The Hopfield model for a neural network is studied in the limit when the number $p$ of stored patterns increases with the size $N$ of the network, as $p=\alpha N$. It is shown that, despite its spin-glass features, the model exhibits associative memory for $\alpha <{\alpha }_c$, ${\alpha }_c\gtrsim 0.14$. This is a result of the existence at low temperature of $2p$ dynamically stable degenerate states, each of which is almost fully correlated with one of the patterns. These states become ground states at $\alpha <0.05\mathrm{}$. The phase diagram of this rich spin-glass is described.

• Received 11 July 1985

DOI:https://doi.org/10.1103/PhysRevLett.55.1530