The mathematical relationship between height and nerve conduction velocity

Many studies have shown an inverse relationship between axon length (or height) and nerve conduction velocity. A linear relationship was assumed, but there is no physiologic indication the relationship is linear. Furthermore, a linear relationship between height and velocity leads to implausibly low velocities for very long nerves. We propose that power regression analysis would produce more accurate results, in line with physiology. In a power regression the goal is to determine exponent x that best fits the curve V = kLx where k is a constant and L is nerve length. In a previous study, the authors established that the product of conduction time T and energy E or TE = kL3. Mathematical derivation from this relationship yields V2/V1 = (L2/L1)(-0.5), or, velocity V is inversely proportional to the square root of length. Data from 22 normal Ulnar Motor Nerve Conductions showed a very high correlation with this formula (exponent x = -0.529 SE = 0.21, theoretical value -0.5). Data from other researchers also supports this relationship. Overall, Ulnar Motor Nerve Motor Conduction Velocity appears to be inversely proportional to the square root of height. We believe this relationship holds for all long nerves.