Mean-field theory for a spin-glass model of neural networks: TAP free energy and the paramagnetic to spin-glass transition

An approach is proposed to the Hopfield model where the mean-field treatment is made for a given set of stored patterns (sample) and then the statistical average over samples is taken. This corresponds to the approach made by Thouless, Anderson and Palmer (TAP) to the infinite-range model of spin glasses. Taking into account the fact that in the Hopfield model there-exist correlations between different elements of the interaction matrix, we obtain its TAP free energy explicitly, which consists of a series of terms exhibiting the cluster effect. The nature of the spin-glass transition in the model is also examined and compared with those given by the replica method as well as the cavity method.