these

Â time constants are larger. And you can see that they're on a similar

Â scale. So let's also remind ourselves what the

Â resting potentials are. Remember that when a potassium current

Â flows, it would be tending to move the membrane voltage toward the potassium

Â potential, down here at minus, minus 80. Well, sodium moves it up here.

Â So let's imagine we're sitting near rest. Rest is about minus 60 milivolts, and

Â then some input comes along that depolarizes the membrane, that is move it

Â to larger, larger voltages. So because the time constant for m is the

Â shortest, as we change voltage the first thing to adjust is going to be the m

Â value, its going to approach its steady state value, at the new value of the

Â voltage. That starts to open sodium channels.

Â Sodium current comes in, and starts to move the membrane toward the sodium

Â equilibrium potential. That's going to further increase sodium

Â conductance, and that's a positive feedback.

Â So what's going to counteract that, and stop the voltage from just ending off to

Â this large value? So at a slight delay because of these

Â slower dynamics for, for h and for n, two things are going to happen.

Â One is that h goes to h infinity. So finally the dynamics of h catch up,

Â and h is going to approach its steady state value.

Â And you could see that as voltage increases, that steady state value is

Â going down. And remember that for the sodium channel

Â to be open, we need a combination of m cubed and h, so if h is going towards 0,

Â then those channels are closing. Also, to help things along, the potassium

Â channel also activates more. So now finally n will also catch up, and

Â we'll see that the potassium channel starts to open more and more with larger

Â voltages. Now what does that do?

Â That starts to pull the voltage back down here, toward the equilibrium potential

Â for potassium. So finally the membrane will come back to

Â rest. So this is just to show the time cost of

Â these events. Voltage increases.

Â Here, there's a fast change, you see this very fast slope in m.

Â There, there's a positive feedback in which this increases very rapidly, until

Â its rise is truncated by the delayed effects of h, now going down and, and

Â closing the sodium channels. And n's starting to increase, and allow

Â that potassium current to bring the membrane back toward the potassium

Â reversal potential. So, you see here that the action

Â potential is this exquisitely timed change of molecules and charges.

Â So, what's so wonderful about Hodgkin and Huxley's model, is that they inferred all

Â of these dynamics without any knowledge of ion channels.

Â And particularly without any knowledge of sub-units.

Â All the dynamics here are explained by simple linear equation by a linear

Â circuit, or a simple rate equation, except for two things.

Â The multiplicative factors that relate the sub-unit behavior to the channel

Â conductance's and the voltage dependence of the sub-unit dynamic's.

Â So, from this fundamental basis there are two quite different directions that we

Â could go in as a modeler. So, one can delve into the dynamics of

Â ion channels, understanding how they come about from the microscopic level, and how

Â different signaling cascades influence these dynamics.

Â There are, of course, hundreds of different channel types dependent on

Â calcium and chloride, and even combinations of multiple ions in very

Â different time scales. And this wide range of dynamics

Â influences the way in which information is processed by single cells, and

Â dictates which neuron types carry out different roles in the brain.

Â Furthermore, realistic neurons are not just patches of membrane.

Â As we've looked at here, they're large distributed structures.

Â So how does this figure into our understanding of computation at the

Â neuronal level. The other direction to go in, is rather

Â than to complexify, to simplify. Can we write down simpler models that can

Â capture the essentials of these dynamics, but are maybe analytical tractable, so

Â that we can learn something mathematically.

Â Or at least be rapid enough to be able to put into large scale simulations, that

Â still respect something about the underlying biophysics of neurons.

Â So we're going to head in these two different directions for the rest of the

Â lecture. In the, in the next part of today's

Â lecture, we'll, we'll first deal with these simplified models, where some

Â examples of reduced models that people have developed that, that are based on

Â Hodgkin Huxley like neurons. And in the last part of today, we'll ju,

Â just touch on one of these topics, that's geometry.

Â How do we deal with Exter, such as dendrites, in modeling single neurons?

Â