You should not get on this horse again until you have a vet check him up. This could be a very serious issue. There are many things that it could be but it needs to me checked out right away.
Is he girth-y because that can sometimes cause a horse to lie down especially if they think their girth is to tight.
How old is he/ how long has he been undersaddle? Is this all recent/studden, or an on-going issue? Happens every time, or?
However I would not continue to ride/work this horse until you can get the issues sorted out, regardless of the cause. Get the vet, figure out what's going on ASAP.
Another option is possible sleep deprivation. Horses need to lie down for a deep sleep, if they do not they get sleep deprived to the point they will collapse, however once you know what to expect it's easy to read, if it's something neurological I would expect less of a "tell".
Regardless, vet as said, I would also never ride a horse that has fallen under saddle more then a once ever trip type situation.
This sounds like a very serious issue, altho without more info (vet report), who knows what it is. You should also not ride him, or otherwise work him until he's got the all clear :)