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?