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Periodicity in piecewise-linear switching networks with delay
R Edwards1, P van den Driessche, Lin Wang
1Department of Mathematics and Statistics, University of Victoria, BC, Canada. edwards@math.uvic.ca
Journal of Mathematical Biology
|March 24, 2007
Summary
Delayed Glass networks, modeling gene and neural networks, exhibit unique stable limit cycles due to regulatory delays. This study analyzes their cyclic behavior and stability for all finite delays.
Area of Science:
- Systems biology
- Computational neuroscience
- Dynamical systems theory
Background:
- Gene regulatory networks and neural networks are modeled using piecewise-linear switching systems called Glass networks.
- These networks exhibit delays in regulatory activities, which can significantly influence their dynamical behavior.
Purpose of the Study:
- To introduce and analyze Glass networks with discrete delays.
- To investigate the conditions for cyclic patterns of switching and the existence of periodic orbits in delayed Glass networks.
- To develop an algorithm for locating and assessing the stability and uniqueness of periodic orbits.
Main Methods:
- Analysis of piecewise-linear switching systems with discrete delays.
- Application of fractional linear mapping to demonstrate periodic orbits.
- Development of an algorithm for locating, stability analysis, and uniqueness checking of periodic orbits.
Main Results:
- Delayed Glass networks possess a periodic orbit for all positive finite delays under specific conditions.
- An algorithm is presented to locate, determine stability, and check uniqueness of periodic orbits.
- The complete dynamics of two-dimensional delayed Glass networks are elucidated, revealing unique globally stable limit cycles under different conditions.
Conclusions:
- Discrete delays in Glass networks lead to predictable cyclic behavior and globally stable limit cycles, contrasting with non-delayed systems.
- The findings provide a comprehensive understanding of the dynamics of delayed Glass networks, crucial for modeling biological and neural systems.
- The developed algorithm offers a practical tool for analyzing the stability and uniqueness of periodic orbits in these complex systems.
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