I believe the crystals Miriam collects are the original summoning catalysts for the demons - in her world, the Alchemist Guild sacrificed hundreds (I may have this number off) of children to grow the crystals then used to summon thousands of demons. The demons are bound to the world using those crystals, and so Miriam is channeling their powers through that connection.
Heh. Miriam's tattoos are actually crystals growing out of her skin. I assume Gebel also has that shit growing out of him under his clothing.