Question about oxygen in 5 gal bucket

Hi everyone, I am looking to build a low-pressure aeroponic system in a 5 gallon bucket using a sprayer apparatus such as this one:
http://www.ebay.com/itm/NEW-Aeroponic-Hydroponic-Clone-Mister-Manifold-EZ-Cloning-Root-9-360-Sprayers-/171303136720?pt=US_Hydroponics&hash=item27e2766dd0

This leaves me with a couple of questions:

1. Do I also need to run an air pump with air stones to oxygenate the water?

2. If I don't run an air pump, how would fresh oxygen get to the roots inside the bucket? There would have to be some way to exchange the air, right?

What do you guys think?

the nutrient solution is oxygenated when its shot through the misters. no need for stones.

Thanks daedalux!

Should this type of mister run continuously or on an interval?

1 minute on, 5 minutes off