Climbing Escher’s stairs: A way to approximate stability landscapes in multidimensional systems

Stability landscapes are useful for understanding the properties of dynamical systems. These landscapes can be calculated from the system’s dynamical equations using the physical concept of scalar potential. Unfortunately, it is well known that for most systems with two or more state variables such potentials do not exist. Here we use an analogy with art to provide an accessible explanation of why this happens and briefly review some of the possible alternatives. Additionally, we introduce a novel and simple computational tool that implements one of those solutions: the decomposition of the differential equations into a gradient term, that has an associated potential, and a non-gradient term, that lacks it. In regions of the state space where the magnitude of the non-gradient term is small compared to the gradient part, we use the gradient term to approximate the potential as quasi-potential. The non-gradient to gradient ratio can be used to estimate the local error introduced by our approximation. Both the algorithm and a ready-to-use implementation in the form of an R package are provided.Author summary: The physical concept of potential, also referred to as stability landscape, has proved to be a useful visualization tool for explaining and communicating advanced concepts about dynamical systems to professional communities as diverse as environmental scientists, ecologists, economists, mathematicians and policy makers. It is a particularly popular tool in cell biology, where it is often referred to as Waddington’s epigenetic landscape. Unfortunately, for most biological models with more than one interacting element, such a stability landscape doesn’t exist unless the system under study satisfies very restrictive conditions. Here we use an analogy with art to provide an accessible explanation of the subtle reasons a potential may fail to exist for a given biological system. Additionally, we introduce a remarkably simple and efficient algorithm to compute potentials that takes into account these limitations, providing the best quasi-potential candidate plus an error map. This error map tells us under which conditions can we trust the potential picture.