The common thought experiment goes something along the lines of this:
Replace a single neuron in the brain with an artificial one. Every time you go out drinking you lose plenty more cells, so clearly "you" are still "you". Now do another, and another, and another. At no point do you "break the pattern", but eventually you're left with a completely virtualized mind. Copying the pattern and destroying the original generally counts as "breaking the pattern", which is what you are describing.
Replace a single neuron in the brain with an artificial one. Every time you go out drinking you lose plenty more cells, so clearly "you" are still "you". Now do another, and another, and another. At no point do you "break the pattern", but eventually you're left with a completely virtualized mind. Copying the pattern and destroying the original generally counts as "breaking the pattern", which is what you are describing.
It's not really a new topic, but is still debated: http://en.wikipedia.org/wiki/Ship_of_Theseus