If the duration of two separate powers say "until the beginning of your next turn" is the active player allowed to choice which ends first?

Example:

Heat wave has a power

WALL OF FLAME: Heat Wave can use Barrier and Willpower. When Heat Wave makes an attack and misses, he can immediately use Barrier as a free action but must place one of the blocking terrain markers adjacent to an opposing character that he missed. When these blocking terrain markers are destroyed or removed from the map at the beginning of your next turn, any opposing characters adjacent to one or more of these squares are dealt 1 damage.

Barrier ends at the "beginning of your next turn" so before you can use outwit on the opposing character's defence power. But what happens if you used outwit last turn on their defence power? Outwit also lasts until the beginning of your next turn so outwit is ending at the same time that the barrier tokens are being removed.

You can not use Outwit before removing Barrier tokens.
So you would need to remove Barrier tokens before beginning the beginning of your turn, so to speak.
On the other hand, any Outwit from last turn would still be in effect when the Barrier tokens are removed as the beginning of the next turn has not yet begun.

Outwit and Barrier end at the same time though, they both state "until the beginning of your next turn" so both end before beginning the beginning of your turn. My question is can someone choose which effect ends first?

While he did answer something different than what you were asking, Quebbster answered that question, too.
Simultaneous events are resolved in any order determined by the active player.
The interesting twist with your scenario is that there is not actually an active player at that point in time. You are between turns. There is no active player.

What do the rules say, then?

It goes to the next player, the player who has to end his Barrier and Outwit. He can do so in any order.

Thanks for the clarification I know Quebbster pretty much answered it but just wanted to be 100% clear on what I was asking.