A cellular automaton model of excitable media including curvature and dispersion

Science

Volumn 247, Issue 4950, 1990, Pages 1563-1566

  Gerhardt, Martin ^a   Schuster, Heike ^a   Tyson, John J ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; CELL COMMUNICATION; COMPUTER SIMULATION; HUMAN; MATHEMATICS; PHYSIOLOGY;

ANIMAL; CELL COMMUNICATION; COMPUTER SIMULATION; HUMAN; MATHEMATICS; MODELS, BIOLOGICAL; SUPPORT, NON-U.S. GOV'T; SUPPORT, U.S. GOV'T, NON-P.H.S.;

EID: 0025716350     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.2321017     Document Type: Article

References (43)

1. A. T. Winfree, J. Theor. Biol. 138, 353 (1989).
2. _, When Time Breaks Down (Princeton Univ. Press, Princeton, NJ, 1987).
3. R. J. Field and M. Burger, Eds., Oscillations and Traveling Waves in Chemical Systems (Wiley, New York, 1985).
4. J. R. Brown, G. A. D'Netto, R. A. Schmitz, in Temporal Order, L. Rensing and N. I. Jaeger, Eds. (Springer-Verlag, Berlin, 1985), pp. 86-95; M. Gerhardt and H. Schuster, Physica D 36, 209 (1989).
5. J. R. Brown, G. A. D'Netto, R. A. Schmitz, in Temporal Order, L. Rensing and N. I. Jaeger, Eds. (Springer-Verlag, Berlin, 1985), pp. 86-95; M. Gerhardt and H. Schuster, Physica D 36, 209 (1989).
6. J. P. Keener, Soc. Ind. Appl. Math. J. Appl. Math. 46, 1039 (1986); _ and J. J. Tyson, Physica D 21, 307 (1986).
7. J. P. Keener, Soc. Ind. Appl. Math. J. Appl. Math. 46, 1039 (1986); _ and J. J. Tyson, Physica D 21, 307 (1986).
8. J. J. Tyson and J. P. Keener, ibid. 32, 327 (1988).
9. V. S. Zykov, Simulation of Wave Processes in Excitable Media (Manchester Univ. Press, Manchester, United Kingdom, 1987).
10. _, Biophysics (USSR) 25, 329 (1980); ibid., p. 906; P. C. Fife, in Non-Equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, Eds. (Springer-Verlag, Berlin, 1984), pp. 76-88; J. J. Tyson, J. Chim. Phys. Phys. Chim. Biol. 84, 1359 (1987).
11. _, Biophysics (USSR) 25, 329 (1980); ibid., p. 906; P. C. Fife, in Non-Equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, Eds. (Springer-Verlag, Berlin, 1984), pp. 76-88; J. J. Tyson, J. Chim. Phys. Phys. Chim. Biol. 84, 1359 (1987).
12. _, Biophysics (USSR) 25, 329 (1980); ibid., p. 906; P. C. Fife, in Non-Equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, Eds. (Springer-Verlag, Berlin, 1984), pp. 76-88; J. J. Tyson, J. Chim. Phys. Phys. Chim. Biol. 84, 1359 (1987).
13. _, Biophysics (USSR) 25, 329 (1980); ibid., p. 906; P. C. Fife, in Non-Equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, Eds. (Springer-Verlag, Berlin, 1984), pp. 76-88; J. J. Tyson, J. Chim. Phys. Phys. Chim. Biol. 84, 1359 (1987).
14. P. Foerster, S. C. Müller, B. Hess, Science 241, 685 (1988); Proc. Natl. Acad. Sci. U.S.A. 86, 6831 (1989).
15. P. Foerster, S. C. Müller, B. Hess, Science 241, 685 (1988); Proc. Natl. Acad. Sci. U.S.A. 86, 6831 (1989).
16. 0/D)].
17. N. Wiener and A. Rosenblueth, Arch. Inst. Cordial. Mex. 16, 205 (1946).
18. B. G. Parley, Comput. Biomed. Res. 1, 265 (1965); I. S. Balakhovsky, Biophysics (USSR) 10, 1175 (1965).
19. B. G. Parley, Comput. Biomed. Res. 1, 265 (1965); I. S. Balakhovsky, Biophysics (USSR) 10, 1175 (1965).
20. G. K. Moe and J. A. Abildskov, Am. Heart J. 58, 59 (1959); G. K. Moe, W. C. Rheinboldt, J. A. Abildskov, ibid. 67, 200 (1964); V. I. Krinsky, Biophysics (USSR) 11, 776 (1966); J. M. Smith and R. J. Cohen, Proc. Natl. Acad. Sci. U.S.A. 81, 233 (1984).
