We can model some of the electrical properties of an unmyelinated axon as an ele

Question

We can model some of the electrical properties of an unmyelinated axon as an electric cable covered with defective insulation so that current leaks out of the axon to the surrounding fluid. We assume the axon consists of a cylindrical membrane filled with conducting fluid. A current of ions can travel along the axon in this fluid and can also leak out through the membrane. The inner radius of the cylinder is 5.0 mu m; the membrane thickness is 8.0 nm. If the resistivity of the axon fluid is 2.0 Ohm middot m, calculate the resistance of a 1.0-cm length of axon to current flow along its length. If the resistivity of the porous membrane is 2.5 times 10^7 Ohm middot m, calculate the resistance of the wall of a 1.0-cm length of axon to current flow across the membrane. Find the length of axon for which the two resistances are equal. This length is a rough measure of the distance a signal can travel without amplification.

Explanation / Answer

for finding the resistance we use the formula

R = pl/a

where p stands for resistivity, l for length and a for area of cross section R for resistance

a ) for axon fluid p = 2 ohm-meter

l = 1 cm = 0.01 m   a = pi * r^2 = 3.14 * ( 5 *10^-6)^2 = 78.53 * 10^-12 m^2

R = 2 *0.01/(78.53*10^-12) = 2.55 * 10^8 ohm

b) similarly for porous membrane

R = (2.5 * 10^7) * 0.01 / (3.14 * (8 * 10^-9)^2 = 1.2434 * 10^21 ohms

c) for axon fluid to make its resistance equal to that of porous membrane the length has to be

l = 1.2434 * 10^21 * ( 3.14 * (5 * 10^-6)^2)/ 2 = 48828125000 m

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