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 N^p-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!)N^p-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.