Skip to main content
SpringerLink
Log in

Methods

  • Chapter
  • First Online:
Complex Structure and Dynamics of the Heart

Part of the book series: Springer Theses ((Springer Theses))

  • 943 Accesses

Abstract

An important class of results generated by biophysical research in the twentieth century is that of mathematical models, both of complete biological systems and of their constituting parts.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The molecular basis of this variable permeability of the membrane (ion channels, cf. Sect. 1.2) was not known at the time.

  2. 2.

    Incidentally, this case brings the model of cardiac tissue very close to the neuronal cable equation from which it branched off. There, the assumption of an infinite conductivity in the extracellular space is justified by the large amount of fluid that surrounds the axon.

  3. 3.

    Other solutions are possible, viz. all those for which a vector potential \(\mathbf {A}\) exists with \({\underline{\sigma }}_e^*\nabla v_e=\nabla \times \mathbf {A}\). For \(v_e= -\mathbf {E}\cdot \mathbf {x}\), choose \(\mathbf {A}=-(({\underline{\sigma }}_i\mathbf {E})_3x_2,({\underline{\sigma }}_i\mathbf {E})_1x_3,({\underline{\sigma }}_i\mathbf {E})_2x_1)^T\).

  4. 4.

    \({\underline{D}}\) constant has to be required only in the interface region, because it is reasonable to assume that the dynamical system, uninfluenced by the phase field, only produces finite spatial gradients in the physical domain \(\mathcal D\). Anything else would render numerical simulations (even without the phase field method) impossible in any case. Therefore, here, it is sufficient to show that these gradients are not amplified by the choice of the steepness parameter \(\xi \) of \(\phi \). Furthermore, the assumption of a nearly straight boundary is probably not as stringent as it might first seem, as on the spatial scale of \(\xi \), every boundary with finite curvature becomes straight for \(\xi \rightarrow 0\). However, this generalization will not be shown here.

  5. 5.

    Another example for a two-variable system with a similar phase space structure is the FitzHugh-Nagumo model [35, 36], which is a reduced version of the nerve axon model constructed by Hodgkin and Huxley.

  6. 6.

    This distance is indeed finite and depends on the parameter \(\epsilon \), which determines, for a fixed starting point of the trajectory, whether the \(u\) dynamics will be successful in driving the system even more to the right and thus to performing an action potential, or whether the \(v\) dynamics is dominant and brings the system back over the \(u\)-nullcline and thus directly to the fixed point.

  7. 7.

    In allusion to some of the excitable media models used in this thesis, the letter \(\mathbf {u}\) refers here to a state in phase space (instead of the more widely-used \(\mathbf {x}\)) to avoid confusion with a position \(\mathbf {x}\in \mathbb {R}^d\) in space (with \(d=1,2,3\)).

  8. 8.

    Exceptions are, e.g., unstable fixed points or unstable periodic orbits embedded into an attractor.

  9. 9.

    The fourth-order accuracy of the nine-point stencil sometimes referred to is only present under special circumstances (for example, for the solution of the Poisson equation), because the fourth-order error has a special form that can be corrected for in those cases.

  10. 10.

    See Sect. 1.2.

  11. 11.

    To optimize measurements for certain experimental requirements—e.g., simultaneous imaging of calcium and voltage or recording of optical signals from below the surface of the tissue—there is a search for alternatives, particularly in the long wave length range [76, 77].

  12. 12.

    For further information on optical imaging in general, see Ref. [78–80].

  13. 13.

    Optical properties of di-4-ANEPPS were mainly extracted from a recent publication on ratiometric optical mapping [84], which contains collected and combined figures from a number of earlier studies (see also references therein). The properties of the LEDs can be found on the data sheet available at the manufacturer’s website http://www.luxeonstar.com/v/vspfiles/downloadables/DS30.pdf (note, however, that the product line has been discontinued).

References

  1. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500–544.

    Article  Google Scholar 

  2. Hodgkin, A. L., Huxley, A. F., & Katz, B. (1952). Measurement of current-voltage relations in the membrane of the giant axon of Loligo. Journal of Physiology, 116, 424–448.

    Article  Google Scholar 

  3. Hodgkin, A. L., & Huxley, A. F. (1952). The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. Journal of Physiology, 116, 497–506.

