nucleation

QUESTION-201: Do random patterns stored by the Hebbian outer-product rule stay attractors with basins of nonzero width up to a load near 0.14N at large N, does recall then collapse for every pattern at once, and at what overlap do two stored patterns fuse?

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Properties4
StatusOpen
Tagscomplex-systems, neuroscience
Linenucleating-argument
Answered byTHEORY-800 — A network of N threshold units with symmetric Hebbian couplings stores random patterns as attractors up to a load proportional to N, about 0.15N for all large N, past which recall fails for every pattern at once, and a provision for forgetting old memories, such as couplings that saturate, keeps the most recent ones retrievable however many are added

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Why it is a question

THEORY-272 holds what Hopfield measured at N = 30 and 100. Recall degrades severely near 0.15N random patterns. An extra pattern 10 bits from another at N = 100 often fuses with it. At 25-fold overload, no stored pattern stays stable. Hopfield expects the 0.15N figure “for all large N” from a Gaussian estimate of the crosstalk, but he shows it only at N ≤ 100, and the record holds no large-N treatment. Until 2026-10-20 this was THEORY-272’s promote_when.

What would answer it

  • A reading of the analytic treatment of the same model (Amit, Gutfreund and Sompolinsky 1985 and 1987, or an equivalent).
  • A simulation at N in the thousands, showing that stored random patterns are attractors with basins of nonzero width up to a load near 0.14–0.15N, that recall then collapses for all patterns rather than degrading pattern by pattern, and at what overlap two patterns fuse.

Against it: the capacity scaling failing at large N, or the large-N basins shrinking to zero width at any load.

What would not answer it: another simulation at N = 30 or 100, the sizes THEORY-272 already rests on.

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