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As a preface, let me assume that the entirety of mental states and their corresponding behaviors in a person are entirely reducible to the physical activity of neurons sending a variety of chemicals across synaptic clefts at appointed times. There's no extra-logical, metaphysical sense of being that guides our actions e.g. spirits, soul stuff, etc.

Beyond this provision, which is obvious, is the overarching question of how particular networks in the brain control various structural and behavioral faculties in the mind. For example, if we place a person in an MRI and ask them to count to three, we expect to find activation in a portion of the parietal lobe. Just how fine-grained is this activation? If we ask the same person to do the same counting at a different time, will a different network in the same area of the brain be used?

The first question has a corollary to it; a kind of role reversal: given a sufficiently detailed picture of a person's brain, is it possible to more or less describe what the person is doing or thinking?

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I gave it another attempt. @ArtemKaznatcheev guilted me into doing it! –  user523 Apr 29 '12 at 18:17

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If we stick them in an MRI, the activation will not be very fine-grained. There are roughly 630,000 neurons in a 3x3x3 mm voxel recorded during fMRI. However even this coarse detail is sufficient to perform many impressive mind-reading tasks (e.g. Mitchell et al., 2008)

But what if we were able to record every single neuron in the brain simultaneously, could we then find a unique 1-to-1 mapping between neuronal firing and behavior? Not really. Consider what happens when you count to three-- and let's ignore everything going on in your brain that's not contributing towards this task (e.g. autonomic functions, mind wandering, etc.) Do your tongue, jaw, and lips move in exactly the same way every time? Does the acoustic signal sound the same every time? No; there is some variability in neuronal firing that causes behavioral variability.

OK, well what if your muscles did move in exactly the same pattern. Then, would the neuronal signature be identical between trials? Not necessarily. Let's assume there are 100 billion neurons in the human brain. We can't observe that many degrees of freedom in our behavior. It's likely that if we removed a single neuron, our behavior might remain 'identical'. There's a massive amount of dimension reduction to go from an array from 100 billion neurons to human behavior. But overall, the answer to your question depends on how you characterize 'performance'. If you define performance to be sufficiently complex such that performance is equivalent to the pattern of neuronal firing, then the question becomes a tautology.

The answer to the reverse question is slightly different. It is likely that if we observe the exact same pattern of neuronal firing twice, then the behavior will be exactly the same. However, even then, there is the possibility of noise in the peripheral nervous system once signals leave our brain that leads to some variability.

In a broader sense, cognitive neuroscience is aimed at answering this exact question: How does the activity of the brain correspond with our cognition and behavior? I like to think we've made some decent progress in this area, but we're still nowhere near a complete answer to this question.

Mitchell, T. M., Shinkareva, S. V., Carlson, A., Chang, K. M., Malave, V. L., Mason, R. A., & Just, M. A. (2008). Predicting human brain activity associated with the meanings of nouns. Science, 320(5880), 1191-1195. PDF

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