It seems clear that what Rand is doing is the same as what his kid's with Aviendha seem to be able to do. They seem to be welded to the Power, in some way. They always hold it (which Rand doesn't seem to), and began channeling when very young.

This is clearly not a genetic ability, since none of Elayne's children are reported to have it and none of Rand's ancestors had it.

Which makes me wonder if this is something that could be learned, and perhaps will be learned by Rand's close posse of channelers. Perhaps they all learn to become truly one with the Power, and Aviendha had kids after she learned, and her kid's somehow picked it up as their brains developed in a body that was so fused to the Power.

All this is wild speculation, of course, but Rand now, and his kids, certainly do point to some very fundamental laws of channeling being broken. For one, the kids constantly use the Power, but do not seem to get tired. For another, they started channeling as babes, which is even weirder. Perhaps they just never learned that channeling should be tiring.