= Three-neuron ring stability
{title2=$(1+\tau\lambda)^3=\Gamma$}
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 $\Gamma=\beta^3W_{13}W_{32}W_{21}$. 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 $\sqrt3/\tau$. 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>.
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