In my alternative universe a race called "Reflections" are the counterpart of humans. They are a reflection, hence the name, of actual humans on earth. Each human has a reflection in this alternate world. Essentially each human's reflection is somewhat similar to their guardian angel. It kind of keeps an eye out for them. Their brains are connected somehow, so that they are basically there the same person with the same personality etc. When the human dies, the reflection dies as well. I am trying to figure out how to link them through the brain. Is there like a certain aspect or physical part of the mind or brain that could potentially be linked to another life in an alternative universe?
I don't recommend debating this answer in a science forum or with Douglas Hofstadter but I feel it could be used to explain the fictional characters you describe.
Thoughts and consciousness don't seem to exist in 3 dimensional space with height, width and breadth. One could try and explain this by saying they exist outside of the physical universe as a "soul" that is peeking into our universe using the brain as an interface. This universe is parallel to ours and shares only time.
We also know of the subconscious which is having thoughts and processes outside of conscious attention. You could say this was an observer or "reflection" in the parallel universe. If you needed the reflection to have physicality then you could add another layer and say that our consciousness was a bridge to yet another parallel universe (this time physical) through the parallel one I first mentioned.
This is all wishy washy pseudo-scientific nonsense but possibly a good literary tool as its hard to immediately prove false for most readers.
I'm thinking the best way to connect the brains in this case would be if the brain was able to have something like a wormhole inside it so that thoughts could travel through the wormhole from one brain to another.
I'd consider playing games with chaos theory. The brain is a highly chaotic system, so its tremendously difficult to predict what it will do just a few seconds from now. You could invent a mechanic which connects the universes (any mechanic will do) in a way which allows the reflection to impart tiny changes inside those chaotic regions which could amplify into guardian angel-esque behaviors. You could violate the laws of physics if you were careful not to break them too badly, science would have a hard time observing the laws being broken due to the highly unpredictable nature of the brain. Or you could stick to the laws of physics wholesale, and actually properly handle the transfer of energy, momentum, etc. It'd be like those MUDS where you could go north, east, south, and west, only there's also a new direction "to refleciton" and back which is orthogonal to the other 3 dimensions.
Actually, managing such a system with the laws of physics intact could be very intruiging. Conservation of angular momentum could be quite interesting when dealing with a mirror image, where all chiralities are reversed. It might be possible to get into bad situations where your reflection couldn't help at all because you needed a little neural activity to go clockwise at just the right moment, but all the reflection could manage was counterclockwise.
Alternate dimensions denotes a 5d space (as classically the 4th dimension is time). this allows us to do the following:
Normal person: location, time, dimesnion = "normal earth equivalent"
Their Reflection: location, time, dimension = "the reflection"
as location and time are the same for both, both are synchronized.
now comes the harder bit, how "the reflection" affects "normal earth equivalent." This is probably what James was talking about in his comment. Though for completeness Chaos Theory as Cort Ammon has mentioned or maybe dark matter/energy would be an excellent starting point.
The brain, like all matter, is made up of all kinds of particles, which are in turn theoretically subject to quantum entanglement . This is the perfect science fiction device because the EPR paradox and Bell's inequalities (see links within the above article) demonstrate that something we have yet to explain can link particles instantaneously across infinite distances, but we really have no idea how.
So saying that your reflections' brains exhibit quantum entanglement with our own is a plausible explanation. In fact, properties such as particle spin are actually mirrored between two entangled particles, making their name appropriate.
using spooky quantum interaction the neurons could be entangled or depending on the degree of separation between the two dimensions it could be something funner like a fancy hat... By that means "intuition" could be explained; as there is no way to properly observe without affecting what you are observing. Using the apparent disappearance of intuition and a decreasing number of 'religious' experiences... one could hypothesize that the relationship between mirror dimensions are many to one instead on one to one... The existence apparently removing these warped mirrors might be an interesting character(s) as well as their apparent justifications; my favorite would be a 10 year old trying to get to sleep but it's too noisy?