For the type of effort you're describing, I think it would be easier to coordinate breathing with pedal strokes than breaths per minute.
At a good, hard effort that I can sustain for a long period of time (such as a long TT), I find that my breathing automatically falls on every other pedal stroke. For example, begin inhale on left led stroke and finish inhale during right leg stroke. Begin exhale on left led stroke and finish exhale on left right leg stroke.
For a really hard effort, such as intervals or hill climbing, I find that it's inhale left stroke, exhale right stroke. Basically doubling the breathing rate.
As for the "panting". That would be an effort where you are anaerobic and you're breathing is completely ragged. No pattern other than "as fast as possible".