Memorisation and forgetting in a learning Hopfield neural network: bifurcation mechanisms, attractors and basins

The paper studies a continuous-time Hopfield network with Hebbian learning, using the arctan activation F(x) = (2/π) arctan((λπ x)/2), and parameters N=81, g=0.3, A=30, B=300, λ=1.4, exactly as stated in their model; they report a pitchfork bifurcation at t=6.5, followed by pairs of saddle-node bifurcations (e.g., at t2=22.8783), and provide cross-sectional evidence that basin boundaries coincide with stable manifolds of index-1 saddles; they also argue that catastrophic forgetting arises when the weight trajectory crosses the same bifurcation manifolds in the opposite direction . The candidate solution gives a complementary rigorous sketch: establishing a strict Lyapunov function for the retrieval system, deriving a supercritical pitchfork via center-manifold reduction at the leading eigenvalue crossing, describing saddle-node creation on codimension-one fold manifolds, identifying basin boundaries as stable manifolds of useful saddles, and interpreting catastrophic forgetting as reverse fold crossings; it even produces an explicit formula yielding t_p ≈ 6.45 consistent with the paper’s t ≈ 6.5 under a simplifying symmetry assumption. The proofs are different in style: the paper is primarily computational/constructive with careful visualizations, while the model provides standard dynamical-systems derivations under generic hypotheses. No substantive contradictions were found; the model’s extra assumptions (e.g., equal off-diagonal weights for the closed-form t_p) should be noted as special-case scaffolding rather than claims about the paper’s exact schedule.