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Existence and attraction of a phase-locked oscillation in a delayed network of two neurons. (English) Zbl 1023.34065

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 T. Krisztin, H.-O. Walther and 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.

MSC:

34K19 Invariant manifolds of functional-differential equations
34K13 Periodic solutions to functional-differential equations
37B25 Stability of topological dynamical systems
92C20 Neural biology

Citations:

Zbl 1004.34002
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