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Following Bryan Krause's advice I tried to be more specific and picked as one specific neural theory of consciousness Stephen Grossberg's Adaptive Resonance Theory in which a specific group of resonances play an important role:

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My question is: What characterizes these resonances (as neural processes) and why can they only occur in the cortex?

What distinguishes Grossberg's resonances from general forms of resonances that presumably may occur in every part of the brain?

What are (general and specific, necessary and sufficient) conditions for Grossberg's resonances to occur – in the language of systems of interacting units – and can it be shown that these conditions are not met in other parts of the brain than the cortex?

Note, that any definition of such resonances should not include as a defining criterion that they are found in the cortex. Otherwise it would not help to answer my initial question.

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