On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

M. Zaks

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 6, Issue: 1, page 149-162
  • ISSN: 0973-5348

Abstract

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In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.

How to cite

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Zaks, M.. "On Chaotic Subthreshold Oscillations in a Simple Neuronal Model." Mathematical Modelling of Natural Phenomena 6.1 (2010): 149-162. <http://eudml.org/doc/197663>.

@article{Zaks2010,
abstract = {In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.},
author = {Zaks, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {slow-fast equations; period-doubling bifurcation},
language = {eng},
month = {6},
number = {1},
pages = {149-162},
publisher = {EDP Sciences},
title = {On Chaotic Subthreshold Oscillations in a Simple Neuronal Model},
url = {http://eudml.org/doc/197663},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Zaks, M.
TI - On Chaotic Subthreshold Oscillations in a Simple Neuronal Model
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/6//
PB - EDP Sciences
VL - 6
IS - 1
SP - 149
EP - 162
AB - In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.
LA - eng
KW - slow-fast equations; period-doubling bifurcation
UR - http://eudml.org/doc/197663
ER -

References

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  1. V. I. Arnold (Editor). Dynamical systems V: Bifurcation theory and catastrophe theory. Encyclopaedia of Mathematical Sciences. Springer. New York, Berlin, Heidelberg, 1999.  
  2. M. Brons, M. Krupa, M. Wechselberger. Mixed mode oscillations due to the generalized canard phenomenon. Fields Institute Communications, 49 (2006), 39–63. 
  3. M. Brons, T. J. Kaper, H. G. Rotstein (Editors). Mixed Mode Oscillations: Experiment, Computation, and Analysis. Focus Issue of Chaos, 18 (2008).  
  4. J. L. Callot, F. Diener, M. Diener. Problem of duck hunt. Compt. Rend. Acad. Sci., 286 (1978), 1059–1061.  
  5. P. Collet, J.-P. Eckmann, H. Koch. On universality for area-preserving maps of the plane. Physica D, 3 (1981), 457–467. 
  6. W. Eckhaus. Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math., 985 (1983), 449–494. 
  7. G. B. Ermentrout. Period doublings and possible chaos in neural models. SIAM J. Appl. Math., 44 (1984), 80–95. 
  8. M. J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys., 19 (1978), 25–52. 
  9. J. M. Greene, R. S. MacKay, F. Vivaldi, M. J. Feigenbaum. Universal behaviour in families of area-preserving maps. Physica D, 3 (1981), 468–486.  
  10. J. Keener, J. Sneyd. Mathematical physiology. Springer, New York, 1998.  
  11. A. Milik, P. Szmolyan, H. Löffelmann, E. Gröller. The geometry of mixed-mode oscillations in the 3d-autocatalator. Int. J. Bif. & Chaos, 8 (1998), 505–519. 
  12. J. Rinzel. Formal Classification of bursting mechanisms in excitable systems. Lecture Notes Biomathematics, 71 (1987) 267–281, Springer, New York.  
  13. O. E. Rössler. An equation for continuous chaos. Phys. Lett. A, 57 (1976), 397–398. 
  14. H. G. Rotstein, R. Kuske. Localized and asynchronous patterns via canards in coupled calcium oscillators. Physica D, 215 (2006), 46–61. 
  15. X. Sailer, M. Zaks, L. Schimansky-Geier. Collective dynamics in an ensemble of globally coupled FHN systems. Fluctuation & Noise Lett., 5 (2005), L299–L304. 
  16. T. Verechtchaguina, I. M. Sokolov, L. Schimansky-Geier. First passage time densities in non-Markovian models with subthreshold oscillations. Europhys. Lett., 73 (2006), 691–697. 
  17. M. Wechselberger. Existence and bifurcation of canards in R3 in the case of a folded node. SIAM J. Appl. Dyn. Sys., 4 (2005), 101–139. 
  18. M. A. Zaks, X. Sailer, L. Schimansky-Geier, A. Neiman, Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems. Chaos, 15 (2005), 026117. 
  19. A. B. Zisook. Universal effects of dissipation in two-dimensional mappings. Phys. Rev. A, 24 (1981), 1640–1642. 

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