American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 2651-2665. doi: 10.3934/dcds.2013.33.2651

DAD characterization in electromechanical cardiac models

Paolo Biscari 1, and  Chiara Lelli 1,

1. 

Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy, Italy

Received  April 2012 Revised  November 2012 Published  January 2013

We investigate the possibility of modeling the delayed afterdepolarization (DAD) occurrence in the framework of the classical FitzHugh-Nagumo (FN) dynamical system, as well as in more recent electromechanically-coupled cardiac models. Within the FN model, we identify the domain in the constitutive parameters' space for which orbits exist which exhibit a sufficiently strong secondary impulse. We then address the question whether a locally-induced secondary pulse succeeds or not in originating a self-propagating traveling impulse. Our results evidence that, in the range where secondary impulses exceed the physiological threshold for DAD onset, a local impulse almost certainly causes a traveling impulse (mechanism known as all-or-none). We then consider a recently proposed electromechanically-coupled generalization of the FN model, and show that the mechanical coupling stabilizes the system, in the sense that the more strong the coupling, the less likely is DAD to occur.
Keywords: Electromechanical cardiac models, spike sequence..
Mathematics Subject Classification: Primary: 37N25; Secondary: 92C10, 92C2.
Citation: Paolo Biscari, Chiara Lelli. DAD characterization in electromechanical cardiac models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2651-2665. doi: 10.3934/dcds.2013.33.2651
References:
[1]

C. Antzelevitch and S. Sicouri, Clinical relevance of cardiac arrhythmias generated by afterdepolarizations: Role of M cells in the generation of U waves, triggered activity and torsade de pointes,, J. Am. Coll. Cardiol., 23 (1994), 259. Google Scholar

[2]

J. Mészáros, D. Khananshvili and G. Hart, Mechanisms underlying delayed afterdepolarizations in hypertrophied left ventricular myocytes of rats,, Am. J. Physiol.-Heart. C., 281 (2001). Google Scholar

[3]

D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model,, Biophys. J., 68 (1995), 1752. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. Google Scholar

[5]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar

[6]

J. Keener and J. Sneyd, "Mathematical Physiology,", Ed. Springer Verlag, (1998). Google Scholar

[7]

P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model,, Eur. Phys. J. Plus, 126 (2011), 1. Google Scholar

[8]

A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system,, SIAM J. Appl. Math., 63 (2002), 459. doi: 10.1137/S0036139901393500. Google Scholar

[9]

K. Schlotthauer and D. M. Bers, Sarcoplasmic reticulum $Ca^{2+}$ release causes myocyte depolarization: Underlying mechanism and threshold for triggered action potentials,, Circ. Res., 87 (2000), 774. Google Scholar

[10]

Y. Xie, D. Sato, A. Garfinkel, Z. Qu and J. Weiss, So little source, so much sink: Requirements for afterdepolarizations to propagate in tissue,, Biophys. J., 99 (2010), 1408. Google Scholar

[11]

C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential II. Afterdepolarization, triggered activity, and potentiation,, Circ. Res., 74 (1994), 1097. Google Scholar

[12]

N. A. Wedge, M. S. Branicky and M. C. Cavusoglu, Proc. $26^{th}$ Int. Conf. IEEE engineering in medicine and biology society,, Ed. IEEE, (2004), 3027. Google Scholar

[13]

M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias,, Prog. Biophys. Mol. Bio., 85 (2004), 501. Google Scholar

[14]

C. Lelli, "Attraction Basin of the Equilibrium Configuration in the FitzHugh-Nagumo Model,", Acta Appl. Math., (2012). doi: 10.1007/s10440-012-9744-9. Google Scholar

[15]

M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations,, J. Math. Biol., 1 (1974), 153. Google Scholar

[16]

C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models,", Ph.D. Thesis, (2012). Google Scholar

[17]

S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations,, SIAM J. Appl. Math., 42 (1982), 247. doi: 10.1137/0142018. Google Scholar

[18]

