
-
Previous Article
Two notes on the O'Hara energies
- DCDS-S Home
- This Issue
-
Next Article
Spatio-temporal coexistence in the cross-diffusion competition system
Mathematical model of signal propagation in excitable media
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic |
This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.
References:
[1] |
O. Bernus and E. Vigmond, Asymptotic wave propagation in excitable media, Phys. Rev. E, 92 (2015), 010901.
doi: 10.1103/PhysRevE.92.010901. |
[2] |
Y.-Y. Chen, H. Ninomiya and R. Taguchi,
Travelling spots in multidimensional excitable media, Journal of Elliptic and Parabolic Equations, 1 (2015), 281-305.
doi: 10.1007/BF03377382. |
[3] |
P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, MS & A. Modeling, Simulation and Applications, 13. Springer, Cham, 2014.
doi: 10.1007/978-3-319-04801-7. |
[4] |
P. Colli-Franzone, V. Gionti, S. Scacchi and C. Storti,
Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry, Mathematical Biosciences, 315 (2019), 108-128.
doi: 10.1016/j.mbs.2019.108228. |
[5] |
E. N. Cytrynbaum, V. MacKay, O. Nahman-Lévesque, M. Dobbs, G. Bub, A. Shrier and L. Glass, Double-wave reentry in excitable media, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 073103, 12 pp.
doi: 10.1063/1.5092982. |
[6] |
K. Deckelnick, G. Dziuk and C. M. Elliot,
Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232.
doi: 10.1017/S0962492904000224. |
[7] |
A. J. Durston,
Dictyostelium discoideum aggregation fields as excitable media, J. Theor. Biol., 42 (1973), 483-504.
doi: 10.1016/0022-5193(73)90242-7. |
[8] |
J. Engelbrecht, T. Peets, K. Tamm, M. Laasmaa and M. Vendelin,
On the complexity of signal propagation in nerve fibres, Proceedings of the Estonian Academy of Sciences, 67 (2018), 28-38.
doi: 10.3176/proc.2017.4.28. |
[9] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
M. A. C. Guyton, Textbook of Medical Physiology, W. B. Saunders Company, 1991. |
[11] |
M. Kolář,
Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146.
|
[12] |
O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Izdat. "Nauka'', Moscow, 1967,736 pp. |
[13] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 1996.
doi: 10.1142/3302. |
[14] |
J. L. Lions, Quelques Méthodes aux Rśolution des Problémes Nonlinéaires, Dunod Gauthiers-Villars, Paris, 1969. |
[15] |
J. Ma, F. Q. Wu, T. Hayat, P. Zhou and J. Tang,
Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media, Physica A: Statistical Mechanics and its Applications, 486 (2017), 508-516.
doi: 10.1016/j.physa.2017.05.075. |
[16] |
J. Mach, M. Beneš and P. Strachota,
Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension, Comput. Math. Appl., 73 (2017), 2053-2065.
doi: 10.1016/j.camwa.2017.02.032. |
[17] |
J. D. Murray, Mathematical Biology, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. |
[18] |
J. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proceedings of IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[19] |
A. Yu. Palyanov and A. S. Ratushnyak,
Some details of signal propagation in the nervous system of C. elegans, Russian Journal of Genetics: Applied Research, 5 (2015), 642-649.
doi: 10.1134/S2079059715060064. |
[20] |
O. Pártl, Reaction-Diffusion Systems in Mathematical Biology, Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague, Prague, 2012. |
[21] |
Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. |
[22] |
L. S. Pontryagin, Ordinary Differential Equations, Second, revised edition Izdat. "Nauka'', Moscow, 1965,331 pp. |
[23] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983. |
[24] |
J. Šembera and M. Beneš,
Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions, Journal of Computational and Applied Mathematics, 136 (2001), 163-176.
doi: 10.1016/S0377-0427(00)00582-3. |
[25] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[26] |
N. A. Trayanova and K. C. Chang,
How computer simulations of the human heart can improve anti-arrhythmia therapy, The Journal of Physiology, 594 (2016), 2483-2502.
doi: 10.1113/JP270532. |
[27] |
K. H. W. J. Ten Tusscher and A. V. Panfilov,
Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282.
doi: 10.1137/030602654. |
[28] |
E. Ullner, A. Zaikin, J. García-Ojalvo, R. Báscones and J. Kurths,
Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312 (2003), 348-354.
doi: 10.1016/S0375-9601(03)00681-9. |
[29] |
H. Wang, J. Wang, X. Y. Thow and Ch. Lee, The First Principle of Neural Circuit and the General Circuit–Probability Theory, submitted, 2018, arXiv: 1805.00605. |
[30] |
J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. |
[31] |
L. D. Weise and A. V. Panfilov, Emergence of spiral wave activity in a mechanically heterogeneous reaction-diffusion-mechanics system, Physical Review Letters, 108 (2012), 228104.
