A lot of Mediterranean herbs do that. Especially thyme. Just take some sandy loam soil and mound it lightly over the center of the plant, leaving the leafy stems exposed. Keep slightly moist. In a few weeks all those stems will have rooted and you'll have a bunch of fresh new plants!

Thyme pattern baldness! LOL! Reba is spot on with her advice. Also, if you look closely at the central portions, you can tell which have died out completely and trim those out. That is often enough to stimulate new growth within the center. When I harvest thyme, it is usually the top growth. Cut back no more than 1/3 of the plant.