How does making the axon wider (larger diameter) affect the speed? Let\'s say th

Question

How does making the axon wider (larger diameter) affect the speed? Let's say the diameter of the axon is increased by some factor f g.7 Figure 6: Increased axon diameter without changing the membrane thickness. a) By what factor does Raxon change? (Note: the resistivity of the axoplasm remains constant, as does the 1 mm segment length. b) By what factor does Rmem change? (Note: the resistivity of the membrane remains constant, as does the 1 nm thickness of the membrane.) c) By what factor does the length constant A change? d) Assuming that the time constant doesn't change, by what factor does the speed therefore change?

Explanation / Answer

from the given data

diameter of axon = d

new diameter, d' = fd

a. Resistivity = rho

then old resistance R = rho*l*4/pi*d^2

new reisstance R' = rho*l*4/pi*f^2*d^2

hence

R' = R/f^2

b. resistivity = rho

old resistance = , R = rho*l*4/pi((d + 1nm)^2 - d^2)

new resistance , R' = rho*l*4/pi((fd + 1nm)^2 - f^2d^2)

R = rho*4*l/pi((1nm)^2 + 2d*1nm)

R' = rho*4*l/pi((1nm)^2 + 2fd*1nm)

R/R' = ((1nm)^2 + 2fd*1nm)/((1nm)^2 + 2d*1nm)

now d = 1000 nm ( approximately)

hence

R/R' = (1 + 2000f)/(1 + 2000) = 4.9975*10^-4 + 0.995*f

R' = R/(4.9975*10^-4 + 0.995f)

c. length constant, lambda2 = sqrt(R'2/R'1) = sqrt( R2*f^2/(4.9975*10^-4 + 0.995f)R1) = lambda1*f/sqrt(4.9975*10^-4 + 0.995f)

where lambda2 is new length constnat

lambda1 = old lenght constnat

d. assuming time constant does not change

speed = length constant/time

so, new speed = v'

old speed = v

hence

v'/v = lambda2/lanbda1 = f/sqrt(4.9975*10^-4 + 0.995f)

for 1000 nm diameter axon

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