0:00

[BLANK_AUDIO]

Â Hello, again.

Â We're now moving on from our series of lectures on the

Â cell cycle to another topic, which is mathematical models of action potentials.

Â And, we're going to start by talking about mathematical models of action potentials

Â that consist of a system of ODEs, and then we're going to use this

Â as a way of transitioning from ordinary differential equations models to what are

Â called partial differential equation models, with

Â the difference being that partial differential equations.

Â 0:57

If you remember in the cell cycle model of Novak and Tyson, we showed this plot here,

Â where when when a cell's undergoing these these

Â rapid cell divisions, cycling will go up and down.

Â It will oscillate as function of time.

Â Pre MPF and MPF will also go up as and down as function of time.

Â And when you have several variables that are inter, that are influencing one

Â another, seven variables in the case of the Novak and Tyson cell cycle model.

Â It can be very difficult to really, to develop the model, to

Â choose parameters for the model when

Â they're they're all influencing one another.

Â And one of the, the key simulations and, and

Â experiments that we, that we discussed with how, with

Â respect to how the Novak and Tyson cell cycle

Â model got developed was this one shown down here.

Â Where they used nondegradable cyclin.

Â 1:44

And they varied the amount of nondegradable cyclin systematically.

Â And, in the model, they predicted how MPF

Â activity would, would change the function of nondegradable cyclin.

Â That was in the system.

Â And they generated this prediction here of bistability

Â of MPF activity as a function of non-degradable cyclin.

Â And the conclusion that we've reached, in discussing

Â that particular simulation and the experiments that followed it.

Â Is that the simulation of a simplified experiment, where

Â it was non-degradable cyclin rad, rather than regular cyclin.

Â This simulation was critical both for developing the model and

Â for getting an, an understanding of how the system worked.

Â By using the non degradable cyclin, this was the only

Â way they were able to generate this prediction of, of bi-stability.

Â 2:31

We're going to encounter something really similar

Â when we talk about the action potential model.

Â The output that we're going to get from our action

Â potential model, this is the Hodgkin and Huxley neuronal

Â action potential, consists of four variables and they change

Â as a function of time, as we see here.

Â And we'll define what these variable mean in subsequent lectures.

Â But, the reason I showed this here is just to

Â illustrate that these four variables all have very complex time courses.

Â And because they influence one another, it's hard to get to

Â really get the control from just looking at these time courses.

Â What we're going to, the simulation results

Â and the experimental results that were going to

Â discuss consists of experiments that are like this where voltage you can see,

Â rather than having a complicated time course is going to be, is going

Â to jump to a particular level, it's going to be held at that level.

Â And then it's going to jump down.

Â This is done using a technique that

Â Hodgkin and Huxley developed called the voltage clamp.

Â And by clamping the voltage, by controlling it, they

Â were able to, they were able to isolate particular currents.

Â In this case, the, the, sodium current.

Â And, the argument that we're going to make.

Â The lesson to, that is going to be, that you should take home

Â from this, is similar to what we saw with the Novak and Tyson models.

Â That when you do the simulation of a simplified experiment, air voltage

Â is being controlled rather than, you know, having a complicated time course.

Â Simulating this simplified experiment is also, in the case of

Â the Hodgkin-Huxley model, critical for the model development and in understanding.

Â So, that's going to be a theme that we're going to, that we're going to touch on

Â that's very similar to what we already

Â encountered when we discussed the cell cycle.

Â 4:05

Before we get specifically to the development of the Hodgkin and

Â Huxley model, and before we get to the experiments that were done

Â using the voltage clamp technique, in part one, in our first

Â lecture on action potential models, we

Â have to discuss some biological background.

Â 4:21

And so what we're going to discuss in this first lecture is

Â some interesting nonlinear behavior that is observed in, in excitable cells.

Â Neurons being one example of, of an excitable cell.

Â But skeletal muscle is another excitable

Â cell, smooth muscle, cardiac muscle, beta cells

Â of the pancreas, there are all, all fall under the category of excitable cells.

Â 4:42

We have to define some term in order

Â to understand the biology and then the other topic

Â we have to discuss in this first lecture

Â is concepts of electro, electrochemical potential and driving force.

Â And by this we mean, specifically, driving force for ions.

Â To cross from one side, one side of

Â the membrane to the other side of the membrane.

Â 5:02

I've used the term action potential already, I haven't

Â discussed what that really means, I haven't defined it.

