Ghost! You do have a choice. Next time it is one of your children's birthdays, plan your own thing. Start some new "Dad" traditions. With the older kids tailor it to their interests. Take your daughter on a horse trail ride, or a horse show. Take your sons to a concert or sporting event or camping trip. You don't have to compete with their birthday celebrations with Mom, just do your own thing. Something different than what they do with their mom. Go ahead post it on Facebook or even better don't, because you are creating memories and relationships with your kids, that is the important thing. Instead of feeling left out, look at as a chance to do something new. And it could be simple too, as simple as taking them out for a special breakfast. Take pictures, lots of them, be present and not a "ghost."
Do this for Christmas too. A christmas play, ice skating, a visit with Santa, there are endless opportunities, find something that your wife isn't already doing with them and make it a new "Dad" tradition. If you and your W end up reconciling, keep it up. Your kids need that special bond with you. Don't leave all of the memory making to your W.