Two classes of ODE models with switch-like behavior

• Two classes of ODE models with natural Boolean approximations are constructed.
• Every Boolean system has counterparts in each of the classes.
• A notion of consistency between Boolean and ODE models is studied.
• Separation of timescales implies consistency under certain assumptions.
• These ODE models constitute a convenient framework for studying consistency.

In cases where the same real-world system can be modeled both by an ODE system  DD and a Boolean system  BB, it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that if BB has certain structures, consistency between DD and BB is implied by sufficient separation of timescales in one class of our models. Namely, if the trajectories of BB are “one-stepping” then we prove a strong form of consistency and if BB has a certain monotonicity property then there is a weaker consistency between DD and BB. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.