testosterone should affect certain abilities of the brain. not like you don't have the ability without t, i don't think that is possible, it's probably more like starting to use those parts of your brain constructively. all hormones affect your brain in some way, i think.
there are also receptors in the brain for sex hormones. the number of receptors is made to match a certain hormone level, and if you have too much or too little of one hormone in your body, that will be bad for your mental health. causes some instability. i think finding the right hormone levels, and balance between, as that matters too, will help the brain function better. at least it doesn't have to stress out over those gonads not functioning right.
there has been made research about the effect on the brain, but only on mice. rising the level makes them display more male pattern behavior, also in females and individuals that have been tweaked into the intersex range by scientists. part of that would be territoriality.