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MICHALE FEE: Today
we're going to continue

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building our equivalent
circuit model of a neuron.

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Again, this is the
Hodgkin-Huxley model,

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and the model was
really developed

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around explaining how neurons
generate action potentials.

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There are two key
ion channels that are

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associated with making spikes.

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There's a sodium
channel that we model

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as a conductance in
series with a battery,

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and there's a
potassium conductance

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that we model the same way.

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And again, those
two conductances

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cooperate to produce
an action potential.

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And we saw, essentially,
how those two conductances,

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together with their
batteries-- the sodium battery

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is up at plus 50
or so millivolts.

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The potassium battery is down
at minus 75 or so millivolts.

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And those two conductances then
work to essentially connect

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the inside of the neuron
to the plus battery

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and then to the minus battery
to give you an action potential.

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And you may remember that
we saw what that looks like.

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So here is plotting
membrane potential in blue.

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Here, we turn on this
sodium conductance.

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The voltage of the cell
races up to about plus 55.

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Then we turn off the sodium.

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We turn on the potassium.

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The voltage goes down
to minus 75 or so.

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And then we turn off
both conductances,

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and the cell recovers.

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So you can see that that
basically produces what

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looks like an action potential.

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That's the basis of action
potential production.

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But in order to understand how
these things turn on and off

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in the time course
that they do, we

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need to understand
a little bit more

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about how these sodium and
potassium conductances work.

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And that's what we're
going to focus on today.

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So let's just start
building our model.

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So each one of those
conductances with a battery

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is associated with a current.

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There's a current
that goes through each

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of those conductances.

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The total current
through the membrane,

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the ionic current through the
membrane in the Hodgkin-Huxley

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model is a sum of three
components, actually--

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a sodium current, a
potassium current--

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those two we just talked about--

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and a leak current that
is just a fixed current.

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So the sodium and
potassium currents

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are functions of
time and voltage.

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The leak current, the leak
conductance, is just fixed.

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And it has a battery of
about minus 50 millivolts.

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And it just tends to keep the
cell sort of hyperpolarized.

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And these two currents,
these two conductances,

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do the job of making
an action potential.

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So the total membrane
current is just

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a sum of those three parts.

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And now we can just take
that membrane current

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and plug it into this equation
for the voltage as a function

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of current now and solve
that differential equation

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to get the voltage to calculate
how the voltage evolves

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in time in the presence of
these membrane currents.

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All right, so you recall
from the last lecture

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that you watched on video
that these currents can be

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written down as a conductance.

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So let's just start here.

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This is the most
similar one to the one

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that you saw in the
previous lecture.

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The current is
just a conductance

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times a driving potential.

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And we described
how that equation

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can be summarized in
electrical circuit components

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as a resistor, which is
one over the conductance,

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times a driving potential,
which is basically

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just the voltage drop
across the conductance.

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Now, each of these conductances,
each of these currents,

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is going to be written down
by a very similar equation.

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So the potassium current is
just the potassium conductance

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times the driving
potential for potassium,

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which is just the membrane
potential minus the Ek.

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And the sodium current is
just the sodium conductance

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times the driving
potential for sodium,

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which is the membrane potential
minus the sodium battery.

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Any questions?

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Yes?

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AUDIENCE: How did
the [INAUDIBLE]..

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MICHALE FEE: Yeah.

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So unless I've made a mistake,
I've tried to put these--

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remember that the
potassium battery is minus,

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is negative, right?

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And the sodium
battery is positive.

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So I've tried to
show that by putting

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the batteries in the
opposite direction, right?

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So the battery
symbol has one side

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that's supposed to
indicate positive voltage,

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and the other side is negative.

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So because they have
the opposite sign,

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I've put them in backwards,
in the opposite direction.

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Does that make sense?

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Don't worry about that too much.

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If I ask you to draw
this, I don't really

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care that much which
way these things go.

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I just want you to
know that this one is

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negative on the inside
and that the sodium is

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positive on the inside.

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This is the inside of
the cell here, right?

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The sodium battery
drives the inside

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of the cell toward
positive voltage,

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because you have positive
ions flowing into the cell.

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So you can see now that
the membrane potential here

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depends on the membrane currents
through this differential

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equation.

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But the membrane currents--
the sodium, potassium, and leak

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currents-- all depend on
these conductances, right?

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And these conductances for
the sodium and potassium

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are voltage dependent
and time dependent.

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So you can see that
the conductances

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depend on the voltage, right?

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So the membrane potential
depends on current.

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Current depends on conductances,
but the conductances

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depend on the
membrane potential.

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So it goes around and
around and around, right?

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So those things all
depend on each other.

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And so what we are
setting out to do

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is to write down the way those
things depend on each other,

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the way you can think about
that system evolving in time.

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So let me just show
you what the plan is.

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So the plan is to write down
an algorithm-- basically,

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a for loop--

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that describes how the neuron
generates an action potential.

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Let me just walk you
through the steps of that,

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and then I'll get
to your question.

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So we're going to start with
some membrane potential V,

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and we're going
to calculate that

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the voltage-dependent parameters
of the sodium and potassium

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conductance using that
membrane potential.

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And once we have
those parameters,

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we can actually calculate the
conductance for each of those,

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for the sodium and potassium.

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Once you know the conductance,
you can get the currents.

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Once you have the
currents, you can

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compute the total
membrane current, just

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by adding them all together.

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Then, you can compute--

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once you have all
those currents,

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you just compute V infinity,
which is just the current times

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the resistance, the
effective resistance.

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Then we're going to
integrate our first order

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linear differential equation to
get a new voltage as a function

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of time and V infinity.

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And we're just going to
go back and start again.

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The so that's the
algorithm that a neuron

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uses to generate a spike.

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And that's what we're
going to work out.

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Now, we've talked
about these things

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over the last few
lectures, how you

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can relate total
current to V infinity

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and then integrate a first
order linear differential

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equation, which is just
relaxing exponentially

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toward V infinity.

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But now we're going to put
these things in, figure out

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the voltage and time
dependence of the sodium

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and potassium conductances.

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Yes?

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AUDIENCE: [INAUDIBLE].

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MICHALE FEE: Yes.

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It's primarily potassium.

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It has a negative potential.

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But it's just constant,
so we're not really

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going to pay much
attention to it.

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So if the sodium and
potassium currents are off,

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then the leak current still
keeps the cell hyperpolarized.

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All right, any questions?

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That's the big picture.

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So here are our
learning objectives.

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I'd like you to be able to
draw that circuit diagram,

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not worrying about the long
and short sides of the battery.

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We're going to talk
about how we measure

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the properties of ion channels.

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That's called a voltage clamp.

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So I want you to be able
to describe what that is.

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I'd like you to be able to
plot the voltage and time

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dependence of the potassium
current for today.

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The next lecture, we'll talk
about the sodium current,

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so we'll add that to our list
of things that we need to know.

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But for today, I'd
like you to able

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to plot the voltage and time
dependence of the potassium

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current and conductance.

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And be able to
explain, biophysically,

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where the time and voltage
dependence of that potassium

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conductance comes from and
be able to write it down

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in terms of quantities
that are called

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the Hodgkin-Huxley
gating variables.

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So that's the plan.

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All right, so let's come
back to our circuit.

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Again, we have a
sodium current that's

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sodium conductance times the
sodium driving potential.

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The conductance is voltage
and time dependent.

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The equilibrium
potential for sodium,

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again, is plus 55 millivolts.

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The potassium current is
just potassium conductance

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times the potassium
driving potential.

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The driving potential
is reference

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to a battery in equilibrium
potential at minus 75.

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And the leak has a battery
at minus 50 millivolts.

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So those are the parts
we're going to use.

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We're going to describe now
the experiments that Hodgkin

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and Huxley did to extract
the parameters of the sodium

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and potassium conductances.

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Today, we're going to focus
on the potassium conductances.

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And then on next
Tuesday, I guess,

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we're going to do the sodium.

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All right, so the reason
Hodgkin and Huxley studied--

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so they studied these
channels, the potassium

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and sodium channels, in
the giant squid axon.

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Now, most of our
axons are about a--

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most of the axons in our brain
are about a micron across.

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This axon is about
a millimeter across.

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Action potentials propagate
much faster in large axons,

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and this axon is involved
in transmitting an action

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potential from the
brain to the tail that

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drives an escape reflex.

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The squid squirts water out
of sort of a chamber that

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has water in it.

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If the squid senses
danger, it contracts

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muscles that squeeze water out
of that, and it makes a jet.

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And it squirts the squid
forward away from danger.

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So that action potential has
to propagate very quickly

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from the brain to the
tail, and it does that

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through this enormous axon.

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That axon is so big you can
now put multiple wires inside

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of it.

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You don't even need to pull
these glass electrodes.

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You can just take little
wires and stick them in--

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00:12:04,730 --> 00:12:06,230
chop out a little
piece of the axon

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and stick wires inside
of it and study it.

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Yes?

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AUDIENCE: So if the
body of this squid

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is like a giant [INAUDIBLE]
it has arms coming out

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of its head?

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MICHALE FEE: Yes.

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You eat that part,
and you eat that part.

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Not that.

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You throw away the
most interesting part.

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OK, any other questions?

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00:12:32,320 --> 00:12:35,530
All right, so now
we're setting out

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to measure these sodium and
potassium conductances, OK?

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So how do we do that?

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So what we really
want to do is to set--

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we want to measure
conductance, which

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is the relation between
voltage and current.

252
00:12:50,670 --> 00:12:52,580
So we'd like to do
is to be able to set

253
00:12:52,580 --> 00:12:56,660
the voltage at a certain level
and measure the current that

254
00:12:56,660 --> 00:13:00,150
flows through these channels.

255
00:13:00,150 --> 00:13:03,690
So you really want to
plot the IV curve, right?

256
00:13:03,690 --> 00:13:06,240
You want to set the voltage,
measure the current,

257
00:13:06,240 --> 00:13:08,400
and do that at a bunch
of different voltages.

258
00:13:08,400 --> 00:13:11,250
And you recall that the
conductance is basically

259
00:13:11,250 --> 00:13:18,580
just the slope of that
curve, right, that line.

260
00:13:18,580 --> 00:13:21,580
So the job is set
voltage, measure current,

261
00:13:21,580 --> 00:13:23,740
extract conductance.

262
00:13:23,740 --> 00:13:26,590
Now, the problem with
that is that as soon

263
00:13:26,590 --> 00:13:30,220
as you set the
voltage of the axon

264
00:13:30,220 --> 00:13:32,650
somewhere up here in
an interesting range,

265
00:13:32,650 --> 00:13:36,020
the thing begins to spike.

266
00:13:36,020 --> 00:13:40,390
And then the voltage is
no longer constant, right?

267
00:13:40,390 --> 00:13:43,360
So it becomes really
hard to make measurements

268
00:13:43,360 --> 00:13:46,737
like this if you depolarize
the cell a little bit

269
00:13:46,737 --> 00:13:48,820
to set try to set the
voltage and all of a sudden,

270
00:13:48,820 --> 00:13:49,737
it's [MIMICS BUZZING].

271
00:13:49,737 --> 00:13:52,180
It's generating spikes.

272
00:13:52,180 --> 00:13:53,510
So what do you do?

273
00:13:53,510 --> 00:13:57,670
So the trick is to develop a
device called a voltage clamp.

274
00:13:57,670 --> 00:14:00,160
This thing basically
holds the-- so,

275
00:14:00,160 --> 00:14:04,190
look, if the action potential
were really, really slow,

276
00:14:04,190 --> 00:14:06,880
then you could actually
set the voltage.

277
00:14:06,880 --> 00:14:10,330
You could change the current
being injected into the cell

278
00:14:10,330 --> 00:14:11,110
by hand.

279
00:14:11,110 --> 00:14:13,735
You say, OK, I'm trying to
set the voltage at zero.

280
00:14:13,735 --> 00:14:15,110
Oh, it got a little
bit too high,

281
00:14:15,110 --> 00:14:16,450
so I turn the current down.

282
00:14:16,450 --> 00:14:18,280
Now the voltage
has gone too low,

283
00:14:18,280 --> 00:14:20,080
so now I turn the current up.

284
00:14:20,080 --> 00:14:23,590
You could do it by hand
if the action potential

285
00:14:23,590 --> 00:14:27,160
were super slow, if it took
a minute to generate, right?

286
00:14:27,160 --> 00:14:29,500
But the action potential
takes a millisecond.

287
00:14:29,500 --> 00:14:31,720
So that's just too
fast for you to follow.

288
00:14:31,720 --> 00:14:34,150
So you just make a
little electrical circuit

289
00:14:34,150 --> 00:14:36,550
that does that job for you.

290
00:14:36,550 --> 00:14:39,760
It uses feedback to set
the voltage where you want.

291
00:14:39,760 --> 00:14:40,840
So you put a--

292
00:14:40,840 --> 00:14:42,010
here's your cell.

293
00:14:42,010 --> 00:14:45,370
Here's your membrane
resistance or conductance

294
00:14:45,370 --> 00:14:47,600
that you're trying to measure.

295
00:14:47,600 --> 00:14:49,190
You put a electrode in the cell.

296
00:14:49,190 --> 00:14:51,050
You put it into a
little amplifier called

297
00:14:51,050 --> 00:14:54,470
an operational amplifier.

298
00:14:54,470 --> 00:14:56,660
And then on the other
side of that amplifier,

299
00:14:56,660 --> 00:15:00,050
that differential
amplifier, you put

300
00:15:00,050 --> 00:15:03,510
the command voltage that
you're trying to set.

301
00:15:03,510 --> 00:15:06,030
So here's the way it works.

302
00:15:06,030 --> 00:15:09,590
Basically, this thing tries
to set the membrane potential.