Y. M. Usachev and S. A. Thayer, All-or-None $Ca^{2+}$ release from intracellular stores triggered by $Ca^{2+}$ influx through voltage-gated $Ca^{2+}$ channels in rat sensory neurons,, J. Neuosci., 17 (1997), 7404. Google Scholar

[19]

C. Cherubini, S. Filippi, P. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects,, Progr. Biophys. Molec. Biol., 97 (2008), 562. Google Scholar

[20]

D. Ambrosi, G. Arioli, F. Nobile and A. Quarteroni, Electromechanical coupling in cardiac dynamics: The active strain approach,, SIAM J. Appl. Math., 71 (2011), 605. doi: 10.1137/100788379. Google Scholar

[21]

J. M. Rogers and A. D. McCulloch, A Collocation-Galerkin finite element model of cardiac action potential propagation,, IEEE Trans. Biomed. Eng., 41 (1994), 743. Google Scholar

[22]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation,, Chaos Sol. Fract., 7 (1996), 293. Google Scholar

[23]

J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation,, Biophys. J., 13 (1973), 1313. Google Scholar

[24]

M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua,", Ed. Cambridge University Press, (2010). Google Scholar

[25]

S. M. Pogwizd, K. Schlotthauer, L. Li, W. Yuan and D. M. Bers, Arrhythmogenesis and contractile dysfunction in heart failure: Roles of sodium-calcium exchange, inward rectifier potassium current, and residual $\beta$-adrenergic responsiveness,, Circ. Res., 88 (2001), 1159. Google Scholar

[26]

N. P. Smith, D. P. Nickerson, E. J. Crampin and P. J. Hunter, Multiscale computational modelling of the heart,, Acta Numer., 13 (2004), 371. doi: 10.1017/S0962492904000200. Google Scholar

show all references

References:
[1]

C. Antzelevitch and S. Sicouri, Clinical relevance of cardiac arrhythmias generated by afterdepolarizations: Role of M cells in the generation of U waves, triggered activity and torsade de pointes,, J. Am. Coll. Cardiol., 23 (1994), 259. Google Scholar

[2]

J. Mészáros, D. Khananshvili and G. Hart, Mechanisms underlying delayed afterdepolarizations in hypertrophied left ventricular myocytes of rats,, Am. J. Physiol.-Heart. C., 281 (2001). Google Scholar

[3]

D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model,, Biophys. J., 68 (1995), 1752. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. Google Scholar

[5]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar

[6]

J. Keener and J. Sneyd, "Mathematical Physiology,", Ed. Springer Verlag, (1998). Google Scholar

[7]

P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model,, Eur. Phys. J. Plus, 126 (2011), 1. Google Scholar

[8]

A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system,, SIAM J. Appl. Math., 63 (2002), 459. doi: 10.1137/S0036139901393500. Google Scholar

[9]

K. Schlotthauer and D. M. Bers, Sarcoplasmic reticulum $Ca^{2+}$ release causes myocyte depolarization: Underlying mechanism and threshold for triggered action potentials,, Circ. Res., 87 (2000), 774. Google Scholar

[10]

Y. Xie, D. Sato, A. Garfinkel, Z. Qu and J. Weiss, So little source, so much sink: Requirements for afterdepolarizations to propagate in tissue,, Biophys. J., 99 (2010), 1408. Google Scholar

[11]

C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential II. Afterdepolarization, triggered activity, and potentiation,, Circ. Res., 74 (1994), 1097. Google Scholar

[12]

N. A. Wedge, M. S. Branicky and M. C. Cavusoglu, Proc. $26^{th}$ Int. Conf. IEEE engineering in medicine and biology society,, Ed. IEEE, (2004), 3027. Google Scholar

[13]

M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias,, Prog. Biophys. Mol. Bio., 85 (2004), 501. Google Scholar

[14]

C. Lelli, "Attraction Basin of the Equilibrium Configuration in the FitzHugh-Nagumo Model,", Acta Appl. Math., (2012). doi: 10.1007/s10440-012-9744-9. Google Scholar

