You're thinking of time dilation as being related to your "brain clock" or the way your brain recognizes time. It isn't. Time itself is relative, and it works at a level below body chemistry. If you're traveling at near light speed, time will appear to pass normally to you (and to your cells, and to a digital clock that you're carrying, and to anything else with you), but in fact external observers will appear to age faster. To them, you're aging slowly and they're aging normally.
The key word is time. The progression of time itself changes, not the time it takes for biological processes to happen. One second is still one second, and the same things happen to you in one second as they would otherwise, it's just that your second isn't equal to the outside world's second.