303
00:15:09,590 --> 00:15:12,690
It tries to make this
value, this voltage,

304
00:15:12,690 --> 00:15:13,920
equal to that voltage.

305
00:15:13,920 --> 00:15:17,640
And it does that by feeding
current back into the cell.

306
00:15:17,640 --> 00:15:19,770
Does that make sense?

307
00:15:19,770 --> 00:15:22,420
OK, so you use an
operational amplifier.

308
00:15:22,420 --> 00:15:24,270
An op amp has two inputs--

309
00:15:24,270 --> 00:15:26,230
a plus input, a minus input.

310
00:15:26,230 --> 00:15:30,370
The output is just a gain
times the plus input minus--

311
00:15:30,370 --> 00:15:32,680
the positive input minus
the negative input.

312
00:15:32,680 --> 00:15:34,070
And the gain is really big.

313
00:15:34,070 --> 00:15:35,380
It's about a million.

314
00:15:35,380 --> 00:15:37,690
So if this input is a
little bit above that input,

315
00:15:37,690 --> 00:15:40,610
this output is big and positive.

316
00:15:40,610 --> 00:15:43,510
If this input is
less than that input,

317
00:15:43,510 --> 00:15:44,900
you can see this is negative.

318
00:15:44,900 --> 00:15:47,230
And so the output
is big and negative.

319
00:15:47,230 --> 00:15:49,970
Any questions?

320
00:15:49,970 --> 00:15:51,250
So don't get confused here.

321
00:15:51,250 --> 00:15:58,570
That G is gain, not conductance,
just for the next few slides.

322
00:15:58,570 --> 00:15:59,810
So how does this work?

323
00:15:59,810 --> 00:16:02,450
You can see that if the
membrane potential is

324
00:16:02,450 --> 00:16:06,440
less than the command
voltage, then the output

325
00:16:06,440 --> 00:16:09,320
voltage is positive and big.

326
00:16:09,320 --> 00:16:11,540
That drives current
into the cell, which

327
00:16:11,540 --> 00:16:14,090
increases the membrane
potential and makes it

328
00:16:14,090 --> 00:16:16,980
approach the command voltage.

329
00:16:16,980 --> 00:16:20,640
If the membrane potential
is larger than the command

330
00:16:20,640 --> 00:16:22,980
voltage, then this thing--

331
00:16:22,980 --> 00:16:24,250
this is bigger than this.

332
00:16:24,250 --> 00:16:25,920
So this is negative.

333
00:16:25,920 --> 00:16:28,590
And that pulls current
out of the neuron

334
00:16:28,590 --> 00:16:30,720
and decreases the
membrane potential.

335
00:16:30,720 --> 00:16:33,390
And in both cases,
the membrane potential

336
00:16:33,390 --> 00:16:36,090
is being pulled toward
the command voltage.

337
00:16:36,090 --> 00:16:41,850
And you can show, if you
just plug in these variables

338
00:16:41,850 --> 00:16:44,310
into a couple of
equations, that as long

339
00:16:44,310 --> 00:16:47,550
as the gain is big enough,
the membrane potential

340
00:16:47,550 --> 00:16:51,720
is forced to be very close
to the command voltage.

341
00:16:51,720 --> 00:16:54,150
All right, so that's
the voltage clamp.

342
00:16:54,150 --> 00:16:58,020
It drives whatever current
is necessary to clamp

343
00:16:58,020 --> 00:17:00,990
the voltage of the cell
at the command voltage.

344
00:17:00,990 --> 00:17:03,630
And then what we do is
during an experiment,

345
00:17:03,630 --> 00:17:06,750
we step the command
voltage around.

346
00:17:06,750 --> 00:17:08,430
The cell tries to spike.

347
00:17:08,430 --> 00:17:09,569
Those currents turn on.

348
00:17:09,569 --> 00:17:12,119
The cell tries to
spike, but this thing

349
00:17:12,119 --> 00:17:15,390
keeps the voltage locked
at whatever it is that you

350
00:17:15,390 --> 00:17:17,849
want the voltage to be.

351
00:17:17,849 --> 00:17:20,763
And then you measure the
amount of current required.

352
00:17:20,763 --> 00:17:22,680
You just measure the
amount of current flowing

353
00:17:22,680 --> 00:17:26,339
through this
resistor here that's

354
00:17:26,339 --> 00:17:30,120
required to hold the
cell at any voltage.

355
00:17:30,120 --> 00:17:31,350
All right, any questions?

356
00:17:31,350 --> 00:17:33,150
Voltage clamp-- very cool.

357
00:17:37,160 --> 00:17:37,660
Yes?

358
00:17:37,660 --> 00:17:39,390
AUDIENCE: Can you
explain gain again?

359
00:17:39,390 --> 00:17:41,110
MICHALE FEE: Gain is
just the multiplier

360
00:17:41,110 --> 00:17:42,890
here in this equation.

361
00:17:42,890 --> 00:17:47,110
So if there's a tiny difference
between the two inputs,

362
00:17:47,110 --> 00:17:48,970
the output is bigger.

363
00:17:48,970 --> 00:17:50,490
That's what gain means, right?

364
00:17:56,190 --> 00:17:59,220
If the gain is a million, if
there's a microvolt difference

365
00:17:59,220 --> 00:18:04,220
between the two inputs, the
output will be about a volt.

366
00:18:04,220 --> 00:18:07,030
And it would be a plus or minus,
depending on which of those two

367
00:18:07,030 --> 00:18:08,870
was more positive.

368
00:18:11,710 --> 00:18:14,650
OK, so now let's get to the
actual voltage clamp experiment

369
00:18:14,650 --> 00:18:16,990
that Hodgkin and Huxley did.

370
00:18:16,990 --> 00:18:18,970
Here, we have two
wires in our cell-- one

371
00:18:18,970 --> 00:18:21,580
to measure voltage and the
other one to inject current.

372
00:18:21,580 --> 00:18:23,240
That's exactly what they did.

373
00:18:23,240 --> 00:18:24,500
There's one wire here.

374
00:18:24,500 --> 00:18:25,750
That's a little piece of axon.

375
00:18:25,750 --> 00:18:29,050
You can literally cut the
squid open, find that axon.

376
00:18:29,050 --> 00:18:32,740
It's a big, white-looking tube
about a millimeter across.

377
00:18:32,740 --> 00:18:37,480
Cut two pieces of it, take two
wires, stick one in each end.

378
00:18:37,480 --> 00:18:39,850
I drew it like this, but
if you did it that way,

379
00:18:39,850 --> 00:18:41,840
they'd probably short together.

380
00:18:41,840 --> 00:18:44,140
So then one of those
measures the voltage.

381
00:18:44,140 --> 00:18:46,750
You set a command,
and the other wire

382
00:18:46,750 --> 00:18:49,180
allows you to inject
current inside the axon.

383
00:18:49,180 --> 00:18:53,230
And then you seal the ends
with a little bit of Vaseline.

384
00:18:53,230 --> 00:18:53,730
Yes?

385
00:18:53,730 --> 00:18:55,550
AUDIENCE: Sorry, what
is the command VC?

386
00:18:55,550 --> 00:18:57,410
MICHALE FEE: VC is
the command voltage

387
00:18:57,410 --> 00:19:01,640
that you're trying to set
the inside of the cell to.

388
00:19:01,640 --> 00:19:05,310
Remember, here,
we're setting the--

389
00:19:05,310 --> 00:19:10,040
VC is what you're controlling
as the experimenter.

390
00:19:10,040 --> 00:19:12,755
You're setting the
voltage with that command.

391
00:19:17,020 --> 00:19:19,360
So the voltage clamp
then holds the inside

392
00:19:19,360 --> 00:19:22,800
of the cell at that
command voltage,

393
00:19:22,800 --> 00:19:27,460
and then you're measuring
the current with this device.

394
00:19:27,460 --> 00:19:30,370
There's a readout that tells
you how much current it's

395
00:19:30,370 --> 00:19:33,100
putting into the cell, or it
goes onto an oscilloscope,

396
00:19:33,100 --> 00:19:34,747
because it's time dependent.

397
00:19:38,610 --> 00:19:39,780
So let's do an experiment.

398
00:19:39,780 --> 00:19:41,610
Here's an example
of an experiment.

399
00:19:41,610 --> 00:19:44,220
They hold the
command at minus 65.

400
00:19:44,220 --> 00:19:48,300
And suddenly, they drop the
command voltage to minus 130.

401
00:19:48,300 --> 00:19:50,460
What does the cell do?

402
00:19:50,460 --> 00:19:51,310
What is the current?

403
00:19:51,310 --> 00:19:51,810
Nothing.

404
00:19:55,600 --> 00:19:59,300
There's a little transient here,
which is the amount of current.

405
00:19:59,300 --> 00:20:03,640
It took to charge that capacitor
up to minus 130 millivolts,

406
00:20:03,640 --> 00:20:04,900
and then nothing happens.

407
00:20:07,555 --> 00:20:09,180
All right, let's do
another experiment.

408
00:20:09,180 --> 00:20:12,660
Now we're going to start
our cell at minus 65

409
00:20:12,660 --> 00:20:17,070
and suddenly jump the
voltage up to zero.

410
00:20:17,070 --> 00:20:19,900
So we're going to
depolarize our cell.

411
00:20:19,900 --> 00:20:22,625
And now something happens.

412
00:20:22,625 --> 00:20:27,450
We get a big pulse of
current that's negative.

413
00:20:27,450 --> 00:20:29,340
What does negative mean?

414
00:20:29,340 --> 00:20:31,110
Anybody remember
what negative current

415
00:20:31,110 --> 00:20:35,260
means by our definition?

416
00:20:35,260 --> 00:20:39,340
Negative means that there
are positive ions going

417
00:20:39,340 --> 00:20:41,090
into this cell.

418
00:20:41,090 --> 00:20:44,770
So it's charging the cell up.

419
00:20:44,770 --> 00:20:47,890
This is membrane current now.

420
00:20:47,890 --> 00:20:50,020
So you just have to
remember that definition.

421
00:20:50,020 --> 00:20:53,560
Negative membrane current
means that positive charges

422
00:20:53,560 --> 00:20:58,990
are going into the cell, and
that's depolarizing the cell.

423
00:20:58,990 --> 00:21:02,270
And then that negative current
lasts for a few milliseconds,

424
00:21:02,270 --> 00:21:05,500
and then the current reverses
sign and becomes positive

425
00:21:05,500 --> 00:21:08,450
and stays on.

426
00:21:08,450 --> 00:21:11,790
So what is that?

427
00:21:11,790 --> 00:21:14,510
So the first thing that
Hodgkin and Huxley did was they

428
00:21:14,510 --> 00:21:20,840
tried to figure out what causes
that pattern of currents.

429
00:21:20,840 --> 00:21:23,310
So here's what they did.

430
00:21:23,310 --> 00:21:29,452
They had this idea that part
of this might be due to sodium.

431
00:21:29,452 --> 00:21:31,410
And so what they did was
they did an experiment

432
00:21:31,410 --> 00:21:36,270
where they replaced
sodium outside the cell,

433
00:21:36,270 --> 00:21:40,585
outside the axon, with an ion.

434
00:21:43,420 --> 00:21:45,700
They kept the chloride, but
they replaced the sodium

435
00:21:45,700 --> 00:21:46,420
with choline.

436
00:21:46,420 --> 00:21:50,440
So they used choline chloride,
so it's a salt solution,

437
00:21:50,440 --> 00:21:53,020
but it has no sodium in it.

438
00:21:53,020 --> 00:21:54,740
And then they redid
that experiment.

439
00:21:54,740 --> 00:21:55,865
And here's what they found.

440
00:21:58,590 --> 00:22:02,070
What they found was-- so
here it is-- with sodium,

441
00:22:02,070 --> 00:22:03,570
and if they replace
the sodium, they

442
00:22:03,570 --> 00:22:06,250
find that they get
almost the same thing,

443
00:22:06,250 --> 00:22:09,510
except that initial
negative pulse goes away.

444
00:22:13,550 --> 00:22:17,230
And so they hypothesized that
that part is due to sodium.

445
00:22:17,230 --> 00:22:22,390
And so now you just
subtract this from this

446
00:22:22,390 --> 00:22:24,110
to get the difference.

447
00:22:24,110 --> 00:22:27,245
And that is the sodium current.

448
00:22:27,245 --> 00:22:28,120
Does that make sense?

449
00:22:30,648 --> 00:22:31,690
And then one other thing.

450
00:22:31,690 --> 00:22:33,190
Through another
set of experiments,

451
00:22:33,190 --> 00:22:35,200
they were able to show
that this part that's

452
00:22:35,200 --> 00:22:38,380
left after you block
or remove sodium

453
00:22:38,380 --> 00:22:40,900
is actually due to potassium.

454
00:22:40,900 --> 00:22:43,540
So this is the
potassium current.

455
00:22:43,540 --> 00:22:46,320
That is a sodium current.

456
00:22:46,320 --> 00:22:49,020
So by doing different
kinds of experiments--

457
00:22:49,020 --> 00:22:52,148
so one of the things that you're
able to do that they didn't do

458
00:22:52,148 --> 00:22:54,190
initially, but later, they
were able to do things

459
00:22:54,190 --> 00:22:56,470
like take that
little piece of axon,

460
00:22:56,470 --> 00:23:01,570
take out a little miniature
paint roller, squish the axon,

461
00:23:01,570 --> 00:23:05,080
roll the roller over the axon,
squish its guts out, and then

462
00:23:05,080 --> 00:23:08,410
fill it up again with solutions
that they control that have

463
00:23:08,410 --> 00:23:09,800
different ions in them.

