Localized and asynchronous patterns via canards in coupled calcium oscillators

Physica D: Nonlinear Phenomena

Volumn 215, Issue 1, 2006, Pages 46-61

  Rotstein, Horacio G ^a   Kuske, Rachel ^b  

^a Boston University   (United States)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Calcium oscillators; Canard phenomenon; Coupled oscillators; Localized oscillations; Mixed mode oscillations

Indexed keywords

CALCIUM; MATHEMATICAL MODELS; OSCILLATIONS;

CALCIUM OSCILLATORS; COUPLED OSCILLATORS; LOCALIZED OSCILLATIONS; MIXED-MODE OSCILLATIONS;

OSCILLATORS (ELECTRONIC);

EID: 33644925550     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.01.007     Document Type: Article

References (46)

1. V.K. Vanag, L. Yang, M. Dolnik, A.M. Zhabotinsky, and I.R. Epstein Oscillatory cluster patterns in a homogeneous chemical system with global feedback Nature 406 6794 2000 389 391
2. V.K. Vanag, A.M. Zhabotinsky, and I.R. Epstein Pattern formation in the Belousov-Zhabotinsky reaction with photochemical global feedback J. Phys. Chem. 104A 2000 11566 11577
3. L. Yang, M. Dolnik, A.M. Zhabotinsky, and I.R. Epstein Oscillatory clusters in a model of the photosensitive Belousov-Zhabotinsky reaction system with global feedback Phys. Rev. E 62 5 2000 6414 6420
4. R. Kuske, and T. Erneux Localized synchronization of two coupled solid state lasers Opt. Commun. 139 1997 125 131
5. R. Kuske, and T. Erneux Bifurcation to localized oscillations European J. Appl. Math. 8 1997 389 402
6. J. Rubin, and A. Bose Localized activity patterns in excitatory neuronal networks Network: Comput. Neural Sys. 15 2004 133 158
7. H.G. Rotstein, N. Kopell, A. Zhabotinsky, and I.R. Epstein A canard mechanism for localization in systems of globally coupled oscillators SIAM J. Appl. Math. 63 2003 1998 2019
8. H.G. Rotstein, N. Kopell, A. Zhabotinsky, and I.R. Epstein Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback J. Chem. Phys. 119 17 2003 8824 8832
9. F. Dumortier, and R. Roussarie Canard cycles and center manifolds Mem. Amer. Math. Soc. 121 577 1996
10. W. Eckhaus Relaxation oscillations including a standard chase on french ducks Lecture Notes in Mathematics vol. 985 1983 Springer-Verlag 449 497
11. E. Benoit, J.L. Callot, F. Dienner, and M. Dienner Chasse au Canard 1980 IRMA Strasbourg
12. M. Krupa, and P. Szmolyan Relaxation oscillation and canard explosion J. Differential Equations 174 2001 312 368
13. M. Krupa, and P. Szmolyan Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions SIAM J. Math. Anal. 33 2 2001 286 314
14. 2+ oscillations and for their frequency encoding through protein phosphorylation Proc. Natl. Acad. Sci. USA 87 1990 1461 1465
15. J. Keener, J. Sneyd, Mathematical Physiology, Springer-Verlag
16. S.M. Baer, and T. Erneux Singular Hopf bifurcation to relaxation oscillations SIAM J. Appl. Math. 52 1992 1651 1664
17. S.M. Baer, and T. Erneux Singular Hopf bifurcation to relaxation oscillations II SIAM J. Appl. Math. 46 1986 721 739
18. M. Wechselberger Existence and bifurcation of canards in R 3 in the case of a folded node SIAM J. Appl. Dyn. Syst. 4 2005 101 139
19. J. Drover, J. Rubin, J. Su, and B. Ermentrout Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies SIAM J. Appl. Math. 65 2004 69 92
20. G.S. Medvedev, and J.E. Cisternas Multimodal regimes in a compartmental model of the dopamine neuron Physica D 194 2004 333 356
21. M. Brons, M. Krupa, M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, 2005 (submitted for publication)
