Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Bioelectricity: A Quantitative Approach

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Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Axial and Membrane Current in the Core-Conductor Model

This week we will examine axial and transmembrane currents within and around the tissue structure: including how these currents are determined by transmembrane voltages from site to site within the tissue, at each moment. The learning objectives for this week are: (1) Select the characteristics that distinguish core-conductor from other models; (2) Identify the differences between axial and trans-membrane currents; (3) Given a list of trans-membrane potentials, decide where axial andtrans-menbrane currents can be found; (4) Compute axial currents in multiple fiber segments from trans-membrane potentials and fiber parameters; (5) Compute membrane currents at multiple sites from trans-mebrane potentials.

- Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics

Biomedical Engineering, Pediatrics

So hello again. This is Roger Coke Barr for the

bioelectricity course, we're in week five and this is lecture segment ten.

In this segment we've been in this whole week, we've been talking about laying the

track and now I am going to do something unusual and attack the work that we just

dug and tell you why it really does not make sense.

Let's look at it in more detail. So in our preceding sections That's, those

sections pretended to show, how, I will say that membrane currents can be

determined directly from VM values without computing the on occurrence.

Now, we said in an earlier slide, something like IM is equal to Vm1 minus

two VM2 plus VM3. And this was divided by some stuff down

there in the denominator. That involved the resistance.

And we claimed that you could be, you could actually find a value of membrane

current in that fashion. So now lemme tell you why one should be

skeptical of this claim And as a matter of history, I can tell you I have worked with

many people, especially students, who did not feel that this was the right way to do

it. And they were smart, thoughtful people.

And what they said was the following. They said, look here, the way that we do

this is based on the fundamental definition that we've used over and over.

If we want to find the membrane current. We could find membrane current as the

capacity current plus the ionic current. So therefore, what we should do is we

should get the ionic current according to the Hodgkin-Huxley model or some other

membrane model. We should add it to the capacity current.

And then we will have the membrane current.

It's as obvious as 1,2,3. And at the same time it's equally obvious

that you must find the membrane current by first finding the ionic current, because

only by first finding the ionic current can you do this addition.

Well that seems pretty, pretty sound argument when you listen to it that way.

And if you accept the argument as being true you would say whatever all this stuff

is on the left, this ain't really right. I mean, really, think about it.

But now let's look at it the other way around.

More from the point of view of this week, rather than the week that talked about

Hodgkin and Huxley. If we accept.

If we accept the definition that we have, that means we also accept the equation

that turns these around. The capacity of current is the membrane

current minus the ionic current. There is a little bit of culture shock

here. Because each of us has become accustomed,

in recent years, especially. Of reading equations not as mathematical

equations. But as equations written into a computer

program. So if you accept the top equation as an

equation written into a computer program. You accept this equation as a directive as

to how to do a computation. And the directive is, first, you start

with the quantities on the right. And then you compute the quantity on the

left. However, if you just accept the equation

as an equation, in the mathematical sense, then of course, if you'd accepted the top

one, you've accepted this one too. And the bottom one turns out to be the one

that we would rather use. So let's work with it in a mathematical

sense, if you do that, then you can do the following thing, you can get the audit

current from the state variables. Just as the argument as been made that one

should do. You can get the membrane current.

From the neighboring cells, as well as the home cell.

And you can put that here as the second term.

And then the consequence is that you can compute, you can actually find the value

for the [inaudible] current. Which, it turns out, is the hardest one to

find. Because there's no other way to get the

capacity current, except through use of the equation on the left in one of its

forms. So, while we started out skeptical, we see

that what is. Been described in this weeks work with the

track, is not only correct, it's correct and essential, because we have to use it

as a way to find the capacity of current and thereby to find dvmdt or any of the

other things that we want to compute from time to time.

Let's look at this more informally in yet another way.

If we start with the equation, I C equals I M minus I ion.

It is as if the following train analogy was true.

Suppose some robbers stop the train. They stop the train and they demand a

ransom of $100,000. In that case the membrane finds that I am

as forced upon it by its neighbors. Or in this case the train finds, that

these robbers have stopped the train and demanded a payment of $100,000 dollars to

let it go forward. Now, of the total IM, some of the current

flows as ionic current. So, we've got to have the membrane current

because that is what is demanded by our neighbors.

We can, we can compensate and provide that to some degree by the ionic current.

Therefore, we subtract it from the total that's required.

But if the train company and the train engineers cannot come up with the whole

payment of $100000, they just don't have that much in the mail car say, which is

what's always robbed in the movies. Then the robbers will demand the remainder

from the rest of the train, from the passengers on the train and they'll have

to get out their purses and empty their pockets and come up with the rest of it.

And in a certain analogous way that's what happens in the equation.

If the total in the membrane current If the membrane current total.

This total. If that membrane current total cannot be

fulfilled by the ionic current. The money and the postal car.

Then the difference has to be made up by the passengers on the train, from the

money in their pockets and in their purses.

So the capacitive current is that residual, and as a result, the passengers

on the train become richer or poorer as a result of that robbery.

And in the case of the membrane, the charge on the capacitance becomes greater

or becomes less as a result of that difference.

Now I'll make a big deal of this and talk about it as much as I have for a whole ten

minutes because it is really a confusing point.

And it is worth taking ten minutes and thinking about it a while in your mind.

So when it comes up again in some other context you can say to yourself yes it

really is true. We begin with IM, we don't end with it.

It is the demand made on the membrane by it's neighbors that then the membrane has

to fulfill through the ionic current and through the capacity current for any

residual. So with this thought and within a week of

working on the track I show you this picture of the dormitory sidewalk, this is

in front of one of the dorms at Duke university and I put this picture up here

because of the sidewalk. The track, if you will.

Down hooch hundreds of students flow each day, as they go back and forth to their

dormitory ramps. If you looked at this building, this set

of items in the picture and said what's most important?

Well, of course it's all important. And the buildings are gonna wail up the

train. And the sidewalk is in no way like the

track. And both are essential.

This system is a whole. What run without having both of those

elements present. Thank you for watching.

We'll move on to the next segment.

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