Damiam, I am in NE Mississippi and grow asparagus. When we begin to have temps in the 40's, the asparagus will begin to yellow and with colder weather, all the "heads" will turn brown. That will happen this week, since we are supposed to get down to the mid-upper 20's for two mornings. I trim those now dormant stalks back to the ground, and the plants come back in the spring. I do mulch my dormant plants with chopped oak leaves, perhaps 4-6" of mulch.
My plants started out as three plants, in 6" pots, and they have just finished their second year. There were perhaps 50 spears and the plants were 6' tall and almost as wide. I will begin to "harvest" the spears next spring/summer and will have enough for the entire summer. I did not cut a single spear the first two years.
'Mary Washington' asparagus, June 2013.