In models for networks of regulatory interactions of biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has led to the use of continuous-time ‘switching’ systems, sometimes referred to as Glass networks, which result from a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties, such as non-uniqueness, in certain cases, due to the discontinuities of the system. Here, the use of ramp functions is explored as an alternative approximation to the sigmoid, which restores continuity to the vector field and removes the assumption of infinitely steep switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for describing a ramp system using the ‘focal points’ of the corresponding Glass network is given. Solutions of two-dimensional networks are explored, and then higher-dimensional networks under certain restrictions. Periodic behaviour is explored using mappings between threshold boundaries. Limitations in these methods are explored, and a general proof of the existence of periodic solutions in negative feedback loops with ramp interactions is given.

在生物分子调节相互作用网络的模型中，调节体浓度与其控制的生产率之间的S形关系导致了连续时间“转换”系统的使用，有时称为“玻璃网络”，这是由于简化假设，即开关行为在特定阈值下立即发生。尽管此假设产生了易于处理的模型，但在某些情况下，由于系统的不连续性，还会引起分析困难，例如非唯一性。在这里，我们探索使用斜坡函数作为S形的替代近似值，通过线性插值相应Glass网络中使用的焦点值，可以恢复矢量场的连续性并消除无限陡峭切换的假设。给出了使用相应的Glass网络的“焦点”描述斜坡系统的一般框架。探索二维网络的解决方案，然后探索在一定限制下的高维网络。使用阈值边界之间的映射来探索周期性行为。探索了这些方法的局限性，并给出了具有斜坡相互作用的负反馈回路中周期解的存在的一般证明。然后在某些限制下使用高维网络。使用阈值边界之间的映射来探索周期性行为。探索了这些方法的局限性，并给出了具有斜坡相互作用的负反馈回路中周期解的存在的一般证明。然后在某些限制下使用高维网络。使用阈值边界之间的映射来探索周期性行为。探索了这些方法的局限性，并给出了具有斜坡相互作用的负反馈回路中周期解的存在的一般证明。