Ipomoea batatas has a perennial root , so each season it grows , the root is taking in nutrients and building new tissue with the vital assistance of the aerial parts , so 'old' tubers shouldn't be an issue unless the tubers were somehow being starved...

The question of whether the heating mat would help to awaken the tubers is a good one and my thoughts are if done carefully , it should help, but be sure the tubers are watered and do not become too dried out from the mat as the skin on many Ipomoea batatas roots is relatively thin...

Thank you, Ron!

My mat only raises the soil temp by 5-10 degrees over the ambient temp, and by that time of year, the basement is in the upper 60's. I may place them in some loose potting mix and keep them watered.