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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. Google Scholar |
[11] |
M. Kolář, Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. Google Scholar |
[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. Google Scholar |
[30] |
J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[11] |
M. Kolář, Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. Google Scholar |
[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. Google Scholar |
[30] |
J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |




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 |
|
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