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If you are asked to build a nerve fiber of a certain outer diameter d_out, what fraction alpha of the diameter should you devote to axon core vs.
If you are asked to build a nerve fiber of a certain outer diameter d_out, what fraction alpha of the diameter should you devote to axon core vs. myelin sheath to optimize action potential propagation speed (see the picture below)?
The formula for the speed of active propagation we saw in class was theta = sqrt(K/(ri*cm)),
where .47/msec. For this problem, you should assume ri is the value that would apply if the entire outer diameter were consumed by the axon, with no myelin. But you will want to re-express ri to capture its dependence on alpha. For cm, you will want to dig up the formula for a cylindrical capacitor, expressed in the usual units (F/length), but again, you will want to incorporate the appropriate dependence on alpha.
Once you have the speed formula expressed with the appropriate dependences on alpha, you will be in a position to calculate the value of alpha that maximizes speed. Remember, the hint that myelinated axons found in nature tend to have alphas around 60%. Look at the picture I grabbed from the web below (carefully chosen of course! ...it seems that most pictures actually show axons with smaller values of alpha…).
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