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Since the derivative of $\tanh (\beta x)$ at zero is $\beta$, the Jacobian matrix of this directed rate network isWith $a=1+\tau \lambda$, its characteristic determinant isThus the eigenvalues satisfyThis establishes three-neuron ring stability from the loop product. The ring is directed, so the symmetric-weight Hopfield network energy argument does not apply. Assume the physical relaxation time $\tau >0$.