The brain (I'm theorizing now, I have no background in neural chemistry) has a similar set of basic gates that can be organized into a Turing machine, and the gate I described previously is one of them.

No.
You can represent logic gates using neural circuits, and use them to describe arbitrary finite-state automata that generalize into Turing-complete automata in the limit of infinite size (or by adding an infinite external memory), but that's not how the brain is organized, and it would be difficult to have any learning in a system constucted in this way.