Metabolic energy cost of action potential velocity

The action potential of the unmyelinated nerve is metabolically expensive. Using the energetic cost per unit length for the biophysically modeled action potential of the squid giant axon, we analyze this cost and identify one possible optimization. The energetic cost arising from an action potential is divided into three separate components: 1) the depolarization of the rising phase; 2) the hyperpolarization of the falling phase; and 3) the largest component, the overlapping of positive and negative currents, which has no electrical effect. Using both the Hodgkin-Huxley (HH) model and an improved version of the HH model (HHSFL), we investigate the variation of these three components as a function of easily evolvable parameters, axon diameter and ion channel densities. Assuming conduction velocity is well designed for each organism, the energy component associated with the rising phase attains a minimum near the biological values of the diameter and channel densities. This optimization is explained by the membrane capacitance per unit length. The functional capacitance is the sum of the intrinsic membrane capacitance and the gating capacitance associated with the sodium channel, and this capacitance minimizes at nearly the same values of diameter and channel density. Because capacitance is temperature independent and because this result is independent of the assumed velocity, the result generalizes to unmyelinated mammalian axons. That is, channel density is arguably an evolved property that goes hand-in-hand with the evolutionary stability of the sodium channel.