Out of Equilibrium Dynamics of the Hopfield Model in its spin-glass phase

Abstract

In this work we study numerically the out of equilibrium dynamics of the Hopfield model for associative memory inside its spin-glass phase. Besides its interest as a neural network model it can also be considered as a prototype of fully connected magnetic systems with randomness and frustration. By adjusting the ratio between the number of stored configurations $p$ and 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 properly 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 work we analyse the two-time auto-correlation function, the decay of the magnetization and the distribution of overlaps between states. The results show that within the spin-glass phase of the model the dynamics exhibits ageing phenomena and presents features that suggest a non trivial breaking of replica symmetry.