Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

交换系统使用分段恒定非线性来对基因调控网络进行建模。该选择在行为分析中提供了优势，并允许根据与参数图的节点关联的摩尔斯图对动力学进行全局描述。参数图捕获参数空间分解为具有相同摩尔斯图的域的空间特征。但是，有许多蜂窝过程没有表现出类似阈值的行为，因此交换系统无法很好地描述它们。我们考虑一类开关系统的扩展，它是由开关相互作用和线性微分方程控制的变量链的混合形成的。我们表明与交换系统及其任何扩展相关的参数图是相同的。对于每个参数图节点，从交换系统的莫尔斯图到其任何扩展的莫尔斯图都有一个保序映射。我们提供了反例，以说明为什么莫尔斯图之间可能更强的关系无效。