464
00:23:09,800 --> 00:23:12,010
And so they're able to study--

465
00:23:12,010 --> 00:23:15,580
just do multiple different
kinds of experiments

466
00:23:15,580 --> 00:23:18,910
to be sure that this
slow thing that turns on,

467
00:23:18,910 --> 00:23:21,550
this slow positive
current is potassium,

468
00:23:21,550 --> 00:23:25,102
and this fast negative
current is sodium.

469
00:23:25,102 --> 00:23:26,074
All right?

470
00:23:29,380 --> 00:23:30,820
So now what you
can do is you can

471
00:23:30,820 --> 00:23:36,250
do this experiment at
different voltages here, right?

472
00:23:36,250 --> 00:23:40,000
We want to measure how
these currents depend

473
00:23:40,000 --> 00:23:42,220
on voltage, right?

474
00:23:42,220 --> 00:23:44,810
We can see here how
they depend on time.

475
00:23:44,810 --> 00:23:47,380
We can also see how
they depend on voltage,

476
00:23:47,380 --> 00:23:50,440
by doing this experiment
at different voltages.

477
00:23:50,440 --> 00:23:52,570
You start at some
negative potential.

478
00:23:52,570 --> 00:23:55,930
You step the voltage up to minus
40, or you step it up to zero,

479
00:23:55,930 --> 00:23:57,580
or you step it up to 40.

480
00:23:57,580 --> 00:24:00,670
And now you can measure
that potassium current

481
00:24:00,670 --> 00:24:03,190
as a function of
time, or the sodium

482
00:24:03,190 --> 00:24:06,300
current as a function of time.

483
00:24:06,300 --> 00:24:08,032
All right?

484
00:24:08,032 --> 00:24:09,190
At different voltages.

485
00:24:14,670 --> 00:24:17,530
All right, so those look kind
of weird, especially that one.

486
00:24:17,530 --> 00:24:19,250
That looks kind of
scary, like what

487
00:24:19,250 --> 00:24:21,910
the heck is going on there?

488
00:24:21,910 --> 00:24:24,880
But it turns out that
both of these things

489
00:24:24,880 --> 00:24:27,310
are actually pretty simple.

490
00:24:27,310 --> 00:24:31,060
Once we dig a little bit more
into how these currents are

491
00:24:31,060 --> 00:24:34,690
produced, you're going to see
that there's a very simple way

492
00:24:34,690 --> 00:24:38,060
to understand what's
happening there.

493
00:24:38,060 --> 00:24:40,388
All right, any questions?

494
00:24:40,388 --> 00:24:40,930
Bear with me.

495
00:24:43,900 --> 00:24:46,630
So now what we want
to do is we want

496
00:24:46,630 --> 00:24:49,420
to measure the voltage
dependence of these things

497
00:24:49,420 --> 00:24:51,680
kind of separately from
the time dependence.

498
00:24:51,680 --> 00:24:53,560
So what we're going
to do is we're

499
00:24:53,560 --> 00:24:57,490
going to measure the peak
potassium current, kind

500
00:24:57,490 --> 00:24:59,470
of this steady state
potassium current

501
00:24:59,470 --> 00:25:01,900
as a function of voltage.

502
00:25:01,900 --> 00:25:04,840
So here's our IV
curve that I promised

503
00:25:04,840 --> 00:25:07,030
that we were going to plot.

504
00:25:07,030 --> 00:25:10,590
Peak current as a
function of voltage.

505
00:25:10,590 --> 00:25:13,830
You can see that it's
approximately linear

506
00:25:13,830 --> 00:25:19,540
above minus 50 or so millivolts.

507
00:25:19,540 --> 00:25:22,440
The sodium current
looks kind of weird.

508
00:25:22,440 --> 00:25:25,730
We're going to plot
the peak sodium current

509
00:25:25,730 --> 00:25:28,940
as a function of voltage.

510
00:25:28,940 --> 00:25:32,330
And we see that the peak sodium
current has this weird shape.

511
00:25:32,330 --> 00:25:35,730
It's sort of linear up
here at positive voltages,

512
00:25:35,730 --> 00:25:39,600
and then it crashes down to
zero at negative voltages.

513
00:25:43,580 --> 00:25:44,970
All right?

514
00:25:44,970 --> 00:25:48,015
Still kind of weird and scary.

515
00:25:48,015 --> 00:25:49,390
So let's see if
we can understand

516
00:25:49,390 --> 00:25:52,330
where this comes from.

517
00:25:52,330 --> 00:25:53,680
I just replotted them here.

518
00:25:53,680 --> 00:25:57,160
So now, remember that we
use this voltage clamp

519
00:25:57,160 --> 00:25:59,470
to measure current as
a [AUDIO OUT] voltage.

520
00:25:59,470 --> 00:26:03,010
But what is it that we're
really trying to understand?

521
00:26:03,010 --> 00:26:05,590
We're really trying to
understand the conductances,

522
00:26:05,590 --> 00:26:07,690
those resistors, right?

523
00:26:07,690 --> 00:26:10,690
We're under trying to understand
the voltage and time dependence

524
00:26:10,690 --> 00:26:13,830
of those conductances.

525
00:26:13,830 --> 00:26:15,720
So we're trying to
extract conductance

526
00:26:15,720 --> 00:26:17,580
as a function of voltage.

527
00:26:17,580 --> 00:26:19,920
And remember that [AUDIO OUT]
is just conductance

528
00:26:19,920 --> 00:26:23,580
times a driving
potential for potassium.

529
00:26:23,580 --> 00:26:26,490
Sodium current is just sodium
conductance times the sodium

530
00:26:26,490 --> 00:26:29,860
driving potential.

531
00:26:29,860 --> 00:26:34,530
So we you could imagine
extracting the conductance

532
00:26:34,530 --> 00:26:37,740
as current divided by
the driving potential.

533
00:26:41,750 --> 00:26:43,580
This is what we're
really trying to find.

534
00:26:47,380 --> 00:26:49,630
Rather than doing this
division, because this one

535
00:26:49,630 --> 00:26:52,780
over-- this thing goes
to zero at the places

536
00:26:52,780 --> 00:26:55,633
where the voltage is equal
to the equilibrium potential.

537
00:26:55,633 --> 00:26:56,800
So we don't want to do that.

538
00:26:56,800 --> 00:26:59,650
We're going to solve
this problem graphically.

539
00:26:59,650 --> 00:27:05,230
So here's the driving
potential, right, V minus Ek.

540
00:27:05,230 --> 00:27:09,790
It goes to zero at Ek, right?

541
00:27:12,460 --> 00:27:15,370
And we want to find
a conductance that

542
00:27:15,370 --> 00:27:17,620
makes this look like this.

543
00:27:20,310 --> 00:27:22,910
So if this is kind
of a straight line,

544
00:27:22,910 --> 00:27:25,100
if the current as a
function of voltage

545
00:27:25,100 --> 00:27:28,890
is kind of a straight line
and this is a straight line,

546
00:27:28,890 --> 00:27:32,210
what does that tell us about
the conductance in this region

547
00:27:32,210 --> 00:27:34,750
up here?

548
00:27:34,750 --> 00:27:36,100
It's constant.

549
00:27:36,100 --> 00:27:37,540
Excellent.

550
00:27:37,540 --> 00:27:41,920
Now, if the driving potential
is very negative here

551
00:27:41,920 --> 00:27:45,460
but the current is zero,
then what's the conductance?

552
00:27:48,400 --> 00:27:50,980
Driving potential
is big and negative.

553
00:27:50,980 --> 00:27:52,420
Current is zero.

554
00:27:52,420 --> 00:27:54,940
What does that tell us
about the conductance?

555
00:27:54,940 --> 00:27:55,820
It's zero.

556
00:27:55,820 --> 00:27:58,090
OK, so not so hard, right?

557
00:27:58,090 --> 00:28:00,700
We have [AUDIO OUT] zero
here and constant here.

558
00:28:00,700 --> 00:28:03,340
So can anyone just show me
what that might look like?

559
00:28:06,010 --> 00:28:06,520
Good.

560
00:28:06,520 --> 00:28:08,920
It could be like a jump.

561
00:28:08,920 --> 00:28:10,960
It could be kind of smooth.

562
00:28:10,960 --> 00:28:13,940
And that's exactly
what it looks like.

563
00:28:13,940 --> 00:28:16,730
So the conductance is zero
here, which it has to be,

564
00:28:16,730 --> 00:28:19,430
because the current is zero,
even though the the potential

565
00:28:19,430 --> 00:28:20,690
is negative.

566
00:28:20,690 --> 00:28:22,820
And the conductance
here is constant,

567
00:28:22,820 --> 00:28:25,550
because the driving
potential is constant here.

568
00:28:25,550 --> 00:28:28,230
The driving potential is linear,
and the current is linear,

569
00:28:28,230 --> 00:28:34,430
so this needs to be
constant, all right?

570
00:28:34,430 --> 00:28:37,220
So the conductance
is very simple.

571
00:28:37,220 --> 00:28:39,010
It turns off at
negative voltages

572
00:28:39,010 --> 00:28:44,040
and turns on and then stays
on at higher voltages.

573
00:28:44,040 --> 00:28:45,420
Let's do that for sodium.

574
00:28:45,420 --> 00:28:47,700
This thing looks
crazy, weird, right?

575
00:28:47,700 --> 00:28:50,230
But let's go through
the same operation.

576
00:28:50,230 --> 00:28:52,260
Here's our driving
potential for sodium.

577
00:28:52,260 --> 00:28:57,390
Remember, it's got a
reversal at plus 55.

578
00:28:57,390 --> 00:28:59,280
But it's a straight line, right?

579
00:28:59,280 --> 00:29:01,380
That's a battery and a resistor.

580
00:29:01,380 --> 00:29:03,240
So there's our
driving potential.

581
00:29:03,240 --> 00:29:05,710
Now, this is linear.

582
00:29:05,710 --> 00:29:06,550
This is linear.

583
00:29:06,550 --> 00:29:10,740
So what does the conductance
look like up here?

584
00:29:10,740 --> 00:29:11,780
Good.

585
00:29:11,780 --> 00:29:15,110
This is zero, but this
is big and negative, so

586
00:29:15,110 --> 00:29:18,170
what does the conductance
look like down here?

587
00:29:18,170 --> 00:29:19,070
Good .

588
00:29:19,070 --> 00:29:21,620
Starting to look
pretty familiar, right?

589
00:29:21,620 --> 00:29:22,420
Boom.

590
00:29:22,420 --> 00:29:23,600
It's exactly the same.

591
00:29:28,340 --> 00:29:34,270
Both of these conductances are
off at negative potentials,

592
00:29:34,270 --> 00:29:38,155
and they turn on at positive
voltages and remain constant.

593
00:29:45,320 --> 00:29:46,542
Yes?

594
00:29:46,542 --> 00:29:47,970
AUDIENCE: [INAUDIBLE].

595
00:29:50,548 --> 00:29:51,340
MICHALE FEE: Great.

596
00:29:51,340 --> 00:29:52,030
Great question.

597
00:29:52,030 --> 00:29:54,370
Because the sodium
conductance turns on,

598
00:29:54,370 --> 00:29:56,080
and then it shuts
off right away.

599
00:29:56,080 --> 00:29:58,240
And the shutting off is
a different mechanism.

600
00:29:58,240 --> 00:30:01,720
So we're trying to figure out
how to ignore that and just

601
00:30:01,720 --> 00:30:06,020
understand the voltage
depends of how it turns on.

602
00:30:06,020 --> 00:30:07,910
Does that make sense?

603
00:30:07,910 --> 00:30:11,190
And we'll get into
that next Tuesday.

604
00:30:11,190 --> 00:30:13,880
But I'm showing both of
these at the same time

605
00:30:13,880 --> 00:30:17,000
because they look
similar at the level

606
00:30:17,000 --> 00:30:19,350
of the voltage dependence.

607
00:30:19,350 --> 00:30:24,056
In fact, the way they
turn on is very similar.

608
00:30:24,056 --> 00:30:27,090
It's just that the sodium has
some other weird thing that

609
00:30:27,090 --> 00:30:28,680
shuts it off after
a few milliseconds,

610
00:30:28,680 --> 00:30:30,210
and we'll talk about
that next time.

611
00:30:30,210 --> 00:30:30,983
Yes?

612
00:30:30,983 --> 00:30:34,527
AUDIENCE: [INAUDIBLE].

613
00:30:34,527 --> 00:30:35,610
MICHALE FEE: No, it's not.

614
00:30:35,610 --> 00:30:38,040
But any non-linearity
here, we're

615
00:30:38,040 --> 00:30:42,030
going to account for by
changes by voltage dependence

616
00:30:42,030 --> 00:30:43,410
of the conductance.

617
00:30:43,410 --> 00:30:44,610
So great point.

618
00:30:44,610 --> 00:30:50,190
It's a subtlety that
we have kind of imposed

619
00:30:50,190 --> 00:30:53,550
by this way of writing
down the voltage dependents

620
00:30:53,550 --> 00:30:54,720
of the current.

621
00:30:57,840 --> 00:30:59,173
Any other questions?

622
00:30:59,173 --> 00:31:01,340
No?

623
00:31:01,340 --> 00:31:03,680
OK, pushing on.

624
00:31:07,340 --> 00:31:13,850
So this kind of gradual
turning on, this sort of zero

625
00:31:13,850 --> 00:31:16,370
conductance down here
and a [INAUDIBLE] up

626
00:31:16,370 --> 00:31:19,970
here, that's called a
sigmoidal voltage dependence.

627
00:31:19,970 --> 00:31:22,760
And it's the voltage
dependence of activation.

628
00:31:22,760 --> 00:31:26,160
It's how these channels turn on.

