Wow, that's a tricky question alright. But on the plus side I have another TV reference that deals with a very similar thing in the form of a Star Trek episode! Even if strictly speaking it makes no sense because of the conservation of energy and all that.
It's tempting to simply postulate that half of those replacement silicon neurons would be redundant and therefore not actually contributing anything - it would certainly make answering the question much easier if that were true - but that would be intellectually lazy of me since I can't think of any particularly convincing argument why this should be the case. Not to mention that even if this were true, how could anyone determine which ones are redundant and so remove those?
What's particularly vexing about this example is that other identity thought experiments don't do much to help. For example, take the Ship of Theseus. If I replace one plank in the hull with two planks, then either they combined take up the same space as the gap left by the original, or one plank can be removed because it's redundant and not actually fulfilling any function. I don't think I've come across any other examples where one part could be replaced with two parts where both carry out the exact same function equally, and then split apart later to create two objects with the same properties as the original.
I think my answer to this at the moment would be that it depends on how the splitting process is achieved. If I replace every neuron in your brain individually with two chips, then I would suggest that you are still you. If I come along later and take half of these out whilst still maintaining the original form of the brain, you would still be you, and creating another brain out of these removed parts would be the same as creating a biological clone. Or alternatively, if I take your brain out completely and break it down into two piles and then make them into two copies of your brain, this would involve destroying the original consciousness and creating two new ones. The continuity of existence of an object is important here.
Thought-provoking stuff. I need a drink now.