The mind and the brain are closely connected, but they are not one and the same. The mind is to the brain as software is to hardware.
The hardware within the brain is being continuously replaced. Molecules enter neurons, are metabolised and the waste products leave. The actual material making up a neruron is of no great significance: only the pattern matters. If an identical pattern is constructed in some other place, then it would be the same person.
Consider a computer running a program. Clearly there are three levels of logical structure. There is the algorithm, the program that embodies it, and the operations of the hardware substrate.
Imagine that the algorithm is an artificial intelligence. Suppose that it is encoded into a program, and somewhere within that program two numbers must be multiplied. Clearly all the pertinent features of the running of the algorithm will be the same whatever routine is used for the multiplication. Substituting one such subroutine for another will make no difference to the conscious experience of the AI, no matter how different their methods for performing the multiplication. Similarly it will make no difference to the subjective experience if the program is executed on a different computer. All that matters is the data sets representing the mind of the AI and the high level operations performed on those data. So long as it embodies the high level operations, the actual low-level structure of the program and the configuration of the hardware are simply not relevant. No one would claim otherwise.
This means that self is a fuzzy concept. Many different actual physical configurations of matter making up the computer running the program correspond to the execution of the same high-level algorithm. This in turn implies that the subjective experience of the AI is independent of many details of the underlying computational susbtrate.
A similar argument applied to the mind.
The future of Ad Astra