# On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

Mathematical Modelling of Natural Phenomena (2010)

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

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topZaks, 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

top- V. I. Arnold (Editor). Dynamical systems V: Bifurcation theory and catastrophe theory. Encyclopaedia of Mathematical Sciences. Springer. New York, Berlin, Heidelberg, 1999.
- M. Brons, M. Krupa, M. Wechselberger. Mixed mode oscillations due to the generalized canard phenomenon. Fields Institute Communications, 49 (2006), 39–63. Zbl1228.34063
- M. Brons, T. J. Kaper, H. G. Rotstein (Editors). Mixed Mode Oscillations: Experiment, Computation, and Analysis. Focus Issue of Chaos, 18 (2008).
- J. L. Callot, F. Diener, M. Diener. Problem of duck hunt. Compt. Rend. Acad. Sci., 286 (1978), 1059–1061.
- P. Collet, J.-P. Eckmann, H. Koch. On universality for area-preserving maps of the plane. Physica D, 3 (1981), 457–467. Zbl1194.37050
- W. Eckhaus. Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math., 985 (1983), 449–494. Zbl0509.34037
- G. B. Ermentrout. Period doublings and possible chaos in neural models. SIAM J. Appl. Math., 44 (1984), 80–95. Zbl0532.92005
- M. J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys., 19 (1978), 25–52. Zbl0509.58037
- 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. Zbl1194.37011
- J. Keener, J. Sneyd. Mathematical physiology. Springer, New York, 1998. Zbl0913.92009
- 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. Zbl0933.92036
- J. Rinzel. Formal Classification of bursting mechanisms in excitable systems. Lecture Notes Biomathematics, 71 (1987) 267–281, Springer, New York.
- O. E. Rössler. An equation for continuous chaos. Phys. Lett. A, 57 (1976), 397–398.
- H. G. Rotstein, R. Kuske. Localized and asynchronous patterns via canards in coupled calcium oscillators. Physica D, 215 (2006), 46–61. Zbl1136.34308
- X. Sailer, M. Zaks, L. Schimansky-Geier. Collective dynamics in an ensemble of globally coupled FHN systems. Fluctuation & Noise Lett., 5 (2005), L299–L304.
- 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.
- 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. Zbl1090.34047
- 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. Zbl1080.82013
- A. B. Zisook. Universal effects of dissipation in two-dimensional mappings. Phys. Rev. A, 24 (1981), 1640–1642.

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