629
00:31:26,160 --> 00:31:27,710
And as I said
before, we're going

630
00:31:27,710 --> 00:31:32,060
to deal with the other
properties of the sodium

631
00:31:32,060 --> 00:31:33,960
channel that turn it off later.

632
00:31:33,960 --> 00:31:38,060
So both have a sigmoidal voltage
dependence of activation.

633
00:31:44,320 --> 00:31:48,070
And if you plot this,
you can see that--

634
00:31:48,070 --> 00:31:52,420
if you plot this on
a log scale here--

635
00:31:52,420 --> 00:31:56,230
log scale on conductance,
linear here on potential.

636
00:31:56,230 --> 00:31:59,140
You can see that
both of these curves,

637
00:31:59,140 --> 00:32:00,910
both the potassium
and the sodium,

638
00:32:00,910 --> 00:32:05,680
have this very characteristic
exponential turn-on followed

639
00:32:05,680 --> 00:32:11,165
by his saturation and constant
conductance at higher voltages.

640
00:32:13,900 --> 00:32:15,720
All right, any questions?

641
00:32:15,720 --> 00:32:18,270
That's voltage dependence.

642
00:32:18,270 --> 00:32:20,430
Now we're going to turn
to time dependence.

643
00:32:23,280 --> 00:32:26,500
So you can see that
the time dependence--

644
00:32:26,500 --> 00:32:29,220
so this driving potential,
that's just constant.

645
00:32:29,220 --> 00:32:32,590
That just depends on the voltage
and the reversal potential.

646
00:32:32,590 --> 00:32:36,270
And so we can separate out--

647
00:32:36,270 --> 00:32:38,670
this thing is just
any time dependence

648
00:32:38,670 --> 00:32:41,610
this has is dependent on the
time dependence of the voltage.

649
00:32:41,610 --> 00:32:43,920
But in our voltage
clamp experiments,

650
00:32:43,920 --> 00:32:45,420
the voltage is constant.

651
00:32:45,420 --> 00:32:47,070
So this thing is constant.

652
00:32:47,070 --> 00:32:48,630
So any time
dependence [AUDIO OUT]

653
00:32:48,630 --> 00:32:52,780
current has to be due to time
dependence of the conductance,

654
00:32:52,780 --> 00:32:54,353
right?

655
00:32:54,353 --> 00:32:55,770
And what that means
is we can just

656
00:32:55,770 --> 00:33:00,090
look at the shape of this
potassium current-- remember,

657
00:33:00,090 --> 00:33:02,220
this is the potassium
current here.

658
00:33:02,220 --> 00:33:05,070
That time dependence is
just due to time dependence

659
00:33:05,070 --> 00:33:06,840
of the conductance.

660
00:33:06,840 --> 00:33:08,120
Does that make sense?

661
00:33:08,120 --> 00:33:10,830
So what's happening is
the potassium conductance

662
00:33:10,830 --> 00:33:12,360
is starting at zero.

663
00:33:12,360 --> 00:33:14,610
The moment you step
the voltage up,

664
00:33:14,610 --> 00:33:17,520
that thing begins to
grow gradually and then

665
00:33:17,520 --> 00:33:22,170
runs up to a constant
potassium conductance in time.

666
00:33:22,170 --> 00:33:25,110
It starts off, ramps
up, and then becomes

667
00:33:25,110 --> 00:33:26,790
constant, all right?

668
00:33:26,790 --> 00:33:30,570
So that's the time dependence.

669
00:33:30,570 --> 00:33:34,790
Sort of gracefully turns on.

670
00:33:34,790 --> 00:33:40,600
That process of turning
on is called activation.

671
00:33:40,600 --> 00:33:43,500
The sodium conductance--
or current, the same thing.

672
00:33:43,500 --> 00:33:46,917
The sodium current is just
the sodium conductance.

673
00:33:46,917 --> 00:33:48,750
The sodium current is
a function [AUDIO OUT]

674
00:33:48,750 --> 00:33:51,410
is the sodium conductance
as a function of time,

675
00:33:51,410 --> 00:33:52,260
times a constant.

676
00:33:52,260 --> 00:33:54,540
In our voltage clamp
experiment, again, this voltage

677
00:33:54,540 --> 00:33:55,680
is constant.

678
00:33:55,680 --> 00:34:00,450
So the sodium
conductance turns on.

679
00:34:00,450 --> 00:34:01,530
That's activation.

680
00:34:01,530 --> 00:34:06,180
But then it turns off, and
that's called inactivation.

681
00:34:06,180 --> 00:34:08,570
So the sodium current
has [AUDIO OUT]

682
00:34:08,570 --> 00:34:10,800
sodium conductance
has two things going

683
00:34:10,800 --> 00:34:16,110
on-- one, activation, and
the second is inactivation.

684
00:34:16,110 --> 00:34:19,159
And it turns out these are
two separate biophysical

685
00:34:19,159 --> 00:34:20,710
mechanisms.

686
00:34:20,710 --> 00:34:27,300
And we're going to spend
more time on this next week.

687
00:34:27,300 --> 00:34:31,280
So, notice something
interesting.

688
00:34:31,280 --> 00:34:33,620
The sodium conductance turns on.

689
00:34:33,620 --> 00:34:34,850
You depolarize the cell.

690
00:34:34,850 --> 00:34:38,600
Sodium conductance turns on
right away and then shuts off.

691
00:34:38,600 --> 00:34:43,730
The potassium conductance has
a delay, and then it turns on.

692
00:34:43,730 --> 00:34:45,670
Does that look familiar?

693
00:34:45,670 --> 00:34:51,580
It looks an awful
lot like this, right?

694
00:34:51,580 --> 00:34:53,130
Here's the sodium conductance.

695
00:34:53,130 --> 00:34:57,300
Turns on and then shuts off.

696
00:34:57,300 --> 00:34:58,770
And then the
potassium conductance

697
00:34:58,770 --> 00:35:01,930
turns on with a delay.

698
00:35:01,930 --> 00:35:04,070
And that gives us
an action potential.

699
00:35:04,070 --> 00:35:06,790
So you can see that when
you use voltage clamp

700
00:35:06,790 --> 00:35:10,240
and dissected out the time
difference of the sodium

701
00:35:10,240 --> 00:35:12,610
and potassium
conductance, it looks just

702
00:35:12,610 --> 00:35:16,270
like the thing we
concocted earlier, just

703
00:35:16,270 --> 00:35:20,760
sort of our toy example for how
to make an action potential.

704
00:35:20,760 --> 00:35:22,770
Pretty cool, right?

705
00:35:22,770 --> 00:35:25,850
OK, it's starting to come
together piece by piece.

706
00:35:31,440 --> 00:35:35,460
So we're now going to
dig in a little bit

707
00:35:35,460 --> 00:35:39,270
deeper into the biophysics
of how you get these voltage

708
00:35:39,270 --> 00:35:40,500
and time dependencies.

709
00:35:43,810 --> 00:35:48,700
So we're going to
derive the equation

710
00:35:48,700 --> 00:35:49,990
for the voltage dependence.

711
00:35:49,990 --> 00:35:51,970
Anybody want to
take a crazy guess

712
00:35:51,970 --> 00:35:54,110
on how we're going to do that?

713
00:35:54,110 --> 00:35:57,970
Just a wild guess, how you might
derive the voltage dependence

714
00:35:57,970 --> 00:36:00,770
of something?

715
00:36:00,770 --> 00:36:01,330
No?

716
00:36:01,330 --> 00:36:01,720
OK.

717
00:36:01,720 --> 00:36:03,470
We're going to use the
Boltzmann equation.

718
00:36:07,150 --> 00:36:09,280
And we're going to derive
different equations that

719
00:36:09,280 --> 00:36:12,970
describe the way those
channels turn on,

720
00:36:12,970 --> 00:36:15,530
how those conductances turn on.

721
00:36:15,530 --> 00:36:17,643
All right?

722
00:36:17,643 --> 00:36:19,310
And once we do that,
we're going to have

723
00:36:19,310 --> 00:36:22,160
a simple set of equations--

724
00:36:22,160 --> 00:36:23,330
and not just equations.

725
00:36:23,330 --> 00:36:26,510
We're going to have
a set of processes

726
00:36:26,510 --> 00:36:29,060
that we can think of
as happening a loop,

727
00:36:29,060 --> 00:36:30,260
in a for loop.

728
00:36:30,260 --> 00:36:34,460
That's our algorithm
for an action potential.

729
00:36:34,460 --> 00:36:36,680
All right, so let's dive
into single channels

730
00:36:36,680 --> 00:36:39,730
and see how they work.

731
00:36:39,730 --> 00:36:43,470
So, of course, currents
result from ionic flow

732
00:36:43,470 --> 00:36:46,320
through ion channels.

733
00:36:46,320 --> 00:36:49,410
It's actually possible
to record currents

734
00:36:49,410 --> 00:36:51,930
from single ion channels.

735
00:36:51,930 --> 00:36:55,980
We can actually make a
version of our voltage clamp

736
00:36:55,980 --> 00:37:00,540
that we can attach to
a single ion channel.

737
00:37:00,540 --> 00:37:02,190
And the way you do that is--

738
00:37:02,190 --> 00:37:04,980
so when you take this piece
of glass and you pull it,

739
00:37:04,980 --> 00:37:07,350
instead of poking
it through the cell,

740
00:37:07,350 --> 00:37:09,420
instead of making
it really sharp

741
00:37:09,420 --> 00:37:11,400
and poking it through
the cell, what you do

742
00:37:11,400 --> 00:37:14,940
is you make it a
little bit blunter,

743
00:37:14,940 --> 00:37:16,350
so it's got kind of a rough end.

744
00:37:16,350 --> 00:37:17,970
And then you can
fire a polish-- you

745
00:37:17,970 --> 00:37:22,940
can hold the end of that
electrode into a flame.

746
00:37:22,940 --> 00:37:23,440
Not quite.

747
00:37:23,440 --> 00:37:25,875
It's usually a filament
that heats up hot.

748
00:37:25,875 --> 00:37:27,250
You hold the end
of the electrode

749
00:37:27,250 --> 00:37:30,220
near this hot filament
and it melts the tip

750
00:37:30,220 --> 00:37:32,290
into a nice, round--

751
00:37:32,290 --> 00:37:34,630
it's still a tube, but
the edges of the tube

752
00:37:34,630 --> 00:37:37,155
are nice and smooth.

753
00:37:37,155 --> 00:37:39,280
And now when you take that
tube and you press it up

754
00:37:39,280 --> 00:37:41,740
against the cell--

755
00:37:41,740 --> 00:37:44,770
actually, you attach
a little plastic tube

756
00:37:44,770 --> 00:37:48,362
to the end of the glass, and
you press that that electrode up

757
00:37:48,362 --> 00:37:49,070
against the cell.

758
00:37:49,070 --> 00:37:52,210
And you actually literally
suck on it with your mouth

759
00:37:52,210 --> 00:37:53,950
onto that tube.

760
00:37:53,950 --> 00:37:57,780
And it sucks the membrane
up against that smooth end

761
00:37:57,780 --> 00:37:58,780
of the electrode.

762
00:37:58,780 --> 00:38:00,520
And it sticks.

763
00:38:00,520 --> 00:38:04,870
The lipids of the membrane
actually seal themselves

764
00:38:04,870 --> 00:38:07,070
onto the end of the glass.

765
00:38:07,070 --> 00:38:12,070
So now no currents can flow
out through these edges

766
00:38:12,070 --> 00:38:13,330
here, all right?

767
00:38:13,330 --> 00:38:15,130
And then you hook it
up to a very sensitive

768
00:38:15,130 --> 00:38:16,390
current amplifier.

769
00:38:16,390 --> 00:38:18,610
And now you can
control the voltage.

770
00:38:18,610 --> 00:38:20,743
You can actually just
rip that off of the cell,

771
00:38:20,743 --> 00:38:21,910
so now there's no more cell.

772
00:38:21,910 --> 00:38:23,950
You just have an ion
channel sitting there

773
00:38:23,950 --> 00:38:26,300
on a piece [AUDIO OUT]
on the end of your glass.

774
00:38:26,300 --> 00:38:29,460
Now you can do a
voltage clamp experiment

775
00:38:29,460 --> 00:38:31,350
and study the current--

776
00:38:31,350 --> 00:38:33,870
the voltage dependence
of the current

777
00:38:33,870 --> 00:38:37,068
through that ion channel.

778
00:38:37,068 --> 00:38:38,360
So here's what this looks like.

779
00:38:38,360 --> 00:38:39,277
Here's one experiment.

780
00:38:39,277 --> 00:38:40,810
We're going to
start at minus 100.

781
00:38:40,810 --> 00:38:43,710
This is a potassium channel.

782
00:38:43,710 --> 00:38:47,532
You depolarize the potassium
channel up to 50 millivolts,

783
00:38:47,532 --> 00:38:49,990
and you see that that current,
through that single channel,

784
00:38:49,990 --> 00:38:52,990
starts flickering on and off.

785
00:38:52,990 --> 00:38:54,370
Here's another trial.

786
00:38:54,370 --> 00:38:57,795
Turns on, turns off,
turns on, turns off.

787
00:38:57,795 --> 00:38:59,170
You can do that
a bunch of times.

788
00:39:01,750 --> 00:39:03,550
You can see something
interesting.

789
00:39:03,550 --> 00:39:06,010
The current is either off--

790
00:39:06,010 --> 00:39:10,300
doesn't turn on gradually,
doesn't change smoothly.

791
00:39:10,300 --> 00:39:12,340
It just flickers
between on and off.

792
00:39:12,340 --> 00:39:15,790
That's a very important
aspect of ion channels.

793
00:39:15,790 --> 00:39:19,600
But if you average all
those trials together,

794
00:39:19,600 --> 00:39:24,010
you see that you get an
average current that looks just

795
00:39:24,010 --> 00:39:28,940
like the current
that Hodgkin-Huxley

796
00:39:28,940 --> 00:39:33,540
measured in the whole axon.