    Article  Google Scholar 

  4. Hodgkin, A. L., & Huxley, A. F. (1952). Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. Journal of Physiology, 116, 449–472.

    Article  Google Scholar 

  5. Hodgkin, A. L., & Huxley, A. F. (1952). The components of membrane conductance in the giant axon of Loligo. Journal of Physiology, 116, 473–496.

    Article  Google Scholar 

  6. Cole, K. (1949). Dynamic electrical characteristics of the squid axon membrane. Archives des Sciences Physiologiques, 3, 253–258.

    Google Scholar 

  7. Marmont, G. (1949). Studies on the axon membrane: I. A new method. Journal of Cellular Physiology, 34, 351–382.

    Article  Google Scholar 

  8. Noble, D. (1962). A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. Journal of Physiology, 160, 317–352.

    Article  Google Scholar 

  9. Noble, D. (1966). Applications of Hodgkin-Huxley equations to excitable tissues. Physiological Reviews, 46, 1–50.

    Google Scholar 

  10. Zhao, Y., et al. (2009). Patch clamp technique: Review of the current state of the art and potential contributions from nanoengineering. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 222, 1–11.

    Google Scholar 

  11. Tammaro, P., Shimomura, K., & Proks, P. (2008). Xenopus oocytes as a heterologous expression system for studying ion channels with the patch-clamp technique. Methods in Molecular Biology (Clifton, NJ), 491, 127–139.

    Google Scholar 

  12. Heimburg, T., & Jackson, A. D. (2005). On soliton propagation in biomembranes and nerves. Proceedings of the National Academy of Sciences of the United States of America, 102, 9790–9795.

    Article  ADS  Google Scholar 

  13. Heimburg, T., & Jackson, A. D. (2007). On the action potential as a propagating density pulse and the role of anesthetics. Biophysical Reviews and Letters, 02, 57–78.

    Article  Google Scholar 

  14. Heimburg, T. (2012). The capacitance and electromechanical coupling of lipid membranes close to transitions: The effect of electrostriction. Biophysical Journal, 103, 918–929.

    Article  ADS  Google Scholar 

  15. Wang, L. J., & Sobie, E. A. (2008). Mathematical model of the neonatal mouse ventricular action potential. American Journal of Physiology: Heart and Circulatory Physiology, 294, H2565–H2575.

    Google Scholar 

  16. Grant, A. O. (2009). Cardiac ion channels. Circulation: Arrhythmia and Electrophysiology, 2, 185–194.

    Google Scholar 

  17. Fox, R. F. (1997). Stochastic versions of the Hodgkin-Huxley equations. Biophysical Journal, 72, 2068–2074.

    Article  ADS  Google Scholar 

  18. Goldwyn, J. H., Imennov, N. S., Famulare, M., & Shea-Brown, E. (2011). Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons. Physical Review E, 83, 041908.

    Article  ADS  Google Scholar 

  19. Clark, J., & Plonsey, R. (1966). A mathematical evaluation of the core conductor model. Biophysical Journal, 6, 95–112.

    Article  ADS  Google Scholar 

  20. Spach, M. S., & Barr, R. C. (1976). Origin of epicardial ST-T wave potentials in the intact dog. Circulation Research, 39, 475–487.

    Article  Google Scholar 

  21. Miller, W. T., & Geselowitz, D. B. (1978). Simulation studies of the electrocardiogram. I. The normal heart. Circulation Research, 43, 301–315.

    Article  Google Scholar 

  22. Spach, M. S., Miller, W. T., Miller-Jones, E., Warren, R. B., & Barr, R. C. (1979). Extracellular potentials related to intracellular action potentials during impulse conduction in anisotropic canine cardiac muscle. Circulation Research, 45, 188–204.

    Article  Google Scholar 

  23. Roth, B., & Wikswo, J. (1986). A bidomain model for the extracellular potential and magnetic field of cardiac tissue. IEEE Transactions on Biomedical Engineering, 33, 467–469.

    Article  Google Scholar 

  24. Roth, B. (1991). Action potential propagation in a thick strand of cardiac muscle. Circulation Research, 68, 162–173.

    Article  Google Scholar 

  25. Clayton, R. H., et al. (2011). Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progress in Biophysics and Molecular Biology, 104, 22–48.

