If you store memories in the weights of a recurrent neural network, how many can it hold before they interfere and it remembers nothing?
▶ Launch the interactive simulationA Hopfield network (1982): N neurons s=±1, symmetric Hebbian weights W_ij=(1/N)Σ_μ ξ_i^μ ξ_j^μ storing P patterns. The update s_i=sign(Σ_j W_ij s_j) only lowers the energy E=-½Σ W_ij s_i s_j, so the dynamics flow downhill to a fixed point — and the stored patterns are made to BE the minima. Sweep the load α=P/N with random patterns, present each stored pattern, settle, and read the recall overlap m(α). Live, recall a corrupted glyph (pattern completion).
a SHARP capacity cliff. Below the critical load the network is a near-perfect associative memory — present a heavily corrupted cue and it flows back to the clean stored pattern (overlap m≈1). But raise α=P/N past α_c≈0.138 (Amit–Gutfreund–Sompolinsky 1985) and the memories interfere catastrophically: the network shatters into spin-glass states and recall collapses (m→0). The lab's measured m(α) sits at ≈1 up to ≈0.13 then falls off a cliff right at α_c — a genuine phase transition, the boundary set by the chemistry of the interference, not the patterns themselves. So ~0.138 N memories is the hard ceiling: a 400-neuron net holds ~55, no more