2018, 23(2): 975-989. doi: 10.3934/dcdsb.2018051

New approach of controlling cardiac alternans

1. 

Département de Mathématiques, Université Mentouri I, Constantine, Algérie

2. 

Le2i FRE 2005, CNRS, Arts et Métiers, Univ. Bourgogne Franche-Comté, Dijon, France

* Corresponding author: sjacquir@u-bourgogne.fr

Received  February 2017 Revised  September 2017 Published  December 2017

The alternans of the cardiac action potential duration is a pathological rhythm. It is considered to be relating to the onset of ventricular fibrillation and sudden cardiac death. It is well known that, the predictive control is among the control methods that use the chaos to stabilize the unstable fixed point. Firstly, we show that alternans (or period-2 orbit) can be suppressed temporally by the predictive control of the periodic state of the system. Secondly, we determine an estimation of the size of a restricted attraction's basin of the unstable equilibrium point representing the unstable regular rhythm stabilized by the control. This result allows the application of predictive control after one beat of alternans. In particular, using predictive control of periodic dynamics, we can delay the onset of bifurcations and direct a trajectory to a desired target stationary state. Examples of the numerical results showing the stabilization of the unstable normal rhythm are given.

Citation: Mounira Kesmia, Soraya Boughaba, Sabir Jacquir. New approach of controlling cardiac alternans. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 975-989. doi: 10.3934/dcdsb.2018051
References:
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H. ArceA. XuH. Gonzalez and M. R. Guevara, Alternans and higher-order rhythms in an ionic model of a sheet of ishemic ventricular muscle, Chaos, (2000), 1054-1500.

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G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibers, J. Physiol, 268 (1977), 177-210.

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A. Beuter, L. Glass, M. C. Mackey and M. S. Titcombe, Nonlinear Dynamics Physiology and Medicine, Springer, 2003.

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R. R. Bitmead, M. Gevers and V. Wertz, Adaptive Optimal Contro, The Thinking GPC, Prentice-Hall, Englewood Cliffs, NJ, 1990.

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S. BoccalettiC. GrebogiY. C. LaiH. Mancini and D. Maza, The control of chaos: Theory and applications, Physics Reports, 329 (2000), 103-197. doi: 10.1016/S0370-1573(99)00096-4.

[7]

A. BoukabouA. Chebbah and N. Mansouri, Predictive control of continuous chaotic systems, Int. J. Bifurc. Chaos, 18 (2008), 587-592. doi: 10.1142/S0218127408020501.

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A. Boukabou, Méthodes de contrôle des systémes chaotiques d'ordre élevé et leur élevé et leur application pour la synchronisation : Contribution à l'élaboration de nouvelles approches, Thése de doctorat, Université de constantine 1,2006.

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M. E. BrandtH. T. Shih and G. R. Chen, Linear Time-delay Feedback Control of a Pathological Rhythm in a Cardiac Conduction Model, Phys. Revol. E, 56 (1997), 1334-1337.

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J. W. Cain, E. G. Tolkacheva, D. G. Schaeffer and D. J. Gauthier, Rate-dependent propagation of cardiac action potentials in a one-dimensional fiber Phys. Rev. E, 70 (2004), 061906. doi: 10.1103/PhysRevE.70.061906.

[11]

T. de CarvalhoR. D. EuzébioJ. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems -Series B, 21 (2016), 1-11.

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D. R. ChialvoR. F. Gilmour and J. Jalife, Low dimensional chaos in cardiac tissue, Nature, 343 (1990), 653-657. doi: 10.1038/343653a0.

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D. J. ChristiniK. M. SteinS. M. MarkowitzS. MittalJ. D. Slotwiner and B. B. Lerman, The role of non linear dynamics in cardiac arrhythmia control, Heart Dis., 1 (1999), 190-200.

[14]

D. J. Christini and L. Glass, Introduction: Mapping and control of complex cardiac arrhythmias, Chaos, 12 (2002), ⅱ-ⅲ and 732-981.

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G. Corliss, Which root does the bisection algorithm find?, SIAM Review, 19 (1977), 325-327. doi: 10.1137/1019044.

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S. Dai and D. G. Schaeffer, Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans Chaos, 20 (2010), 02313120, 8pp. doi: 10.1063/1.3456058.