797
00:39:33,540 --> 00:39:34,560
How is that possible?

798
00:39:43,075 --> 00:39:44,530
AUDIENCE: [INAUDIBLE].

799
00:39:47,940 --> 00:39:49,150
MICHALE FEE: Yeah.

800
00:39:49,150 --> 00:39:49,650
Good.

801
00:39:49,650 --> 00:39:51,600
So, basically,
what we're doing is

802
00:39:51,600 --> 00:39:55,530
we're measuring one
ion channel many times.

803
00:39:55,530 --> 00:39:59,930
But on a cell, you're
measuring a bunch

804
00:39:59,930 --> 00:40:03,470
of ion channels, each of which
is doing something like this.

805
00:40:03,470 --> 00:40:05,900
But they're happening
all at the same time,

806
00:40:05,900 --> 00:40:08,140
and the current
is being averaged.

807
00:40:08,140 --> 00:40:11,480
So here, we're averaging
the current one at a time.

808
00:40:11,480 --> 00:40:14,510
And on a whole cell, we're
just averaging a bunch of them

809
00:40:14,510 --> 00:40:15,770
at once.

810
00:40:15,770 --> 00:40:17,780
It's called
ergodicity in physics.

811
00:40:22,470 --> 00:40:26,250
It's called the
ensemble average.

812
00:40:26,250 --> 00:40:28,920
OK, you can do the
same thing for sodium.

813
00:40:28,920 --> 00:40:31,620
You take your patch,
a new patch electrode.

814
00:40:31,620 --> 00:40:33,650
Fire polish it.

815
00:40:33,650 --> 00:40:35,450
Push it up to a cell.

816
00:40:35,450 --> 00:40:36,860
Apply some suction.

817
00:40:36,860 --> 00:40:37,730
Glues on.

818
00:40:37,730 --> 00:40:41,360
This time, we had
a sodium channel.

819
00:40:41,360 --> 00:40:43,680
And now you can see
that the thing, again,

820
00:40:43,680 --> 00:40:46,830
flickers on, flickers off,
flickers on, flickers off.

821
00:40:46,830 --> 00:40:49,830
But now, they all flicker
on right at the beginning,

822
00:40:49,830 --> 00:40:52,860
and then they flicker
off and stay off.

823
00:40:52,860 --> 00:40:55,320
And if you average all
those different trials,

824
00:40:55,320 --> 00:40:59,500
you see an ensemble
average sodium current

825
00:40:59,500 --> 00:41:03,680
that looks just like when
you measure the sodium

826
00:41:03,680 --> 00:41:05,520
current on a whole axon, OK?

827
00:41:08,280 --> 00:41:11,870
But the key thing is that these
channels have two states--

828
00:41:11,870 --> 00:41:13,790
on and off-- and they
flicker back and forth

829
00:41:13,790 --> 00:41:17,480
between those two states,
conducting and non-conducting.

830
00:41:17,480 --> 00:41:20,930
So we can now write down--

831
00:41:20,930 --> 00:41:22,820
we could start
working with this idea

832
00:41:22,820 --> 00:41:25,460
that our ion channels are
either open or closed.

833
00:41:25,460 --> 00:41:27,620
And we can think
of a probability

834
00:41:27,620 --> 00:41:30,920
that the channel is being
open, that the channel is open.

835
00:41:30,920 --> 00:41:34,430
And we can have a total
number of channels.

836
00:41:34,430 --> 00:41:36,260
The number of open
channels is just

837
00:41:36,260 --> 00:41:38,780
the number of channels you
have times the probability

838
00:41:38,780 --> 00:41:41,560
that any one of them is open.

839
00:41:41,560 --> 00:41:45,910
If g is the inductance
of one open channel,

840
00:41:45,910 --> 00:41:48,670
then we can write down the
total potassium conductance

841
00:41:48,670 --> 00:41:53,680
as the probability that any
given ion channel is open

842
00:41:53,680 --> 00:41:56,830
times the number of channels
times the inductance of one

843
00:41:56,830 --> 00:41:58,365
open channel.

844
00:41:58,365 --> 00:41:59,240
Does that make sense?

845
00:42:02,650 --> 00:42:05,200
And now the claim
here [AUDIO OUT]

846
00:42:05,200 --> 00:42:08,920
all of the interesting
voltage and time dependence

847
00:42:08,920 --> 00:42:11,740
of these channels happens here.

848
00:42:11,740 --> 00:42:15,040
Obviously, the number of them
isn't changing very rapidly.

849
00:42:15,040 --> 00:42:20,800
The conductance per channel
is constant, per open channel.

850
00:42:20,800 --> 00:42:23,800
So the interesting stuff
is in the probability

851
00:42:23,800 --> 00:42:25,590
that the channel [AUDIO OUT].

852
00:42:30,340 --> 00:42:33,190
And if we want to
get the current,

853
00:42:33,190 --> 00:42:37,030
we're just going to plug this
conductance into here, OK?

854
00:42:37,030 --> 00:42:39,430
All right, so let's start
with a potassium channel.

855
00:42:39,430 --> 00:42:41,830
Let's dig in a little bit
deeper into what the potassium

856
00:42:41,830 --> 00:42:43,090
channel looks like.

857
00:42:43,090 --> 00:42:46,820
Potassium channel is formed
by four identical subunits.

858
00:42:46,820 --> 00:42:53,230
They're produced
separately by ribosomes.

859
00:42:53,230 --> 00:42:57,540
They form a
heteromer, a tetramer.

860
00:42:57,540 --> 00:42:59,360
And that tetramer
has a hole that

861
00:42:59,360 --> 00:43:05,270
runs down the middle of it,
which is where the ions flow.

862
00:43:05,270 --> 00:43:08,000
Each of these subunits
has a voltage sensor that

863
00:43:08,000 --> 00:43:09,960
allows it to turn on and off.

864
00:43:09,960 --> 00:43:15,950
In order for the channel to be
on, all four of those subunits

865
00:43:15,950 --> 00:43:16,670
has to be open.

866
00:43:16,670 --> 00:43:19,840
So each subunit has an open
state and a closed state.

867
00:43:19,840 --> 00:43:22,460
And for the channel to
be open, all four of them

868
00:43:22,460 --> 00:43:23,870
have to be in the open state.

869
00:43:28,350 --> 00:43:34,430
So if n is the probability
that any one subunit is open--

870
00:43:34,430 --> 00:43:37,415
I meant to make you guys
answer this question

871
00:43:37,415 --> 00:43:38,540
before I showed the answer.

872
00:43:38,540 --> 00:43:41,810
But is it clear how
if any one subunit has

873
00:43:41,810 --> 00:43:44,490
a probability of
being open of n,

874
00:43:44,490 --> 00:43:48,170
then the probability that
the whole channel is open

875
00:43:48,170 --> 00:43:49,310
is n to the four?

876
00:43:52,780 --> 00:43:55,540
This n is called
a gating variable.

877
00:43:55,540 --> 00:44:01,720
I would like you to know that
the probability that a sodium

878
00:44:01,720 --> 00:44:04,960
potassium channel is
open is n to the four.

879
00:44:04,960 --> 00:44:09,680
That's an important thing
for you to remember.

880
00:44:09,680 --> 00:44:13,970
That assumes that those four
subunits are independent.

881
00:44:13,970 --> 00:44:15,800
And in potassium
channels, that's

882
00:44:15,800 --> 00:44:19,710
a very good approximation.

883
00:44:19,710 --> 00:44:23,780
So we can now write down the
conductance of our potassium

884
00:44:23,780 --> 00:44:24,830
channel--

885
00:44:24,830 --> 00:44:26,930
something times n
to the four, where

886
00:44:26,930 --> 00:44:33,090
that something is that
inductance of one ion channel.

887
00:44:40,030 --> 00:44:43,560
So we can now write
down the current as n.

888
00:44:43,560 --> 00:44:47,730
Open conductance times n to the
four times a driving potential.

889
00:44:47,730 --> 00:44:50,670
And that n is called
the gating variable

890
00:44:50,670 --> 00:44:53,810
for the potassium conductance.

891
00:44:53,810 --> 00:44:55,310
All right, any questions?

892
00:44:59,150 --> 00:45:01,050
No?

893
00:45:01,050 --> 00:45:01,550
Yes?

894
00:45:01,550 --> 00:45:04,440
AUDIENCE: [INAUDIBLE].

895
00:45:04,440 --> 00:45:06,570
MICHALE FEE: n absolutely
does depend on voltage.

896
00:45:06,570 --> 00:45:07,390
Very good.

897
00:45:07,390 --> 00:45:08,640
That's where we're going next.

898
00:45:16,150 --> 00:45:19,220
But before we go on
to that, I wanted

899
00:45:19,220 --> 00:45:22,120
to add one other thing,
which I think is really cool.

900
00:45:22,120 --> 00:45:25,990
We're going to do the voltage
dependence of a potassium

901
00:45:25,990 --> 00:45:27,970
channel using the
Boltzmann equation.

902
00:45:30,940 --> 00:45:35,490
So here's the way you think
about a potassium channel

903
00:45:35,490 --> 00:45:37,595
working.

904
00:45:37,595 --> 00:45:38,720
Here's a potassium channel.

905
00:45:38,720 --> 00:45:40,940
We're showing a cross-section.

906
00:45:40,940 --> 00:45:41,460
Here it is.

907
00:45:41,460 --> 00:45:43,550
Here's the membrane,
the lipid bilayer.

908
00:45:43,550 --> 00:45:47,370
Here's our potassium channel,
sitting in the membrane.

909
00:45:47,370 --> 00:45:49,920
And we're taking a cross-section
through that tetramer

910
00:45:49,920 --> 00:45:53,180
that shows two subunits.

911
00:45:53,180 --> 00:45:55,280
And I'm showing
the voltage sent--

912
00:45:55,280 --> 00:46:00,320
I'm showing the mechanism
that that opens and closes

913
00:46:00,320 --> 00:46:02,180
one of those subunits.

914
00:46:02,180 --> 00:46:06,530
This subunit we'll also have a
voltage sensor and a gate that

915
00:46:06,530 --> 00:46:08,000
looks the same.

916
00:46:08,000 --> 00:46:11,980
So look, the voltage sensor--
how do you sense voltage?

917
00:46:11,980 --> 00:46:14,710
You sense voltage
with charge, right?

918
00:46:14,710 --> 00:46:20,150
Voltage differences, I should
say, you sense with a charge.

919
00:46:20,150 --> 00:46:23,540
Because voltage gradients
are electric fields,

920
00:46:23,540 --> 00:46:26,310
and electric fields
push on charges.

921
00:46:26,310 --> 00:46:30,050
So if we want to detect
the voltage difference

922
00:46:30,050 --> 00:46:33,720
across this membrane, we put
a charge in the membrane.

923
00:46:33,720 --> 00:46:35,910
When the voltage
difference is zero,

924
00:46:35,910 --> 00:46:40,560
there's little force
on those charges.

925
00:46:40,560 --> 00:46:43,200
Now, if we suddenly
hyperpolarize the [AUDIO OUT]

926
00:46:43,200 --> 00:46:45,530
cell so it's very
negative, now there's

927
00:46:45,530 --> 00:46:50,980
an electric field
inside the membrane that

928
00:46:50,980 --> 00:46:53,230
points toward the inside
of the cell, which

929
00:46:53,230 --> 00:46:57,290
pushes those charges toward
the inside of the cell.

930
00:46:57,290 --> 00:46:59,920
And now you can just have a
little mechanical linkage.

931
00:46:59,920 --> 00:47:02,110
That's not really
what it looks like,

932
00:47:02,110 --> 00:47:07,390
but there's some way that the
amino acids and the protein

933
00:47:07,390 --> 00:47:10,510
are configured so that when
those charges get pushed on,

934
00:47:10,510 --> 00:47:13,030
it closes a gate.

935
00:47:13,030 --> 00:47:14,800
And now the current
can no longer

936
00:47:14,800 --> 00:47:19,450
flow through the ion channel.

937
00:47:26,300 --> 00:47:29,460
OK, so now we're going
to derive how this--

938
00:47:29,460 --> 00:47:32,940
we're going to see how to
derive this voltage dependence

939
00:47:32,940 --> 00:47:35,100
from the Boltzmann equation.

940
00:47:35,100 --> 00:47:39,720
All right, so, everybody,
this is just for fun.

941
00:47:39,720 --> 00:47:42,480
I don't expect you to
know how to do this.

942
00:47:42,480 --> 00:47:46,620
I just want you to see
it, because I personally

943
00:47:46,620 --> 00:47:48,210
get chills when I see this.

944
00:47:48,210 --> 00:47:49,660
It's really cool.

945
00:47:49,660 --> 00:47:52,950
But I'm not expecting you to
be able to reproduce it, OK?

946
00:47:52,950 --> 00:47:56,390
So just watch.

947
00:47:56,390 --> 00:47:58,170
So, again, the
Boltzmann equation

948
00:47:58,170 --> 00:48:01,500
says that the
probability of being

949
00:48:01,500 --> 00:48:04,440
in two states, open or
closed, depends on the energy

950
00:48:04,440 --> 00:48:08,000
difference between them.

951
00:48:08,000 --> 00:48:10,650
So we have an open state
and a closed state.

952
00:48:10,650 --> 00:48:13,530
And when the voltage
inside the cell

953
00:48:13,530 --> 00:48:17,840
is zero-- when the voltage
difference between the inside

954
00:48:17,840 --> 00:48:20,700
and the outside of the
cell is zero, [AUDIO OUT]

955
00:48:20,700 --> 00:48:24,300
know that the sodium
channel likes to be open.

