Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 84 / 1 / c

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 $\tau >0$.