Ponds over the winter

I was just thinking about one or two things & was wondering what happens to
the oxygen levels in the pond over the winter if all the
filtration/circulation is switched off. I know the fish go very quiet & hang
out down the bottom but surely the water down there gets pretty oxygen
depleted? As I understand it water is actually at its most dense at about 4
degrees C (39F) so in the height of winter its quite likely that the pond is
actually warmer at the bottom than the top. If this is so then its obviously
not a good idea to circulate the water out the bottom of the pond in mid
winter (assuming filters use a bottom drain etc) which means the filters
really do need to be turned off....

What's the general concensus - how do the fish survive in an oxygen depleted
environment?

I.