The second time I noticed a linkage is when, during a guided meditation by Sri Sri Ravishankar (Art of Living), guruji asks us to say OM. I noticed that even to say OM silently in my mind, I need to breath in deeply and during exhalation I can utter the OM. I am unable to silently uttter OM if I hold my breath.
I notice that during the "kumbhaks" (pauses between inhalations and exhalations), the mind is typically silent and free of thoughts.
During your pranayam practices, extend these "kumbhaks" pauses to get that inner peace.