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It sounds a lot like she has an issue with her back which may be what your veterinarian is trying to say with it being neurological. It could simply be the result of pain, but another likley possibility here is that there is compression on her spinal cord possibly the result of a disc compressing on a nerve. The prednisone is also a very strong anti-inflammatory which has a side effect of increased thirst and urination. The urinary issues may be the result of pain and not wanting to move to go to the bathroom or it could be that she doesn't feel that she has to go because of the swelling. If her condition is the result of back/spine issues the recovery is very gradual and this may too fall into place especially after she can come off of the prednisone. Strict rest, leash walks for potty breaks only and time is what we need right now. In some cases it can take up to six weeks for complete recovery. If at any time her condition worsens or her urinary issues do not resolve after the doc takes her off the prednisone then there may be something more going on like urinary incontinence in an older pet or a possible bladder infection in conjunction with her current ailment.
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Whether or not you should wait has to be left up to you and your veterinarian since I am unable to evaluate his reflexes, gait, or how much he has improved in those 7 days. My thoughts though are as long as he is improving you should be able to wait up to another week to see if he continues improving. With the prednisone you become more thirsty and what goes in must come out. More frequent potty breaks may help to improve this. Another possibility here is that he may have steroid induced diabetes. I see this more often in cats than in dogs but it might be worth having a blood profile performed to evaluate other possible causes before assuming it is the spinal compression especially since his leg function is improving. If you do decide to go to the neuro at least you will have the bloodwork and won't have to pay the higher fees from the specialist as it is very very likely they will want recent bloodwork on your boy.
Oh yes, that is a very significant detail. Actually somehow he needs to get to an appointment sooner than later in this respect. It is great that he is showing some improvements but the lack of anal tone in conjunction with the bladder issues are a result of spinal compression alone. 3 possibilities here are blood clot in the spinal column, tumor, or herniated disk. The latter is most likely given his age. I would at this point get the appointment scheduled and ask your veterinarian to push to get you in sooner. Sometimes the disk issues do not show up on xray but will with an MRI.