There is many things that could be causing your cat to do this, if your vet has said there is no physical reason for it, then here are your options, and this could be a long tough road (but not always).
1.) Find a qualified cat behaviorist to come out to your house and work with you guys and your cat.
2.) Talk to your vet about medications for this cat that can help modify his behavior, and in turn might also help his litterbox habits get back to normal.
3.) Keep him in a calm quiet area, that is warm, where he feels safe, has toys food/water, litterbox etc. Do not bug him, do not make it worse, spend time with him by sitting down on the floor semi near him and wait for him to make his moves. Let him come to you, don't go to him. [/list]