Â And I, alluded to some, interesting nonlinear behavior in the,

Â the neuron, so let's begin by talking about what this un,

Â un, what I mean by this unusual nonlinear behavior and that

Â will give us a chance to define the term action potential.

Â 5:24

A neuron, in this case a, a squid giant axon, if

Â no, no electrical stimulus is applied to it and you have

Â an electrode in the, in the neuron, you're measuring the voltage

Â from the inside of the neuron to the outside of the neuron.

Â When the neuron is at rest, the voltage will be constant.

Â And it'll be constant at a level of around

Â minus 60 millivolts is, are the units in this case.

Â 5:53

Now, if a single electrical stimulus is applied to this neuron, what will happen?

Â This is a single stimulus, here, called I, little i stem, big I stem rather.

Â I in this case is short for current.

Â If the current stimulus is applied to this neuron, what will happen to

Â the voltage is the voltage will go up, and it will come down.

Â It will go down to a level that's, of greater magnitude than minus 60.

Â It will go down to around minus 70 or so, and then it will come back to minus 60.

Â And this will all ha, happen within a few milliseconds.

Â This voltage wave formed here is what, what we define as action potential.

Â These are seen in, in all of the neurons

Â in your body in, in response to these electrical stimuli.

Â And this is very interesting nonlinear behavior, this

Â this is, happens after the stimulus is applied,

Â so this is autonomous to the, the neuron

Â and it's this characteristic a time course error.

Â 6:47

In contrast, if a sustained stimulus is applied, this is a stimulus

Â that's over and done with in, in, say the course of a millisecond.

Â If a stimulus current is applied for, for many milliseconds.

Â What you can see, in some neurons, are these repeated action potentials.

Â One, and then a second one, and then a second one, and then another one here.

Â [BLANK_AUDIO]

Â Another interesting and unusual nonlinear behavior that's observed

Â in the neuron is the sharp response threshold.

Â This is a paper from Hodgkin.

Â We're going to discuss Hodgin and Huxley.

Â 7:20

So the mathematical model there was published in 1952 but

Â they were working on this for a very long time.

Â And in 1938, Hodgkin published this paper that showed this sharp response threshold.

Â Showing that if a weak stimulus is applied, there

Â will be a, a small change in the voltage,

Â a little bit greater, a little bit greater, a

Â little bit greater causes a little bit bigger of change.

Â And then, as you apply the voltage a little bit

Â more, all of a sudden you see this, this threshold behavior.

Â You see this sharp response.

Â And if this threshold, behavior looks familiar to you, it should.

Â Because this is very similar to what we saw in,

Â in some of the other, biological systems that we encountered.

Â 8:00

Now all of this is, all this background is

Â just to convince you that this is unusual non-linear behavior.

Â It's not very trivial to describe this mathematically to say,

Â okay, how can you get a neuron that rests and under

Â normal conditions, without a stimulus, okay, and it can exhibit either

Â single stimulus or sustained stimulus, and it can exhibit a threshold.

Â So, that's what we want to address in these

Â lectures on action potential models and on the Hodgkin-Huxley

Â model, is how can we account for this

Â type of interesting, nonlinear behavior in a quantitative way?

Â 8:39

You're going to hear me use terms

Â such as depolarization, repolarization, and, and hyperpolarization.

Â And I think it's important to note that we use these in a, in a fairly loose way.

Â That might not be technically correct.

Â [NOISE] If you have a membrane that's at zero millivolts.

Â That doesn't have any voltage.

Â What I mean by that is, if you measure the

Â voltage on the inside, minus the voltage on the outside.

Â This would be an unpolarized membrane.

Â Because, you would have no voltage

Â difference between the inside and the outside.

Â 9:11

Now, what I showed on the previous slide, is on a resting

Â neuron you can measure a voltage of around minus 60 millivolts and

Â so, because there's a voltage difference between the inside and the outside,

Â this is a membrane that we would, we would call a polarized membrane.

Â 9:25

So, where the term depolarization comes from is when the voltage goes up,

Â when it goes from minus 60 millivolts

Â towards zero millivolts, you're losing this polarization.

Â It's moving from being polarized towards being unpolarized,

Â and so that's why we call it a depolarization.

Â It's because, it's losing polarization.

Â 9:53

But where the terms that we use

Â in, in physiology are, are technically not correct.

Â Is that what would happen if you went from minus

Â 60 millivolts to 0 millivolts, and then you kept going?

Â You went above 0 millivolts.

Â Then, you would go from depolarized to unpolarized,

Â and then you would polarize the membrane again.

