The nerve isn’t a single wire, it’s a series of connections that lead from the limb to the brain. If part of the chain ie removed with the limb, the remainder can still send signals to the brain. The brain has associated the signal from this chain as belonging to the end of the limb, so it acts as if the pain originated at the (missing) limb.
You bring a candle to your hand. You feel the heat and get burned.
When that happens, the neurons in your brain corresponding to various stimuli link together:
The sight of the candle coming near to your hand.
The increase in heat.
The actual burning sensation.
Each of those triggered different neurons in your brain to fire and when they fire together connections are established. The cool thing about these connections is, once created, one grouping can be used to activate another. So maybe next time you only see the candle coming toward your hand and the group of neurons associated with seeing that activates. That activation travels down those preestablished connections and activates the group of neurons associated with feeling the increase in heat (whether or not there is an increase of heat) and a burning sensation (whether or not there is actually a burning sensation).
So you can “feel” a burning sensation in an arm that isn’t there because the burning sensation was never created by your arm to begin with, it was created by your brain and that part of your brain can be activated through existing connections to other neurons firing.
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