to lower one's head―bow one's head―(謙遜してなら)―humble oneself
気が狂う.
うなずく.
首を振る.
to make a great effort of the mind in order to understand or find the answer to a difficult question
to make repeated bows with one's head
to hold up one's head―(頭角を現すの意味なら)―make a figure―cut a figure―cut a conspicuous figure―make a prominent figure.