I heard about this in a My Little Pony fanfic: replace a single neuron with a computer-controlled, interoperable equivalent. You feel totally normal. Replace some adjacent neurons. You still feel normal. Keep replacing neurons. Only the neurons at the edge have to interact with physical neurons. The ones in the middle are fully emulated. At no point have you lost consciousness. Eventually, you'll have replaced yr entire brain & you don't feel any different.
Your neurons have been being replaced the whole time. There's only one of you at the end. You can decide that's a copy of you instead of being you. But then you have to decide at what point it stopped being you. That's the real fun of this thought experiment. Replacing one neuron doesn't make you a different person, right? That happens all the time. Does replacing two neurons make it a different person? How many neurons do you have to replace before you're not you?