What about Extra Sensory Perception? At the risk of sounding like a devotee of a cheesy, worn-out sci-fi TV show that begs not to be taken seriously, I would assert that ESP is an example (but not the best example) of a behavior that is independent of the brain. In the generous sense we "agreed on" in class, ESP is a behavior--a behavior even more uncontrollable than subconscious thought, than dreams, than having to draw breath to stay alive. Assuming that it exists (granted I cannot prove that it does or doesn't), it just happens, unpredictably, regardless of the fact that the person who is perceiving is miles away from the source that is sensed, both physically and mentally. It is seemingly independent of both external and internal stimulae; it is just a response. It is awareness of the entrance of a body of knowledge that has traveled at infinite speeds across space into a person's consciousness, and it is pretty hard to explain away.
Fair question/concern. As you say, ESP may or may not exist, so whether we have to account for it itself is a bit up in the air (some of our discussion of sensory systems later in the course will at least help to define the problem of the existence of ESP). Clearly, though, some people (at least) have the experience of ESP, and so we would need at least to be able to account for the experience. Let me know at the end of the course if we don't make it that far. PG