Math. Model. Nat. Phenom.
Volume 14, Number 4, 2019
Singular perturbations and multiscale systems
Article Number 406
Number of page(s) 14
Published online 28 May 2019
  1. S.M. Baer, J. Rinzel and H. Carillo, Analysis of an autonomous phase model for neuronal parabolic bursting. J. Math. Biol. 33 (1995) 309–333. [PubMed] [Google Scholar]
  2. É. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard. Collect. Math. 32 (1981) 37–119. [Google Scholar]
  3. N. Berglund and H. Kunz, Chaotic hysteresis in an adiabatically oscillating double well. Phys. Rev. Lett. 78 (1997) 1691–1694. [Google Scholar]
  4. N. Berglund, Adiabatic dynamical systems and hysteresis. Ph.D. thesis, Department of Physics, École Polytechnique Fédérale de Lausanne (EPFL), no. 1800 (1998). Available at: [Google Scholar]
  5. N. Berglund and H. Kunz, Memory effects and scaling laws in slowly driven systems. J. Phys. A: Math. Gen. 32 (1999) 15–39. [CrossRef] [Google Scholar]
  6. R. Bertram, M.J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57 (1995) 413–439. [Google Scholar]
  7. K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser, The forced van der Pol equation II: Canards in the reduced system. SIAM J. Appl. Dyn. Syst. 2 (2003) 570–608. [Google Scholar]
  8. J.-L. Callot, Bifurcations du portrait de phase pour des équations différentielles linéaires du second ordre ayant pour type l’équation d’Hermite. Ph.D thesis, Université de Strasbourg, France (1981). [Google Scholar]
  9. F. Clément and J.-P. Françoise Mathematical modeling of the GnRH pulse and surge generator. SIAM J. Appl. Dyn. Syst. 6 (2007) 441–456. [Google Scholar]
  10. C.A. Del Negro, C.F. Hsiao, S.H. Chandler and A. Garfinkel, Evidence for a novel bursting mechanism in rodent trigeminal neurons. Biophys. J. 75 (1998) 174–182. [CrossRef] [PubMed] [Google Scholar]
  11. P. De Maesschalck, F. Dumortier and R. Roussarie, Canard-cycle transition at a fast-fast passage through a jump point. C. R. Math. 352 (2014) 27–30. [Google Scholar]
  12. M. Desroches, T.J. Kaper and M. Krupa, Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. Chaos 23 (2013) 046106. [Google Scholar]
  13. M. Desroches, M. Krupa and S. Rodrigues, Spike-adding in parabolic bursters: the role of folded-saddle canards. Phys. D 331 (2016) 58–70. [CrossRef] [Google Scholar]
  14. M. Desroches and V. Kirk, Spike-adding in a canonical three-time-scale model: superslow explosion and folded-saddle canards. SIAM J. Appl. Dyn. Syst. 17 (2018) 1989–2017. [Google Scholar]
  15. M. Diener, Nessie et les canards, Institut de Recherche Mathématique Avancée. Preprint IRMA-76-P-38 (1979). [Google Scholar]
  16. M. Diener, Deux nouveaux “phénomènes-canard”. C. R. Acad. Sci. Paris Ser. A 290 (1980) 541–544. [Google Scholar]
  17. F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Memoirs of the American Mathematical Society, Rhode Island (1996) 577. [Google Scholar]
  18. G.B. Ermentrout and N.J. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46 (1986) 233–253. [Google Scholar]
  19. M. Golubitsky, K. Jǒsić and T.J. Kaper, An unfolding theory approach to bursting in fast-slow systems, in Global Analysis of Dynamical Systems, edited by H.W. Broer, B. Krauskopf and G. Vegter, CRC Press, Boca Rotan, US (2001) 282–313. [Google Scholar]
  20. J. Grasman, H. Nijmeijer and E.J.M. Veling Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator. Phys. D 13 (1984) 195–210. [CrossRef] [Google Scholar]
  21. J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications. Springer, Berlin (1987). [CrossRef] [Google Scholar]
  22. J. Grasman, Relaxation oscillations, in Encyclopedia of Complexity and Systems Science, edited by R.A. Meyers, Springer, New York, (2009) 7602–7616. [CrossRef] [Google Scholar]