Â 10:11

And then, if you went from zero millivolts up to some value, say plus 30

Â millivolts, and then you came down towards zero

Â millivolts, that would be depolarizing the membrane again.

Â And this would be really confusing, because here

Â depolarization means that the voltage is, is going up.

Â And here, depolarization means that the voltage is going down.

Â So, these terms would be technically correct.

Â 10:31

But what we actually use in, in physiology, when we're

Â discussing electrical behavior and, and neurons and myocytes, is whenever the

Â voltage is going up, we call that depolarization, and then

Â when the voltage is coming back down, we call that repolarization.

Â And again, if it becomes even more negative than the

Â resting level we started with, we can call this hyperpolarization.

Â So, these are the terms we're going to use.

Â Depolarization for going up.

Â And then repolarization, for coming back down.

Â Even if these terms are not, precisely correct.

Â [SOUND]

Â Now, we want to ask, where do these changes in voltage come from?

Â What causes the membrane to depolarize, or to repolarize, or to hyperpolarize.

Â And the answer is that voltage changes in,

Â voltages changes in excitable cells result from ion movements.

Â So that, if a cation, a positive

Â ion, flows in, that will depolarize the membrane.

Â But if a cation flows out, that will repolarize, or hyperpolarize the membrane.

Â So, if we look at an action potential in a, in a

Â neuron such as this, where it starts at minus 60, it goes

Â up, it comes down, it goes to a value that's even more

Â negative than minus 60, and then it comes back to minus 60 again.

Â When the voltage is going up, when it's

Â depolarizing, that's because the cation is going in

Â or an anion is, is going out, when

Â it depolarizes, that's because a cation is going

Â out or an anion is going in and then, when it comes back to minus 60

Â again, after the hyperpolarization, that's because a cation

Â is coming in or an anion is going out.

Â 12:07

And this leads us to two questions that we're going to address in these lectures.

Â One is, how can we describe

Â the depolarization or repolarization process quantitatively?

Â And then, the second question we want to address

Â is, which ions are the most likely candidates?

Â So, if I said the voltage is going up here

Â and this is because the cation is, is going in.

Â One, you can imagine several different

Â cations, potassium, magnesium, lithium, calcium, sodium.

Â So, we want to address which ions are the most

Â likely candidates for this phase, this phase or this phase.

Â 12:56

Capacitor's are something that many of you

Â may have encountered in, in say physics class.

Â And what you learned in freshman physics classes for instance.

Â That if you connect the battery to a com, a capacitor

Â is basically two parallel metal plates, that are separated by, by air.

Â If you collect, connect the batteries to these two parallel

Â plates, positive charges will come off the battery onto this plate.

Â Negative charges will go off the negative terminal

Â of the battery onto this bottom plate, here.

Â And the way that you describe voltage changes

Â on this capacitor are through this equation here.

Â Capacitance times the change in voltage with respect to time,

Â is equal to the ionic current that flows onto the capacitor.

Â And we if we think about cell membranes, the structure of

Â cell membranes, we can understand how this can behave as a capacitor.

Â 13:47

and the, the cell membrane is hydrophobic on the inside where the lipid groups are.

Â And that's hydrophilic on the outside, where the charged

Â parts are the, are the phospholipids are, are located.

Â And so, this basically behaves as two parallel plates, where the

Â charges can't easily move across from one side to the other.

Â But the charges kind of collect either on a inside face of a

Â cell membrane or on the outside face of the, of the cell membrane.

Â So, this does have this sort of parallel plate arrangement similar to what

Â you see in an idealized capacitor because the cell membrane is a bilayer.

Â And this, because the cell membrane behaves as a capacitor, this

Â is where we get our differential equation for, for membrane voltage.

Â The equation that we're going to have to voltage in this

Â case looks very similar to the equation we have up here.

Â Capacitance times the change in voltage with respect to time is equal,

Â in this case it's going to be the negative of the ionic current, but

Â that's just convention based on how we define one direction as negative and

Â another direction as positive, and we will discuss that as we go on.

Â 14:49

The first question we addressed is accounting for

Â changes in membrane potential or membrane voltage quantitatively.

Â The second question we said we wanted to

Â address is, which ions are the most likely candidates.

Â And to understand which ions are the most likely candidates,

Â we need to know what the ionic concentrations are in cells.

Â 15:07

In a mammalian ventricular myocyte, for instance, these

Â are typical of the equations you might observe.

