Which rings of this heterocyclic compound are aromatic?
I think that all the four rings are aromatic.

If the top nitrogen donates it's pair of electrons to the ring, and oxygen takes it's pair, the three fused rings would have 14 electrons in the rings' peripheral, causing it to be aromatic:



Even the 3D drawings of this molecule confirm it's planarity of the 3 fused rings:





I think we can at least suspect all the rings in the molecule to be aromatic.