This simulation shows the spatial pattern formation in three-component reaction diffusion systems using Fitzhugh-Nagumo model exhibiting the Turing type instability. 

Expected Result: The simulation should exhibit spatially inhomogeneous, temporally stable patterns under the proper selection of system parameters.

The FitzHugh-Nagumo model is a well-known reaction-diffusion system, introduced in the study of electrical interaction of the nerves. The evolution of self-organizing pattern formation was introduced by Alan Turing [2] providing a detailed explanation of how reaction-diffusion systems could be responsible for the emergence of pattern formation in nature. For such patterns to evolve, a short-range activation and long-range inhibition mechanism is a requirement. This means that the diffusion coefficient of the inhibitor is significantly larger than the diffusion coefficient of the activator [3,4]. One can see how diffusion coefficients are selected as well as how reaction functions emerge as an additional term in the XML file. Specifically, here we consider a 3-component
FitzHugh–Nagumo model arises in the study of excitable wavetrains. For the parameter selection, we refer readers to the work in [5]. Through CompuCell3D, the simulation effectively illustrates the emergence of elegant spot-stripe type of patterns.

References:

[1] FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal, 1(6), 445-466.
[2] Turing, A. M. (1990). The chemical basis of morphogenesis. Bulletin of mathematical biology, 52, 153-197.
[3] Murray J.D. (2001). Mathematical biology II: Spatial models and biomedical applications, Vol. 3, Springer New York.
[4] Gierer A., Meinhardt H. (1972) A theory of biological pattern formation, Kybernetik 12: 30–39.
[5] Villar-Sepúlveda, E., & Champneys, A. R. (2023). General conditions for Turing and wave instabilities in reaction-diffusion systems. Journal of Mathematical Biology, 86(3), 39.


Author: Gülsemay YİĞİT, 2024
