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My question is based on the following excerpt from an article I read recently:

Since the early neurological work of Karl Lashley and Wilder Penfield in the 1950s and 1960s, it has become clear that long-term memories are not stored in just one part of the brain, but are widely distributed throughout the cortex. After consolidation, long-term memories are stored throughout the brain as groups of neurons that are primed to fire together in the same pattern that created the original experience, and each component of a memory is stored in the brain area that initiated it (e.g. groups of neurons in the visual cortex store a sight, neurons in the amygdala store the associated emotion, etc). Indeed, it seems that they may even be encoded redundantly, several times, in various parts of the cortex, so that, if one engram (or memory trace) is wiped out, there are duplicates, or alternative pathways, elsewhere, through which the memory may still be retrieved.

Is it, therefore, theoretically possible to create a machine that perfectly maps out the way in which neurons fire in Patient A regarding a certain memory, and then to stimulate an identical firing of neurons in Patient B, so as to allow them to live that memory or even believe it to be theirs?

By extension, could this machine map out the neuron firing order that occurs while someone studies mathematics or physics, and then to replicate such firing in another person so as to impart that knowledge upon them? Or is there some additional element that arises when living through the experience yourself that cannot be reproduced in such a binary fashion as "firing of neutrons". If so, what does that say about the passage quoted above?

a machine that perfectly maps out the way in which neurons fire in Patient A regarding a certain memory, and then to stimulate an identical firing of neurons in Patient B, so as to allow them to live that memory or even believe it to be theirs?

This can only work if the wiring of the neurons in our brain is standardized and homogeneous like the circuitry in a high end smartphone. Sadly, that's not the case.

It is known that from the moment we are born any experience we have remodel the neurons and their connection: those more used are kept, those unused are discarded. This means that no two persons have the same neurons and connections. Thus, at most, firing pattern(A) into B would result in noise, like opening an encrypted file without unencrypting it.

No, each person brain is different and the map from one will not produce the same results in another.

Due to plasticity and how each brain is individually trained to your sensory organs and experiences each memory in each brain is unique. Each brain a has a unique map of connections called a Connectome. A perfect copy of someone else's memory would require identical brains otherwise the linkages will not match up with the rest of the brain. Your map for concepts is not identical to mine, so the connections will not be to the same concepts. Your spliced inot connections will produce nonsense. Research into artificial senses show the same problem, their solution is an extended process by which the brain and the interface learn each others connections, basically the same way you learn to see with your eyes as a baby.

additional source

To "replicate such firing" is, as noted, impossible in 2 different brains. They are just too different. However, consider that right now, you are reading these words and my thoughts are becoming your thoughts. We have both spent years learning English in order that these formations of black and white pixels fire the correct neurons. This learning process standardized our brains. Not on a neuron basis, but on a concept basis.

If the machine can see these concepts, and create and connect new neurons to hold these concepts, to do "direct neural learning" the machine would need to know how both brains encode concepts and translate from one to the other. So if you wanted to learn physics from Einstein, you'd also need anything tangentially related in his brain: basic math, calculus, German, patent office forms. The machine could match his basic math to your basic math, concept by concept, neuron by neuron, and so on, and create only additional the neurons it needs to encode new concepts. (Or just overwrite vast sections of your brain! You'd know physics, but might be confused as to why you are not in Princeton in 1955.)

Computers do this all the time with emulators, cross compilers, Just In Time compilers, boot loaders, and whatever WINE is (WINE Is Not an Emulator), but computer memory is easily mapped and insignificant by comparison.