What determines the mathematical relations between energy use, axon thickness, and firing rate of action potentials in a neuron? 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:

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*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?
 A: Your model simply shows that there is a maximum firing rate for neurons, which is indeed the case. After each pulse, there is a refractory period during which neurons are simply unable to fire again.
Your assumption about how energy is used however is not exactly how this works in reality. The energy is used to polarize the cells by moving ions across the membranes in a highly out of equilibrium state. It is actually reaching this "rest" state that takes energy. When the action potential propagates, some specialized channels open in the membrane that let ion concentrations across the cell membrane get back to electric neutrality (or even hyperpolarization). This does not require ATP. ATP is required to get the membrane back to a state where another action potential may be propagated. This requires a certain amount of energy and a certain amount of time as ions must be pumped across the cell membrane. During this repolarization, no action potential may be transmitted. This may seem like a small difference, but has some effects when successive pulses are closer to each other. See for example Yi, G., Grill, W.M. Average firing rate rather than temporal pattern determines metabolic cost of activity in thalamocortical relay neurons. Sci Rep 9, 6940 (2019). https://doi.org/10.1038/s41598-019-43460-8. There is also a metabolic cost to just maintaining the cell in a "ready" state, not just when the cell is firing.
However, as a first order approximation, your model indeed shows that there is a limiting factor to firing rate due to energy availability. I don't know if this is the "real" limiting factor to neuron firing in-vivo. For example, transmission between neuron in the synapses involves different mechanisms and these may introduce limitations, especially for continuous prolonged firing.
A: I wrote circumference and then I went on to think about cross sectional area instead. What a sloppy mixup. This basically seems to solve the model and give the correct conclusion: We get $E∝R⋅d$ and $E∝d^2$, which implies that $d^2∝R⋅d$, i.e. $R∝d$, which is confirmed by experiment.