Â On the outside of the cell you might see five millimolar potassium, 140 millimolar

Â sodium and two millimolar calcium, and on the inside you would see, much

Â greater concentration of potassium, a much lower

Â concentration of sodium and then, a much,

Â much, much lower concentration of calcium on the inside compared to on the outside.

Â This is typical of, of mammalian cells the

Â specific numbers come from a ventricular myocyte but, these

Â 15:57

In the model that we're going to discuss, of the squid giant

Â axon, that was derived and, and built by Hodgkin and Huxley.

Â We had, the concentrations are a little,

Â are different, although the pattern's the same.

Â Remember that squids live in the ocean, they live in salt water, so their

Â concentration, their osmolarity is, is much higher

Â on both the inside and the outside.

Â But the pattern is the same, where you see a

Â higher concentration of potassium on the inside than on the outside.

Â Here it's 400 and 20, rather than 140 and five, but much higher in

Â the inside than on the outside with

Â potassium, but the reverse is true with sodium.

Â Much higher sodium on the outside than on the inside.

Â 16:51

And these movement are going to

Â either depolarize or hyperpolarize the membrane respectively.

Â So, if sodium is going in, you have cations

Â going into the cell, that's going to depolarize the membrane.

Â And then if potassium comes out, that potassium coming

Â out is going to depolarize slash hyperpolarize the membrane.

Â 17:23

And so, let's consider what happens when we have a membrane that

Â might be permeable to one ion and not permeable to other ions.

Â In other words, some ions can cross the membrane readily and other ions can't.

Â So, this this is a thought experiment here which we call a concentration cell.

Â Imagine we have 100 millimol of potassium chloride in

Â this chamber here, which we're calling i for inside.

Â And we have 20 millimol of potassium chloride in

Â this chamber here, which we're calling o for outside.

Â So we can say, by diffusion, potassium is going to

Â want to move from this side to this side.

Â 17:57

Let's also imagine that these two chambers,

Â the inside and the outside chamber, are separated

Â by a membrane and the membrane is permeable

Â to potassium but is not permeable to chloride.

Â If the membrane were permeable to both, then

Â both potassium and chloride would, would cross the cell

Â membrane and you would end up with 60 on either side, 100 plus 20 divided by 2.

Â But, in this case, the membrane is

Â permeable to potassium and not permeable to chloride.

Â So, let's think about what happens if you started with 20 and 20 and then you

Â instantly raise this left hand side, the inside

Â chamber, from 20 to 100, what would occur?

Â 18:40

But you would have an excess of positive charge on

Â the right chamber, than you would on the left chamber.

Â So, excess positive ions on the, on the right would reduce the voltage

Â difference because, remember, this membrane is a,

Â is a capacitor, as we already discussed.

Â 18:55

And, because you have a buildup of positive charges on the, on the

Â right hand side here, you've developed

Â a voltage difference across the cell membrane.

Â And the voltage difference, having more positive charges on the

Â right, would oppose the left to right movement of potassium.

Â And eventually, you'd have an equilibrium.

Â You'd have an equilibrium where the voltage difference is

Â trying to push potassium ions from the right to the

Â left would balance the diffusion, which is trying to push

Â potassium ions from the, from the left to the right.

Â So, we can understand this just by going through a thought experiment like this.

Â Now the question becomes, how can we

Â understand this process in more quantitative terms?

Â In order to, to understand ion movement across cell membranes in

Â quantitative terms, we need to introduce

Â this concept called the electrochemical potential.

Â 19:58

Mu zero, in this case, is what

Â we called the standard electochem, electrochemical potential.

Â But we, we're interested in potential differences from the inside

Â membrane or the membrane to the outside of the membrane.

Â And mu zero is going to be the same on both sides of the membrane.

Â So, from now on, we're going to ignore this.

Â 20:16

R times T times natural log of C.

Â This term describes diffusion.

Â R and T, in this case, are the

Â gas constant is R, the absolute temperature is T.

Â And this is taking the natural log of C being the concentration.

Â 20:31

And this term describes diffusion.

Â In other words, a higher concentration

Â leads to a higher electrochemical potential.

Â If you have more concentration, then this term,

Â natural log of C, is going to get bigger.

Â Therefore, your electrochemical potential is going to get bigger.

Â 20:45

In the third term here, little z times F times V.

Â Little z refers to the valence of your ion.

Â So, that would be plus one for,

Â a monovalent cation, such as sodium or potassium.

Â F is a term called Faraday's constant.

Â And V, in this case, it the voltage.

