Geodesic
Non-Classical Relaxation Oscillations in Neurodynamics
Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 2, pp. 71-89.

Voir la notice de l'article provenant de la source Math-Net.Ru

A modification of the well-known FitzHugh–Nagumo model from neuroscience is proposed. This model is a singularly perturbed system of ordinary differential equations with a fast variable and a slow one. The existence and stability of a nonclassical relaxation cycle in this system are studied. The slow component of the cycle is asymptotically close to a discontinuous function, while the fast component is a $\delta$-like function. A one-dimensional circle of unidirectionally coupled neurons is considered. It is shown the existence of an arbitrarily large number of traveling waves for this chain. In order to illustrate the increasing of the number of stable traveling waves numerical methods were involved.
Mots-clés : impuls neuron
Keywords: FitzHugh–Nagumo model, relaxation cycle, asymptotics, stability, buffering. 
@article{MAIS_2014_21_2_a6,
     author = {S. D. Glyzin and A. Yu. Kolesov and N. Kh. Rozov},
     title = {Non-Classical {Relaxation} {Oscillations} in {Neurodynamics}},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {71--89},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2014_21_2_a6/}
}
TY  - JOUR
AU  - S. D. Glyzin
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - Non-Classical Relaxation Oscillations in Neurodynamics
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2014
SP  - 71
EP  - 89
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2014_21_2_a6/
LA  - ru
ID  - MAIS_2014_21_2_a6
ER  - 
%0 Journal Article
%A S. D. Glyzin
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T Non-Classical Relaxation Oscillations in Neurodynamics
%J Modelirovanie i analiz informacionnyh sistem
%D 2014
%P 71-89
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2014_21_2_a6/
%G ru
%F MAIS_2014_21_2_a6
S. D. Glyzin; A. Yu. Kolesov; N. Kh. Rozov. Non-Classical Relaxation Oscillations in Neurodynamics. Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 2, pp. 71-89. http://geodesic.mathdoc.fr/item/MAIS_2014_21_2_a6/

[1] A. L. Hodgkin, A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve”, J. Physiol., 117 (1952), 500–544

[2] J. Nagumo, S. Arimoto, S. Yoshizawa, “An active pulse transmission line simulating nerve axon”, Proc. IRE, 50 (1962), 2061–2070 | DOI

[3] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane”, Biophys. J., 1 (1961), 445–466 | DOI

[4] E. F. Mishchenko, N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Plenum Press, 1980, 228 pp. | MR | MR | Zbl

[5] Mishchenko E. F., Kolesov Yu. S., Kolesov A. Yu., Rozov N. Kh., Periodicheskiye dvizheniya i bifurkatsionnyye protsessy v singulyarno vozmushchennykh sistemakh, Fizmatlit, M., 1995 (in Russian) | MR

[6] S. Glyzin, A. Kolesov, N. Rozov, “Buffer phenomenon in neurodynamics”, Doklady Mathematics, 85:2 (2012), 297–300 | DOI | MR | Zbl

[7] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Discrete autowaves in neural systems”, Computational Mathematics and Mathematical Physics, 2:5 (2012), 702–719 | DOI | MR | Zbl

[8] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation self-oscillations in neuron systems, III”, Differential Equations, 48:2 (2012), 159–175 | DOI | MR | Zbl

[9] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation self-oscillations in hopfield networks with delay”, Izvestiya Mathematics, 77:2 (2013), 271–312 | DOI | DOI | MR | Zbl