"Say we continue this process as neurons die by replacing each neuron with an artificial one."
Your premise is completely loony. You can't replace a neuron, which is living matter, with inanimate matter.
Each neuron is a cell that uses biochemical reactions to receive, process and transmit information.
If you understand, even on a base level, how a neuron works. You can conceptually think that we could build an electronic circuit that mimics a neuron. We're not there yet...note I said yet.
Just the simple fact that the brain works with what are essentially electric charges (energy cannot be created or destroyed [1st law of Thermodynamics] but it can change forms) means that a circuit could be created to mimic it's effects.
If I said to you 50 years ago that computers would be the size of a briefcase and easily accessable to the majority of the population you would have thought I was crazy. Think of everything we've known in the past. First we knew the Earth was flat. I think this gets my point across.
It's a THOUGHT EXPERIMENT. YMMV.