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Oscillatory pulses in FitzHugh-Nagumo type systems with cross-diffusion.

E P Zemskov¹, I R Epstein, A Muntean

¹Department of Chemistry, Brandeis University, MS 015, Waltham, MA 02454, USA. zemskov@brandeis.edu

Mathematical Medicine and Biology : a Journal of the IMA

|August 6, 2010

Summary

This study analyzes FitzHugh-Nagumo reaction-diffusion systems with cross-diffusion. Researchers describe wavy pulse dynamics and oscillatory tails, revealing a key feature of pulse appearance in bistable regimes.

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Area of Science:

Mathematical Biology
Chemical Kinetics
Pattern Formation

Background:

FitzHugh-Nagumo models are fundamental for studying excitable media.
Cross-diffusion introduces complex dynamics not present in standard reaction-diffusion systems.
Wavy pulses and traveling fronts are crucial patterns in biological systems.

Purpose of the Study:

To analyze reaction-diffusion systems with linear cross-diffusion terms.
To describe the occurrence and properties of wavy pulses in prototypical systems.
To understand the role of bistability in pulse dynamics.

Main Methods:

Analytical description using piecewise linear approximations of reaction functions.
Investigation of two prototypical FitzHugh-Nagumo systems.
Characterization of pulse wave profiles and oscillatory tails.

Main Results:

Complete description of wavy pulse occurrence and properties.
Identification of oscillatory tails in pulse wave profiles, analogous to traveling fronts.
Discovery of pulse appearance in the bistable regime as a fundamental feature.

Conclusions:

Cross-diffusion fundamentally alters pulse dynamics in reaction-diffusion systems.
The bistable regime is critical for the emergence of pulses in these systems.
The findings offer insights into pattern formation relevant to biological processes.