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The idea of shutting down Na-K pump is quite interesting, though I don't think it would be possible in practice. The electrochemical gradients of Na+ and K+ are so prevalent in our bodies that it is not quite possible to only shut down the Na-K pumps in certain neurons.  I think typical anesthetics usually works on the synapses because it simulates the way nervous system regulates itself. And also because synapses bind with neurotransmitters, the inhibition/activation can be localized and controlled more easily.