doi: 10.1103/PhysRevLett.108.228104. |
[32] |
D. P. Zipes and J. Jalife, Cardiac Electrophysiology, Saunders Elsevier, Philadelphia, 1995. |
[33] |
V. S. Zykov, Spiral wave initiation in excitable media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018).
doi: 10.1098/rsta.2017.0379. |
[34] |
V. S. Zykov, A. S. Mikhailov and S. C. Müller,
Wave propagation in excitable media with fast inhibitor diffusion, Lecture Notes in Physics, 532 (2007), 308-325.
doi: 10.1007/BFb0104233. |
[35] |
V. S. Zykov and E. Bodenschatz,
Wave propagation in inhomogeneous excitable media, Annual Review of Condensed Matter Physics, 9 (2018), 435-461.
|
show all references
References:
[1] |
O. Bernus and E. Vigmond, Asymptotic wave propagation in excitable media, Phys. Rev. E, 92 (2015), 010901.
doi: 10.1103/PhysRevE.92.010901. |
[2] |
Y.-Y. Chen, H. Ninomiya and R. Taguchi,
Travelling spots in multidimensional excitable media, Journal of Elliptic and Parabolic Equations, 1 (2015), 281-305.
doi: 10.1007/BF03377382. |
[3] |
P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, MS & A. Modeling, Simulation and Applications, 13. Springer, Cham, 2014.
doi: 10.1007/978-3-319-04801-7. |
[4] |
P. Colli-Franzone, V. Gionti, S. Scacchi and C. Storti,
Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry, Mathematical Biosciences, 315 (2019), 108-128.
doi: 10.1016/j.mbs.2019.108228. |
[5] |
E. N. Cytrynbaum, V. MacKay, O. Nahman-Lévesque, M. Dobbs, G. Bub, A. Shrier and L. Glass, Double-wave reentry in excitable media, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 073103, 12 pp.
doi: 10.1063/1.5092982. |
[6] |
K. Deckelnick, G. Dziuk and C. M. Elliot,
Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232.
doi: 10.1017/S0962492904000224. |
[7] |
A. J. Durston,
Dictyostelium discoideum aggregation fields as excitable media, J. Theor. Biol., 42 (1973), 483-504.
doi: 10.1016/0022-5193(73)90242-7. |
[8] |
J. Engelbrecht, T. Peets, K. Tamm, M. Laasmaa and M. Vendelin,
On the complexity of signal propagation in nerve fibres, Proceedings of the Estonian Academy of Sciences, 67 (2018), 28-38.
doi: 10.3176/proc.2017.4.28. |
[9] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
M. A. C. Guyton, Textbook of Medical Physiology, W. B. Saunders Company, 1991. |
[11] |
M. Kolář,
Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146.
|
[12] |
O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Izdat. "Nauka'', Moscow, 1967,736 pp. |
[13] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 1996.
doi: 10.1142/3302. |
[14] |
J. L. Lions, Quelques Méthodes aux Rśolution des Problémes Nonlinéaires, Dunod Gauthiers-Villars, Paris, 1969. |
[15] |
J. Ma, F. Q. Wu, T. Hayat, P. Zhou and J. Tang,
Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media, Physica A: Statistical Mechanics and its Applications, 486 (2017), 508-516.
doi: 10.1016/j.physa.2017.05.075. |
[16] |
J. Mach, M. Beneš and P. Strachota,
Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension, Comput. Math. Appl., 73 (2017), 2053-2065.
doi: 10.1016/j.camwa.2017.02.032. |
[17] |
J. D. Murray, Mathematical Biology, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. |
[18] |
J. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proceedings of IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[19] |
A. Yu. Palyanov and A. S. Ratushnyak,
Some details of signal propagation in the nervous system of C. elegans, Russian Journal of Genetics: Applied Research, 5 (2015), 642-649.
doi: 10.1134/S2079059715060064. |
[20] |
O. Pártl, Reaction-Diffusion Systems in Mathematical Biology, Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague, Prague, 2012. |
[21] |
Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. |
[22] |
L. S. Pontryagin, Ordinary Differential Equations, Second, revised edition Izdat. "Nauka'', Moscow, 1965,331 pp. |
[23] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983. |
[24] |
J. Šembera and M. Beneš,
Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions, Journal of Computational and Applied Mathematics, 136 (2001), 163-176.
doi: 10.1016/S0377-0427(00)00582-3. |
[25] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[26] |
N. A. Trayanova and K. C. Chang,
How computer simulations of the human heart can improve anti-arrhythmia therapy, The Journal of Physiology, 594 (2016), 2483-2502.
doi: 10.1113/JP270532. |
[27] |
K. H. W. J. Ten Tusscher and A. V. Panfilov,
Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282.
doi: 10.1137/030602654. |
[28] |
E. Ullner, A. Zaikin, J. García-Ojalvo, R. Báscones and J. Kurths,
Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312 (2003), 348-354.
doi: 10.1016/S0375-9601(03)00681-9. |
[29] |
H. Wang, J. Wang, X. Y. Thow and Ch. Lee, The First Principle of Neural Circuit and the General Circuit–Probability Theory, submitted, 2018, arXiv: 1805.00605. |
[30] |
J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. |
[31] |
L. D. Weise and A. V. Panfilov, Emergence of spiral wave activity in a mechanically heterogeneous reaction-diffusion-mechanics system, Physical Review Letters, 108 (2012), 228104.