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B. Echebarria and A. Karma, Spatiotemporal control of cardiac alternans, Chaos, 12 (2002), 923-930.

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A. GarfinkelM. L. SpanoW. L. Ditto and J. N. Weiss, Controlling cardiac chaos, Science, 257 (1992), 1230-1235. doi: 10.1126/science.1519060.

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A. GarfinkelJ. N. WeissW. L. Ditto and M. L. Spano, Chaos control of cardiac arrhythmias, Elsevier Science Inc, 94 (1995), 76-80. doi: 10.1016/1050-1738(94)00083-2.

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A. GarfinkelP. S. ChenD. O. WalterH. S. KaragueuzianB. KoganS. J. EvansM. KarpoukhinC. HwangT. UchidaM. GotohO. NwasokwaP. Sager and J. N. Weiss, Quasiperiodicity and chaos in cardiac fibrillation, J. Clin. Invest., 99 (1997), 305-314. doi: 10.1172/JCI119159.

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M. R. Guevara, F. Alonso, D. Jeandupeux, A. G. Antoni and V. Ginneken, Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model, In: Cell to Cell Signalling: From Experiments to Theoretical Models, edited by Goldbeter A. Academic Press, London, 1989.

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K. HallD. J. ChristiniM. TremblayJ. J. CollinsL. Glass and J. Billette, Dynamic control of cardiac alternans, Phys. Rev. Lett., 78 (1997), 4518-4521.

[23]

G. M. Hall and D. J. Gauthier, Experimental control of cardiac muscle alternans Phys. Rev. Lett. , 88 (2002), 198102. doi: 10.1103/PhysRevLett.88.198102.

[24]

X. HanY. ChenW. GaoJ. XueX. HanZ. FangC. Yang and X. Wu, Study of the restitution of action potential duration using the artificial neural network, Mathematical Bioscience, 207 (2007), 78-88. doi: 10.1016/j.mbs.2006.09.019.

[25]

H. M. Hastings, F. H. Fenton, S. J. Evans, O. Hotomaroglu, J. Geetha, K. Gittelson and J. Nilson, A. Garfinkel, Alternans and the onset of ventricular fibrillation, Phys. Rev. E, 62 (2000), p4043.

[26]

S. Jin YooJ. Bac Park and Y. Hochoi, Stable predictive Control of chaotic system using self -Recurrent wavelet Neural Network, International Journal of Control, Automation, and Systems, 3 (2005), 43-55.

[27]

P. N. Jordan and D. J. Christini, Adaptive diastolic interval control of cardiac action potential Duration alternans, J. Cardiovasc. Electrophysiol, 15 (2004), 1177-1185. doi: 10.1046/j.1540-8167.2004.04098.x.

[28]

D. M. Le, A. V. Dvornikov, P. Y. Lai and C. K. Chan, Predicting Self-terminating Ventricular Fibrillation in an Isolated Heart Europhys. Lett. , 104 (2013), 48002. doi: 10.1209/0295-5075/104/48002.

[29]

T. J. Lewis and M. R. Guevara, Chaotic dynamics in an ionic model of the propagated cardiac action potential, J. Theor. Biol., 146 (1990), 407-432. doi: 10.1016/S0022-5193(05)80750-7.

[30]

S. W. Morgan, I. V. Biktasheva and V. N. Biktashev, Control of scroll wave turbulence using resonant perturbations, Phys. Rev. E, 78 (2008), 046207, 13pp.

[31]

J. B. Nolasco and R. W. Dahlen, A graphic method for the study of alternation of cardiac action potentials", J. Appl. Physiol., 25 (1968), 191-196.

[32]

E. Ott, C. Grebogi and J. A. York, Controlling chaos Phys. Rev. Lett. , 64 (1990), 2837. doi: 10.1103/PhysRevLett.64.1196.

[33]

E. Ott, C. Grebogi and J. A. Yorke, Controlling chaotic dynamical systems, D. Campbell (Ed. ), Chaos/XAOC, Soviet-American Perspective on Nonlinear Science, American Institute of Physics, New York, 8 (1990).

[34]

R. Pandit, A. Pande, S. Sinha and A. Sen, Spiral turbulence and spatiotemporal chaos: characterization and control in two excitable media, Physica A, 306 (2002), 211-21.

