When a rotating external field larger than a critical strength is applied to a meandering spiral with frequency close to the spiral frequency, the spiral may phase-lock to the applied field and perform rigid rotation instead. We show that this conversion happens by stabilization of an unstable circular-core spiral due to the external field. From calculating overlap integrals of adjoint critical modes (response functions), the Arnold tongue for phase-locking is predicted, matching the outcome from direct numerical simulations. © 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft. Teng-Chao Li Bing-Wei Li Bo Zheng Hong Zhang Alexander Panfilov Hans Dierckx Teng-Chao Li Bing-Wei Li Bo Zheng Hong Zhang Alexander Panfilov Hans Dierckx Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, People's Republic of China Department of Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000, Gent, Belgium Department of Physics, Hangzhou Normal University, Hangzhou 311121, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, People's Republic of China Laboratory of Computational Biology and Medicine, Ural Federal University, Ekaterinburg, Russia Author to whom any correspondence should be addressed. Teng-Chao Li, Bing-Wei Li, Bo Zheng, Hong Zhang, Alexander Panfilov and Hans Dierckx 2019-04-01 2019-04-08 07:10:30 cgi/release: Article released cgi/article: new .article bin/incoming: New from .zip Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence . Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Natural Science Foundation of Zhejiang Province https://doi.org/10.13039/501100004731 LY16A050003 National Natural Science Foundation of China https://doi.org/10.13039/501100001809 11675141 11775186 China Postdoctoral Science Foundation https://doi.org/10.13039/501100002858 No. 2018M632444 yes When a rotating external field larger than a critical strength is applied to a meandering spiral with frequency close to the spiral frequency, the spiral may phase-lock to the applied field and perform rigid rotation instead. We show that this conversion happens by stabilization of an unstable circular-core spiral due to the external field. From calculating overlap integrals of adjoint critical modes (response functions), the Arnold tongue for phase-locking is predicted, matching the outcome from direct numerical simulations. General Scientific Summary Introduction and background . Spiral waves have been observed in diverse systems, from the well-known Belousov-Zhabotinsky reaction to cardiac tissue, where they cause life-threatening cardiac arrhythmias. The dynamics of a spiral wave is determined by the motion of its tip, also known as a "topological defect" or "phase singularity". Tip orbits are often not circular but follow complex cycloidal orbits, which is called "meander". It was hypothesized that spiral wave meander is the cause of a potentially lethal cardiac arrhythmia "Torsade de Pointes", which often occurs as a result of adverse drug action or genetic mutations. Thus, control of spiral wave meander is an important scientific question. Until now, several control schemes have been proposed to control or suppress meandering of spiral waves. Nevertheless, most