Then they apply electric current and study the connectivity.
Nope, that's no longer possible on such a brain. What you do is you inject special tracer substances (while the mouse is still alive). These substances will stain neurons at the site of injection and then cross the synapse to connected cells, either in the direction of the information flow or opposed to it, depending on the system used. These tracers are then imaged using the method that this article is about. To further aid you, you can do different stainings to see what type/subtype of neurons you are looking at. By combining this information with known functional properties of the found neuronal types, you can try to infer what is actually happening in the area you examine.