The competitive threshold linear networks have been recently developed and are typical examples of nonsmooth systems that can be easily constructed. Due to their flexibility for manipulation, they are used in several applications, but their dynamics (both local and global) are not completely understood. In this work, we take some recently developed threshold systems and by a simple modification in the parameter space, we obtain new global dynamic behavior. Heteroclinic cycles and other remarkable scenarios of global bifurcation are reported.