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Related Experiment Video

Updated: Feb 9, 2026

Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy

Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy

Published on: June 15, 2022

Global dynamics for switching systems and their extensions by linear differential equations.

Zane Huttinga¹, Bree Cummins¹, Tomáš Gedeon¹

¹Department of Mathematical Sciences, Montana State University, Bozeman, MT 59715.

Physica D. Nonlinear Phenomena

|June 6, 2018

Summary

This study extends switching systems for modeling gene regulatory networks. The research demonstrates that these extended models retain identical parameter graphs, enhancing the analysis of complex cellular processes.

Keywords:

Morse graphs gene regulation switching systems transcription/translation model

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Area of Science:

Systems biology
Mathematical modeling of biological systems
Gene regulatory network analysis

Background:

Switching systems, using piecewise constant nonlinearities, are effective for modeling gene regulatory networks and analyzing dynamics via Morse graphs and parameter graphs.
However, many biological processes lack threshold-like behavior, limiting the applicability of standard switching systems.

Purpose of the Study:

To introduce and analyze extensions of switching systems that incorporate linear differential equations alongside switching interactions.
To investigate the impact of these extensions on the underlying parameter graph and Morse graph structures.

Main Methods:

Developing a theoretical framework for hybrid systems combining switching dynamics and linear differential equations.
Analyzing the topological properties of the parameter graph for both the original switching system and its extensions.
Establishing relationships between the Morse graphs of the switching system and its extended versions using order-preserving maps.

Main Results:

Demonstrating that the parameter graphs of the switching system and its extensions remain identical.
Proving the existence of an order-preserving map from the Morse graph of the switching system to that of any extension for each parameter graph node.
Providing counterexamples to illustrate the limitations of stronger potential relationships between the Morse graphs.

Conclusions:

The parameter graph structure is invariant under the considered extensions of switching systems, preserving global dynamical organization.
While specific Morse graph relationships are confirmed, stronger direct equivalences are shown to be invalid, highlighting the nuanced impact of extensions on local dynamics.