956
00:48:24,300 --> 00:48:29,570
So what that means is that the
open state has a lower energy

957
00:48:29,570 --> 00:48:31,970
than the closed state, right?

958
00:48:31,970 --> 00:48:34,970
Sodium channel likes to open
when the cell is depolarized.

959
00:48:34,970 --> 00:48:36,770
That means the voltage
inside and outside

960
00:48:36,770 --> 00:48:40,670
are close to each other, right?

961
00:48:40,670 --> 00:48:43,120
Open state has a lower
energy than the closed state.

962
00:48:43,120 --> 00:48:46,940
Let's call that energy
difference delta u.

963
00:48:46,940 --> 00:48:50,630
And it's close to kt,
because when it's open,

964
00:48:50,630 --> 00:48:53,270
the channel kind of flickers
back and forth between open

965
00:48:53,270 --> 00:48:54,380
and closed.

966
00:48:54,380 --> 00:48:57,160
Does that make sense?

967
00:48:57,160 --> 00:48:58,180
Now let's put on--

968
00:49:01,740 --> 00:49:04,260
let's hyperpolarize
the inside of our cell.

969
00:49:04,260 --> 00:49:07,090
So now the voltage
inside is low.

970
00:49:07,090 --> 00:49:09,760
There's a voltage
gradient, an electric field

971
00:49:09,760 --> 00:49:13,980
that is trying to
push those charges in.

972
00:49:13,980 --> 00:49:16,650
Now, you can see that
those charges here

973
00:49:16,650 --> 00:49:19,290
are sitting at a lower voltage.

974
00:49:19,290 --> 00:49:22,500
So in the closed
state, those charges

975
00:49:22,500 --> 00:49:24,660
are down here at
a lower potential.

976
00:49:24,660 --> 00:49:29,520
What does that mean for the
energy of the closed state

977
00:49:29,520 --> 00:49:32,640
when the cell is hyperpolarized?

978
00:49:32,640 --> 00:49:34,050
Its lower.

979
00:49:34,050 --> 00:49:36,240
The energy of the
closed state is low

980
00:49:36,240 --> 00:49:39,270
because those charges are
toward the inside of the cell,

981
00:49:39,270 --> 00:49:41,640
and the voltage is low.

982
00:49:44,170 --> 00:49:47,590
Now, what happens if the
cell is hyperpolarized,

983
00:49:47,590 --> 00:49:49,390
but it happens to be
in the open state?

984
00:49:49,390 --> 00:49:51,760
You can see those charges
are closer this way.

985
00:49:51,760 --> 00:49:56,800
So you can see that the
energy of-- these charges

986
00:49:56,800 --> 00:50:00,500
are still sitting in a voltage
that's lower than outside.

987
00:50:00,500 --> 00:50:03,370
So that open state has
a slightly lower energy.

988
00:50:03,370 --> 00:50:05,560
But you can see that
the closed state still

989
00:50:05,560 --> 00:50:08,620
has a much lower energy
than the open state, OK?

990
00:50:08,620 --> 00:50:10,780
And we can write down
that voltage difference

991
00:50:10,780 --> 00:50:17,840
as a gating charge times
this voltage difference.

992
00:50:20,480 --> 00:50:22,460
So now let's just take--

993
00:50:22,460 --> 00:50:25,220
here is an open state.

994
00:50:25,220 --> 00:50:28,850
It has an energy difference
of a little amount, w.

995
00:50:28,850 --> 00:50:32,180
Open state is lower than
closed by an amount w.

996
00:50:32,180 --> 00:50:38,850
When the voltage
inside the cell is low,

997
00:50:38,850 --> 00:50:42,300
we've decreased the
energy of the closed state

998
00:50:42,300 --> 00:50:46,620
by this amount-- gating charge
times membrane potential.

999
00:50:46,620 --> 00:50:50,760
And now we have an energy
difference in the open state

1000
00:50:50,760 --> 00:50:52,770
and the closed, the
energy difference

1001
00:50:52,770 --> 00:50:54,510
between the open
state and closed state

1002
00:50:54,510 --> 00:50:57,130
as a function of voltage.

1003
00:50:57,130 --> 00:51:00,490
We have a simple equation that
describes the energy difference

1004
00:51:00,490 --> 00:51:02,860
between the open and
closed state as a function

1005
00:51:02,860 --> 00:51:04,660
of the membrane potential.

1006
00:51:04,660 --> 00:51:09,190
And now we can just plug that
into the Boltzmann equation

1007
00:51:09,190 --> 00:51:14,560
and derive the probability
of being open and closed.

1008
00:51:14,560 --> 00:51:17,650
So we just plug that
delta u into here,

1009
00:51:17,650 --> 00:51:21,380
w minus gating
charge times voltage.

1010
00:51:21,380 --> 00:51:24,150
Now let's calculate the
probability of being open.

1011
00:51:24,150 --> 00:51:27,990
This gives us the ratio
of open to closed.

1012
00:51:27,990 --> 00:51:31,370
How do we calculate the
probability of being open?

1013
00:51:31,370 --> 00:51:35,020
Well, n is the
probability of being open.

1014
00:51:35,020 --> 00:51:37,560
That's just probability
of being open divided

1015
00:51:37,560 --> 00:51:39,790
by open plus closed.

1016
00:51:39,790 --> 00:51:41,400
What's the probability
of being open

1017
00:51:41,400 --> 00:51:43,110
plus the probability
of being closed?

1018
00:51:43,110 --> 00:51:46,980
Well, if it's in one or the
other, then the sum of those

1019
00:51:46,980 --> 00:51:49,650
has to be one, OK?

1020
00:51:52,370 --> 00:51:55,650
And now divide both
top and bottom by p0.

1021
00:51:55,650 --> 00:52:00,120
The probability of being open
it's just 1 over 1 plus p

1022
00:52:00,120 --> 00:52:03,600
closed over p open, which
is just the inverse of this.

1023
00:52:03,600 --> 00:52:04,670
And that's equal to that.

1024
00:52:07,590 --> 00:52:11,430
All right, that may have gone
by a little bit too fast.

1025
00:52:11,430 --> 00:52:14,740
And I wasn't very
smooth on that.

1026
00:52:14,740 --> 00:52:16,770
But you can see the idea, right?

1027
00:52:20,970 --> 00:52:23,430
It's estimating how
the energy difference

1028
00:52:23,430 --> 00:52:24,870
between the open
and closed state

1029
00:52:24,870 --> 00:52:27,210
depends on the
voltage of the cell,

1030
00:52:27,210 --> 00:52:29,250
and it's just an
energy difference.

1031
00:52:29,250 --> 00:52:33,770
So it has to be a charge
times a voltage, yeah?

1032
00:52:33,770 --> 00:52:37,750
And that's right [AUDIO OUT]
charge times a voltage.

1033
00:52:37,750 --> 00:52:40,060
And now we're just doing
a little bit of algebra

1034
00:52:40,060 --> 00:52:41,890
to extract the
probability of open

1035
00:52:41,890 --> 00:52:45,990
from open divided by closed.

1036
00:52:45,990 --> 00:52:50,730
And now if we just plug that
into there, we get this.

1037
00:52:50,730 --> 00:52:51,230
All right?

1038
00:52:51,230 --> 00:52:54,900
So now let's see how that
compares to the actual answer.

1039
00:52:54,900 --> 00:53:01,200
Probability of open is just 1
over 1 plus this exponential.

1040
00:53:01,200 --> 00:53:02,880
Here's what that
data looked like.

1041
00:53:02,880 --> 00:53:06,570
Remember, that was the
data for the conductance

1042
00:53:06,570 --> 00:53:07,635
as a function of voltage.

1043
00:53:11,050 --> 00:53:16,160
Here's a fit to a functional
form that looks like that.

1044
00:53:16,160 --> 00:53:19,774
And here is the
prediction from Boltzmann.

1045
00:53:22,450 --> 00:53:26,000
You can see that it
almost exactly fits.

1046
00:53:26,000 --> 00:53:29,650
And you can actually
extract, biophysically,

1047
00:53:29,650 --> 00:53:33,490
what the gating charge is
inside this tiny, little protein

1048
00:53:33,490 --> 00:53:37,060
simply by fitting
this to the data.

1049
00:53:40,270 --> 00:53:41,260
Pretty cool, right?

1050
00:53:41,260 --> 00:53:41,760
Yes?

1051
00:53:41,760 --> 00:53:44,495
AUDIENCE: What is w?

1052
00:53:44,495 --> 00:53:46,120
MICHALE FEE: It's
the energy difference

1053
00:53:46,120 --> 00:53:50,820
between the open and closed
state when the voltage is zero.

1054
00:53:50,820 --> 00:53:54,630
So you kind of have
to fit that, too.

1055
00:53:54,630 --> 00:53:59,590
If the voltage is zero,
it's the energy difference

1056
00:53:59,590 --> 00:54:04,120
between the open and closed
state when the voltage is zero.

1057
00:54:04,120 --> 00:54:08,500
And then you subtract from
that the energy of the gating

1058
00:54:08,500 --> 00:54:12,240
charge as a function of
voltage inside the cell, OK?

1059
00:54:15,120 --> 00:54:17,040
Yes

1060
00:54:17,040 --> 00:54:19,030
AUDIENCE: So is that
the [INAUDIBLE]..

1061
00:54:21,890 --> 00:54:24,590
MICHALE FEE: Yes,
each has the sensor,

1062
00:54:24,590 --> 00:54:28,820
and they all have to be open
for the ion channel to be open.

1063
00:54:28,820 --> 00:54:29,653
Yes?

1064
00:54:29,653 --> 00:54:32,840
AUDIENCE: Do you not need to
put it to the power of four?

1065
00:54:32,840 --> 00:54:34,840
MICHALE FEE: No, because
this is the probability

1066
00:54:34,840 --> 00:54:38,310
that one subunit is open.

1067
00:54:38,310 --> 00:54:39,350
But that's a good point.

1068
00:54:39,350 --> 00:54:40,950
If you want to
compare that to the--

1069
00:54:40,950 --> 00:54:41,940
so you're right.

1070
00:54:41,940 --> 00:54:43,920
If you want to compare
that to the conductance

1071
00:54:43,920 --> 00:54:45,600
of the whole
channel, then it has

1072
00:54:45,600 --> 00:54:49,050
to be raised to
the power of four.

1073
00:54:49,050 --> 00:54:51,270
And that's been
accounted for here.

1074
00:54:51,270 --> 00:54:51,930
Good question.

1075
00:54:55,970 --> 00:54:58,860
Any other questions?

1076
00:54:58,860 --> 00:55:00,560
Boltzmann men equation
is pretty cool.

1077
00:55:07,376 --> 00:55:12,800
If you know the mass
of a nitrogen molecule

1078
00:55:12,800 --> 00:55:14,900
and the acceleration
due to gravity,

1079
00:55:14,900 --> 00:55:18,140
what can you calculate with
the Boltzmann equation?

1080
00:55:18,140 --> 00:55:18,650
Any idea?

1081
00:55:22,060 --> 00:55:26,800
The mass of a nitrogen
molecule and the acceleration

1082
00:55:26,800 --> 00:55:28,568
due to gravity.

1083
00:55:28,568 --> 00:55:29,860
AUDIENCE: Pressure of nitrogen?

1084
00:55:29,860 --> 00:55:33,180
The partial pressure
of nitrogen?

1085
00:55:33,180 --> 00:55:35,060
MICHALE FEE: Close.

1086
00:55:35,060 --> 00:55:39,990
You can calculate the
height of the atmosphere.

1087
00:55:39,990 --> 00:55:42,173
You can do all kinds
of really cool stuff

1088
00:55:42,173 --> 00:55:43,340
with the Boltzmann equation.

1089
00:55:48,280 --> 00:55:51,605
OK, there was another
question here.

1090
00:55:51,605 --> 00:55:52,105
No?

1091
00:55:56,330 --> 00:55:58,760
So you can extract,
actually, these quantities--

1092
00:55:58,760 --> 00:56:03,440
the gating charge and this
energy difference in the zero

1093
00:56:03,440 --> 00:56:04,190
voltage state.

1094
00:56:06,950 --> 00:56:10,600
And the fit is very good.

1095
00:56:10,600 --> 00:56:14,980
OK, so that's
voltage dependence.

1096
00:56:14,980 --> 00:56:17,260
I highlighted
these slides that I

1097
00:56:17,260 --> 00:56:27,490
don't expect you to be
able to reproduce in blue,

1098
00:56:27,490 --> 00:56:31,180
just to make it more
clear for your review

1099
00:56:31,180 --> 00:56:34,270
what you have to focus on.

1100
00:56:34,270 --> 00:56:36,040
OK, let's look at
the time dependence.

1101
00:56:36,040 --> 00:56:38,090
The time dependence
is pretty simple.

1102
00:56:38,090 --> 00:56:40,600
It's going to just involve a
linear first order differential

1103
00:56:40,600 --> 00:56:41,120
equation.

1104
00:56:41,120 --> 00:56:45,930
You guys are all super
experts on that now, right?

1105
00:56:45,930 --> 00:56:49,960
So we have an ion channel--

1106
00:56:49,960 --> 00:56:53,540
sorry, a subunit that's
either open or closed, right?

1107
00:56:53,540 --> 00:56:55,850
We have an open
state, closed state.

1108
00:56:55,850 --> 00:56:59,420
What we're going to do is--

1109
00:56:59,420 --> 00:57:02,510
so the way to think about
this is the ion channel,

1110
00:57:02,510 --> 00:57:05,090
the subunit, if the
cell is polarized,

1111
00:57:05,090 --> 00:57:08,000
is sitting in the
closed state, right?

1112
00:57:08,000 --> 00:57:14,330
When you depolarize the neuron,
that changes the energy levels.

