Transition of spiral solutions according to the time and space steps discretization of reaction-diffusion system of FitzHugh-Nagumo type
Main Article Content
Abstract
Spiral solutions or spiral waves can be found in many natural systems. Spiral waves were observed in studies about the potential in brain and heart cells. Their appearance in the human heart is a presentation of arrhythmia. The paper showed how to create spiral solutions of diffusion-reaction system of FitzHugh-Nagumo type and the transition of spiral solutions according to the time step and space step discretization of finite difference method. Decreasing the value of space step discretization makes the spiral wave grow bigger, but if the value of time step discretization is increased at the same given space step, the finite difference method will be explosive, meaning that spiral wave no longer exists.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
reaction-diffusion equations of FitzHugh-Nagumo, space step discretization, time step discretization
References
Ambrosio, B., & Aziz-Alaoui, M. A. (2013). Synchronization and control of a network of coupled reaction-diffusion systems of generalized FitzHugh-Nagumo type. ESAIM: Proceedings, (39), 15-24. https://doi.org/10.1051/proc/201339003.
Ermentrout, G. B., & Terman, D. H. (2009). Mathematical Foundations of Neurosciences. Springer.
Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., (117), 500-544. https://doi.org/10.1113/jphysiol.1952.sp004764.
Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Terrence J. Sejnowski and Tomaso A. Poggio The MIT Press, Cambridge.
Keener, J. P., & Sneyd, J. (2009). Mathematical Physiology: Systems Physiology (2nd ed.). Antman S.S., Marsden J.E., and Sirovich L. Springer.
Murray, J. D. (2002). Mathematical Biology. I. An Introduction (3rd ed.). Springer.
Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proc. IRE., (50), 2061–2070. https://doi.org/10.1109/jrproc.1962.288235.
Phan, V. L. E. (2019). Synchronization on the network of two Fitzhugh-Nagumo systems with the effect of the external electrical stimulation. Dong Thap University Journal of Science, 38, 74-77. https://doi.org/10.52714/dthu.38.6.2019.701
Most read articles by the same author(s)
- Van Long Em Phan, Synchronization in complete networks of ordinary differential equations of Fitzhugh – Nagumo type with nonlinear coupling , Dong Thap University Journal of Science: Vol. 10 No. 5 (2021): Natural Sciences Issue (English)
- Van Long Em Phan, Tan Dat Vo, Sufficient condition for generalized synchronization in the networks of two ordinary differential equations of FitzHugh-Nagumo type with bidirectionally linear coupling , Dong Thap University Journal of Science: Vol. 12 No. 8 (2023): Natural Sciences Issue (Vietnamese)
- Van Long Em Phan, Synchronization in complete networks of reaction-diffusion equations of Fitzhugh-Nagumo wiht spiral solutions , Dong Thap University Journal of Science: No. 37 (2019): Part B - Natural Sciences
- TS. Van Long Em Phan, Sinh viên Tan Dat Nguyen, Sinh viên Minh Phuc Nguyen, Sinh viên Thi Ngoc Lan Nguyen, Identical synchronization controller between the Hindmarsh -Rose 2D and the FitzHugh-Nagumo type model , Dong Thap University Journal of Science: Vol. 13 No. 8 (2024): Natural Sciences Issue (Vietnamese)