publication . Other literature type . Preprint . Part of book or chapter of book . Article . 2013

Voltage Interval Mappings for an Elliptic Bursting Model

Jeremy Wojcik; Andrey Shilnikov;
  • Published: 19 Oct 2013
  • Publisher: Unpublished
Abstract We performed a thorough bifurcation analysis of a mathematical elliptic bursting model, using a computer-assisted reduction to equationless, one-dimensional Poincare mappings for a voltage interval. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally mixed-mode oscillations and quiescence in the FitzHugh–Nagumo–Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincare mappings, for the neuronal models and for similar (electro)chemica...
arXiv: Quantitative Biology::Neurons and CognitionAstrophysics::High Energy Astrophysical Phenomena
free text keywords: Nonlinear Sciences - Chaotic Dynamics, Mathematical analysis, Bursting, Oscillation, Poincaré conjecture, symbols.namesake, symbols, Lyapunov exponent, Topological entropy, Mathematics, Theta model, Hopf bifurcation, Homoclinic orbit, Statistical and Nonlinear Physics, Condensed Matter Physics, Biological neuron model, Bifurcation, Mathematical model, Voltage, Tonic (music)
Related Organizations
40 references, page 1 of 3

1. Wojcik J. and Shilnikov A. Voltage interval mappings for dynamics transitions in elliptic bursters. Physica D, 240:1164-1180, 2011.

2. F. Argoul and J.C. Roux. Quasiperiodicity in chemistry: an experimental path in the neibourhood of a codimension-two bifurcation. Physics Letters, 108A(8):426-430, 1985.

3. J. Rinzel. A formal classification of bursting mechanisms in excitable systems. In Proc. International Congress of Mathematics ed A M Gleason (AMS) 157893, 1987. [OpenAIRE]

4. J. Rinzel and Y. S. Lee. Dissection of a model for neuronal parabolic bursting. J Math Biol, 25(6):653-675, 1987.

5. R. Bertram, M.J. Butte, T. Kiemel, and A. Sherman. Topological and phenomenological classification of bursting oscillations. Bull.Math.Biol., 57(3):413-439, May 1995. [OpenAIRE]

6. J. Guckenheimer. Towards a global theory of singularly perturbed systems. Progress in Nonlinear Differential Equations and Their Applications, 19:214-225, 1996.

7. V.I. Arnold, V.S. Afraimovich, Yu.S. Ilyashenko, and L.P. Shilnikov. Bifurcation Theory, volume V of Dynamical Systems. Encyclopaedia of Mathematical Sciences. Springer, 1994.

8. E.F. Mischenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov. Asymptotic methods in singularly perturbed systems. Monographs in Contemporary Mathematics. Consultants Bureau, New York, 1994. [OpenAIRE]

9. A.N. Tikhonov. On the dependence of solutions of differential equations from a small parameter. Mat. Sbornik, 22(64):193-204, 1948.

10. A. I. Neishtadt. On delayed stability loss under dynamical bifurcations i. Dfferential Equations, 23:1385-1390, 1988.

11. L.P. Shilnikov, A.L. Shilnikov, D. Turaev, and L.O. Chua. Methods of qualitative theory in nonlinear dynamics, volume 1 and 2. World Scientific, Singapore, 1998, 2001. [OpenAIRE]

12. F.N. Albahadily, J. Ringland, and M. Schell. Mixed-mode oscillations in an electrochemical system. i. a farey sequence which does not occur on a torus. J. Chemical Physics, 90(2):813- 822, 1989. [OpenAIRE]

13. P. Gaspard and XJ Wang. Homoclinic orbits and mixed-mode oscillations in far-fromequilibrium. J. of Statistical Physics, 48(1/2):151-199, 1987.

14. J.L. Hudson and D Marinko. An experimental study of multiple peak periodic and nonperiodic oscillations in the belousov-zhabotinskii reaction. J. Chem. Phys., 71(4):1600-1606, 1979. [OpenAIRE]

15. R.E. Griffiths and M.C. Pernarowski. Return map characterizations for a model of bursting with two slow variables. SIAM J.Appl.Math., 66(6):1917-1948, 2006. [OpenAIRE]

40 references, page 1 of 3
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Other literature type . Preprint . Part of book or chapter of book . Article . 2013

Voltage Interval Mappings for an Elliptic Bursting Model

Jeremy Wojcik; Andrey Shilnikov;