
Application of the method of quasinormal forms to the mathematical model of a single neuron
M. M. Preobrazhenskaya^{} ^{} P. G. Demidov Yaroslavl State University
Abstract:
We consider a scalar nonlinear differentialdifference equation with two delays, which models the behavior of a single neuron. Under some additional suppositions for this equation it is applied a wellknown method of quasinormal forms. Its essence lies in the formal normalization of the Poincare–Dulac, the production of a quasinormal form and the subsequent application of the conformity theorems. In this case, the result of the application of quasinormal forms is a countable system of differentialdifference equations, which manages to turn into a boundary value problem of the Korteweg–de Vries equation. The investigation of this boundary value problem allows to make the conclusion about the behavior of the original equation. Namely, for a suitable choice of parameters in the framework of this equation it is implemented the buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.
Keywords:
buffering, differentialdifference equation, asymptotic form, stability, Korteweg–de Vries equation, quasinormal forms.
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UDC:
517.9 Received: 17.07.2014
Citation:
M. M. Preobrazhenskaya, “Application of the method of quasinormal forms to the mathematical model of a single neuron”, Model. Anal. Inform. Sist., 21:5 (2014), 38–48
Citation in format AMSBIB
\Bibitem{Pre14}
\by M.~M.~Preobrazhenskaya
\paper Application of the method of quasinormal forms to the mathematical model of a single neuron
\jour Model. Anal. Inform. Sist.
\yr 2014
\vol 21
\issue 5
\pages 3848
\mathnet{http://mi.mathnet.ru/mais397}
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