[35]

K. D. PhungG. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete and Continuous Dynamical Systems -Series B, 8 (2007), 925-941. doi: 10.3934/dcdsb.2007.8.925.

[36]

K. Pyragas, Continues control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428.

[37]

Z. Qu, Nonlinear dynamic Control of Irregular Rythms, J. Cardiovasc. Electrophysiol., 15 (2004), 1186-1187.

[38]

Z. Qu, J. N. Weiss and A. Garfinkel, Spatiotemporal Chaos in a Simulated Ring of Cardiac Cells Phys. Rev. Lett. , 78 (1997), p1387. doi: 10.1103/PhysRevLett.78.1387.

[39]

W. J. RappelF. Fenton and A. Karma, Spatiotemporal control of wave instabilities in cardiac tissue, Phys. Rev. Lett, 83 (1999), 456-459. doi: 10.1103/PhysRevLett.83.456.

[40]

F. J. RomeirasC. GrebogiE. Ott and W. P. Dayawansa, Controlling chaotic dynamical systems, Physica D, 58 (1992), 165-192. doi: 10.1016/0167-2789(92)90107-X.

[41]

T. ShinbrotC. GrebogiE. Ott and J. Yorke, Using chaos to target stationary states of flows, Physics Letters A, 169 (1992), 349-354. doi: 10.1016/0375-9601(92)90239-I.

[42]

J. Smallwood, Chaos Control of the Hénon map and an Impact Oscillator by the Ott-Grebogi-York Method, The Nonlinear Journal, 2 (2000), 37-44.

[43]

E. G. Tolkacheva, M. M. Romeo, M. Guerraty and J. Daniel, condition for alternans and its controln in a two-dimensional mapping model of paced cardiac dynamics, Phys. Rev. E, 69 (2004), 031904.

[44]

T. Ushio and S. Yamamoto, Prediction based control of chaos, Phy. Lett. A, 264 (1999), 30-35. doi: 10.1016/S0375-9601(99)00782-3.

[45]

M. Watanabe and Jr. Gilmour, Strategy for control of complex lowdimensional dynamics in cardiac tissue, J Math. Biol., 35 (1996), 73-87. doi: 10.1007/s002850050043.

[46]

J. N. WeissA. GarfinkelM. L. Spano and W. L. Ditto, Chaos and chaos control in biology. The American society for clinical investigation, Inc, (1994), 1355-1360.

[47]

B. XuS. JacquirG. LaurentJ.-M. Bilbault and S. Binczak, Analysis of an experimental model of in vitro cardiac tissue using phase space reconstruction, Biomedical Signal Processing and Control, 13 (2014), 313-326.

[48]

B. XuS. JacquirG. LaurentJ.-M. Bilbault and S. Binczak, A hybrid stimulation strategy for suppression of spiral waves in cardiac tissue, Chaos, Solitons and Fractals, 44 (2011), 633-639.

[49]

G. Zheng-Ning and C. Xin-Ming, Distributed predictive control of spiral wave in cardiac excitable media Chin. Phys. B, 19 (2010), 050514. doi: 10.1088/1674-1056/19/5/050514.

show all references

References:
[1]

S. AlonsoF. Sagués and A. S. Mikhailov, Taming Winfree turbulence of scroll waves in excitable media, Science, 299 (2003), 1722-1725. doi: 10.1126/science.1080207.

[2]

H. ArceA. XuH. Gonzalez and M. R. Guevara, Alternans and higher-order rhythms in an ionic model of a sheet of ishemic ventricular muscle, Chaos, (2000), 1054-1500.

[3]

G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibers, J. Physiol, 268 (1977), 177-210.

[4]

A. Beuter, L. Glass, M. C. Mackey and M. S. Titcombe, Nonlinear Dynamics Physiology and Medicine, Springer, 2003.

[5]

R. R. Bitmead, M. Gevers and V. Wertz, Adaptive Optimal Contro, The Thinking GPC, Prentice-Hall, Englewood Cliffs, NJ, 1990.

[6]

S. BoccalettiC. GrebogiY. C. LaiH. Mancini and D. Maza, The control of chaos: Theory and applications, Physics Reports, 329 (2000), 103-197. doi: 10.1016/S0370-1573(99)00096-4.

