The Phase Space of Interactions and the Hebb Rule in the Neural Network Models

We study the phase space of couplings in neural network models by introducing the overlap with the Hebb rule. Replica calculations and numerical calculations have been performed for both spherical and Ising constraints of the couplings. For small capacity, we find that the Hebb couplings are very close to the Maximally Stable Network (MSN) in both couplings. Further, in the Ising coupling, we study both the replica symmetric and the one-step replica symmetry breaking solutions and discuss which of these solutions is appropriate in several regions in the space of the temperature and the capacity by analyzing numerical results obtained by the enumeration method.