Â This terms describes the electrical effects.

Â Greater voltage means you have a greater electrochemical,

Â electrochemical potential when you have a positively charged species.

Â For instance, z is greater than, zero.

Â 21:13

So.

Â What you can conclude from looking at this equation for electrochemical potential,

Â is that when the concentration gets

Â bigger, your, your potential's going to go up.

Â And then, when the voltage gets bigger, your potential

Â goes up, assuming that you have a positively charged, ion.

Â And the reason that we care about this is we

Â can understand how ions move

Â by calculating this electrochemical potential.

Â And it's analogous to what you have with a ball rolling down a hill.

Â 21:39

You probably, you may have learned about potential energy in,

Â in the context of things like, you know, lifting something up.

Â When you raise, when you, you know, take a ball,

Â take a ball or a rock and you life it up

Â to the top of the hill, you are increasing it's

Â potential energy, because then you can roll it down the hill.

Â Just like a ball rolling down the hill, an ion will want

Â to move from higher electrochemical potential

Â to lower electrochemical potential, just like

Â the potential energy that you you impart to a rock or to

Â a ball when you lift it up to the top of a hill.

Â 22:12

When we compute electrochemical potential, the units

Â in this case are in joules per mole.

Â It's a way of calculating energy, but it's energy normalized to how much is present.

Â Let's go back now to our example of the concentration cell, where we

Â had high potassium on the inside

Â compartment, low potassium on the outside compartment.

Â 22:31

We said that potassium would move from left to right,

Â due to diffusion, we'd get a build up of charge and

Â the electrical field, the voltage difference this created, would put,

Â would be acting to push potassium ions from right to left.

Â So, we can compute that at equilibrium the potential on the

Â inside compartment has to equal the potential on the outside compartment.

Â And therefore we can set R times T

Â times natural log of concentration on the inside,

Â plus zF times voltage on the inside equals

Â this term here, with inside replaced with, with outside.

Â 23:05

And remember, we've ignored the standard electrochemical potential, mu naught.

Â Because we decided that that was the same on both the inside and on the outside.

Â So, in this case, we can rearrange terms.

Â We can take the two voltage terms, and move them to one side of the equation.

Â And the two concentration terms, and move them to the other side of the equation.

Â And then we can compute voltage on the inside

Â minus voltage on the outside is equal to RT

Â divided by little z times F times natural log

Â of concentration on the outside over concentration on the inside.

Â This is a definition of the equilibrium potential,

Â or what is commonly called the Nernst potential.

Â What this means is that this is the voltage

Â that you can be at where the diffusion term pushing

Â the potassium from left to right exactly, exactly balances the

Â electrical term that's pushing ions from, from right to left.

Â So, when you're at this particular voltage, that's calculated

Â as follows, then you're at equilibrium, and because this was

Â this was originally def, derived by someone, a man

Â named Nernst, this is what we call the Nernst potential.

Â >> Now, if we go back to the squid giant axon, which

Â we said had electric, ionic concentrations like this, high potassium on the

Â inside, high sodium on the outside, each ion, both sodium and potassium,

Â are associated with their own Nernst potentials which we can compute like this.

Â E sub X equals RT over zF times the natural log of whatever the

Â species X is on the outside, over whatever the species X is on the inside.

Â 24:33

And if we plug in these numbers that

Â we have for potassium and, and sodium up here,

Â we can compute that, E sub K, or

Â the equilibrium voltage for potassium is minus 72 millivolts.

Â Th equilibrium voltage for sodium is plus 55 millivolts

Â 24:51

and what this means is that if you're exactly at

Â mi, minus 72 millivolts, potassium doesn't want to move in

Â or move out and if you're exactly at plus 55

Â millivolts, sodium doesn't want to move in or move out.

Â You can take that a step further and say that however far away you are

Â from minus 72 millivolts is however much potassium wants to move in or move out.

Â In other words, the distance away from the

Â reversal potential, if you take the current voltage minus

Â the reversal potential, E sub X, that is what we call the driving force for ion X.

Â What I mean by that in, in simple quali, qualitative terms is, if we went from

Â minus 73, from minus 72 to minus 73, how much would potassium want to move in?

Â Well, it might want to move in a little bit,

Â but not that much, because you're almost at equilibrium, right?

Â What if we went from minus 72 all the way to say, minus 172?

Â What if we went if we hyperpolarized the voltage by

Â 100 millivolts, well then we would be very far away

Â from equilibrium, and then potassium would want to move in

Â and it would want to move in quite a bit.