[7]

A. BoukabouA. Chebbah and N. Mansouri, Predictive control of continuous chaotic systems, Int. J. Bifurc. Chaos, 18 (2008), 587-592. doi: 10.1142/S0218127408020501.

[8]

A. Boukabou, Méthodes de contrôle des systémes chaotiques d'ordre élevé et leur élevé et leur application pour la synchronisation : Contribution à l'élaboration de nouvelles approches, Thése de doctorat, Université de constantine 1,2006.

[9]

M. E. BrandtH. T. Shih and G. R. Chen, Linear Time-delay Feedback Control of a Pathological Rhythm in a Cardiac Conduction Model, Phys. Revol. E, 56 (1997), 1334-1337.

[10]

J. W. Cain, E. G. Tolkacheva, D. G. Schaeffer and D. J. Gauthier, Rate-dependent propagation of cardiac action potentials in a one-dimensional fiber Phys. Rev. E, 70 (2004), 061906. doi: 10.1103/PhysRevE.70.061906.

[11]

T. de CarvalhoR. D. EuzébioJ. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems -Series B, 21 (2016), 1-11.

[12]

D. R. ChialvoR. F. Gilmour and J. Jalife, Low dimensional chaos in cardiac tissue, Nature, 343 (1990), 653-657. doi: 10.1038/343653a0.

[13]

D. J. ChristiniK. M. SteinS. M. MarkowitzS. MittalJ. D. Slotwiner and B. B. Lerman, The role of non linear dynamics in cardiac arrhythmia control, Heart Dis., 1 (1999), 190-200.

[14]

D. J. Christini and L. Glass, Introduction: Mapping and control of complex cardiac arrhythmias, Chaos, 12 (2002), ⅱ-ⅲ and 732-981.

[15]

G. Corliss, Which root does the bisection algorithm find?, SIAM Review, 19 (1977), 325-327. doi: 10.1137/1019044.

[16]

S. Dai and D. G. Schaeffer, Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans Chaos, 20 (2010), 02313120, 8pp. doi: 10.1063/1.3456058.

[17]

B. Echebarria and A. Karma, Spatiotemporal control of cardiac alternans, Chaos, 12 (2002), 923-930.

[18]

A. GarfinkelM. L. SpanoW. L. Ditto and J. N. Weiss, Controlling cardiac chaos, Science, 257 (1992), 1230-1235. doi: 10.1126/science.1519060.

[19]

A. GarfinkelJ. N. WeissW. L. Ditto and M. L. Spano, Chaos control of cardiac arrhythmias, Elsevier Science Inc, 94 (1995), 76-80. doi: 10.1016/1050-1738(94)00083-2.

[20]

A. GarfinkelP. S. ChenD. O. WalterH. S. KaragueuzianB. KoganS. J. EvansM. KarpoukhinC. HwangT. UchidaM. GotohO. NwasokwaP. Sager and J. N. Weiss, Quasiperiodicity and chaos in cardiac fibrillation, J. Clin. Invest., 99 (1997), 305-314. doi: 10.1172/JCI119159.

[21]

M. R. Guevara, F. Alonso, D. Jeandupeux, A. G. Antoni and V. Ginneken, Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model, In: Cell to Cell Signalling: From Experiments to Theoretical Models, edited by Goldbeter A. Academic Press, London, 1989.

[22]

K. HallD. J. ChristiniM. TremblayJ. J. CollinsL. Glass and J. Billette, Dynamic control of cardiac alternans, Phys. Rev. Lett., 78 (1997), 4518-4521.

[23]

G. M. Hall and D. J. Gauthier, Experimental control of cardiac muscle alternans Phys. Rev. Lett. , 88 (2002), 198102. doi: 10.1103/PhysRevLett.88.198102.

[24]

X. HanY. ChenW. GaoJ. XueX. HanZ. FangC. Yang and X. Wu, Study of the restitution of action potential duration using the artificial neural network, Mathematical Bioscience, 207 (2007), 78-88. doi: 10.1016/j.mbs.2006.09.019.