of previous works are purely phenomenological and lack quantitative explanation. Main results . We propose a new way to control spiral wave meander, by application of a rotating external field which converts the spiral to a rigidly rotating spiral via phase-locking. Based on a response functions approach, we develop a quantitative theory to predict the stabilization regime and the conditions of phase locking and show that our theory is in excellent agreement with numerical simulations. Wider implications . We believe that the methodology presented here will be applicable to control spiral wave dynamics not only in chemical systems but also in other excitable systems in physics, chemistry and biology (e.g. cardiac tissue). We also provide one of rare examples where analytical approaches can not only qualitatively but also quantitatively predict the dynamics of a complex non-linear system. We hope it will be interesting for a wide group of experimental and theoretical scientists working on non-linear processes in excitable media. � 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft [1] Winfree A T 2001 The Geometry of Biological Time (New York: Springer) 10.1007/978-1-4757-3484-3 Winfree A T The Geometry of Biological Time 0939-6047 12 2001 [2] Cross M C and Hohenberg P C 1993 Rev. Mod. Phys. 65 851 10.1103/RevModPhys.65.851 Cross M C and Hohenberg P C Rev. Mod. Phys. 0034-6861 65 1993 851 [3] Winfree A T 1972 Science 175 634 10.1126/science.175.4022.634 Winfree A T Science 175 1972 634 [4] Ouyang Q and Flesselles J-M 1996 Nature 379 143 10.1038/379143a0 Ouyang Q and Flesselles J-M Nature 379 1996 143 [5] Jacubith S, Rotermund H, Engel W, Von Oertzen A and Ertl G 1990 Phys. Rev. Lett. 65 3013 10.1103/PhysRevLett.65.3013 Jacubith S, Rotermund H, Engel W, Von Oertzen A and Ertl G Phys. Rev. Lett. 0031-9007 65 1990 3013 [6] Nettesheim S, von Oertzen A, Rotermund H and Ertl G 1993 J. Chem. Phys. 98 9977 10.1063/1.464323 Nettesheim S, von Oertzen A, Rotermund H and Ertl G J. Chem. Phys. 98 1993 9977 [7] Morris S W, Bodenschatz E, Cannell D S and Ahlers G 1993 Phys. Rev. Lett. 71 2026 10.1103/PhysRevLett.71.2026 Morris S W, Bodenschatz E, Cannell D S and Ahlers G Phys. Rev. Lett. 0031-9007 71 1993 2026 [8] Lee K J, Cox E C and Goldstein R E 1996 Phys. Rev. Lett. 76 1174 10.1103/PhysRevLett.76.1174 Lee K J, Cox E C and Goldstein R E Phys. Rev. Lett. 76 1996 1174 [9] Sawai S, Thomason P A and Cox E C 2005 Nature 433 323 10.1038/nature03228 Sawai S, Thomason P A and Cox E C Nature 433 2005 323 [10] Gorelova N A and Bures J J 1983 J. Neurobiol. 14 353 10.1002/neu.480140503 Gorelova N A and Bures J J J. Neurobiol. 0022-3034 14 1983 353 [11] Pertsov A M, Davidenko J M, Salomontsz R, Baxter W and Jalife J 1993 Circ. Res. 72 631 10.1161/01.RES.72.3.631 Pertsov A M, Davidenko J M, Salomontsz R, Baxter W and Jalife J Circ. Res. 72 1993 631 [12] Gray R, Jalife J, Panfilov A, Baxter W, Cabo C, Davidenko J and Pertsov A 1995 Science 270 1222 10.1126/science.270.5239.1222 Gray R, Jalife J, Panfilov A, Baxter W, Cabo C, Davidenko J and Pertsov A Science 270 1995 1222 [13] Clayton R H, Zhuchkova E A and Panfilov A V 2006 Prog. Biophys. Mol. Biol. 90 378 