Â So, the way that we would specify that quantitatively is

Â we compute the voltage minus the Nernst potential, and that

Â tells us what our driving force is, how much does

Â this, this ion want to move in or move out.

Â And the answer is, it depends on how far away you are from equilibrium.

Â 26:08

What this means in, in, in physical terms,

Â is we're basically converting electrochemical potential from units

Â of joules per mole into units of volts,

Â and volts are defined as joules per, per coulomb.

Â So, we take our voltage difference, our, our driving force, voltage minus

Â Nernst potential, that's our dif, difference

Â in electrochemical potential divided by Faraday's constant.

Â And if you work out the, the units there, you'll see that that's

Â going to be units of joules per coulomb, rather than in joules per mole.

Â Finally, why do we go to all this trouble

Â of defining the Nernst potential, defining the driving force?

Â Well, the answer is, knowing the driving force is

Â going to help us to mathematically compute ionic currents.

Â We said that driving force is defined

Â as a voltage minus a particular Nersnt potential.

Â So, you have one driving force for potassium.

Â Another driving for for sodium.

Â Here, we're just using sodium as an example.

Â But the same considerations would apply, to potassium.

Â 27:35

Now, let's think about this in terms of units, we

Â have a voltage here and we want to compute the current.

Â The way we can compute a current from a voltage difference,

Â from voltage minus a Nernst potential, what the term you need in

Â order to get the units to work out is a conductance,

Â a conductance in this case is the reciprocal of, of a resistance.

Â 27:56

So, this is equation we're going to use for our, calculate our sodium current

Â in the Hodgkin-Huxley model, where the current

Â resulting from sodium, is a conductance for sodium,

Â g sub NA times the voltage minus the Nernst potential for sodium, V minus

Â E Na and conductance times voltage is going to give us currant, in this case.

Â 28:21

And so, this is just to, to review how the units are going to work out.

Â Usually voltage and Nernst potentials are given in millvolts in, in physiology.

Â Our currant, in this case, is in fact going

Â to be a currant density, which is going to be

Â in units of micro amp here is in per

Â centimeters squared and this is a way of normalizing it.

Â To the membrane area.

Â Conductance is, therefore, also going to have to be normalized.

Â This is going to be in units of millisiemens per centimeters squared.

Â The other thing we see here is, we said that if V minus ENa is less than zero.

Â Sodium is going to move into the cell.

Â In that case, if V minus ENa is less than

Â 0, than our current is also going to be less than 0.

Â And this a convention that we use in physiology.

Â By convention, inward current is negative.

Â 29:06

So, this seems like a really simple equation here.

Â If you know what your voltage you're at,

Â you know how that relates to the Nernst potential.

Â You can compute the current just by multiplying it by the conductance.

Â What we're going to see in the next lectures

Â on this topic is that this is simple

Â in principle, but the things makes it complicated

Â is your conductance, your g sub X, in

Â other words, your g sub Na or your g sub K can be dependent on both

Â voltage and time, and it's this voltage dependence

Â and this time dependence that makes this complicated.

Â And this voltage dependence and time dependence is also what gives

Â us the interesting non-linear behaviors that

Â we already illustrated at the beginning.

Â 29:41

Now, to summarize this first lecture on the,

Â action potential model development by Hodgkin and Huxley.

Â We see that neurons can exhibit complex non-linear behavior.

Â And this is complex non-linear behavior

Â that is very challenging to describe mathematically.

Â So, that's what we're going to go through in the next couple of lectures

Â is how can we develop a

Â mathematical representation of this complex non-linear behavior.

Â The second thing, thing we've seen is changes in the

Â membrane potential or voltage, we're going to use these terms interchangeably.

Â Membrane voltage and membrane potential.

Â These change can result from ion movements across the cell membrane.

Â [BLANK_AUDIO].

Â The final thing we've seen is how can we decide if

Â an ion is going to move in or it's going to move out?

Â Well, that is determined by electrochemical potentials,

Â electrochemical potentials which determine which direction ions move.

Â And we can compute a Nernst potential for each ion.

Â And the Nernst potential represents the equilibrium where

Â the ion doesn't want to move in or move out.

Â And it's when you get away from that Nernst potential

Â that the ion is going to move either in or out.

Â And these concepts that we've talked about in

Â this first section are going to be critical.

Â For understanding the Hodgkin Huxley model in, in more quantitative terms,

Â which is what we're going to do in the next couple of lectures.

Â