21. G. K. Moe and J. A. Abildskov, Am. Heart J. 58, 59 (1959); G. K. Moe, W. C. Rheinboldt, J. A. Abildskov, ibid. 67, 200 (1964); V. I. Krinsky, Biophysics (USSR) 11, 776 (1966); J. M. Smith and R. J. Cohen, Proc. Natl. Acad. Sci. U.S.A. 81, 233 (1984).
22. G. K. Moe and J. A. Abildskov, Am. Heart J. 58, 59 (1959); G. K. Moe, W. C. Rheinboldt, J. A. Abildskov, ibid. 67, 200 (1964); V. I. Krinsky, Biophysics (USSR) 11, 776 (1966); J. M. Smith and R. J. Cohen, Proc. Natl. Acad. Sci. U.S.A. 81, 233 (1984).
23. G. K. Moe and J. A. Abildskov, Am. Heart J. 58, 59 (1959); G. K. Moe, W. C. Rheinboldt, J. A. Abildskov, ibid. 67, 200 (1964); V. I. Krinsky, Biophysics (USSR) 11, 776 (1966); J. M. Smith and R. J. Cohen, Proc. Natl. Acad. Sci. U.S.A. 81, 233 (1984).
24. J. M. Greenberg and S. P. Hastings, Soc. Ind. Appl. Math. J. Appl. Math. 34, 515 (1978); J. M. Greenberg, B. D. Hassard, S. P. Hastings, Bull. Am. Math. Soc. 84, 1296 (1978); V. S. Zykov and A. S. Mikhailov, Sov. Phys. Dokl. 31, 51 (1986); M. Markus and B. Hess, in Dissipative Structures in Transport Processes and Combustion, D. Meinköhn, Ed. (Springer-Verlag, Berlin, in press). Markus and Hess have incorporated curvature effects into their automaton by introducing a randomized spatial grid.
25. J. M. Greenberg and S. P. Hastings, Soc. Ind. Appl. Math. J. Appl. Math. 34, 515 (1978); J. M. Greenberg, B. D. Hassard, S. P. Hastings, Bull. Am. Math. Soc. 84, 1296 (1978); V. S. Zykov and A. S. Mikhailov, Sov. Phys. Dokl. 31, 51 (1986); M. Markus and B. Hess, in Dissipative Structures in Transport Processes and Combustion, D. Meinköhn, Ed. (Springer-Verlag, Berlin, in press). Markus and Hess have incorporated curvature effects into their automaton by introducing a randomized spatial grid.
26. J. M. Greenberg and S. P. Hastings, Soc. Ind. Appl. Math. J. Appl. Math. 34, 515 (1978); J. M. Greenberg, B. D. Hassard, S. P. Hastings, Bull. Am. Math. Soc. 84, 1296 (1978); V. S. Zykov and A. S. Mikhailov, Sov. Phys. Dokl. 31, 51 (1986); M. Markus and B. Hess, in Dissipative Structures in Transport Processes and Combustion, D. Meinköhn, Ed. (Springer-Verlag, Berlin, in press). Markus and Hess have incorporated curvature effects into their automaton by introducing a randomized spatial grid.
27. J. M. Greenberg and S. P. Hastings, Soc. Ind. Appl. Math. J. Appl. Math. 34, 515 (1978); J. M. Greenberg, B. D. Hassard, S. P. Hastings, Bull. Am. Math. Soc. 84, 1296 (1978); V. S. Zykov and A. S. Mikhailov, Sov. Phys. Dokl. 31, 51 (1986); M. Markus and B. Hess, in Dissipative Structures in Transport Processes and Combustion, D. Meinköhn, Ed. (Springer-Verlag, Berlin, in press). Markus and Hess have incorporated curvature effects into their automaton by introducing a randomized spatial grid.
28. max)].
29. The spatial domain is uniform and isotropic by assumption. An anisotropic medium could be modeled with a rectangular neighborhood.
30. 2/s and the speed of planar solitary waves is roughly 30 cm/s, so with r = 3, one cell ≅ 0.3 mm and one time step ≅ 3 ms.
31. Using the space and time scales estimated for the Belousov-Zhabotinskii reaction in (17), we find that the spiral wave in Fig. 4 has a period of 11 s, a wave speed of 75 μm/s, and a wavelength of 0.8 mm. These values are comparable to the data observed for spirals in the Belousov-Zhabotinskii reaction, for example, S. C. Müller, T. Plesser, B. Hess, Science 230, 661 (1985).
