1). Analytically find out (by taking a derivative) the maximum speed as a function of alpha. Show your work. Plot the speed as a function of alpha for d_outside = {1, 10, and 100 µm} as you increase alpha from 0 to 1. Plot all three curves on the same piece of paper. Label the plot fully and correctly.

Background Information:
If you are asked to build a nerve fiber of a certain diameter, how much of the diameter should you devote to myelin to optimize action potential propagation speed? The formula for the speed of active propagation we saw in class was theta = sqrt(K*d/(4*Ri*Cm)). (You can assume Ri = 50 Ohm cm, Cm = 1uF/cm^2, and K=10.47/msec).

This formula is blind to whether there is myelin (except insofar as myelin affects Cm), but note that d in this formula refers to the diameter of the raw axon itself (i.e. the cytoplasmic tube, not including myelin). Now consider a nerve fiber with a total outside diameter d_outside =10µm. Let alpha be the fraction of d_outside that is taken up by the axon itself (i.e. d_axon = alpha * d_outside), and (1-alpha) be the fraction of d_outside that is taken up by myelin.

We are told by wise people that alpha should be about 60% in order to maximize speed of propagation for a given d_outside (until the total fiber diameter gets down to about 1um, after which myelin starts hurting rather than helping, which is why it is rare to find myelinated fibers below 1um).

Your job is to see if you can derive this 60% number (or something of that order of magnitude) from the speed formula in the notes. Assume that the plasma membrane is 20nm thick, and this also corresponds to the thickness of a single wrap within the myelin sheath. So if you have a 1 µm thickness of myelin, that would be 50 wraps.

This will increase the separation of the "conducting plates" 50 times compared to a bare axon (perhaps it should be 51 if you include the original membrane thickness, but you can ignore this), which would reduce membrane capacitance correspondingly. So you have a tradeoff between using the available diameter for the axon itself, which increases the d in the speed formula, or using the available diameter for myelin, which decreases the d, but also decreases Cm.