doi: 10.1103/PhysRevLett.108.228104. |
[32] |
D. P. Zipes and J. Jalife, Cardiac Electrophysiology, Saunders Elsevier, Philadelphia, 1995. |
[33] |
V. S. Zykov, Spiral wave initiation in excitable media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018).
doi: 10.1098/rsta.2017.0379. |
[34] |
V. S. Zykov, A. S. Mikhailov and S. C. Müller,
Wave propagation in excitable media with fast inhibitor diffusion, Lecture Notes in Physics, 532 (2007), 308-325.
doi: 10.1007/BFb0104233. |
[35] |
V. S. Zykov and E. Bodenschatz,
Wave propagation in inhomogeneous excitable media, Annual Review of Condensed Matter Physics, 9 (2018), 435-461.
|




Model characteristics | |
Fixed point location | |
Diffusion coefficients | |
where |
|
Initial state and boundary conditions | |
for each experiment. | |
Numerical characteristics | |
Total length of simulation |
|
Internal time step |
|
Mesh |
Model characteristics | |
Fixed point location | |
Diffusion coefficients | |
where |
|
Initial state and boundary conditions | |
for each experiment. | |
Numerical characteristics | |
Total length of simulation |
|
Internal time step |
|
Mesh |
Mesh | time step | ||||||
error of |
error of |
error of |
error of |
error of |
error of |
||
0.0625 | 0.001953 | 0.056240 | 0.161228 | 0.108336 | 0.265209 | 0.363029 | 0.779687 |
0.0313 | 0.000488 | 0.021852 | 0.049984 | 0.043336 | 0.094244 | 0.235573 | 0.651494 |
0.0156 | 0.000122 | 0.004384 | 0.008584 | 0.008694 | 0.014913 | 0.053264 | 0.107053 |
0.0078 | 0.000031 | 0.000884 | 0.001813 | 0.001997 | 0.002953 | 0.013961 | 0.021748 |
0.0039 | 0.000008 | 0.000217 | 0.000472 | 0.000505 | 0.000698 | 0.003454 | 0.004255 |
Mesh | time step | ||||||
error of |
error of |
error of |
error of |
error of |
error of |
||
0.0625 | 0.001953 | 0.056240 | 0.161228 | 0.108336 | 0.265209 | 0.363029 | 0.779687 |
0.0313 | 0.000488 | 0.021852 | 0.049984 | 0.043336 | 0.094244 | 0.235573 | 0.651494 |
0.0156 | 0.000122 | 0.004384 | 0.008584 | 0.008694 | 0.014913 | 0.053264 | 0.107053 |
0.0078 | 0.000031 | 0.000884 | 0.001813 | 0.001997 | 0.002953 | 0.013961 | 0.021748 |
0.0039 | 0.000008 | 0.000217 | 0.000472 | 0.000505 | 0.000698 | 0.003454 | 0.004255 |
Mesh | Mesh | EOC u | EOC v | EOC |
EOC |
EOC |
EOC |
0.0625 | 0.0313 | 1.363831 | 1.689564 | 1.321875 | 1.492657 | 0.623911 | 0.259143 |
0.0313 | 0.0156 | 2.317446 | 2.541744 | 2.317474 | 2.659830 | 2.144942 | 2.605427 |
0.0156 | 0.0078 | 2.310130 | 2.243271 | 2.122186 | 2.336317 | 1.931758 | 2.299371 |
0.0078 | 0.0039 | 2.026351 | 1.941520 | 1.983479 | 2.080882 | 2.015062 | 2.353652 |
Mesh | Mesh | EOC u | EOC v | EOC |
EOC |
EOC |
EOC |
0.0625 | 0.0313 | 1.363831 | 1.689564 | 1.321875 | 1.492657 | 0.623911 | 0.259143 |
0.0313 | 0.0156 | 2.317446 | 2.541744 | 2.317474 | 2.659830 | 2.144942 | 2.605427 |
0.0156 | 0.0078 | 2.310130 | 2.243271 | 2.122186 | 2.336317 | 1.931758 | 2.299371 |
0.0078 | 0.0039 | 2.026351 | 1.941520 | 1.983479 | 2.080882 | 2.015062 | 2.353652 |
Initial state of the second wave at time |
|
Initial state of the second wave at time |
|
[1] |
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 |
[2] |
Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 |
[3] |
B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 |
[4] |
Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks and Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001 |
[5] |
Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 |
[6] |
Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203 |
[7] |
Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022063 |
[8] |
Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations and Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027 |
[9] |
Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118 |
[10] |
Mostafa Bendahmane, Kenneth H. Karlsen. Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Networks and Heterogeneous Media, 2006, 1 (1) : 185-218. doi: 10.3934/nhm.2006.1.185 |
[11] |
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 |
[12] |
Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083 |
[13] |
Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203 |
[14] |
Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021303 |
[15] |
Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377 |
[16] |
Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173 |
[17] |
Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028 |
[18] |
John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 |
[19] |
Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 |
[20] |
Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]