Excellent find! I figured it was some type of guitar synthesizer based on all the extra knobs, I just had no idea where to go from there. The closest thing I had found was the Gibson Les Paul Recording model.
On a semi-related note, that's actually the second guitar synthesizer Brad's used in a music video. The guitar in the Bleed it Out video is a 1970s "Guitorgan" guitar (a combination of a guitar and an organ). I had always assumed it was a Gibson ES-345 (the body style, control layout, hardware, and inlays are all consistent with a vintage ES-345), but he talks about the guitar in Part 2 of the making of Bleed it Out on LPUTV.