This looks like it's going to be a "No, it's not possible, because . . ." answer, but hopefully it answers your question. Your proposal is that the Milky Way is a brain, and that the stars are like neurons, correct? I'm not a biologist or neuroscientist, but I do know one characteristic of neurons that helps them do their job: They rely on chemical and electrical impulses.
Neurons work by transmitting electrical signals along pathways called synapses. These signals can be either (nearly) continuous or in regular or irregular "pulses" - i.e. the strength of the signal goes up, down, or disappears. The point here, though, is that signals between neurons are generally not continuous. The gravitational force between two stars, on the other hand, is generally the same, barring a catastrophic event. In other words, the gravitational "signals" from stars don't function like those from neurons.
There are other differences between stars and neurons, too. One is that neurons cannot regrow in significant quantities, whereas stars are forming all the time. Generally in an organism, the neurons in a brain are of similar ages; this is clearly not the case for stars. Another difference is that neural networks are relatively fixed, whereas stars are constantly moving around.
I'll conclude by repeating what I mentioned in a comment, which is that the idea that the universe is a living organism has been around for some time. Another idea is that the universe is actually a giant computer (or just Earth, if you're a sci-fi fan). The problem is that while most people would disagree with these large-scale hypotheses, we may never know for sure if they are right or wrong.
I hope this helps.
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