[25]

H. M. Hastings, F. H. Fenton, S. J. Evans, O. Hotomaroglu, J. Geetha, K. Gittelson and J. Nilson, A. Garfinkel, Alternans and the onset of ventricular fibrillation, Phys. Rev. E, 62 (2000), p4043.

[26]

S. Jin YooJ. Bac Park and Y. Hochoi, Stable predictive Control of chaotic system using self -Recurrent wavelet Neural Network, International Journal of Control, Automation, and Systems, 3 (2005), 43-55.

[27]

P. N. Jordan and D. J. Christini, Adaptive diastolic interval control of cardiac action potential Duration alternans, J. Cardiovasc. Electrophysiol, 15 (2004), 1177-1185. doi: 10.1046/j.1540-8167.2004.04098.x.

[28]

D. M. Le, A. V. Dvornikov, P. Y. Lai and C. K. Chan, Predicting Self-terminating Ventricular Fibrillation in an Isolated Heart Europhys. Lett. , 104 (2013), 48002. doi: 10.1209/0295-5075/104/48002.

[29]

T. J. Lewis and M. R. Guevara, Chaotic dynamics in an ionic model of the propagated cardiac action potential, J. Theor. Biol., 146 (1990), 407-432. doi: 10.1016/S0022-5193(05)80750-7.

[30]

S. W. Morgan, I. V. Biktasheva and V. N. Biktashev, Control of scroll wave turbulence using resonant perturbations, Phys. Rev. E, 78 (2008), 046207, 13pp.

[31]

J. B. Nolasco and R. W. Dahlen, A graphic method for the study of alternation of cardiac action potentials", J. Appl. Physiol., 25 (1968), 191-196.

[32]

E. Ott, C. Grebogi and J. A. York, Controlling chaos Phys. Rev. Lett. , 64 (1990), 2837. doi: 10.1103/PhysRevLett.64.1196.

[33]

E. Ott, C. Grebogi and J. A. Yorke, Controlling chaotic dynamical systems, D. Campbell (Ed. ), Chaos/XAOC, Soviet-American Perspective on Nonlinear Science, American Institute of Physics, New York, 8 (1990).

[34]

R. Pandit, A. Pande, S. Sinha and A. Sen, Spiral turbulence and spatiotemporal chaos: characterization and control in two excitable media, Physica A, 306 (2002), 211-21.

[35]

K. D. PhungG. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete and Continuous Dynamical Systems -Series B, 8 (2007), 925-941. doi: 10.3934/dcdsb.2007.8.925.

[36]

K. Pyragas, Continues control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428.

[37]

Z. Qu, Nonlinear dynamic Control of Irregular Rythms, J. Cardiovasc. Electrophysiol., 15 (2004), 1186-1187.

[38]

Z. Qu, J. N. Weiss and A. Garfinkel, Spatiotemporal Chaos in a Simulated Ring of Cardiac Cells Phys. Rev. Lett. , 78 (1997), p1387. doi: 10.1103/PhysRevLett.78.1387.

[39]

W. J. RappelF. Fenton and A. Karma, Spatiotemporal control of wave instabilities in cardiac tissue, Phys. Rev. Lett, 83 (1999), 456-459. doi: 10.1103/PhysRevLett.83.456.

[40]

F. J. RomeirasC. GrebogiE. Ott and W. P. Dayawansa, Controlling chaotic dynamical systems, Physica D, 58 (1992), 165-192. doi: 10.1016/0167-2789(92)90107-X.

[41]

T. ShinbrotC. GrebogiE. Ott and J. Yorke, Using chaos to target stationary states of flows, Physics Letters A, 169 (1992), 349-354. doi: 10.1016/0375-9601(92)90239-I.

[42]

J. Smallwood, Chaos Control of the Hénon map and an Impact Oscillator by the Ott-Grebogi-York Method, The Nonlinear Journal, 2 (2000), 37-44.

[43]

E. G. Tolkacheva, M. M. Romeo, M. Guerraty and J. Daniel, condition for alternans and its controln in a two-dimensional mapping model of paced cardiac dynamics, Phys. Rev. E, 69 (2004), 031904.

[44]

T. Ushio and S. Yamamoto, Prediction based control of chaos, Phy. Lett. A, 264 (1999), 30-35. doi: 10.1016/S0375-9601(99)00782-3.

