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Segregation and pursuit waves in activator-inhibitor systems
Vicenç Méndez1, Werner Horsthemke, Evgeny P Zemskov
1Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edicifi Cc. Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2007
Summary
This study explores how cross-diffusion impacts wave propagation in activator-inhibitor systems. We found exact solutions for traveling fronts and solitary pulses, offering new insights into wave dynamics.
Area of Science:
- Computational neuroscience
- Mathematical biology
- Chemical kinetics
Background:
- Activator-inhibitor systems are fundamental models for pattern formation and wave propagation.
- FitzHugh-Nagumo kinetics provide a simplified yet robust framework for studying neuronal excitability.
- Cross-diffusion, unlike self-diffusion, involves coupled movement of species, potentially altering wave behavior.
Purpose of the Study:
- To analyze the influence of cross-diffusion on traveling waves in a piecewise linear activator-inhibitor model.
- To derive and examine exact analytic solutions for wave phenomena under cross-diffusion.
- To compare the findings with established models, specifically the Rinzel-Keller model with self-diffusion.
Main Methods:
- Utilizing a piecewise linear approximation of FitzHugh-Nagumo kinetics.
- Incorporating a cross-diffusion term for either the activator or the inhibitor.
- Deriving exact analytic solutions for traveling fronts and solitary pulses.
- Analyzing speed diagrams associated with these solutions.
Main Results:
- Exact analytic solutions for traveling fronts and solitary pulses were obtained under cross-diffusion.
- The speed diagrams for these wave solutions were derived and analyzed.
- The behavior of waves with cross-diffusion was compared to the self-diffusion case using the Rinzel-Keller model.
Conclusions:
- Cross-diffusion significantly affects the dynamics and speed of propagating waves in activator-inhibitor systems.
- The derived analytic solutions provide a precise mathematical description of these effects.
- This work offers a valuable comparison to existing models and deepens the understanding of diffusion's role in biological pattern formation.
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