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 & 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. doi: 10.1016/j.physd.2009.12.003. Google Scholar

[2]

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

[3]

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

[4]

B. P. Belousov, Aperiodic reaction and its mechanism,, Collection of short papers on radiation medicine for 1958, 145 (1958). Google Scholar

[5]

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

[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). doi: 10.1007/s10231-004-0145-1. Google Scholar

[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. Google Scholar

[8]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lect. Notes Biomath., 28 (1979). Google Scholar

[9]

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

[10]

R. Finn, "Equlibrium Capillary Surfaces,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 284 (1986). Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984). Google Scholar

[17]

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

[18]

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

[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. doi: 10.1016/0167-2789(94)90054-X. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

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. doi: 10.1016/j.physd.2009.12.003. Google Scholar

[2]

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

[3]

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

[4]

B. P. Belousov, Aperiodic reaction and its mechanism,, Collection of short papers on radiation medicine for 1958, 145 (1958). Google Scholar

[5]

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

[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). doi: 10.1007/s10231-004-0145-1. Google Scholar

[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. Google Scholar

[8]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lect. Notes Biomath., 28 (1979). Google Scholar

[9]

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

[10]

R. Finn, "Equlibrium Capillary Surfaces,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 284 (1986). Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984). Google Scholar

[17]

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

[18]

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

[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. doi: 10.1016/0167-2789(94)90054-X. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[1]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

[2]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[3]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[4]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216

[5]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203

[6]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028

[7]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072

[8]

Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118

[9]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

[10]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643

[11]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

[12]

Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203

[13]

Miguel A. Herrero, Leandro Sastre. Models of aggregation in dictyostelium discoideum: On the track of spiral waves. Networks & Heterogeneous Media, 2006, 1 (2) : 241-258. doi: 10.3934/nhm.2006.1.241

[14]

C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872

[15]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[16]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851

[17]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

[18]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[19]

Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23

[20]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

[Back to Top]