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Recent generalizations of the Hopfield model of associative memories are able to store a number $P$ of random patterns that grows exponentially with the number $N$ of neurons, $P=\exp(\alpha N)$.
Random-energy model: An exactly solvable model of disordered systems
Bernard Derrida · 1981
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Neural networks and physical systems with emergent collective computational abilities
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Daniel J. Amit, Hanoch Gutfreund, and Haim Sompolinsky · 1985
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