Relaxation self-oscillations in neuron systems. I
type: Item
instance of: scholarly article
title: Relaxation self-oscillations in neuron systems. I
zbMATH Open document ID: 1276.34068
DOI: 10.1134/S0012266111070020
author: S. D. Glyzin
author: A. Yu. Kolesov
author: N. Kh. Rozov
published in: Differential Equations
publication date: 2011-10-25T00:00:00Z
review text: As a model for the electrical activity of a single neuron, the authors consider the scalar differential-delay equation \[ {du\over dt}= \lambda[-1+\alpha f(u(t- 1))-\beta g(u(t))]\,u(t),\tag{\(*\)} \] where the parameter \(\lambda\) is sufficiently large (that means that \((*)\) is a singularly perturbed equation), the positive parameters \(\alpha\) and \(\beta\) satisfy the condition \(\alpha> 1+\beta\), and the continuously differentiable functions \(f\) and \(g\) satisfy the relations \[ \begin{aligned} f(0)= g(0)= 1,\quad 0<\beta g(u)+ 1<\alpha\quad &\text{for }u\geq 0,\\ f(u),\;g(u),\;uf'(u),\;ug'(u)= O\Biggl({1\over u}\Biggr)\quad &\text{as }u\to +\infty.\end{aligned} \] They prove that under these assumptions equation \((*)\) has for sufficiently large \(\lambda\) an exponentially stable cycle \(u= u_*(t,\lambda)\) with period \(T_*(\lambda)\) obeying \[ \lim_{\lambda\to+\infty} T_*(\lambda)= \alpha+ 1+(\beta+ 1)/(\alpha- \beta- 1), \] moreover, the spiking behavior of \(u_*(b,\lambda)\) is described.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K26
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 5963591
Wikidata QID: Q59448480
MaRDI profile type: Publication
cites work: Q3258424
cites work: Q3262973
cites work: Q3244598
cites work: Q4884302
cites work: Q4864744
cites work: Q3964370
cites work: Q3841712
full work available at URL: s0012266111070020
OpenAlex ID: W2161876804
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description: scientific article; zbMATH DE number 5963591