10.1016/j.pbiomolbio.2005.06.011 Clayton R H, Zhuchkova E A and Panfilov A V Prog. Biophys. Mol. Biol. 0079-6107 90 2006 378 [14] Jahnke W, Skaggs W E and Winfree A T 1989 J. Phys. Chem. 93 740 10.1021/j100339a047 Jahnke W, Skaggs W E and Winfree A T J. Phys. Chem. 93 1989 740 [15] Li G, Ouyang Q, Petrov V and Swinney H L 1996 Phys. Rev. Lett. 77 2105 10.1103/PhysRevLett.77.2105 Li G, Ouyang Q, Petrov V and Swinney H L Phys. Rev. Lett. 77 1996 2105 [16] Zykov V S 1986 Biophysics 31 862 Zykov V S Biophysics 31 1986 862 [17] Winfree A T 1990 SIAM Rev. 32 1 10.1137/1032001 Winfree A T SIAM Rev. 32 1990 1 [18] Barkley D 1994 Phys. Rev. Lett. 72 164 10.1103/PhysRevLett.72.164 Barkley D Phys. Rev. Lett. 0031-9007 72 1994 164 [19] Winfree A T 1973 Science 181 937 10.1126/science.181.4103.937 Winfree A T Science 181 1973 937 [20] Steinbock O, Zykov V S and Müller S C 1993 Nature 366 322 10.1038/366322a0 Steinbock O, Zykov V S and Müller S C Nature 0028-0836 366 1993 322 [21] Yamazaki M et al 2012 Cardiovasc. Res. 94 48 10.1093/cvr/cvr357 Yamazaki M et al Cardiovasc. Res. 0008-6363 94 2012 48 [22] Panfilov A and Hogeweg P 1993 Phys. Lett. A 176 295 10.1016/0375-9601(93)90921-L Panfilov A and Hogeweg P Phys. Lett. 0375-9601 176 A 1993 295 [23] Bär M and Eiswirth M 1993 Phys. Rev. E 48 R1635 10.1103/PhysRevE.48.R1635 Bär M and Eiswirth M Phys. Rev. 1063-651X 48 E 1993 R1635 [24] Ouyang Q, Swinney H L and Li G 2000 Phys. Rev. Lett. 84 1047 10.1103/PhysRevLett.84.1047 Ouyang Q, Swinney H L and Li G Phys. Rev. Lett. 84 2000 1047 [25] Gray R A, Jalife J, Panfilov A V, Baxter W T, Cabo C and Pertsov A M 1995 Circulation 91 2454 10.1161/01.CIR.91.9.2454 Gray R A, Jalife J, Panfilov A V, Baxter W T, Cabo C and Pertsov A M Circulation 91 1995 2454 [26] Qu Z and Garfinkel A 2004 Cardiac Electrophysiology. From Cell to Bedside 4th edn ed D P Zipes and J Jalife (Philadelphia, PA: Saunders) pp 327–34 10.1016/B0-7216-0323-8/50039-7 Qu Z and Garfinkel A ed Zipes D P and Jalife J Cardiac Electrophysiology. From Cell to Bedside 2004 327 334 [27] Braune L and Engel H 1993 Chem. Phys. Lett. 204 257 10.1016/0009-2614(93)90005-L Braune L and Engel H Chem. Phys. Lett. 0009-2614 204 1993 257 [28] Braune M, Schrader A and Engel H 1994 Chem. Phys. Lett. 222 358 10.1016/0009-2614(94)87075-6 Braune M, Schrader A and Engel H Chem. Phys. Lett. 0009-2614 222 1994 358 [29] Grill S, Zykov V S and Müller S C 1995 Phys. Rev. Lett. 75 3368 10.1103/PhysRevLett.75.3368 Grill S, Zykov V S and Müller S C Phys. Rev. Lett. 75 1995 3368 [30] Mantel R M and Barkley D 1996 Phys. Rev. E 54 4791 10.1103/PhysRevE.54.4791 Mantel R M and Barkley D Phys. Rev. 1063-651X 54 E 1996 4791 [31] Braune M and Engel H 2000 Phys. Rev. E 62 5986 10.1103/PhysRevE.62.5986 Braune M and Engel H Phys. Rev. 1063-651X 62 E 2000 5986 [32] Zykov V S, Bordiougov G, Brandtstädter H, Gerdes I and Engel H 2003 Phys. Rev. E 68 016214 10.1103/PhysRevE.68.016214 Zykov V S, Bordiougov G, Brandtstädter H, Gerdes I and Engel H Phys. Rev. 1063-651X 68 E 016214 2003 [33] Schlesner J, Zykov V S, Engel H and Schöll E 2006 Phys. Rev. E 74 046215 10.1103/PhysRevE.74.046215 Schlesner J, Zykov V S, Engel H and Schöll E Phys. Rev. 1063-651X 74 E 046215 2006 [34] Chen