September  2012, 17(6): 1605-1638. doi: 10.3934/dcdsb.2012.17.1605

The structure of the quiescent core in rigidly rotating spirals in a class of excitable systems

1. 

Departament d'Informàtica i Matemàtica Aplicada. Universitat de Girona, Campus Montilivi, EPS-P4, Girona, 17071, Spain

2. 

ICMAT (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 15, Madrid, 28049, Spain

3. 

Institute for Applied Mathematics. University of Bonn., Endenicher Allee 60, Bonn, D-53115, Germany

Received  May 2011 Revised  August 2011 Published  May 2012

We consider a class of excitable system whose dynamics is described by Fitzhugh-Nagumo (FN) equations. We provide a description for rigidly rotating spirals based on the fact that one of the unknowns develops abrupt jumps in some regions of the space. The core of the spiral is delimited by these regions. The description of the spiral is made using a mixture of asymptotic and rigorous arguments. Several open problems whose rigorous solution would provide insight in the problem are formulated.
Citation: Maria Aguareles, Marco A. Fontelos, Juan J. Velázquez. The structure of the quiescent core in rigidly rotating spirals in a class of excitable systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1605-1638. doi: 10.3934/dcdsb.2012.17.1605
References:
[1]

M. Aguareles, S. J. Chapman and T. Witelski, Motion of spiral waves in the complex Ginzburg-Landau equation, Phys. D, 239 (2010), 348-365. doi: 10.1016/j.physd.2009.12.003.

[2]

F. Alcantara and M. Monk, Signal propagation during aggregation in the slime mold Dictyostelium discoideum, J. Gen. Microbiology, 85 (1974), 321-334.

[3]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[4]

B. P. Belousov, Aperiodic reaction and its mechanism, Collection of short papers on radiation medicine for 1958, Med. Publ., 145, Moscow, 1959.

[5]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827.

[6]

B. Fiedler, J.-S. Guo and J.-C. Tsai, Rotating spirals of curvature flows: A center manifold approach, Ann. Mat. Pure Appl. (4), 185 (2006), S259-S291. doi: 10.1007/s10231-004-0145-1.

[7]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusions equations to travelling front solutions, Arch. of Rat. Mech. Anal., 65 (1977), 335-361.

[8]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lect. Notes Biomath., 28, Springer-Verlag, Berlin-New York, 1979.

[9]

P. C. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys., 39 (1985), 687-703. doi: 10.1007/BF01008360.

[10]

R. Finn, "Equlibrium Capillary Surfaces," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 284, Springer-Verlag, New York, 1986.

[11]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[12]

J. P. Keener, A geometrical theory for spiral waves in excitable media, SIAM Journal on Applied Mathematics, 46 (1986), 1039-1056. doi: 10.1137/0146062.

[13]

J. P. Keener, The core of the spiral, SIAM Journal on Applied Mathematics, 52 (1992), 1370-1390. doi: 10.1137/0152079.

[14]

J. P. Keener and J. J. Tyson, The Dynamics of scroll waves in excitable media, SIAM Review, 34 (1992), 1-39. doi: 10.1137/1034001.

[15]

D. A. Kessler, H. Levine and W. N. Reynolds, Theory of the spiral core in excitable media, Phys. D, 70 (1994), 115-139. doi: 10.1016/0167-2789(94)90060-4.

[16]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984.

[17]

J. Lechleiter, S. Girard, E. Peralta and D. Clapham, Spiral calcium wave propagation and annihilation in Xenopus-Laevis Oocytes, Science, 252 (1991), 123-126. doi: 10.1126/science.2011747.

[18]

D. Margerit and D. Barkley, Large-excitability asymptotics for scroll waves in three-dimensional excitable media, Phys. Rev. E (3), 66 (2002), 036214, 13 pp. doi: 10.1103/PhysRevE.66.036214.

[19]

A. S. Mikhailov, V. A. Davydov and V. S. Zykov, Complex dynamics of spiral waves and motion of curves, Phys. D, 70 (1994), 1-39. doi: 10.1016/0167-2789(94)90054-X.

[20]

J. V. Moloney and A. C. Newell, Nonlinear optics, Phys. D, 44 (1990), 1-37. doi: 10.1016/0167-2789(90)90045-Q.

[21]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse trasmission line simulating nerve axon, Proceeding of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[22]

J. C. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.

[23]

B. Sandstede and A. Scheel, Defects in oscillatory media: Towards a classification, SIAM J. Appl. Dyn. Syst., 3 (2004), 1-68.

[24]

A. Scheel, Bifurcation to spiral waves in reaction-diffusion systems, SIAM J. Math. Anal., 29 (1998), 1399-1418. doi: 10.1137/S0036141097318948.

[25]

F. Siegert and C. J. Weijer, Three-dimensional scroll waves organize Dictyostelium slugs, Proc. Nat. Acad. Sci., 89 (1992), 6433-6437. doi: 10.1073/pnas.89.14.6433.

[26]

J. J. Tyson and J. P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Phys. D, 32 (1988), 327-361. doi: 10.1016/0167-2789(88)90062-0.

[27]

A. T. Winfree, Electrical instability in cardiac muscle: Phase singularities and rotors, J. Theor. Biol., 138 (1989), 353-405. doi: 10.1016/S0022-5193(89)80200-0.