  23. J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation I: the slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst. 2 (2003) 1–35. [Google Scholar]
  24. J. Guckenheimer and Y. Ilyashenko, The duck and the devil: canards on the staircase. Moscow Math. J. 1 (2001) 27–47. [CrossRef] [Google Scholar]
  25. A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500–544. [CrossRef] [PubMed] [Google Scholar]
  26. E.M. Izhikevich, Neural Excitability, Spiking, and Bursting. Int. J. Bifurc. Chaos 10 (2000) 1171–1266. [CrossRef] [Google Scholar]
  27. E.M. Izhikevich, N.S. Desai, E.C. Walcott and F.C. Hoppensteadt, Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci. 26 (2003) 161–167. [CrossRef] [PubMed] [Google Scholar]
  28. E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT press. Cambridge (2007). [Google Scholar]
  29. A. Kepecs and X.J. Wang, Analysis of complex bursting in cortical pyramidal neuron models. Neurocomputing 32 (2000) 181–187. [Google Scholar]
  30. M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion. J. Differ. Equ. 174 (2001) 312–368. [Google Scholar]
  31. J.E. Lisman, Bursts as a unit of neural information: making unreliable synapses reliable. Trends Neurosci. 20 (1997) 38–43. [CrossRef] [PubMed] [Google Scholar]
  32. J.E. Littlewood, On non-linear differential equations of the second order: III. The equation ÿ − k(1 − y2) + y = bμk cos(μt + α) for large k, and its generalizations. Acta Math. 97 (1957) 267–308. [CrossRef] [Google Scholar]
  33. J.E. Littlewood, On non-linear differential equations of the second order: IV. The general equation ÿ + kf(y) + g(y) = bkp(φ), φ = t + α. Acta Math. 98 (1957) 1–110. [CrossRef] [Google Scholar]
  34. S. Moran, S.M. Moenter and A. Khadra, A unified model for two modes of bursting in GnRH neurons. J. Comput. Neurosci. 40 (2016) 297–315. [CrossRef] [PubMed] [Google Scholar]
  35. R.E. Plant and M. Kim, On the mechanism underlying bursting in the Aplysia abdominal ganglion R15 cell. Math. Biosci. 26 (1975) 357–375. [Google Scholar]
  36. R.E. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys. J. 16 (1976) 227–244. [CrossRef] [PubMed] [Google Scholar]
  37. R.E. Plant, The effects of calcium++ on bursting neurons: a modelling study. Biophys. J. 21 (1978) 217–237. [CrossRef] [PubMed] [Google Scholar]
  38. R.E. Plant, Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biol. 11 (1981) 15–32. [CrossRef] [PubMed] [Google Scholar]
  39. J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in Proc. of the International Congress of Mathematicians. Berkeley, California (1986) 1578–1593. [Google Scholar]
  40. J. Rinzel and Y.S. Lee, Dissection of a model of neuronal parabolic bursting. J. Math. Biol. 25 (1987) 653–675. [CrossRef] [PubMed] [Google Scholar]
  41. I.V. Schurov, Ducks on the torus: existence and uniqueness. J. Dyn. Control Syst. 16 (2010) 267–300. [Google Scholar]
  42. I.V. Schurov, Canard cycles in generic fast-slow systems on the torus. Trans. Moscow Math. Soc. 71 (2010) 175–207. [CrossRef] [Google Scholar]
  43. I.V. Schurov and N. Solodovnikov, Duck factory on the two-torus: multiple canard cycles without geometric constraints. J. Dyn. Control Syst. 23 (2017) 481–498. [Google Scholar]
  44. P. Smolen, D. Terman and J. Rinzel, Properties of a bursting model with two slow inhibitory variables. SIAM J. Appl. Math. 53 (1993) 861–892. [Google Scholar]
  45. C. Soto-Treviño, N. Kopell and D. Watson, Parabolic bursting revisited. J. Math. Biol. 35 (1996) 114–128. [CrossRef] [PubMed] [Google Scholar]
  46. P. Szmolyan and M. Wechselberger, Canards in ℝ3. J. Differ. Equ. 177 (2001) 419–453. [Google Scholar]
  47. D. Terman Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51 (1991) 1418–1450. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.