Slowly oscillating periodic solutions for a delayed frustrated network of two neurons.

*(English)*Zbl 0998.34058Here, the authors discuss the dynamics for a network of two neurons with negative feedback of the form

$$\dot{x}\left(t\right)=-{\mu}_{1}x\left(t\right)+F\left(y(t-{\tau}_{1})\right)+{I}_{1},\phantom{\rule{1.em}{0ex}}\dot{y}\left(t\right)=-{\mu}_{2}y\left(t\right)-G\left(x(t-{\tau}_{2})\right)+{I}_{2},$$

where ${\mu}_{1}>0,{\mu}_{2}>0,{\tau}_{1}\ge 0,{\tau}_{2}\ge 0$ are constants, ${\tau}_{1}+{\tau}_{2}>0$, ${I}_{1}$ and ${I}_{2}$ are constants, $F$ and $G$ are bounded ${C}^{1}$-functions with ${F}^{\text{'}}\left(\xi \right)>0,{G}^{\text{'}}\left(\xi \right)>0$. The authors show a two-dimensional closed disk bordered by a slowly oscillating periodic orbit. A description on the dynamics of the flow restricted to this closed disk is given. This paper is related to the authors work [Existence and attraction of a phase-locked oscillation in a delayed network of two neurons (to appear)].

Reviewer: Yuan Rong (Beijing)