1113
00:57:14,330 --> 00:57:18,330
[AUDIO OUT] Which way was it?

1114
00:57:18,330 --> 00:57:19,610
I forget.

1115
00:57:19,610 --> 00:57:22,020
The closed state
has a lower energy.

1116
00:57:22,020 --> 00:57:25,040
Now, when you
depolarize the cell,

1117
00:57:25,040 --> 00:57:27,440
the closed state suddenly
has a much higher energy,

1118
00:57:27,440 --> 00:57:31,780
and it's close to
the open state.

1119
00:57:31,780 --> 00:57:35,680
And so, at some
point, that subunit

1120
00:57:35,680 --> 00:57:39,670
will jump over to the
open state, right?

1121
00:57:39,670 --> 00:57:40,930
But that takes time.

1122
00:57:40,930 --> 00:57:43,090
You change the energy
levels, but it takes time

1123
00:57:43,090 --> 00:57:47,830
for the system to jump
into the open state.

1124
00:57:47,830 --> 00:57:48,520
Why is that?

1125
00:57:51,980 --> 00:57:57,330
Because that transition is
caused by thermal fluctuations.

1126
00:57:57,330 --> 00:57:59,540
And so you have to wait for
one of those fluctuations

1127
00:57:59,540 --> 00:58:02,030
to kick you over
into the open state.

1128
00:58:04,950 --> 00:58:06,660
So we're going to
model those transitions

1129
00:58:06,660 --> 00:58:10,830
between open and closed states
with a simple rate equation

1130
00:58:10,830 --> 00:58:12,360
that's voltage dependent.

1131
00:58:12,360 --> 00:58:14,100
We have an open
state, and we imagine

1132
00:58:14,100 --> 00:58:18,230
that n is the probability
of being in the open state.

1133
00:58:18,230 --> 00:58:20,340
And we can equivalently
think of it

1134
00:58:20,340 --> 00:58:24,120
as if we have a
population of subunits.

1135
00:58:24,120 --> 00:58:28,080
And let's think of it
more as the fraction.

1136
00:58:28,080 --> 00:58:29,970
It's also equivalent to--

1137
00:58:29,970 --> 00:58:33,160
just whichever way you
want to think about it,

1138
00:58:33,160 --> 00:58:34,400
either works well.

1139
00:58:34,400 --> 00:58:36,960
But you can also think
of it as the probability

1140
00:58:36,960 --> 00:58:39,290
of being in the open
state, or the number

1141
00:58:39,290 --> 00:58:44,980
of subunits that are in the
open state in a population.

1142
00:58:44,980 --> 00:58:46,360
You also have a closed state.

1143
00:58:46,360 --> 00:58:47,920
So if n is the
probability of being

1144
00:58:47,920 --> 00:58:50,980
in the open state,
[AUDIO OUT] of a closed state

1145
00:58:50,980 --> 00:58:56,923
with probability
1 minus n, right?

1146
00:58:56,923 --> 00:58:58,340
If you're in the
closed state, you

1147
00:58:58,340 --> 00:59:02,750
have some transition rate,
probability per unit time,

1148
00:59:02,750 --> 00:59:05,570
of going from the closed
state to the open state.

1149
00:59:05,570 --> 00:59:07,070
And if you're in
the open state, you

1150
00:59:07,070 --> 00:59:10,380
have some probability
per unit time beta

1151
00:59:10,380 --> 00:59:11,900
of going into the closed state.

1152
00:59:16,490 --> 00:59:19,340
So those things have
units of per second,

1153
00:59:19,340 --> 00:59:20,840
probability per second.

1154
00:59:20,840 --> 00:59:21,880
Yes?

1155
00:59:21,880 --> 00:59:22,770
Rebecca, right?

1156
00:59:22,770 --> 00:59:23,467
AUDIENCE: Yeah.

1157
00:59:23,467 --> 00:59:25,255
What's the cause for
the fluctuations?

1158
00:59:25,255 --> 00:59:26,600
Just regular [INAUDIBLE]?

1159
00:59:26,600 --> 00:59:28,010
MICHALE FEE: Just warmth.

1160
00:59:36,030 --> 00:59:40,190
And these things are
voltage dependent, remember?

1161
00:59:40,190 --> 00:59:43,220
Those depend on the
energy difference

1162
00:59:43,220 --> 00:59:45,390
between that open
and closed state.

1163
00:59:49,550 --> 00:59:53,700
All right, let's develop our
first order linear equation.

1164
00:59:53,700 --> 00:59:55,710
It's going be very simple.

1165
00:59:55,710 --> 00:59:58,370
We have a closed
state, an open state.

1166
00:59:58,370 --> 01:00:01,010
The change in the
number of open states

1167
01:00:01,010 --> 01:00:04,400
is just going to be the number
of closed states, the number

1168
01:00:04,400 --> 01:00:08,060
of closed subunits that
open, minus the number

1169
01:00:08,060 --> 01:00:10,610
of open subunits that close.

1170
01:00:10,610 --> 01:00:13,740
That makes sense?

1171
01:00:13,740 --> 01:00:16,300
All right, that's simple enough.

1172
01:00:16,300 --> 01:00:21,230
The change in the number
open subunits per unit time

1173
01:00:21,230 --> 01:00:24,800
is going to be the
number of closed subunits

1174
01:00:24,800 --> 01:00:27,260
that there are times
the probability

1175
01:00:27,260 --> 01:00:31,540
that a closed subunit
opens per unit time--

1176
01:00:31,540 --> 01:00:33,960
that's alpha-- minus--

1177
01:00:33,960 --> 01:00:36,120
remember, the number
of closed subunits

1178
01:00:36,120 --> 01:00:39,120
that open is the number
of closed subunits

1179
01:00:39,120 --> 01:00:41,460
times the probability
per unit time

1180
01:00:41,460 --> 01:00:44,310
that a closed subunit
opens, all right?

1181
01:00:44,310 --> 01:00:46,245
And the number of open
subunits that close

1182
01:00:46,245 --> 01:00:50,340
is just the number of open
subunits times the probability

1183
01:00:50,340 --> 01:00:53,245
that any one of them
closes per unit time.

1184
01:00:53,245 --> 01:00:54,120
Does that make sense?

1185
01:00:56,700 --> 01:01:01,040
A lot of words, but the equation
ends up being very simple.

1186
01:01:01,040 --> 01:01:07,920
The change per unit
time of n is just

1187
01:01:07,920 --> 01:01:10,470
the number of close
subunits, one [AUDIO OUT],,

1188
01:01:10,470 --> 01:01:13,950
times the probability that
those open per unit time

1189
01:01:13,950 --> 01:01:18,270
alpha minus beta times n.

1190
01:01:18,270 --> 01:01:21,420
Alpha times 1 minus
n minus beta times n.

1191
01:01:28,080 --> 01:01:29,520
Any questions about that?

1192
01:01:29,520 --> 01:01:32,040
Alpha, beta are
voltage dependent.

1193
01:01:34,880 --> 01:01:38,150
So I've rewritten that equation.

1194
01:01:38,150 --> 01:01:41,480
n is the probability
that a subunit is open.

1195
01:01:45,720 --> 01:01:46,790
Let's just rewrite this.

1196
01:01:46,790 --> 01:01:47,690
Let's expand this.

1197
01:01:47,690 --> 01:01:51,160
Alpha minus alpha times
n minus beta times n.

1198
01:01:51,160 --> 01:01:52,700
Factor out the n.

1199
01:01:52,700 --> 01:02:00,720
So you have dn dt equals alpha
minus alpha plus beta times n.

1200
01:02:00,720 --> 01:02:06,060
Divide both sides by 1
over alpha plus beta.

1201
01:02:06,060 --> 01:02:08,460
What's the steady state
of this-- the steady state

1202
01:02:08,460 --> 01:02:09,780
solution of this equation?

1203
01:02:16,800 --> 01:02:19,230
That is the steady
state solution, right?

1204
01:02:19,230 --> 01:02:24,000
If you set dn dt equal to
zero, then n is equal to that.

1205
01:02:24,000 --> 01:02:25,885
[AUDIO OUT] just n infinity.

1206
01:02:28,560 --> 01:02:30,640
And what's that?

1207
01:02:30,640 --> 01:02:34,070
Alpha and beta have
units of per unit time.

1208
01:02:34,070 --> 01:02:37,270
So what is one-- what units do
1 over alpha plus beta have?

1209
01:02:40,680 --> 01:02:41,340
Time.

1210
01:02:41,340 --> 01:02:44,583
So what might that be?

1211
01:02:44,583 --> 01:02:45,940
AUDIENCE: Tau.

1212
01:02:45,940 --> 01:02:46,730
MICHALE FEE: Tau.

1213
01:02:46,730 --> 01:02:47,660
It's a time constant.

1214
01:02:50,370 --> 01:02:53,430
So, after all of this,
what we end up with

1215
01:02:53,430 --> 01:03:01,230
is an equation that looks
exactly like what we had for--

1216
01:03:01,230 --> 01:03:04,440
we have a first order linear
differential equation exactly

1217
01:03:04,440 --> 01:03:06,990
the same form as
the equation we used

1218
01:03:06,990 --> 01:03:10,260
to understand the way
the voltage changes

1219
01:03:10,260 --> 01:03:13,620
in a cell in response
to current injection.

1220
01:03:13,620 --> 01:03:18,220
So if we change n infinity,
what is this thing going to do?

1221
01:03:18,220 --> 01:03:20,410
What is n going to do?

1222
01:03:20,410 --> 01:03:26,950
It's going to relax [AUDIO OUT]
n infinity with a time

1223
01:03:26,950 --> 01:03:30,310
constant tau.

1224
01:03:30,310 --> 01:03:32,500
In all of these
things, the tau is

1225
01:03:32,500 --> 01:03:35,810
tau sub n, because it's
for the n gating variable.

1226
01:03:35,810 --> 01:03:37,670
So that's why this
has an n here.

1227
01:03:41,210 --> 01:03:46,380
So n infinity and tau
are voltage dependent,

1228
01:03:46,380 --> 01:03:49,740
because they come from
alpha and beta, which

1229
01:03:49,740 --> 01:03:51,218
are voltage dependent.

1230
01:03:54,030 --> 01:03:58,800
But we actually just derived the
steady state voltage dependence

1231
01:03:58,800 --> 01:04:02,178
of the potassium
conductance, right,

1232
01:04:02,178 --> 01:04:03,345
from the Boltzmann equation?

1233
01:04:09,160 --> 01:04:13,636
What is n infinity for
very negative voltages?

1234
01:04:13,636 --> 01:04:17,390
Do you remember it,
just approximately?

1235
01:04:17,390 --> 01:04:19,346
Big, small, [AUDIO OUT]?

1236
01:04:22,840 --> 01:04:24,320
What is the steady state?

1237
01:04:24,320 --> 01:04:28,060
What's the probability that
a potassium channel is open,

1238
01:04:28,060 --> 01:04:30,340
that a subunit is open,
at very negative voltages?

1239
01:04:30,340 --> 01:04:31,780
Do you remember?

1240
01:04:31,780 --> 01:04:32,380
Zero.

1241
01:04:32,380 --> 01:04:33,410
It's off.

1242
01:04:33,410 --> 01:04:37,250
For big voltage, n
infinity has to be--

1243
01:04:37,250 --> 01:04:42,210
if we think of it as a
probability, it's close to one.

1244
01:04:42,210 --> 01:04:46,550
So n infinity goes from
zero at negative voltages,

1245
01:04:46,550 --> 01:04:52,280
sigmoidal activation up
to one at high voltages.

1246
01:04:52,280 --> 01:04:58,550
OK, so now let's look at how n
changes as a function of time.

1247
01:04:58,550 --> 01:05:01,030
So here's our
[AUDIO OUT] potential.

1248
01:05:01,030 --> 01:05:02,980
We're going do a voltage
clamp experiment.

1249
01:05:02,980 --> 01:05:07,790
We're going to start at minus 80
millivolts and step up to zero.

1250
01:05:07,790 --> 01:05:11,380
So what is n
infinity going to do?

1251
01:05:11,380 --> 01:05:15,310
n infinity is just a
function of voltage, right?

1252
01:05:15,310 --> 01:05:17,200
It's like those energy levels.

1253
01:05:17,200 --> 01:05:19,510
They change immediately.

1254
01:05:19,510 --> 01:05:21,250
So what is n
infinity going to do?

1255
01:05:24,730 --> 01:05:25,230
Good.

1256
01:05:25,230 --> 01:05:30,690
It's going to start at close
to zero, jump up to one,

1257
01:05:30,690 --> 01:05:35,010
and then jump back to
close to zero immediately

1258
01:05:35,010 --> 01:05:37,890
following the voltage.

1259
01:05:37,890 --> 01:05:39,400
But what is n going to do?

1260
01:05:39,400 --> 01:05:41,152
Now let's plot n.

1261
01:05:41,152 --> 01:05:43,560
n is going to start at zero.

1262
01:05:46,340 --> 01:05:49,200
When you step the voltage
up, n infinity will jump up,

1263
01:05:49,200 --> 01:05:55,030
and n will relax exponentially
to a high n infinity

1264
01:05:55,030 --> 01:05:59,350
close to one, right?

1265
01:05:59,350 --> 01:06:02,010
And then when we
turn back, n infinity

1266
01:06:02,010 --> 01:06:07,620
jumps back down to zero, and
n relaxes exponentially back.

1267
01:06:07,620 --> 01:06:08,454
Yes?

1268
01:06:08,454 --> 01:06:10,350
AUDIENCE: On the n
[INAUDIBLE],, that's not

1269
01:06:10,350 --> 01:06:12,246
relaxing [INAUDIBLE] to one.