    Article  Google Scholar 

  26. Fenton, F. H., Cherry, E. M., Karma, A., & Rappel, W. (2005). Modeling wave propagation in realistic heart geometries using the phase-field method. Chaos, 15, 013502.

    Article  ADS  MathSciNet  Google Scholar 

  27. Bueno-Orovio, A., Pérez-García, V. M., & Fenton, F. H. (2006). Spectral methods for partial differential equations in irregular domains: The spectral smoothed boundary method. SIAM Journal on Scientific Computing, 28, 886–900.

    Article  MATH  MathSciNet  Google Scholar 

  28. Bueno-Orovio, A., Cherry, E. M., & Fenton, F. H. (2008). Minimal model for human ventricular action potentials in tissue. Journal of Theoretical Biology, 253, 544–60.

    Article  MathSciNet  Google Scholar 

  29. Li, X., Lowengrub, J., Rätz, A., & Voigt, A. (2009). Solving PDEs in complex geometries: A diffuse domain approach. Communications in Mathematical Sciences, 7, 81–107.

    Article  MATH  MathSciNet  Google Scholar 

  30. Baudoin, F. (2012). Diffusion semi-group lectures, (Wordpress.com). Lecture 3: The heat equation associated to a diffusion operator. http://fabricebaudoin.wordpress.com/2012/03/28/lecture-3-the-heat-equation-associated-to-a-diffusion-operator/.

  31. Fenton, F. H. & Cherry, E. M. (2008). Models of cardiac cell. Scholarpedia, 3, 1868. http://www.scholarpedia.org/article/Models_of_cardiac_cell.

    Google Scholar 

  32. Barkley, D., Kness, M., & Tuckerman, L. (1990). Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation. Physical Review A, 42, 2489–2492.

    Article  ADS  MathSciNet  Google Scholar 

  33. Fenton, F. H., & Karma, A. (1998). Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos, 8, 20–47.

    Article  ADS  MATH  Google Scholar 

  34. Barkley, D. (1991). A model for fast computer simulation of waves in excitable media. Physica D, 49, 61–70.

    Article  ADS  Google Scholar 

  35. FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1, 445–466.

    Article  ADS  Google Scholar 

  36. Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50, 2061–2070.

    Article  Google Scholar 

  37. Bär, M., & Eiswirth, M. (1993). Turbulence due to spiral breakup in a continuous excitable medium. Physical Review E, 48, R1635–R1637.

    Article  ADS  Google Scholar 

  38. Fenton, F. H., Cherry, E. M., Hastings, H., & Evans, S. (2002). Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos, 12, 852–892.

    Article  ADS  Google Scholar 

  39. Gray, R., Pertsov, A., & Jalife, J. (1998). Spatial and temporal organization during cardiac fibrillation. Nature, 392, 75–78.

    Article  ADS  Google Scholar 

  40. Witkowski, F., et al. (1998). Spatiotemporal evolution of ventricular fibrillation. Nature, 392, 78–82.

    Article  ADS  Google Scholar 

  41. Iyer, A., & Gray, R. (2001). An experimentalist’s approach to accurate localization of phase singularities during reentry. Annals of Biomedical Engineering, 29, 47–59.

    Article  Google Scholar 

  42. Bray, M., Lin, S.-F., Aliev, R., Roth, B., & Wikswo, J. (2001). Experimental and theoretical analysis of phase singularity dynamics in cardiac tissue. Journal of Cardiovascular Electrophysiology, 12, 716–722.

    Article  Google Scholar 

  43. Clayton, R., Zhuchkova, E., & Panfilov, A. (2006). Phase singularities and filaments: Simplifying complexity in computational models of ventricular fibrillation. Progress in Biophysics and Molecular Biology, 90, 378–398.

    Article  Google Scholar 

  44. Mermin, N. (1979). Topological theory of defects in ordered media. Reviews of Modern Physics, 51, 591–648.

    Article  ADS  MathSciNet  Google Scholar 

  45. Oseledec, V. I. A. (1968). multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Transaction of the Moscow Mathematical Society, 19, 197–231.

    Google Scholar 

  46. Ruelle, D. (1979). Analycity properties of the characteristic exponents of random matrix products. Advances in Mathematics, 32, 68–80.

    Article  MATH  MathSciNet  Google Scholar 

  47. Ruelle, D. (1979). Ergodic theory of differentiable dynamical systems. Publications Mathématiques de l’IHÉS, 50, 27–58.

