Don't think so.
I have hardly any serious understanding in the workings of the brain, but analogies of computer networks to neural ones [and it's well known that analogies are reliable sources, and meant to be taken at face value (is like->is equal to)] make me assume that no single brain cell takes the role center, but rather, your consciousness is the sum, or average, or any kind of function of all the cells.
Certain agglomerations may be weighted [just think about how much role the stomach-acid regulation center has in your consciousness] , but the removal of a single cell does'nt result in loss of consciousness. [as evidenced by their dying every single...just lets make up some kind of rate here]
I see no reason why the results of addition of a single artificial neuron [which is assumed to be functionally identical to your originals] would be different.
Now, given that, cell-by-cell replacement should'nt result in loss of consciousness. This reminds me of some ark replaced part-by-part, in the end having none of the original. I will go with 'it is still the same ship'.
Obviously, if we took the flag, and built a ship around it, we would'nt come to the same conclusions. Thusly, there must be some amount, that if replaced at once, is considered to be a different entity, and therefore a rate at which it's parts can be replaced.
If we copied it, we would'nt call it the same: so we can't be merely the layout, the hardware of our brain. Therefore, we must be the process. [how did it follow? by the scientific process of 'wild mass guessing'...sans mass]
Thus, we can't interrupt the process. Though, we may some invidual child processes, and with the passage of time, they will be reinitiated by the parent...which, per se, does'nt exist:it is the sum of all child processes, and the layout is not assumed to be hierarchical.
Then, we can go on to the next process. I see some parallels with reinstalling on a server, for which off time is too big a risk.
_________________________________
I don't know what I am talking about