Convergence and periodicity of solutions for a neural network of two neurons (Q1883534)

The authors consider the following system \[ \dot{x}(t)=-x+f(y(t-\tau)),\quad \dot{y}(t)=-y+f(x(t-\tau)), \] with a piecewise constant nonlinearity \[ f(\psi)=\begin{cases} 1& \psi\in [-a,a],\\ 0& \psi\in (-\infty,-a)\cup (a,\infty). \end{cases} \] For a set of initial functions, they obtain several results on the asymptotic behavior of solutions of this system.