Oscillatory dynamics for a coupled membrane-bulk diffusion model with Fitzhugh-Nagumo membrane kinetics.

*(English)*Zbl 1347.35032##### MSC:

35B32 | Bifurcations in context of PDEs |

35B20 | Perturbations in context of PDEs |

35B35 | Stability in context of PDEs |

92B25 | Biological rhythms and synchronization |

35K51 | Initial-boundary value problems for second-order parabolic systems |

35K58 | Semilinear parabolic equations |

35B25 | Singular perturbations in context of PDEs |

##### Keywords:

bulk diffusion; active membranes; Hopf bifurcation; winding number; synchronous oscillations; slow-fast membrane dynamics##### Software:

XPPAUT
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\textit{J. Gou} and \textit{M. J. Ward}, SIAM J. Appl. Math. 76, No. 2, 776--804 (2016; Zbl 1347.35032)

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##### References:

[1] | J. C. Alexander, Spontaneous oscillations in two 2-component cells coupled by diffusion, J. Math. Biol., 23 (1986), pp. 205–219. · Zbl 0588.92002 |

[2] | P. Ashwin, S. Coombes, and R. Nicks, Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience, arXiv:1506.05828, 2015. · Zbl 1356.92015 |

[3] | S. M. Baer, T. Erneux, and J. Rinzel, The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance, SIAM J. Appl. Math., 49 (1989), pp. 55–71. · Zbl 0683.34039 |

[4] | R. Bertram, M. J. Butte, T. Kiemel, and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), pp. 413–439. · Zbl 0813.92010 |

[5] | P. C. Bressloff and S. D. Lawley, Escape from subcellular domains with randomly switching boundaries, Multi. Model. Simul., 13 (2015), pp. 1420–1445. · Zbl 1329.82092 |

[6] | W. Y. Chiang, Y. X. Li, and P. Y. Lai, Simple models for quorum sensing: Nonlinear dynamical analysis, Phys. Rev. E., 84 (2011), 041921. |

[7] | G. B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. · Zbl 1003.68738 |

[8] | A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0837.92009 |

[9] | A. Gomez-Marin, J. Garcia-Ojalvo, and J. M. Sancho, Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling, Phys. Rev. Lett., 98 (2007), 168303. |

[10] | J. Gou, Y. X. Li, and W. Nagata, Interactions of in-phase and anti-phase synchronies in two cells coupled by a spatially diffusing chemical: Double-Hopf bifurcations, IMA J. Appl. Math., accepted. |

[11] | J. Gou, W. Y. Chiang, P. Y. Lai, M. J. Ward, and Y. X. Li, A theory of synchrony by coupling through a diffusive medium, Phys. Rev. E., submitted. · Zbl 1376.92020 |

[12] | J. Gou, Y. X. Li, W. Nagata, and M. J. Ward, Synchronized oscillatory dynamics for a 1-D model of membrane kinetics coupled by linear bulk diffusion, SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 2096–2137. · Zbl 1331.35038 |

[13] | J. Gou and M. J. Ward, An asymptotic analysis of a 2-D model of dynamically active compartments coupled by bulk diffusion, J. Nonlinear Sci., to appear. · Zbl 1439.92024 |

[14] | S. D. Lawley, J. C. Mattingly, and M. C. Reed, Stochastic switching in infinite dimensions with application to random parabolic PDEs, SIAM J. Math. Anal., 47 (2015), pp. 3035–3063. · Zbl 1338.35515 |

[15] | B. N. Kholodenko, Cell-signalling dynamics in time and space, Nat. Rev. Mol. Cell Biol., 7 (2006), pp. 165–176. |

[16] | D. Kulginov, V. P. Zhdanov, and B. Kasemo, Oscillatory surface reaction kinetics due to coupling of bistability and diffusion limitations, J. Chem. Phys., 106 (1997), p. 3117. |

[17] | H. Levine and W. J. Rappel, Membrane bound turing patterns, Phys. Rev. E., 72 (2005), 061912. |

[18] | P. Mandel and T. Erneux, The slow passage through a steady bifurcation: Delay and memory effects, J. Stat. Phys., 48 (1987), pp. 1059–1070. |

[19] | J. Müller and H. Uecker, Approximating the dynamics of communicating cells in a diffusive medium by ODEs: Homogenization with localization, J. Math. Biol., 67 (2013), pp. 1023–1065. · Zbl 1277.35036 |

[20] | J. Müller, C. Kuttler, B. A. Hense, M. Rothballer, and A. Hartmann, Cell-cell communication by quorum sensing and dimension-reduction, J. Math. Biol., 53 (2006), pp. 672–702. · Zbl 1113.92022 |

[21] | F. Naqib, T. Quail, L. Musa, H. Vulpe, J. Nadeau, J. Lei, and L. Glass, Tunable oscillations and chaotic dynamics in systems with localized synthesis, Phys. Rev. E., 85 (2012), 046210. |

[22] | H. Nakao, Phase reduction approach to synchronization of nonlinear oscillators, Contemp. Phys., Oct. 2015. |

[23] | A. P. Peirce and H. Rabitz, Effect of Defect structures on chemically active surfaces: A continuum approach, Phys. Rev. B., 38 (1998), pp. 1734–1753. |

[24] | M. Remme, M. Lengyel, and B. Gutkin, The role of ongoing dendritic oscillations in single-neuron dynamics, PLoS Comput. Biol., 5 (2009), e1000493. |

[25] | H. Riecke and L. Kramer, Surface-induced chemical oscillations and their influence on space- and time-periodic patterns, J. Chem. Phys., 83 (1985), p. 3941. |

[26] | M. A. Schwemmer and T. J. Lewis, The robustness of phase-locking in neurons with dendro-dendritic electrical coupling, J. Math. Biol., 68 (2014), pp. 303–340. · Zbl 1402.92117 |

[27] | S. Y. Shvartsman, E. Schütz, R. Imbihl, and I. G. Kevrekidis, Dynamics on microcomposite catalytic surfaces: The effect of active boundaries, Phys. Rev. Lett., 83 (1999), 2857. |

[28] | S. Smale, A Mathematical model of two cells via Turing’s equation, in Some Mathematical Questions in Biology. V J. D. Cowan, ed., Ameri. Math. Soc. Lectures on Math. Life Sci. 6, AMS, Providence, RI, 1974, pp. 15–26. |

[29] | M. J. Ward, Asymptotics for strong localized perturbations: Theory and applications, Lecture notes, Fourth Winter School on Applied Mathematics, Hong Kong, 2010. |

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