Out-of-equilibrium dynamics of the Hopfield model in its spin-glass phase

In this paper, we study numerically the out-of-equilibrium dynamics of the Hopfield model for associative memory inside its spin-glass phase. Aside fr om its interest as a neural network model, it can also be considered as a p rototype of a fully connected magnetic system with randomness and frustrati on. By adjusting the ratio between the number of stored configurations p an d the total number of neurons N, one can control the phase-space structure, whose complexity can vary between the simple mean-field ferromagnet (when p = 1) and that of the Sherrington Kirkpatrick spin-glass model (for a prop erly taken limit of an Infinite number of patterns). In particular, little attention has been devoted to the spin-glass phase of this model. In this p aper, we analyze the two-time autocorrelation function, the decay of the ma gnetization and the distribution of overlaps between states. The results sh ow that within the spin-glass phase of the model, the dynamics exhibits agi ng phenomena and presents features that suggest a non trivial breaking of r eplica symmetry.