Automatic Generation of Interactive Multidimensional Phase Portraits

Abstract

Mathematical models formally and precisely represent biological mechanisms with complex dynamics. To understand the possible behaviors of such systems, phase portrait diagrams can be used to visualize their overall global dynamics across a domain. However, producing these phase portrait diagrams is a laborious process reserved to mathematical experts. Here, we developed a computational methodology to automatically generate phase portrait diagrams to study biological dynamical systems based on ordinary differential equations. The method only needs as input the variables and equations describing a multidimensional biological system and it automatically outputs for each pair of dependent variables a complete phase portrait diagram, including the critical points and their stability, the nullclines of the system, and a vector space of the trajectories. Crucially, the portraits generated are interactive, and the user can move the visualized planar slice, change parameters with sliders, and add trajectories in the phase and time domains, after which the diagrams are updated in real time. The method is available as a user-friendly graphical interface or can be accessed programmatically with a Mathematica package. The generated portraits and particular views can be saved as computable notebooks preserving the interactive functionality, an approach that can be adopted for reproducible science and interactive pedagogical materials. The method, code, and examples are freely-available at https://lobolab.umbc.edu/autoportrait.