[Paper Review] Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology

M. G. Blyth, Ludovic Renson+1 more|arXiv (Cornell University)|Aug 12, 2020

Gene Regulatory Network Analysis4 citations

TL;DR

This tutorial introduces numerical continuation and bifurcation analysis as powerful computational tools for studying nonlinear dynamics in systems and synthetic biology. It guides researchers through key concepts, demonstrates their application using a neuron model, and reviews major software packages to help users select appropriate tools based on coding experience, performance needs, and system complexity, with a focus on reproducibility and usability in biological modeling.

ABSTRACT

Mathematical modelling allows us to concisely describe fundamental principles in biology. Analysis of models can help to both explain known phenomena, and predict the existence of new, unseen behaviours. Model analysis is often a complex task, such that we have little choice but to approach the problem with computational methods. Numerical continuation is a computational method for analysing the dynamics of nonlinear models by algorithmically detecting bifurcations. Here we aim to promote the use of numerical continuation tools by providing an introduction to nonlinear dynamics and numerical bifurcation analysis. Many numerical continuation packages are available, covering a wide range of system classes; a review of these packages is provided, to help both new and experienced practitioners in choosing the appropriate software tools for their needs.

Motivation & Objective

• To make numerical continuation and bifurcation analysis accessible to researchers in systems and synthetic biology without a background in nonlinear dynamics.
• To demonstrate how bifurcation analysis can reveal critical transitions in biological models, such as the onset of bursting in neuron dynamics.
• To provide a practical comparison of existing numerical continuation software packages to guide tool selection based on user expertise, performance, and system type.
• To promote the use of computational methods for analyzing nonlinear differential equations that are analytically intractable but essential for modeling complex biological systems.
• To support reproducible and extensible research by highlighting tools with scripting capabilities and specialized toolboxes for biological applications.

Proposed method

• Introduces foundational concepts of nonlinear dynamics, including state variables, time evolution, and qualitative system behaviors (e.g., steady-state, oscillatory, chaotic).
• Defines bifurcations as qualitative changes in system dynamics due to parameter variations, illustrated through examples like the canard explosion in food-chain models.
• Uses a conductance-based neuron model to demonstrate how numerical continuation can track bifurcations and explain bursting behavior.
• Reviews 15+ numerical continuation software packages, categorizing them by system type (ODEs, PDEs, maps, hybrid, nonsmooth systems) and interface (GUI, scripting, library-based).
• Compares core tools (XPPAUTO, MatCont, PyDSTool, CoCo) based on usability, performance, bifurcation detection capability, and extensibility.
• Emphasizes the use of scripting for reproducibility and collaboration, and highlights specialized toolboxes in PyDSTool for accelerating analysis in systems biology.

Experimental results

Research questions

• RQ1 How can numerical continuation be used to detect and analyze bifurcations in nonlinear biological models, such as those describing neuronal activity?
• RQ2 What are the key differences in capabilities, usability, and performance among major numerical continuation software packages for systems biology?
• RQ3 Which software tools are best suited for researchers with minimal coding experience versus those with advanced programming skills?
• RQ4 How do different software packages handle complex bifurcation structures, such as those involving periodic orbits or sliding bifurcations in hybrid systems?
• RQ5 In what ways can numerical continuation tools improve the design and analysis of synthetic gene networks in systems biology?

Key findings

• Numerical continuation enables the detection of bifurcations in nonlinear biological models, such as the transition to bursting in a neuron model, which cannot be captured by standard simulation alone.
• XPPAUTO and PyDSTool are recommended for users with minimal coding experience due to their graphical user interfaces and built-in tools for phase plane analysis.
• For high-performance computation and complex bifurcation tracking, MatCont is best suited due to its ability to detect and track a wide range of bifurcation types.
• PyDSTool and CoCo are ideal for advanced users needing extensibility and integration into larger workflows, with PyDSTool offering specialized toolboxes for systems biology applications.
• Software with scripting interfaces, such as PyDSTool and CoCo, support reproducible and collaborative research, enabling easier adaptation and automation of analyses.
• The tutorial identifies that PDECONT and pde2path are suitable for PDE-based models, while WAVETRAIN and SlideCont are tailored for PDEs and nonsmooth systems, respectively, highlighting the diversity of available tools.