As I understand it, ions flow down their electrochemical potentials through ion channels during a neuron's action potential. Otherwise, ion pumps work to restore and maintain the resting membrane potential. What is the energy cost of a neuron's action potential? That is, how much work must the ion pumps perform to restore the resting membrane potential after an action potential occurs?
There's a few [1, 2, 3] sources that claim it's on the order of $10^8$ ATPs per action potential. The first paper (which is a review that cites the second paper) also has some equations for converting from ATPs to free energy, although that's going to be very context dependent.
If you can't access the third paper, someone has helpfully entered the relevant figures into the bionumbers site.