This may be fairy ring or center clump dieback of hosta. Old and established clumps are the ones most commonly affected. The symptoms sound similar to your description: a perimeter of buds emerging in spring with the center buds dead or rotted. Fast-growing clumps of hosta should be divided every five to six years to help prevent this. We recommend that you dig up both clumps and discard the one that is completely dead. Carefully examine the roots for signs of other possible problems (e.g., voles, gophers, etc.). Divide the surviving hosta, discarding the dead center parts, and replant the healthy sections in fresh soil that has been amended with organic matter (well-aged manure, compost, and/or peat moss) and fertilizer (5-10-5 or 10-10-10). You can replant in the same place if the centers are not rotten; if rot is present, definitely choose a new spot to plant in. You will want to water them well. Mulch is helpful in preventing "heaving" and the resulting winter-kill of exposed crowns and roots. If you do mulch, keep an eye out for slugs and their damage.
A good reference for hosta enthusiasts is Iowa State University's publication: Hosta Diseases and Pests.
