# 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:

• 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.

• Do you mean "proportional to cross-sectional area $A \propto d^2$"? Circumference and diameter are linearly related, $C\propto d$.
– rob
Jun 8 '20 at 5:11
• @rob, oh, yes, 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 $E\propto R\cdot d$ and $E\propto d^2$, which implies that $d^2\propto R\cdot d$, i.e. $R\propto d$, which is confirmed by experiment. Do you agree? Do you want to write it as an answer so I can give you the bounty? Jun 9 '20 at 4:50
• I don't think I understand the problem well enough to actually answer the question, and I'm not lusting after the bounty. We encourage people to self-answer if they have solved their own problem, so that the question is most useful for people who find it by searching later.
– rob
Jun 9 '20 at 5:29
• @rob, ok thanks. I'll wait for someone to give a reliable answer rather than answering myself. Jun 9 '20 at 5:37

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.