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Spiral waves are common in nature and have received a lot of attention. Spiral wave is the source of ventricular tachycardia and fibrillation, and pinned spiral wave is less likely to be eliminated than free spiral wave. Therefore, it is important to find an effective method to control the pinned spiral wave. In this work, we investigate the feedback control approach to eliminating pinned spiral wave based on the lattice Boltzmann method, by using the FitzHugh-Nagumo model as an object. The numerical results show that the feedback control method has a good control effect on the pinned spiral wave no matter whether it is pinned on a circular or rectangular obstacle. In addition, the excitability coefficient, amplitude of feedback control, recording feedback signal time and obstacle size are systematically investigated by numerical simulation. The study shows that there are three cases of pinned spiral wave cancellation. Firstly, the amplitude of feedback control and excitability coefficient are related to the time required to eliminate the pinned spiral wave, and the larger the amplitude of feedback control signal or the smaller the excitability coefficient, the faster the cancellation of the pinned spiral waveis. Secondly, the size of the obstacle and the excitability coefficient affect the time interval between the time of recording the feedback signal and the time of adding the feedback control that can successfully control the pinned spiral wave. Finally, the recorded feedback signal time affects the minimum amplitude of feedback control required to successfully eliminate the pinned spiral wave, while the added feedback control time is constant. According to the discussion in this paper, it can be seen that the feedback control method has a good control effect on the pinned spiral wave.
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Keywords:
- feedback control approach /
- pinned spiral waves /
- FitzHugh-Nagumo model /
- lattice Boltzmann method
[1] Winfree A T 1972 Science 175 634
Google Scholar
[2] Rotermund H H, Jakubith S, Von Oertzen A, Ertl G 1991 Phys. Rev. Lett. 66 3083
Google Scholar
[3] Witkowski F X, Leon L J, Penkoske P A, Giles W R, Spano M L, Ditto W L, Winfree A T 1998 Nature 392 78
Google Scholar
[4] Zakin A N, Zhabotinsky A M 1970 Nature 225 535
Google Scholar
[5] Valderrábano M, Kim Y, Yashima M, Wu T, Karagueuzian H S, Chen P S 2000 J. Am. Coll. Cardiol. 36 2000
Google Scholar
[6] Zou X, Levine H, Kessler D A 1993 Phys. Rev. E 47 R800
Google Scholar
[7] Steinbock O, Müller S C 1993 Phys. Rev. E 47 1506
[8] Fu Y Q, Zhang H, Cao Z 2005 Phys. Rev. E 72 046206
Google Scholar
[9] Ponboonjaroenchai B, Srithamma P, Kumchaiseemak N, et. al. 2017 J. Phys. Conf. Ser. 901 012027
Google Scholar
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Google Scholar
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Google Scholar
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[13] Hou Z M, Shi B C, Chai Z H 2017 Comput. Math. Appl. 74 2330
Google Scholar
[14] Chai Z H, Shi B C 2020 Phys. Rev. E 102 023306
Google Scholar
[15] Shi B C, Guo Z L 2009 Phys. Rev. E 79 016701
Google Scholar
[16] Higuera F J, Succi S 1989 EPL 8 517
Google Scholar
[17] Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
Google Scholar
[18] FitzHugh R 1961 Biophys. J 1 445
Google Scholar
[19] Courtemanche M, Skaggs W, Winfree A T 1990 Phys. D 41 173
Google Scholar
[20] Liang H, Wu H 2008 J. Am. Stat. Assoc. 103 1570
Google Scholar
[21] Concha A, Garrido R 2015 J. Comput. Nonlinear Dyn. 10 021023
Google Scholar
[22] 何雅玲, 王勇, 李庆 2009 格子Boltzmann 方法的理论及应用(北京: 科学出版社) 第123—126页
He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) pp123–126
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图 1 (a) $ t = 15 $时, $ u = 0.5 $的等值线情况; (b) $ t = 15 $ 时, $ y = 15 $处u值情况
Figure 1. (a) Contour case for $ u = 0.5 $ at $ t = 15 $; (b) value of u for $ y = 15 $ at $ t = 15 $
图 3 在点(4, 20)处u和v的值 (a) u与时间步t 的关系; (b) u和v的相图
Figure 3. Values of u and v at the point (4, 20): (a) Relationship between u and the time step t; (b) phase diagram of u and v
图 4 利用反馈法控制圆形障碍物下钉扎螺旋波的情况
Figure 4. Suppression of a pinned spiral wave under a circular obstacle by feedback control approach
图 5 $ \varepsilon = 0.004,\; R = 20,\; k_{{\mathrm{fb}}} = 0.5 $下反馈法控制钉扎螺旋波过程中$ u_{{\mathrm{all}}} $的变化值
Figure 5. Change of $ u_{{\mathrm{all}}} $ of unpinning spiral waves with $ \varepsilon = 0.004,\; R = 20,\; k_{{\mathrm{fb}}} = 0.5 $.
