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

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Из курса от партнера Duke University

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

Hello. This is Roger Coke Barr for the

Bioelectricity course. We're in Week five and this is Lecture

number four. We're going to begin talking about some of

the features of a one-day cable. It's called one-day because the spatial

change all occurs along the x-axis. Let's just talk about the dimensions of

our cable. So, if we think about this cable along one

axis, we'll call that axis x. At different times, people sometimes call

X. Let's say that our fiber has only three important geometrical dimensions.

It has a radius, Call that a. It has the radius to an outer

structure that is to say, the limit of where currents can be conducted.

We'll call that b. And here, we are assuming that b is very

large indeed. There is a limit, but it's, it's a big

number. And then finally, we are on some occasions

concerned with the length of the fiber. So, if it goes from here to here, we'll

call that distance l. But more often, we subdivide our cable

into segments and where it's subdivided into segments, we'll think of each

segment, We subdivide into segments.

Let's say, each segment has a length delta x Along axis x, there are currents.

There's an intracellular current. I'll call that Ii..

And when the intracellular current is positive, that means it's going in the

same direction as x. There's an extracellular current.

Sometimes called the interstitial current. But here are es so that it is easily

distinguished from the number zero, e for extracellualar and when there is

extracellular current, it likewise falls in the direction of x.

It's important to realize that, that is a restriction because the extracellular

volume is very large. So, in general, currents outside the fiber

can flow on all sorts of pathways. Right now, we are forcing our

extracellular current to just flow in the x direction alone.

Then, there is membrane current. Membrane current flows across the

boundary. You're familiar with that, because we've

talked about it so much in various other connections.

When the membrane current is positive, we'll say that it's flowing from the

inside to the outside. One-day uniform cable model has resistance

that's specified in various fashions. So, we consider the bulk conductivity of

the interior is row i The bulk conductivity of the extracellular is row

E. These are the conductivities in ohm

centimeters. And we were talking way back there about

seawater and how it is in the range of 25 ohm centimeters.

That's what we're talking about here with row i and row e.

Often, the bulk resistivity, as it is called, is transformed into another form

called ri An ri is row i divided by the cross-sectional area.

So here, that would be row i divided by ia squared.

It's done that way because now ri can be converted to the resistance r The

intracellular side can be equal to ri times delta x That is to say, people

create this entity, ri, so that some of the calculation that otherwise would be

necessary, row i delta x over pi a squared.

This is already done in advance. A similar thing is done for row e uwing

the radius b, but often, then re is rather small.

This model of a fiber, where there's one dimension, we've called it x Currents

along the fiber, intracellular, extracellular, they bring current crossing

over. This is called the core conductor model.

It's a simple model, but it was not made up by somebody in kindergarten.

It is a model that was created or at least perfected by Lord Kelvin in England.

This is the Kelvin for whom the units degrees Kelvin is named.

Kelvin made use of the core conductor model because he was a member of the

scientific establishment in England. When at huge cost a cable is laid from one

side of the Atlantic to the other and the cable failed.

It did not work. So, a scientific committee or a review

committee was put together. Kelvin was a member of the committee to

figure out what had happened. And in the course of doing that work, the

work on the transatlantic cable, the mathematical model that was frequently

used came to be called the core conductor model because in that connection this.

This cylinder was a physical cable that went from England to the United States.

Of course, we have adapted that work into an entirely different context.

But the basic idea is much the same. In our case, we're thinking perhaps of a

nerve axon that is possibly cylindrical. It's within a salt solution that surrounds

it. So, although much smaller and not going

nearly as far, it is fundamentally similar in the currents that it carries to the

transatlantic cable that was studied by Calvin and thus, we use the core conductor

model here, just as he did there. Thank you for watching.

I'll see you at the next segment.

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