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

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

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

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

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

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

We can represent that as these different branches.

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

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

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

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

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

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

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

resistor symbol stands for, they depend on the voltage.

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

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

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

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

The open probability increases when the membrane is depolarized.

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

configuration in order for ions to flow through.

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

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

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

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

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

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

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

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

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

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

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

current to flow. A trans, a transition between states

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

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

state, and probability 1 minus n of being closed.

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

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

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

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

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

rate beta, that also is voltage dependent.

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

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

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

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

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

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

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

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

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

quantities tau and infinity in the following way.

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

We'll come back and use this relationship shortly.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

channels. So the probably of the potassium channel

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

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

conductance. Similarly the probability of the sodium

channel being open is given by m cubed of h.

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

the voltage dependent and time dependent conductance for sodium.

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

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

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

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

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

conductance. So we can write down that equation here.

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

branch. We have the Ionic currents, which come

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

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

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

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

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

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

three equations for the different activation and inactivation variables,

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

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

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

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

be open. Here's how these activation steady states

depend on voltage. They all have this kind of sigmoidal

form. As we see from the behavior of n

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

larger voltage. While the sodium channel first has an

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

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

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

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

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

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

volt. Remember the exponential solution.

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

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

adjust. Each activation variable will tend toward

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

So, which of these variables here reacts fastest?

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

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

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

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

scale. So let's also remind ourselves what the

resting potentials are. Remember that when a potassium current

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

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

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

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

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

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

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

voltage. That starts to open sodium channels.

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

equilibrium potential. That's going to further increase sodium

conductance, and that's a positive feedback.

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

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

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

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

and h is going to approach its steady state value.

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

going down. And remember that for the sodium channel

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

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

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

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

voltages. Now what does that do?

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

for potassium. So finally the membrane will come back to

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

these events. Voltage increases.

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

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

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

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

that potassium current to bring the membrane back toward the potassium

reversal potential. So, you see here that the action

potential is this exquisitely timed change of molecules and charges.

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

of these dynamics without any knowledge of ion channels.

And particularly without any knowledge of sub-units.

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

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

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

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

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

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

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

different signaling cascades influence these dynamics.

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

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

different time scales. And this wide range of dynamics

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

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

Furthermore, realistic neurons are not just patches of membrane.

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

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

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

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

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

that we can learn something mathematically.

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

still respect something about the underlying biophysics of neurons.

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

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

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

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

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

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

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