I was doing some reading about anti-epileptic sodium channel blockers, then wondered how many sodium channels are actually conducting (actively passing ions) at any given time, that is, in an "average, awake" brain?
Unfortunately, there is no way to know how many sodium channels are active (or even present) in one person's brain due to the extremely high variability of channel and neuron density in a population and the lack of methodology around quantifying total number of healthy active synapses (1). Diseases like epilepsy can occur due various points of the sodium signaling process going wrong. It could be sodium channels are deformed and not accepting the proper signal, it could a low density of channel proteins, or it could be damage to the neuron themselves (2-4). No one has examined the "normal/healthy range" of sodium channels in the brain. When categorizing sodium channel deficiency scientists will often compare the number of sodium channels in a healthy brain vs. a disease brain using bioluminescence imaging or some other visualization technique. Bioluminescence specially quantifies the number of active sodium channels by binding causing healthy sodium channels light up and then quantifies density by the brightness of the synapse. This is not a direct quantification of the number of proteins, but I can not think of any technique that allows you to count the number of proteins on a cell without damaging the cell. I'm sorry this isn't technically an answer to your question, but hopefully it provides some helpful references to understand molecular brain protein imaging.
This is a difficult question because we are talking about properties of channels-molecules which can have stochastic opening and closing rates. The question is made even more difficult by the fact that there are multiple types of sodium channels. One can begin to quantify this by recording from many many cells in the awake animal using in-vivo electrophysiology by washing in sodium channel blockers and looking at sodium currents. Conversely you can start doing antibody staining for sodium channels in a brain area -- then take acute slices from that area and do single-channel recordings so you have some sense of number* probability of opening.