 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-84/1/c/solution

 Top articles  Latest articles New article in topic

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

Since the derivative of

tanh (β x)

at zero is

β

, the Jacobian matrix of this directed rate network is

J = \frac{1}{τ} - 1 β W_{21} 0 0 - 1 β W_{32} β W_{13} 0 - 1 .

With

a = 1 + τ λ

, its characteristic determinant is

τ^{3} det (λ I - J) = det a - β W_{21} 0 0 a - β W_{32} - β W_{13} 0 a = a^{3} - β^{3} W_{13} W_{32} W_{21} .

Thus the eigenvalues satisfy

(1 + τ λ)^{3} = Γ, Γ = β^{3} W_{13} W_{32} W_{21} .

This 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

τ > 0

.

 Read the full article

 New to topics? Read the docs here!