hopefully you used a very low gravity starter, like 1.010. I would suggest this any time you are collecting bottle dregs, sour or not. Sour dregs can be stored a lot longer since they will not die as quickly as sacch. I typically keep them at cellar temps and not in the fridge, just due to the lag time after being in the cold. As far as growing a starter, let it spend at least a week in the small 1.010 starter. Keeping that in a jar is perfect, only shaking it up a few times here and there. After that, you can certainly move that to a higher gravity starter, around 1.030 or so, and throw it on a stir plate for another week. You should have a healthy pitch at that point. If the slurry isnt large enough, just do another step.