I have a naive model of action potential energy use and I’m unsure where the model is wrong. Clearly the model is wrong because its conclusion is wrong:
When an action potential moves along an axon, it moves along the entire surface area (circumference) of the axon. Hence, in order to maintain the desired amount of voltage along the axon, to a first approximation, it requires an amount of energy proportional to the circumference of the axon, i.e. proportional to its diameter squared $d^2$. Since this applies for each spike, the energy per second $E$ is proportional to the spike rate $R$ and energy per spike which is proportional to $d^2$, so that $E\propto R\cdot d^2$.
In order to supply the sufficient amount of energy to allow for a certain spike rate over a longer timespan, there need to be an amount of mitochondria present in the axon proportional to the energy needs per second. Moreover, since the mitochondria per segment of the axon take space that is proportional to a first approximation to the circumference $d^2$ of the axon, we also have $E\propto d^2$.
However, clearly these equations are simultaneously possible only if $R$ is a constant: if $R$ rises above a certain level, the amount of energy needed to maintain an action potential explodes: we need more and more energy to maintain the action potential across the entire circumference of the axon, and we need more and more axon circumference to store all the mitochondria, and because these effects increase in the same amount, there can't ever be a mean firing rate above this threshold, and there is a unique energy-efficient firing rate.
What is wrong about this?