Multiconnected neural network models

A generalisation of the Hopfield model which includes interactions between p()2) Ising spins is considered. The exact storage capacity behaves as Np-1/2(p-1)! ln N when the number of nodes, N, is large. In the limit p to infinity , the thermodynamics of the model can be solved exactly without using the replica method; at zero temperature, a solution which is completely correlated with the input pattern exists for alpha < alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin-glass solution if alpha < alpha 1=1/4 ln 2 where the number of patterns n=(2 alpha /p!)Np-1. For finite values of p, the correlation with the input pattern is not complete; for p=3 and 4, approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p/4 ln p.