[28]

A. T. Winfree, Spiral waves of chemical activity, Science, 175 (1972), 634-636. doi: 10.1126/science.175.4022.634.

[29]

A. N. Zaikin and A. M. Zhabotinksy, Concentration wave propagation in two-dimensional liquid phase self-oscillating system, Nature, 225 (1970), 535-537. doi: 10.1038/225535b0.

[30]

A. M. Zhabotinksy, Periodic oscillation reactions in liquid phase, Doklady Academii Nauka SSSR, 157 (1964), 392-393.

[31]

A. M. Zhabotinksy, Periodic processes of malonic acid oscillation in a liquid phase, Biofizika, 9 (1964), 306-311.

show all references

References:
[1]

M. Aguareles, S. J. Chapman and T. Witelski, Motion of spiral waves in the complex Ginzburg-Landau equation, Phys. D, 239 (2010), 348-365. doi: 10.1016/j.physd.2009.12.003.

[2]

F. Alcantara and M. Monk, Signal propagation during aggregation in the slime mold Dictyostelium discoideum, J. Gen. Microbiology, 85 (1974), 321-334.

[3]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[4]

B. P. Belousov, Aperiodic reaction and its mechanism, Collection of short papers on radiation medicine for 1958, Med. Publ., 145, Moscow, 1959.

[5]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827.

[6]

B. Fiedler, J.-S. Guo and J.-C. Tsai, Rotating spirals of curvature flows: A center manifold approach, Ann. Mat. Pure Appl. (4), 185 (2006), S259-S291. doi: 10.1007/s10231-004-0145-1.

[7]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusions equations to travelling front solutions, Arch. of Rat. Mech. Anal., 65 (1977), 335-361.

[8]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lect. Notes Biomath., 28, Springer-Verlag, Berlin-New York, 1979.

[9]

P. C. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys., 39 (1985), 687-703. doi: 10.1007/BF01008360.

[10]

R. Finn, "Equlibrium Capillary Surfaces," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 284, Springer-Verlag, New York, 1986.

[11]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[12]

J. P. Keener, A geometrical theory for spiral waves in excitable media, SIAM Journal on Applied Mathematics, 46 (1986), 1039-1056. doi: 10.1137/0146062.

[13]

J. P. Keener, The core of the spiral, SIAM Journal on Applied Mathematics, 52 (1992), 1370-1390. doi: 10.1137/0152079.

[14]

J. P. Keener and J. J. Tyson, The Dynamics of scroll waves in excitable media, SIAM Review, 34 (1992), 1-39. doi: 10.1137/1034001.

[15]

D. A. Kessler, H. Levine and W. N. Reynolds, Theory of the spiral core in excitable media, Phys. D, 70 (1994), 115-139. doi: 10.1016/0167-2789(94)90060-4.

[16]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984.

[17]

J. Lechleiter, S. Girard, E. Peralta and D. Clapham, Spiral calcium wave propagation and annihilation in Xenopus-Laevis Oocytes, Science, 252 (1991), 123-126. doi: 10.1126/science.2011747.

[18]

D. Margerit and D. Barkley, Large-excitability asymptotics for scroll waves in three-dimensional excitable media, Phys. Rev. E (3), 66 (2002), 036214, 13 pp. doi: 10.1103/PhysRevE.66.036214.

[19]

A. S. Mikhailov, V. A. Davydov and V. S. Zykov, Complex dynamics of spiral waves and motion of curves, Phys. D, 70 (1994), 1-39. doi: 10.1016/0167-2789(94)90054-X.

[20]

J. V. Moloney and A. C. Newell, Nonlinear optics, Phys. D, 44 (1990), 1-37. doi: 10.1016/0167-2789(90)90045-Q.

[21]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse trasmission line simulating nerve axon, Proceeding of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[22]

J. C. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.

[23]

B. Sandstede and A. Scheel, Defects in oscillatory media: Towards a classification, SIAM J. Appl. Dyn. Syst., 3 (2004), 1-68.

[24]

A. Scheel, Bifurcation to spiral waves in reaction-diffusion systems, SIAM J. Math. Anal., 29 (1998), 1399-1418. doi: 10.1137/S0036141097318948.

[25]

F. Siegert and C. J. Weijer, Three-dimensional scroll waves organize Dictyostelium slugs, Proc. Nat. Acad. Sci., 89 (1992), 6433-6437. doi: 10.1073/pnas.89.14.6433.

[26]

J. J. Tyson and J. P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Phys. D, 32 (1988), 327-361. doi: 10.1016/0167-2789(88)90062-0.

[27]

A. T. Winfree, Electrical instability in cardiac muscle: Phase singularities and rotors, J. Theor. Biol., 138 (1989), 353-405. doi: 10.1016/S0022-5193(89)80200-0.

[28]

A. T. Winfree, Spiral waves of chemical activity, Science, 175 (1972), 634-636. doi: 10.1126/science.175.4022.634.

[29]

A. N. Zaikin and A. M. Zhabotinksy, Concentration wave propagation in two-dimensional liquid phase self-oscillating system, Nature, 225 (1970), 535-537. doi: 10.1038/225535b0.

[30]

A. M. Zhabotinksy, Periodic oscillation reactions in liquid phase, Doklady Academii Nauka SSSR, 157 (1964), 392-393.

[31]

A. M. Zhabotinksy, Periodic processes of malonic acid oscillation in a liquid phase, Biofizika, 9 (1964), 306-311.

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