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

PLoS Comput Biol. 2020 Apr 10;16(4):e1007788. doi: 10.1371/journal.pcbi.1007788. eCollection 2020 Apr.

Abstract

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.

Publication types

  • Research Support, Non-U.S. Gov't
  • Review

MeSH terms

  • Algorithms
  • Computational Biology / methods*
  • Ecosystem
  • Models, Biological*
  • Models, Statistical
  • Systems Analysis

Grants and funding

This work was supported by funding from the European Union’s Horizon 2020 research and innovation programme for the ITN CRITICS (CRItical Transitions In Complex Systems) under Grant Agreement Number 643073 (https://app.dimensions.ai/details/grant/grant.3938426). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.