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.

From the lesson

Electricity in Solutions

This week's theme focuses on the foundations of bioelectricity including electricity in solutions. The learning objectives for this week are: (1) Explain the conflict between Galvani and Volta; (2) Interpret the polarity of Vm in terms of voltages inside as compared to outside cells; (3) Interpret the polarity of Im in terms of current flow into or out of a cell.; (4) Determine the energy in Joules of an ordinary battery, given its specifications; (5) State the “big 5” electrical field variables (potentials, field, force, current, sources) and be able to compute potentials from sources (the basis of extracellular bioelectric measurements such as the electrocardiogram) or find sources from potentials.

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

Biomedical Engineering, Pediatrics

Hello again.

This is Roger Cook Barr for the bioelectricity course.

This is section one, subsection seven.

You'll recall that the name of our course is that,

it is Bioelectricity, A Quantitative Approach.

The core conductor model is an historic.

Quantitative model of how bioelectric membrane, and in particular,

bioelectric nerves work was first started in the 1800's and

has been used since then for many problems within bioelectricity,

but also more broadly, such as the Transatlantic cable.

The core conductor model has a cylindrical geometry.

The thought is their cylindrical geometry is

sufficient to separate an inside volume from an outside volume.

That is, it's a model that's, geometrically very simple, but

retains the most critical features of real structures.

If you think of it as a uniform cylindrical surface of radius H.

You may, you might also think of it as having a long and

here undefined overall length, but certainly much longer than the radius.

In this particular illustration have identified a diameter 2h.

And over here on the right hand side, set the diam,

the radius should be, is 5 micrometers,

and the length between these green segments is 100 micrometers.

The green segments are not actually partitions there,

they're simply mathematical references put in in order to

have some reference points along the axis of the cylinder.

The letters A through F that are here present on the slide.

So here's the letter B for example.

You simply identify different points that are interest,

of interest in this particular sequence of slides.

They don't have any long term interest, and they're not a part of the core

conductor model itself, they're just some points here that I've identified.

It is a characteristic of the core conductor model that one

assumes cylindrical symmetry.

So, I've drawn this dotted line going around the nerve on the outside,

sort of symbolically there with the dotted line,

to indicate whatever happens around the axis.

Is the same at every point around the axis, all the voltages are the same, or

anything else is the same that has no differentiation is

made between one point or another as you go around on the outside.

The most famous use of the core conductor model was for

the analysis of the second Transatlantic telephone cable.

That analysis was done by Lord Kelvin, associated with temperature that's in

Kelvin, and it was, done after the failure of the first Transatlantic cable.

However, it's been used most frequently in bioelectric context such as this one.

In the Core Conductor Model, one thinks of the interior as having a resistivity

Row I, here, greek letter row,

subscript I, or row E.

For the volume external to the fiber.

Here, we are thinking of the conductivity or

resistivity as being uniform in the interior and

uniform in the exterior but a different value between those two regions.

Sometimes, the characteristics are specified in terms of resistivity, so

here we've specified Row E 25 ohm centimeters

as the exterior resistivity, and if you do the inverse of Row E.

You find the conductivity, which is in

this case one twenty-fifth or .04, seamans per centimeter.

A similar thing is done for the conductivity's inside,

here I've chose 50 ohm centimeters inside, or

conductivity of one over fifty .02 seamans per centimeter.

These values are not to far from the resistivity of ocean water

which is in the range of you know round about 25 ohm centimeters.

Now, that we have the values for the resistivity so I know the geometry.

It's possible to compute the axial resistance from one place to another and

know how resistive this nerve is.

A lot of times, people think of nerves as being similar to wires,

let's just look and see what the actual resistance is of this nerve model,

see how that might relate to a wire.

The formula that's used is well known, it says the resistant,

or the resistance, the actual resistance between two places in ohms

is the resistivity times the length divided by the cross-sectional area.

Cross-sectional area is the area.

Through which a current might flow, that is to say, there's, there's this, this,

green area, a cross sectional area, so if we, spell

out that in terms of a formula, it's, rowi times L divided by Pi times H squared.

Repeated the formula over here on the left and put in numbers here on the right.

You'll notice that in so far as units are concerned, we have centimeters,

centimeters in the numerator, centimeters squared in the denominator.

So that the result comes out in ohms, as we would expect that it would do.

Now, if one carries out the computation,

the numerical computation, we find that the resistance between these two.

[SOUND] Cylindrical cross-sectional disk.

These two disk blocking off the cylinders is 636,620 ohms approximately,

has to be approximation because you have pi in there.

So, 636,620 ohms in a, enter our parameters that

are reasonable for this particular nerve example.

And if we compared that to the axial resistance of a copper or silver wire we'd

find for the same dimensions that the wire is less than one ohm.

So nerves are not good wires.

Or saying that a little bit differently, nerves and

wires are not very much like each other.

They're just different and they function differently, even though sort of

at a very first glance, they might have some geometrical similarities.

I hope on this calculation that you will repeat it and

see if you get the same thing, I frequently make numerical mistakes, so

if you get a different number, please post that in the forum so

that we can get it corrected for the whole class.

Similarly, I hope you will look up the resistance of copper or

silver wire, you can do it in a few minutes with Google.

And, see what you think the resistance is for

a wire of the same length and diameter.

So, that concludes this section, our first section with a core conductor model,

there's a lot more that can be done.

We'll see some more in the next section.

Thank you.

Talk to you again, soon

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