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Correspondence between discrete and continuous models of excitable media: trigger waves
Y B Chernyak1, A B Feldman, R J Cohen
1Division of Health Sciences and Technology, Harvard University-Massachusetts Institute of Technology, Cambridge, USA. yurich@atrium.mit.edu
Summary
We developed a framework linking continuous partial differential equation (PDE) models to discrete cellular automata (CA) models for excitable media. This allows precise parameter determination for CA models, accurately simulating trigger wave behavior.
Area of Science:
- Computational Biology
- Mathematical Modeling
- Physics of Complex Systems
Background:
- Continuous partial differential equation (PDE) models and discrete cellular automata (CA) models are used to simulate excitable media.
- Establishing a quantitative link between these modeling approaches is crucial for accurate predictions.
Purpose of the Study:
- To present a theoretical framework for relating continuous PDE models of excitable media to discrete CA models.
- To derive expressions for CA model parameters based on physical system properties.
Main Methods:
- Deriving expressions for CA plane wave speed, critical curvature, and effective diffusion constant.
- Equating these to corresponding PDE quantities to solve for CA parameters.
- Analyzing trigger wave solutions in a 2D excitable medium with no recovery.
Main Results:
- A unique solution for CA parameter values was found by equating PDE and CA model quantities.
- The framework was tested against FitzHugh-Nagumo and sawtooth PDE models, showing good agreement.
- Trigger wave behavior in excitable media is governed by a limited set of parameters.
Conclusions:
- The developed framework provides a quantitative link between PDE and CA models for excitable media.
- This enables accurate parameterization of CA models for simulating physical systems.
- The study confirms that trigger wave dynamics are controlled by a small, identifiable set of parameters.