    Article  MATH  MathSciNet  Google Scholar 

  48. Raghunathan, M. S. (1979). A proof of Oseledec’s multiplicative ergodic theorem. Israel Journal of Mathematics, 32, 356–362.

    Article  MATH  MathSciNet  Google Scholar 

  49. Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 15, 9–20.

    Article  ADS  MATH  Google Scholar 

  50. Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2: Numerical application. Meccanica, 15, 21–30.

    Article  Google Scholar 

  51. Eckmann, J., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617–656.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  52. Kuptsov, P. V., & Parlitz, U. (2012). Theory and computation of covariant Lyapunov vectors. Journal of Nonlinear Science, 22, 727–762.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Ginelli, F., et al. (2007). Characterizing dynamics with covariant Lyapunov vectors. Physical Review Letters, 99, 130601.

    Article  ADS  Google Scholar 

  54. Wolfe, C. L., & Samelson, R. M. (2007). An efficient method for recovering Lyapunov vectors from singular vectors. Tellus Series A: Dynamic Meteorology and Oceanography, 59, 355–366.

    Article  Google Scholar 

  55. Szendro, I. G., Pazo, D., Rodriguez, M. A., & Lopez, J. M. (2007). Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices. Physical Review E, 76, 025202.

    Article  ADS  Google Scholar 

  56. Pazo, D., Szendro, I. G., Lopez, J. M., & Rodriguez, M. A. (2008). Structure of characteristic Lyapunov vectors in spatiotemporal chaos. Physical Review E, 78, 016209.

    Article  ADS  Google Scholar 

  57. Yang, H., Takeuchi, K. A., Ginelli, F., Chate, H., & Radons, G. (2009). Hyperbolicity and the effective dimension of spatially extended dissipative systems. Physical Review Letters, 102, 074102.

    Article  ADS  Google Scholar 

  58. Hoover, W., & Hoover, C. G. (2012). Local Gram-Schmidt and covariant Lyapunov vectors and exponents for three harmonic oscillator problems. Communications in Nonlinear Science and Numerical Simulation, 17, 1043–1054.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  59. Strikwerda, J. (2004). Finite difference schemes and partial differential equations (2nd ed.). Philadelphia, PA, USA: Society for Industrial and Applied Mathematics (SIAM). http://books.google.de/books?id=SH8R_flZBGIC.

  60. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. Numerical recipes in C: The art of scientific computing (2nd ed.). New York, NY, USA: Cambridge University Press.

    Google Scholar 

  61. Cherry, E. M., Greenside, H. S., & Henriquez, C. S. (2003). Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method. Chaos, 13, 853–865.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  62. Hornung, D. (2013). Cardiac arrhythmia termination on the vascular and organ scale. Ph.D. Thesis, University of Göttingen.

    Google Scholar 

  63. Behrend, A. (2011). Interaction of pinned spiral waves with a pulsed electric field in heterogeneous excitable media. Diploma Thesis, University of Göttingen.

    Google Scholar 

  64. Linke, P. (2012). Segmentation of a human whole heart scan and simulation of the cardiac electrical conduction system. Bachelor’s Thesis, University of Göttingen.

    Google Scholar 

  65. Mayer, A. (2012). Spatio-temporal dynamics in a numerical model for ventricular tissue of neonatal mice. Bachelor’s Thesis, University of Göttingen.

    Google Scholar 

  66. Tom Wörden, H. (2012). Simulation of the electrical activity of heart tissue using the bidomain model. Bachelor’s Thesis, University of Göttingen.

    Google Scholar 

  67. MATLAB Natick, Massachusetts, USA: The MathWorks Inc. http://www.mathworks.com/.

  68. Cohen, L. B., Keynes, R. D., & Hille, B. (1968). Light scattering and birefringence changes during nerve activity. Nature, 218, 438–441.

    Article  ADS  Google Scholar 

  69. Tasaki, I., Watanabe, A., Sandlin, R., & Carnay, L. (1968). Changes in fluorescence, turbidity, and birefringence associated with nerve excitation. Proceedings of the National Academy of Sciences of the United States of America, 61, 883–888.

    Article  ADS  Google Scholar 

  70. Salzberg, B. M., Davila, H. V., & Cohen, L. B. (1973). Optical recording of impulses in individual neurones of an invertebrate central nervous system. Nature, 246, 508–509.