1270
01:06:12,246 --> 01:06:14,150
It's to n infinity,
right, at the top?

1271
01:06:14,150 --> 01:06:17,510
MICHALE FEE: Yes, but for
a voltage around zero,

1272
01:06:17,510 --> 01:06:19,668
n infinity is going
to be close to one .

1273
01:06:22,360 --> 01:06:23,630
Any questions?

1274
01:06:30,200 --> 01:06:34,600
That is called activation.

1275
01:06:34,600 --> 01:06:36,660
That's the activation.

1276
01:06:36,660 --> 01:06:38,560
Now, this looks a
little funny, right?

1277
01:06:38,560 --> 01:06:41,050
This thing is turning
on immediately.

1278
01:06:41,050 --> 01:06:43,180
It doesn't have that
nice sigmoidal shape

1279
01:06:43,180 --> 01:06:47,620
that the potassium current had,
or the potassium conductance.

1280
01:06:47,620 --> 01:06:48,653
Why is that?

1281
01:06:48,653 --> 01:06:49,820
What are we looking at here?

1282
01:06:49,820 --> 01:06:51,340
We're plotting n.

1283
01:06:51,340 --> 01:06:54,310
What is the potassium
conductance or the current?

1284
01:06:54,310 --> 01:06:57,610
How does that relate to n?

1285
01:06:57,610 --> 01:06:59,090
n to the fourth.

1286
01:06:59,090 --> 01:07:02,810
So what do we do to plot
the potassium current

1287
01:07:02,810 --> 01:07:04,940
or the potassium conductance?

1288
01:07:04,940 --> 01:07:07,820
We just take this
[AUDIO OUT],, right?

1289
01:07:07,820 --> 01:07:10,500
So what does that look like?

1290
01:07:10,500 --> 01:07:12,960
This process of turning off--

1291
01:07:12,960 --> 01:07:15,840
the gating variable getting
bigger is called activation.

1292
01:07:15,840 --> 01:07:18,870
The gating variable getting
smaller, n getting smaller,

1293
01:07:18,870 --> 01:07:22,180
is called disactivation.

1294
01:07:22,180 --> 01:07:25,450
So now let's plot
this to the fourth.

1295
01:07:25,450 --> 01:07:29,020
So this conductance
turns on gradually,

1296
01:07:29,020 --> 01:07:31,020
but the conductance
is proportional to n

1297
01:07:31,020 --> 01:07:31,780
to the fourth.

1298
01:07:31,780 --> 01:07:33,470
So let's plot that.

1299
01:07:33,470 --> 01:07:34,900
So if we plot n
to the fourth, you

1300
01:07:34,900 --> 01:07:40,030
can see that that function
now turns on smoothly in time.

1301
01:07:40,030 --> 01:07:42,000
This is time now, right?

1302
01:07:45,250 --> 01:07:49,780
So that gating variable
n relaxes exponentially,

1303
01:07:49,780 --> 01:07:52,960
but the conductance goes as the
gating variable to the fourth.

1304
01:07:52,960 --> 01:08:01,600
And so it has this nice,
graceful turn-on, right?

1305
01:08:01,600 --> 01:08:03,700
Because it's the
gating [AUDIO OUT]

1306
01:08:03,700 --> 01:08:08,950
exponential to the fourth
looks exactly like this.

1307
01:08:08,950 --> 01:08:13,740
In fact, that's how Hodgkin
and Huxley figured out

1308
01:08:13,740 --> 01:08:19,649
that it's n to the four, because
they knew that if they assume

1309
01:08:19,649 --> 01:08:22,410
that it's an exponentially
decaying gating

1310
01:08:22,410 --> 01:08:25,930
variable, that the only way
they could fit the turn on

1311
01:08:25,930 --> 01:08:29,340
of the conductance was by
raising it to the fourth power.

1312
01:08:29,340 --> 01:08:34,640
If they raised it to the second
power, it was still too sudden.

1313
01:08:34,640 --> 01:08:36,830
If they raised it
to the third power,

1314
01:08:36,830 --> 01:08:38,630
it was still not quite right.

1315
01:08:38,630 --> 01:08:40,819
But if they raised it to
the fifth power, oops,

1316
01:08:40,819 --> 01:08:41,816
it's too delayed.

1317
01:08:41,816 --> 01:08:43,399
If they raised it
to the fourth power,

1318
01:08:43,399 --> 01:08:51,160
it exactly fits the shape of
the conductance turning on.

1319
01:08:51,160 --> 01:08:53,120
And so they inferred--

1320
01:08:53,120 --> 01:08:56,040
they didn't know about subunits.

1321
01:08:56,040 --> 01:08:59,910
They just had a piece of
axon, a piece of squid lying

1322
01:08:59,910 --> 01:09:01,850
on a table in front of them.

1323
01:09:01,850 --> 01:09:04,109
And they were able to
figure out that there

1324
01:09:04,109 --> 01:09:07,319
were four independent processes
that turn on the potassium

1325
01:09:07,319 --> 01:09:08,040
conductance.

1326
01:09:11,330 --> 01:09:13,250
Pretty cool, right?

1327
01:09:13,250 --> 01:09:15,800
That's what you get by
doing things quantitatively.

1328
01:09:21,992 --> 01:09:23,450
So they could
[AUDIO OUT] the shape

1329
01:09:23,450 --> 01:09:25,189
of that potassium
conductance turning

1330
01:09:25,189 --> 01:09:27,890
on by this exponential
gating variable raised

1331
01:09:27,890 --> 01:09:28,790
to the fourth power.

1332
01:09:32,640 --> 01:09:34,420
And from that, they
were able to infer

1333
01:09:34,420 --> 01:09:40,700
that it's four independent first
order processes that combine

1334
01:09:40,700 --> 01:09:44,260
to produce that activation.

1335
01:09:44,260 --> 01:09:49,370
OK, the offset also fits if you
raise it to the fourth power.

1336
01:09:49,370 --> 01:09:52,580
They were able to measure
the size of the potassium

1337
01:09:52,580 --> 01:09:55,790
conductance to measure
n infinity directly.

1338
01:09:55,790 --> 01:09:58,340
So we derived it using
the Boltzmann equation,

1339
01:09:58,340 --> 01:09:59,900
but they measured
it directly just

1340
01:09:59,900 --> 01:10:03,770
by the size of the conductance.

1341
01:10:03,770 --> 01:10:06,482
And you can also
measure the time course.

1342
01:10:06,482 --> 01:10:07,940
You don't need to
worry about this.

1343
01:10:07,940 --> 01:10:10,880
I'm not expecting--
you should know this.

1344
01:10:10,880 --> 01:10:12,563
I'm just showing you
that just for fun.

1345
01:10:12,563 --> 01:10:13,730
You don't need to know that.

1346
01:10:20,160 --> 01:10:22,120
And you can extract
these tau's just

1347
01:10:22,120 --> 01:10:27,430
by measuring this exponential
decay at different voltages,

1348
01:10:27,430 --> 01:10:33,190
or measuring the inferred
first order process.

1349
01:10:33,190 --> 01:10:36,230
You can infer the time constant
of the first order process

1350
01:10:36,230 --> 01:10:41,100
on the onset and the offset
to extract these tau's

1351
01:10:41,100 --> 01:10:42,360
as a function of voltage.

1352
01:10:42,360 --> 01:10:44,145
This is tau as a
function of voltage.

1353
01:10:49,900 --> 01:10:53,530
From these two quantities,
you can actually

1354
01:10:53,530 --> 01:10:56,390
extract alpha and beta.

1355
01:10:56,390 --> 01:11:01,240
And so you can write down a
simple algebraic expression

1356
01:11:01,240 --> 01:11:02,630
for alpha and beta.

1357
01:11:02,630 --> 01:11:05,500
And that's the way they actually
wrote those things down.

1358
01:11:05,500 --> 01:11:07,930
They wrote them down as
alpha and beta, rather than n

1359
01:11:07,930 --> 01:11:11,270
infinity and tau.

1360
01:11:11,270 --> 01:11:14,080
Those are simple expressions
for alpha and beta

1361
01:11:14,080 --> 01:11:17,680
in units of per millisecond
as a function of voltage

1362
01:11:17,680 --> 01:11:21,010
in units of millivolts.

1363
01:11:21,010 --> 01:11:23,840
I think-- yeah, it's millivolts.

1364
01:11:23,840 --> 01:11:25,830
There it is right there.

1365
01:11:25,830 --> 01:11:30,090
So you can actually just
take those parameters

1366
01:11:30,090 --> 01:11:34,860
and calculate n
infinity and tau n

1367
01:11:34,860 --> 01:11:39,300
and calculate the gating
variable from that using

1368
01:11:39,300 --> 01:11:40,710
that differential equation.

1369
01:11:40,710 --> 01:11:43,020
So we have these
nice expressions

1370
01:11:43,020 --> 01:11:48,180
for what the steady state,
n infinity, and tau n are.

1371
01:11:52,080 --> 01:11:54,225
Now, why did we-- yes?

1372
01:11:54,225 --> 01:11:56,900
AUDIENCE: [INAUDIBLE]?

1373
01:11:56,900 --> 01:11:59,170
MICHALE FEE: Yes,
they're per unit time.

1374
01:11:59,170 --> 01:12:02,501
Yeah, so they have units
of per millisecond.

1375
01:12:06,430 --> 01:12:09,570
OK, so now let's come
back to our picture.

1376
01:12:09,570 --> 01:12:13,320
We have n infinity and tau
n as a function of voltage.

1377
01:12:13,320 --> 01:12:16,550
Now we just can plug those
into this differential equation

1378
01:12:16,550 --> 01:12:17,450
and solve for n.

1379
01:12:17,450 --> 01:12:19,820
Well, we already
know what that does.

1380
01:12:19,820 --> 01:12:22,400
n relaxes exponentially
toward n infinity

1381
01:12:22,400 --> 01:12:25,360
with a time constant tau.

1382
01:12:25,360 --> 01:12:29,650
But you can integrate
that numerically.

1383
01:12:29,650 --> 01:12:32,905
You get the potassium
conductance as n to the four,

1384
01:12:32,905 --> 01:12:34,780
g times n to the four.

1385
01:12:34,780 --> 01:12:38,140
You get the potassium
current as g n to the four

1386
01:12:38,140 --> 01:12:43,310
times the driving
potential, or V minus Ek.

1387
01:12:43,310 --> 01:12:49,000
And now let's come
back to our algorithm

1388
01:12:49,000 --> 01:12:53,000
for making an action potential.

1389
01:12:53,000 --> 01:12:55,870
So we have the parts related
to the potassium current.

1390
01:12:55,870 --> 01:12:58,120
We still have to add the
parts related to sodium,

1391
01:12:58,120 --> 01:13:00,230
but it's going to
look very similar.

1392
01:13:00,230 --> 01:13:01,370
So here's the idea.

1393
01:13:01,370 --> 01:13:04,420
We start with the membrane
potential at time step t.

1394
01:13:04,420 --> 01:13:07,890
We compute n infinity and tau n.

1395
01:13:07,890 --> 01:13:12,690
We integrate dn dt one time
step to get the next n.

1396
01:13:15,530 --> 01:13:18,650
Plug n into our equation to
get the potassium current.

1397
01:13:21,970 --> 01:13:24,670
We then add that to
all the other currents

1398
01:13:24,670 --> 01:13:26,950
to get the total
membrane current.

1399
01:13:26,950 --> 01:13:30,520
We compute V
infinity of the cell.

1400
01:13:30,520 --> 01:13:35,830
We integrate dv dt one time
step to get the next voltage.

1401
01:13:35,830 --> 01:13:39,100
And you plug that in and
calculate the next n infinity,

1402
01:13:39,100 --> 01:13:41,410
all right?

1403
01:13:41,410 --> 01:13:43,690
So we still have to
add the sodium parts,

1404
01:13:43,690 --> 01:13:45,850
but you can see we've
gone through all

1405
01:13:45,850 --> 01:13:49,390
of these steps
for the potassium.

1406
01:13:49,390 --> 01:13:53,110
And so we're just
this shy of having

1407
01:13:53,110 --> 01:13:55,930
a full-blown algorithm
for [AUDIO OUT] an action

1408
01:13:55,930 --> 01:13:58,120
potential in a neuron.

1409
01:13:58,120 --> 01:14:00,550
And not only do you understand
all the little steps,

1410
01:14:00,550 --> 01:14:03,190
but you understand the
fundamental biophysics

1411
01:14:03,190 --> 01:14:06,250
that leads to that voltage
and time dependence.

1412
01:14:10,100 --> 01:14:13,350
All right, so, again, what
I'd like you to be able to do

1413
01:14:13,350 --> 01:14:17,790
is to draw that circuit,
the Hodgkin-Huxley model.

1414
01:14:17,790 --> 01:14:21,630
I'd like you to be able to
explain, at a basic level,

1415
01:14:21,630 --> 01:14:25,065
what a voltage clamp
is and how it works.

1416
01:14:25,065 --> 01:14:27,690
I'd like you to be able to plot
the voltage and time dependence

1417
01:14:27,690 --> 01:14:29,070
of the potassium current--

1418
01:14:29,070 --> 01:14:31,560
remember, this
sigmoidal activation

1419
01:14:31,560 --> 01:14:34,830
of the potassium current--
and the conductance, voltage

1420
01:14:34,830 --> 01:14:37,170
and time dependence.

1421
01:14:37,170 --> 01:14:39,450
And be able to explain the
time and voltage dependence

1422
01:14:39,450 --> 01:14:41,310
of the potassium
conductance in terms

1423
01:14:41,310 --> 01:14:43,890
of the Hodgkin-Huxley
gating variables.

1424
01:14:43,890 --> 01:14:45,440
OK?