I have a question about action potentials in a neuron.
Do voltage-gated sodium and potassium channels use ATP? I mean when they are closed or when they want to open the gate, do they use ATP?
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No, they change conformation in response to voltage. Wikipedia has some general discussion of the gating process, as would any basic neuroscience textbook.
The primary energy (ATP) cost of neurotransmission is in establishing ion concentration gradients, via the sodium/potassium ATPase. You can think of this as a special case of secondary active transport, where the energy costs are in establishing a gradient that is then used for a later process which does not cost any direct energy.