22. H.G. Rotstein, R. Clewley, M. Wechselberger, N. Kopell, Reduction of dimensions and dynamics of a medial entorhinal cortex layer II stellate cell model, 2005 (in preparation)
23. M. Brons, and K. Bar-Eli Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction J. Phys. Chem. 95 1991 8706 8713
24. M. Brons, P. Gross, and K. Bar-Eli Circle maps and the devil's staircase in a periodically perturbed Oregonator Internat. J. Bifur. Chaos 7 11 1997 2621 2628
25. F. Buchholtz, M. Dolnik, and I.R. Epstein Diffusion-induced instabilities near a canard J. Phys. Chem. 99 1995 15093 15101
26. M. Brons, and K. Bar-Eli Asymptotic analysis of canards in the EOE equations and the role of the inflection line J. Phys. Chem. 445 1994 305 322
27. M. Brons, and J. Sturis Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system Phys. Rev. E 64 2001 026209
28. A. Milik, and P. Szmolyan Multiple time scales and canards in a chemical oscillator C.K.R.T. Jones A. Khibnik Multiple Time-Scale Dynamical Systems (IMA Volume) 2000 Springer New York 122
29. A. Milik, P. Szmolyan, H. LOffelmann, and E. Groller Geometry of mixed-mode oscillations in the 3d-autocatalator Internat. J. Bifur. Chaos 8 1998 505 519
30. V. Petrov, S.K. Scott, and K. Showalter Mixed-mode oscillations in chemical systems J. Chem. Phys. 97 1992 6191 6198
31. B.D. Aguda, R. Larter, and B. Clarke Dynamic elements of mixed-mode oscillations and chaos in a peroxidase-oxidase model network J. Chem. Phys. 90 1989 4168 4175
32. M.T.M. Koper Bifurcations of mixed-mode oscillations in a three variable autonomous van der Pol-Duffing model with a cross-shaped phase diagram Physica D 80 1995 72 94
33. E.I. Volkov, and D.V. Volkov Multirhythmicity generated by slow variable diffusion in a ring of relaxation oscillators and noise-induced abnormal interspike variability Phys. Rev. E 65 2002 046232
34. C.J. Wilson, and J.C. Callaway A coupled oscillator model of the dopaminergic neurons of the substantia nigra J. Neurophysiol. 83 2000 3084 3100
35. P. Szmolyan, and M. Wechselberger Canards in R 3 J. Differential Equations 177 2001 419 453
36. J. Guckenheimer, K. Hoffman, and W. Weckesser Numerical computation of canards Internat. J. Bifur. Chaos 10 12 2000 2669 2687
37. K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva, and W. Weckesser The forced van der Pol oscillator II: Canard in the reduced flow SIAM J. Appl. Dyn. Syst. 2 2003 570 608
38. J. Guckenheimer, K. Hoffman, and W. Weckesser The forced van der Pol oscillator I: The slow flow and its bifurcations SIAM J. Appl. Dyn. Syst. 2 2003 1 35
39. R. FitzHugh Impulses and physiological states in models of nerve membrane Biophys. J. 1 1961 445 466
40. V.K. Vanag, A.M. Zhabotinsky, and I.R. Epstein Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction Phys. Rev. Lett. 86 2001 3
41. J.D. Murray Mathematical Biology 1989 Springer Berlin
42. S.H. Strogatz Nonlinear Dynamics and Chaos 1994 Addison Wesley Reading MA
43. E.F. Mishchenko, and N.Kh. Rozov Differential Equations with Small Parameters and Relaxation Oscillations 1980 Plenum Press New York, London
44. J. Grasman Asymptotic Methods for Relaxation Oscillations and Applications 1986 Springer-Verlag NY
45. F. Dumortier Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations D. Schlomiuk Bifurcations and Periodic Orbits of Vector Fields 1993 Kluwer Academic Press Dordrecht 19 73
46. J. Moehlis Canards in a surface oxidation reaction J. Nonlinear Sci. 12 2002 319 345