J X, Zhang H and Li Y Q 2009 J. Chem. Phys. 130 124510 10.1063/1.3098543 Chen J X, Zhang H and Li Y Q J. Chem. Phys. 130 124510 2009 [35] Chen J X, Zhang H and Li Y Q 2006 J. Chem. Phys. 124 014505 10.1063/1.2145754 Chen J X, Zhang H and Li Y Q J. Chem. Phys. 124 014505 2006 [36] Ji L, Zhou Y, Li Q, Qiao C and Ouyang Q 2013 Phys. Rev. E 88 042919 10.1103/PhysRevE.88.042919 Ji L, Zhou Y, Li Q, Qiao C and Ouyang Q Phys. Rev. 1063-651X 88 E 042919 2013 [37] Li B W, Cai M C, Zhang H, Panfilov A and Dierckx H 2014 J. Chem. Phys. 140 184901 10.1063/1.4874645 Li B W, Cai M C, Zhang H, Panfilov A and Dierckx H J. Chem. Phys. 140 184901 2014 [38] Li B W, Deng L Y and Zhang H 2013 Phys. Rev. E 87 042905 10.1103/PhysRevE.87.042905 Li B W, Deng L Y and Zhang H Phys. Rev. 1063-651X 87 E 042905 2013 [39] Li T C, Gao X, Zheng F F, Cai M C, Li B W, Zhang H and Dierckx H 2016 Phys. Rev. E 93 012216 10.1103/PhysRevE.93.012216 Li T C, Gao X, Zheng F F, Cai M C, Li B W, Zhang H and Dierckx H Phys. Rev. 1063-651X 93 E 012216 2016 [40] Biktasheva I V and Biktashev V N 2003 Phys. Rev. E 67 026221 10.1103/PhysRevE.67.026221 Biktasheva I V and Biktashev V N Phys. Rev. 1063-651X 67 E 026221 2003 [41] Biktasheva I V, Holden A V and Biktashev V N 2006 Int. J. Bifurcation Chaos 16 1547 10.1142/S0218127406015490 Biktasheva I V, Holden A V and Biktashev V N Int. J. Bifurcation Chaos 0218-1274 16 2006 1547 [42] Biktasheva I V, Barkley D, Biktashev V N, Bordyuogov G V and Foulkes A J 2009 Phys. Rev. E 79 056702 10.1103/PhysRevE.79.056702 Biktasheva I V, Barkley D, Biktashev V N, Bordyuogov G V and Foulkes A J Phys. Rev. 1063-651X 79 E 056702 2009 [43] Keener J 1988 Physica D 31 269 10.1016/0167-2789(88)90080-2 Keener J Physica 0167-2789 31 D 1988 269 [44] Marcotte C D and Grigoriev R O 2016 Chaos 26 093107 10.1063/1.4962644 Marcotte C D and Grigoriev R O Chaos 26 093107 2016 [45] Dierckx H, Biktasheva I V, Verschelde H, Panfilov A V and Biktashev V N 2017 Phys. Rev. Lett. 119 258101 10.1103/PhysRevLett.119.258101 Dierckx H, Biktasheva I V, Verschelde H, Panfilov A V and Biktashev V N Phys. Rev. Lett. 119 258101 2017 [46] Barkley D 1991 Physica D 49 61 10.1016/0167-2789(91)90194-E Barkley D Physica 0167-2789 49 D 1991 61 [47] Steinbock O, Schütze J and Müller S 1992 Phys. Rev. Lett. 68 248 10.1103/PhysRevLett.68.248 Steinbock O, Schütze J and Müller S Phys. Rev. Lett. 0031-9007 68 1992 248 [48] Agladze K and De Kepper P 1992 J. Phys. Chem. 96 5239 10.1021/j100192a015 Agladze K and De Kepper P J. Phys. Chem. 96 1992 5239 [49] Krinsky V, Hamm E and Voignier V 1996 Phys. Rev. Lett. 76 3854 10.1103/PhysRevLett.76.3854 Krinsky V, Hamm E and Voignier V Phys. Rev. Lett. 76 1996 3854 [50] Feynman R 1948 Rev. Mod. Phys. 20 367 10.1103/RevModPhys.20.367 Feynman R Rev. Mod. Phys. 0034-6861 20 1948 367 [51] Dierckx H, Selsil O, Verschelde H and Biktashev V N 2012 Phys. Rev. Lett. 109 174102 10.1103/PhysRevLett.109.174102 Dierckx H, Selsil O, Verschelde H and Biktashev V N Phys. Rev. Lett. 109 174102 2012 [52] Biktashev V, Holden A and Nikolaev E 1996 Int. J. Bifurcation Chaos 6 2433 10.1142/S0218127496001582 Biktashev V, Holden A and Nikolaev E Int. J. Bifurcation Chaos 0218-1274 6 1996 2433