Storage capacity of two-dimensional neural networks.

We investigate the maximum number of embedded patterns in the two-dimensional Hopfield model. The grand state energies of two specific network states, namely, the energies of the pure-ferromagnetic state and the state of specific one stored pattern are calculated exactly in terms of the correlation function of the ferromagnetic Ising model. We also investigate the energy landscape around them and the stability of the pure retrieval state. Taking into account the qualitative features of the phase diagrams obtained by Nishimori, Whyte, and Sherrington [Phys. Rev. E 51, 3628 (1995)], we conclude that the network cannot retrieve more than three patterns.