    Article  ADS  Google Scholar 

  71. Salama, G., & Morad, M. (1976). Merocyanine 540 as an optical probe of transmembrane electrical activity in the heart. Science, 191, 485–487.

    Article  ADS  Google Scholar 

  72. Salama, G. (1979). Optical probes of membrane potential in heart muscle. Journal of Physiology, 292, 267–295.

    Article  Google Scholar 

  73. Loew, L. M. (1982). Design and characterization of electrochromic membrane probes. Journal of Biochemical and Biophysical Methods, 6, 243–260.

    Article  Google Scholar 

  74. Fluhler, E., Burnham, V. G., & Loew, L. M. (1985). Spectra, membrane binding, and potentiometric responses of new charge shift probes. Biochemistry, 24, 5749–5755.

    Article  Google Scholar 

  75. Montana, V. (1992). A naphthyl analog of the aminostyryl pyridinium class of potentiometric membrane dyes shows consistent sensitivity in a variety of tissue, cell, and model membrane preparations. Journal of Membrane Biology, 130, 1–10.

    Google Scholar 

  76. Salama, G., et al. (2005). Properties of new, long-wavelength, voltage-sensitive dyes in the heart. Journal of Membrane Biology, 208, 125–140.

    Article  Google Scholar 

  77. Matiukas, A. (2006). New near-infrared optical probes of cardiac electrical activity. American Journal of Physiology: Heart and Circulatory Physiology, 290, H2633–H2643.

    Google Scholar 

  78. Efimov, I. R. (2004). Optical imaging of the heart. Circulation Research, 95, 21–33.

    Article  Google Scholar 

  79. Efimov, I., & Salama, G. (2012). The future of optical mapping is bright: RE: Review on: "Optical imaging of voltage and calcium in cardiac cells and tissues" by Herron, Lee, and Jalife. Circulation Research, 110, e70–e71.

    Article  Google Scholar 

  80. Herron, T. J., Lee, P., & Jalife, J. (2012). Optical imaging of voltage and calcium in cardiac cells & tissues. Circulation Research, 110, 609–623.

    Article  Google Scholar 

  81. Luther, S., et al. (2011). Low-energy control of electrical turbulence in the heart. Nature, 475, 235–239.

    Article  ADS  Google Scholar 

  82. Fenton, F. H., et al. (2009). Termination of atrial fibrillation using pulsed low-energy far-field stimulation. Circulation, 120, 467–474.

    Article  Google Scholar 

  83. Fedorov, V. V., et al. (2007). Application of blebbistatin as an excitation-contraction uncoupler for electrophysiologic study of rat and rabbit hearts. Heart Rhythm, 4, 619–626.

    Article  Google Scholar 

  84. Bachtel, A. D., et al. (2011). A novel approach to dual excitation ratiometric optical mapping of cardiac action potentials with di-4-ANEPPS using pulsed led excitation. IEEE Transactions on Biomedical Engineering, 58, 2120–2126.

    Article  Google Scholar 

  85. Morley, G., et al. (1999). Characterization of conduction in the ventricles of normal and heterozygous cx43 knockout mice using optical mapping. Journal of Cardiovascular Electrophysiology, 10, 1361–1375.

    Article  Google Scholar 

  86. Nygren, A., et al. (2000). Voltage-sensitive dye mapping of activation and conduction in adult mouse hearts. Annals of Biomedical Engineering, 28, 958–967.

    Article  Google Scholar 

  87. Mourad, A., & Nash, M. P. (2007). Method for quantifiying conduction velocity during ventricular fibrillation. Physical Review E, 75, 011914.

    Article  ADS  Google Scholar 

  88. Petitprez, S., et al. (2010). SAP97 and dystrophin macromolecular complexes determine two pools of cardiac sodium channels Nav1.5 in cardiomyocytes. Circulation Research, 108, 294–304.

    Article  Google Scholar 

  89. Zemlin, C., Mironov, S., & Pertsov, A. (2006). Near-threshold field stimulation: Intramural versus surface activation. Cardiovascular Research, 69, 98–106.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

    Philip Bittihn

Authors
  1. Philip Bittihn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philip Bittihn .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bittihn, P. (2015). Methods. In: Complex Structure and Dynamics of the Heart. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-12232-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation