Sure pieces of your brain are actually doing the mechanics of controlling your body and thus the bicycle... but you aren't thinking about it consciously (mostly).

Imagine posing a dummy walking down the street in stop-motion animation. You are very consciously controlling every aspect, every tiny movement is subject to your awareness and decision. Compared to walking down the street yourself. You don't think about balancing, placing each step, contracting and releasing dozens of muscles in sync... it just happens. Sure sometimes you make very explicit decisions on where to step, to avoid a puddle or dog shit or whatever, but most of the time all of that is way outside of conscious control, you just do it.