The heart is the vitality+endurance, and yes, determine if boxer go down.
The fact that goes back up during the game isn't random, but based on what CPU does in the break, and other factors.
you beat opponent heart (which is endurance) down to 10. During break the CPU choose the strategy to heal the boxer (like the one you can choose too), so that adds a multiplier to the boxer endurance regain (vitality).
What can happen, is that a boxer has endurance 10, but vitality 90, so he gets a huge regain of the endurance, thus the "heart" value goes up again.
Same things happens when he gets knocked down: if manages to get up, regain some endurance. is hard to explain by word, but after trying it, I though that would be cool if a boxer goes back up, filled with "rage", increasing his heart value again, ready to counterattack.
Of course sometime this doesn't happen, like in real boxe, some boxers just go down and if they get up again are a wreck