[45]

M. Watanabe and Jr. Gilmour, Strategy for control of complex lowdimensional dynamics in cardiac tissue, J Math. Biol., 35 (1996), 73-87. doi: 10.1007/s002850050043.

[46]

J. N. WeissA. GarfinkelM. L. Spano and W. L. Ditto, Chaos and chaos control in biology. The American society for clinical investigation, Inc, (1994), 1355-1360.

[47]

B. XuS. JacquirG. LaurentJ.-M. Bilbault and S. Binczak, Analysis of an experimental model of in vitro cardiac tissue using phase space reconstruction, Biomedical Signal Processing and Control, 13 (2014), 313-326.

[48]

B. XuS. JacquirG. LaurentJ.-M. Bilbault and S. Binczak, A hybrid stimulation strategy for suppression of spiral waves in cardiac tissue, Chaos, Solitons and Fractals, 44 (2011), 633-639.

[49]

G. Zheng-Ning and C. Xin-Ming, Distributed predictive control of spiral wave in cardiac excitable media Chin. Phys. B, 19 (2010), 050514. doi: 10.1088/1674-1056/19/5/050514.

Figure 1.  The time evolution of transmembrane action potential using cable simulations (Beeler-Reuter model) for a periodically paced cell: (a) the response $1:1$, (b) the presence of alternans $2:2$, (c) the response $2:1$, (d) the irregular response. APD means Action Potential Duration, DI means Diastolic Interval.
Figure 2.  Bifurcation diagram from [29]
Figure 3.  At $t_s = 302$ $ms$ the alternans without control evolution over 20000 iterations starting from initial condition $APD_1 = 240$ $ms.$
Figure 4.  Initiation of predictive control of the alternans $2:2$, (a) after $838$ beats of alternans, (b) after only one beat of alternans. After control is initiated in figure (a) or (b), $APD_{n}$ alternates around $APD^{\ast}$ as the asymptotic stability of $1:1$ rhythm is performed.
Figure 5.  Bifurcation diagram ($APD_i$ vs. $t_s$) without control for $t_s = 200-400$ $ms$. At each $t_s$, the map (1) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $t_s$ was $0.1$ $ms$, $APD_1 = 198$ $ms$.
Figure 6.  Bifurcation diagram ($APD_i$ vs. $t_s$) with predictive control for $t_s = 200-400$ $ms$. At each $t_s$, the controlled map (3) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $t_s$ was $0.1$ $ms$. $APD_1 = 198$ $ms$, $\varepsilon = 7.7$ $ms$, $K = -0.2.$
Figure 7.  Initiation of predictive control of periodic rhythm $6:2$, (a) after 2 beats, (b) after 1 beat. After control is initiated in Fig(a) or (b), $APD_{n}$ alternates about $APD^{\ast}$ as the asymptotic stability of $6:1$ rhythm is performed.
Figure 8.  The suppression of noisy alternans with the predictive control.
Figure 9.  Bifurcation diagram ($APD_i$ vs. $\varepsilon$) with predictive control for $\varepsilon = 0-15$ $ms$. At each $\varepsilon$, the controlled map (3) was iterated 20000 times and the first 19800 iterates discarded to suppress transients due to initial conditions. Increment in $\varepsilon$ was $0.1$ $ms$, $t_s = 302$ $ms$, $APD_1 = 240$ $ms$, $K = -0.1$.
Table 1.  Example of parameter values $K$, $\varepsilon$ and $APD_{1}$ for the stabilisation of the unstable equilibrium point (or $1:1$ unstable rhythm).
K $\varepsilon$ $APD_{1}$
-0.1 9 ms 240 ms
-0.1 7.7 ms 198 ms
K $\varepsilon$ $APD_{1}$
-0.1 9 ms 240 ms
-0.1 7.7 ms 198 ms
Table 2.  Example of parameter values $K$, $\varepsilon$ and $APD_{1}$ for for controlling the periodic rhythm $6:2$.
K $\varepsilon$ $APD_{1}$
-0.4 1.96 ms 240 ms
-0.4 1.96 ms 202 ms
K $\varepsilon$ $APD_{1}$
-0.4 1.96 ms 240 ms
-0.4 1.96 ms 202 ms
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