For sensitive persons, the combination of: a) Dry air at altitude, combined with b) limited nasal mucus production (for whatever reason), c) combined with ANY sort of inhaled irritant can lead to a somewhat exaggerated immune reaction in the nasal passages, resulting in brief but dramatic nosebleeds.
A bit of napkin or tissue stuffed under the upper lip so as to put some pressure on the surfaces there seems to stop the bleeding fairly quickly and reliably... at least it does for me.
As a child (with a standard thin-skinned blue-blood Anglo-Saxon MK 2 nose) in a high, dry mountain place, a month or so in Spring always seemed to bring many of these.. but seldom did they occur otherwise. Eventually I became acquainted with others so afflicted, with a similar pattern.
Higher humidity in the air system might be helpful on the 777. For the short term, a bit of salty mist in the nose does a lot. Avoid antihistamines, decongestants, and other meds that are vasoconstrictors when flying, if possible.. these tend to dry out the membranes and predispose one to low-humidity sensitivity.