32. O. E. Rössler and C. Kahlert, Z. Naturforsch. Teil A 34, 565 (1979); W. Jahnke, W. E. Skaggs, A. T. Winfree, J. Phys. Chem. 93, 740 (1989).
33. O. E. Rössler and C. Kahlert, Z. Naturforsch. Teil A 34, 565 (1979); W. Jahnke, W. E. Skaggs, A. T. Winfree, J. Phys. Chem. 93, 740 (1989).
34. Y. Kuramoto and S. Koga, Prog. Theor. Phys. 66, 1081 (1981); A. T. Winfree, in Cardiac Electrophysiology, from Cell to Bedside, D. P. Zipes and J. Jalife, Eds. (Saunders, Philadelphia, in press).
35. Y. Kuramoto and S. Koga, Prog. Theor. Phys. 66, 1081 (1981); A. T. Winfree, in Cardiac Electrophysiology, from Cell to Bedside, D. P. Zipes and J. Jalife, Eds. (Saunders, Philadelphia, in press).
36. M. A. Allessie, F. I. M. Bonke, F. J. G. Schopman, Circ. Res. 33, 54 (1973); E. Downar et al., J. Am. Coll. Cardiol. 4 (no. 4), 703 (1984); N. Shibata et al., Am J. Physiol. 255, 891 (1988); D. W. Frazier et al., J. Clin. Invest. 83, 1039 (1989).
37. M. A. Allessie, F. I. M. Bonke, F. J. G. Schopman, Circ. Res. 33, 54 (1973); E. Downar et al., J. Am. Coll. Cardiol. 4 (no. 4), 703 (1984); N. Shibata et al., Am J. Physiol. 255, 891 (1988); D. W. Frazier et al., J. Clin. Invest. 83, 1039 (1989).
38. M. A. Allessie, F. I. M. Bonke, F. J. G. Schopman, Circ. Res. 33, 54 (1973); E. Downar et al., J. Am. Coll. Cardiol. 4 (no. 4), 703 (1984); N. Shibata et al., Am J. Physiol. 255, 891 (1988); D. W. Frazier et al., J. Clin. Invest. 83, 1039 (1989).
39. M. A. Allessie, F. I. M. Bonke, F. J. G. Schopman, Circ. Res. 33, 54 (1973); E. Downar et al., J. Am. Coll. Cardiol. 4 (no. 4), 703 (1984); N. Shibata et al., Am J. Physiol. 255, 891 (1988); D. W. Frazier et al., J. Clin. Invest. 83, 1039 (1989).
40. This annihilation process is not necessarily an artifact of the discrete nature of our cellular automaton, because spontaneous annihilation of excitation waves has been observed in the Belousov-Zhabotinskii reaction [M. L. Smoes, in Dynamics of Synergelic Systems, H. Haken, Ed. (Springer-Verlag, Berlin, 1980), pp. 80-96] and in computations on models of heart tissue (J. P. Keener, personal communication).
41. This annihilation process is not necessarily an artifact of the discrete nature of our cellular automaton, because spontaneous annihilation of excitation waves has been observed in the Belousov-Zhabotinskii reaction [M. L. Smoes, in Dynamics of Synergelic Systems, H. Haken, Ed. (Springer-Verlag, Berlin, 1980), pp. 80-96] and in computations on models of heart tissue (J. P. Keener, personal communication).
42. One revolution of the spiral wave in Fig. 4 takes 2 s of central processing unit (cpu) time on a VAXstation 3200. A comparable calculation of one revolution of a spiral wave in a PDE model of the Belousov-Zhabotinskii reaction requires about 40 min of cpu time on the same computer. Most of the acceleration can be attributed to the fact that, because of the rapid changes registered by the excitation variable, the PDE integration routine must take temporal steps 150 times shorter than the size of the temporal step of the cellular automaton.
43. This work was supported by the National Science Foundation, grant DMS-8810456 (to J.J.T.), and by the Deutsche Forschungsgemeinschaft (to H.S.). The State Council of Higher Education in Virginia purchased the VAXstation 3200, on which all calculations were carried out. We acknowledge helpful discussion over several years with J. P. Keener, A. T. Winfree, G. B. Ermentrout, and A. Dress.