Three-neuron ring stability ($(1+\tau \lambda {)}^3=\Gamma$)

For a directed three-neuron rate ring $\tau {\dot{x}}_i=-x_i+W_{i,i-1}\tanh (\beta x_{i-1})$, the linearization has characteristic equation $(1+\tau \lambda {)}^3=\Gamma$, where $\Gamma ={\beta }^3{W}_{13}{W}_{32}{W}_{21}$. Positive loop gain has strict linear asymptotic stability for $\Gamma <1$. Negative loop gain has it for $-8<\Gamma <0$; at $\Gamma =-8$ a complex pair crosses the imaginary axis at frequency $\sqrt{3}/\tau$. The saturating cubic gives a supercritical Hopf bifurcation under an ordinary loop-gain variation. This directed system does not inherit the energy-descent property of a symmetric Hopfield network.