图 6 利用反馈法控制正方形障碍物下钉扎螺旋波的情况
Figure 6. Suppression of a pinned spiral wave under a square obstacle by feedback control approach
图 7 不同$ k_{{\mathrm{fb}}} $下消除钉扎螺旋波的时间步
Figure 7. Time step of unpinning spiral wave with different $ k_{{\mathrm{fb}}} $
图 8 (a)钉扎螺旋波的周期随可激性系数$ \varepsilon $的变化; (b)记录反馈信号时间$ t_0 $随可激性系数$ \varepsilon $的变化; (c)$ k_{{\mathrm{fbmin}}} $随可激性系数$ \varepsilon $的变化; (d)钉扎螺旋波的控制时间t随可激性系数$ \varepsilon $的变化
Figure 8. (a) Variation of the period of the pinned spiral wave with $ \varepsilon $; (b) variation of the time of recording the feedback signal with $ \varepsilon $; (c) variation of $ k_{{\mathrm{fbmin}}} $ with $ \varepsilon $; (d) variation of the time required to eliminate the pinned spiral wave with $ \varepsilon $
图 9 不同$ \varepsilon $下, 能够成功控制钉扎螺旋波的记录反馈信号时间$ t_0 $与加入反馈控制时间$ t_1 $之间时间间隔对应的范围
Figure 9. Corresponding range of the time interval between the recording feedback signal time $ t_0 $ and the adding feedback control time $ t_1 $ with different $ \varepsilon $
图 10 在不同的$ t_0 $下, 能成功控制钉扎螺旋波所需的$ k_{{\mathrm{fbmin}}} $的变化情况
Figure 10. Change of $ k_{{\mathrm{fbmin}}} $ of unpinning spiral wave successfully with different $ t_0 $
图 11 不同R下, 能够成功控制钉扎螺旋波的记录反馈信号时间$ t_0 $与加入反馈控制时间$ t_1 $之间时间间隔对应的范围
Figure 11. Corresponding range of the time interval between the recording feedback signal time $ t_0 $ and the adding feedback control time $ t_1 $ with different R
表 1 不同半径下钉扎螺旋波的波长及相对误差(1200×1200网格和1600×1600网格下)
Table 1. Wavelengths and relative errors of pinned spiral waves at different radius (under 1200×1200 grids and 1600×1600 grids).
半径 3 4 5 6 7 8 9 10 1200×1200网格下的波长 5.031 5.552 6.055 6.575 7.602 8.275 8.871 9.451 1600×1600网格下的波长 5.041 5.582 6.071 6.602 7.633 8.345 8.912 9.502 相对误差 0.0020 0.0054 0.0026 0.0041 0.0041 0.0081 0.0046 0.0047 
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[1] Winfree A T 1972 Science 175 634
Google Scholar
[2] Rotermund H H, Jakubith S, Von Oertzen A, Ertl G 1991 Phys. Rev. Lett. 66 3083
Google Scholar
[3] Witkowski F X, Leon L J, Penkoske P A, Giles W R, Spano M L, Ditto W L, Winfree A T 1998 Nature 392 78
Google Scholar
[4] Zakin A N, Zhabotinsky A M 1970 Nature 225 535
Google Scholar
[5] Valderrábano M, Kim Y, Yashima M, Wu T, Karagueuzian H S, Chen P S 2000 J. Am. Coll. Cardiol. 36 2000
Google Scholar
[6] Zou X, Levine H, Kessler D A 1993 Phys. Rev. E 47 R800
Google Scholar
[7] Steinbock O, Müller S C 1993 Phys. Rev. E 47 1506
[8] Fu Y Q, Zhang H, Cao Z 2005 Phys. Rev. E 72 046206
Google Scholar
[9] Ponboonjaroenchai B, Srithamma P, Kumchaiseemak N, et. al. 2017 J. Phys. Conf. Ser. 901 012027
Google Scholar
[10] Chen J X, Peng L, Ma J 2014 EPL 107 38001
Google Scholar
[11] Yuan G Y, Gao Z, Yan S 2021 Nonlinear Dyn. 104 2583
Google Scholar
[12] Yuan G Y, Yang S P, Wang G R, Chen S G 2008 Chin. Phys. B 17 1674
[13] Hou Z M, Shi B C, Chai Z H 2017 Comput. Math. Appl. 74 2330
Google Scholar
[14] Chai Z H, Shi B C 2020 Phys. Rev. E 102 023306
Google Scholar
[15] Shi B C, Guo Z L 2009 Phys. Rev. E 79 016701
Google Scholar
[16] Higuera F J, Succi S 1989 EPL 8 517
Google Scholar
[17] Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
Google Scholar
[18] FitzHugh R 1961 Biophys. J 1 445
Google Scholar
[19] Courtemanche M, Skaggs W, Winfree A T 1990 Phys. D 41 173
Google Scholar
[20] Liang H, Wu H 2008 J. Am. Stat. Assoc. 103 1570
Google Scholar
[21] Concha A, Garrido R 2015 J. Comput. Nonlinear Dyn. 10 021023
Google Scholar
[22] 何雅玲, 王勇, 李庆 2009 格子Boltzmann 方法的理论及应用(北京: 科学出版社) 第123—126页
He Y L, Wang Y, Li Q 2009 Lattice Boltzmann Method: Theory and Applications (Beijing: Science Press) pp123–126
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