Existence and attraction of a phase-locked oscillation in a delayed network of two neurons
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.
scientific article; zbMATH DE number 1832864