Existence and attraction of a phase-locked oscillation in a delayed network of two neurons
type: Item
instance of: scholarly article
title: Existence and attraction of a phase-locked oscillation in a delayed network of two neurons
zbMATH Open document ID: 1023.34065
author: Jianhong Wu
author: Yuming Chen
published in: Differential and Integral Equations
publication date: 2003-11-02T00:00:00Z
review text: The authors study the global dynamics of the following coupled system of delay differential equations \[ \dot x = - \mu x (t) + f(y(t-\tau)),\quad \dot y = - \mu y (t) + f(x(t-\tau)), \] where \(\tau\) and \(\mu\) are positive constants, \(f\) is a \(C^1\) map and \(f\) satisfies the condition of monotone positive feedback \(f(0)=0\), \(f'(\xi)>0\) for \(\xi\in \mathbb{R}\). The paper gives conditions for the existence of asynchronous solutions (i.e., solutions with different components \(x\) and \(y\)), describes their basin of attraction and heteroclinic connections from synchronous solutions to the asynchronous solutions. The reduced system \(\dot x = \mu x(t) + f(x(t-1))\), which explains the dynamics of synchronized solutions, was studied by \textit{T. Krisztin, H.-O. Walther} and \textit{J. Wu} [Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback. Fields Institute Monographs. 11. Providence, RI: AMS (1999; Zbl 1004.34002)]. The technique developed in the above mentioned book and discrete Lyapunov functionals are used to derive the main results.
reviewed by: S. Yanchuk
Mathematics Subject Classification ID: 34K19
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 1832864
zbMATH Keywords: synchronization
zbMATH Keywords: neurons
zbMATH Keywords: delayed feedback
zbMATH Keywords: discrete Lyapunov functional
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1832864