 Mutually excitatory saturating rate network

The symmetric two-neuron rate model has nonnegative activity and saturating recurrent input. Every equilibrium has $x=y$: otherwise monotonicity of $g$ gives a contradiction. With positive parameters it has three distinct equilibria exactly when $am>2\sigma$, namely zero and $z_{\pm }=(m\pm \sqrt{m^2-4{\sigma }^2/a^2})/2$. The lower positive equilibrium is a saddle and the origin and upper equilibrium are stable nodes. The resulting two attracting activity levels model a persistent switch rather than a spiking waveform.