We propose in this paper a method to hierarchically organize a certain type of piecewise affine differential system. This specific class of dynamical systems has been extensively studied for the past few years, as it provides a good framework to model gene regulatory networks. Using the hierarchical organization of a piecewise affine system, we present a technique to qualitatively analyze the asymptotic behavior of the whole system thanks to the analysis of several smaller subsystems. Specifically adapted to these networks, an algorithm of threshold elimination is presented, that refines in certain cases the hierarchical decomposition and therefore improves the analysis.