Thank you for the additional information. It helps.
It sounds like he is dealing with a combination of the effects of his disorder and a psychological issue.
Depression can be caused by neurological damage. Your husband's doctor is right to be concerned about his mental state. Your husband needs to be seen for an evaluation of his mental state related to his disorder. Since he refuses treatment however, there is little that can be done to help him. Unless he expresses a desire to hurt himself or someone else, he cannot be forced into treatment. But you can talk with his doctor, not for information about your husband (unless he signed a consent) but to at least get his opinion on how much of your husband's problem is as a result of the disorder and how much is behavioral.
If your husband has left your family and continues to have this affair against your request to stop, then it sounds like he has no intention at this point to try and repair the damage he has done to your marriage. Continuing to remove his personal belongings from your home says that he is continuing with his behavior, for now. However, the fact that he keeps in contact with you and your boys is a positive sign. He has not let go of the relationship he has with you and the children. Whether or not he will be able to work out his feelings and come back to the family is a matter of time.
Since your husband will not respond to your requests to repair the marriage, you are left to respond to it yourself. Start by going to counseling yourself (your children may benefit as well from counseling). You need the support right now and someone who can help you decide how to handle this situation.
Also, decide if you want to continue trying to work on your marriage or you want to move on. Here are some resources to help you:
Should I Stay or Should I Go?: A Guide to Sorting Out Whether Your Relationship Can--and should--be Saved by Lundy Bancroft and Judith Patrissi
My Husband's Affair Became the Best Thing That Ever Happened to Me by Anne Bercht
Should You Stay or Should You Go? Compelling Questions and Insights to Help You Make that Difficult Relationship Decision by Susie Collins and Otto Collins
How to Know If It's Time to Go: A 10-Step Reality Test for Your Marriage by Dr. Lawrence Birnbach and Dr. Beverly Hyman
You can find these books on Amazon.com or your local library may have them for you.
Either way you decide, be sure to put you and your children first for now. Until your husband can sort out what he is doing, there is no reason to put your life on hold. Keep moving forward and trying to work on this issue. Hopefully, your husband will realize he needs help and make his way back to you and your kids.
I hope this has helped you,