[15]

M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations,, J. Math. Biol., 1 (1974), 153. Google Scholar

[16]

C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models,", Ph.D. Thesis, (2012). Google Scholar

[17]

S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations,, SIAM J. Appl. Math., 42 (1982), 247. doi: 10.1137/0142018. Google Scholar

[18]

Y. M. Usachev and S. A. Thayer, All-or-None $Ca^{2+}$ release from intracellular stores triggered by $Ca^{2+}$ influx through voltage-gated $Ca^{2+}$ channels in rat sensory neurons,, J. Neuosci., 17 (1997), 7404. Google Scholar

[19]

C. Cherubini, S. Filippi, P. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects,, Progr. Biophys. Molec. Biol., 97 (2008), 562. Google Scholar

[20]

D. Ambrosi, G. Arioli, F. Nobile and A. Quarteroni, Electromechanical coupling in cardiac dynamics: The active strain approach,, SIAM J. Appl. Math., 71 (2011), 605. doi: 10.1137/100788379. Google Scholar

[21]

J. M. Rogers and A. D. McCulloch, A Collocation-Galerkin finite element model of cardiac action potential propagation,, IEEE Trans. Biomed. Eng., 41 (1994), 743. Google Scholar

[22]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation,, Chaos Sol. Fract., 7 (1996), 293. Google Scholar

[23]

J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation,, Biophys. J., 13 (1973), 1313. Google Scholar

[24]

M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua,", Ed. Cambridge University Press, (2010). Google Scholar

[25]

S. M. Pogwizd, K. Schlotthauer, L. Li, W. Yuan and D. M. Bers, Arrhythmogenesis and contractile dysfunction in heart failure: Roles of sodium-calcium exchange, inward rectifier potassium current, and residual $\beta$-adrenergic responsiveness,, Circ. Res., 88 (2001), 1159. Google Scholar

[26]

N. P. Smith, D. P. Nickerson, E. J. Crampin and P. J. Hunter, Multiscale computational modelling of the heart,, Acta Numer., 13 (2004), 371. doi: 10.1017/S0962492904000200. Google Scholar

[1]

Per Danzl, Ali Nabi, Jeff Moehlis. Charge-balanced spike timing control for phase models of spiking neurons. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1413-1435. doi: 10.3934/dcds.2010.28.1413
[2]

Erik Grandelius, Kenneth H. Karlsen. The cardiac bidomain model and homogenization. Networks & Heterogeneous Media, 2019, 14 (1) : 173-204. doi: 10.3934/nhm.2019009
[3]

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
[4]

Mostafa Bendahmane, Fatima Mroue, Mazen Saad, Raafat Talhouk. Mathematical analysis of cardiac electromechanics with physiological ionic model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4863-4897. doi: 10.3934/dcdsb.2019035
[5]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[6]

Laurence Cherfils, Alain Miranville, Shuiran Peng, Chuanju Xu. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1601-1621. doi: 10.3934/dcdss.2019109
[7]

Andrea Malchiodi. Construction of multidimensional spike-layers. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 187-202. doi: 10.3934/dcds.2006.14.187
[8]

Shinsuke Koyama, Ryota Kobayashi. Fluctuation scaling in neural spike trains. Mathematical Biosciences & Engineering, 2016, 13 (3) : 537-550. doi: 10.3934/mbe.2016006
[9]

Changfeng Gui, Zhenbu Zhang. Spike solutions to a nonlocal differential equation. Communications on Pure & Applied Analysis, 2006, 5 (1) : 85-95. doi: 10.3934/cpaa.2006.5.85
[10]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[11]

Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial & Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451
[12]

Alexandre Cornet. Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1055-1076. doi: 10.3934/mbe.2018047
[13]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[14]

Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217
[15]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[16]

Andrea Malchiodi. Perturbative techniques for the construction of spike-layers. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020055
[17]

Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005
[18]

Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135
[19]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533
[20]

Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences