tart of the current initiative's next turn.
There should be nothing that triggers off the "turning of the round".

There is nothing in 4e RAW -- per round abilities reset at the start of a particular combatant's turn if they are per-round.

As a toy example why, someone whose initiative is last in a round can become first in the next round without changing anything in the in-world fiction. But with that rule, they become able to apply a new iteration of Frost.

Adding mechanics like that is just icky.

(The closest in RAW 4e I know of are the monsters who act on fixed initiative values)