High-dimensional switches have been proposed as a way to model cellular differentiation, particularly in the context of basic Helix-Loop-Helix (bHLH) competitive heterodimerization networks. A previous study derived a simple rule showing how many elements can be co-expressed, depending on the rate of competition within the network. A limitation to that rule, however, is that many biochemical parameters were considered to be identical. Here, we derive a generalized rule. This in turns allows one to study more ways in which these networks could be regulated, linking intrinsic cellular differentiation determinants to extracellular cues.