  We present results for two different kinds of high order connections between
neurons acting as corrections to the Hopfield model. Equilibrium properties are
analyzed using the replica mean-field theory and compared with numerical
simulations. An optimal learning algorithm for fourth order connections is
given that improves the storage capacity without increasing the weight of the
higher order term. While the behavior of one of the models qualitatively
resembles the original Hopfield one, the other presents a new and very rich
behavior: depending on the strength of the fourth order connections and the
temperature, the system presents two distinct retrieval regions separated by a
gap, as well as several phase transitions. Also, the spin glass states seems to
disappear above a certain value of the load parameter $\alpha$, $\alpha_g$.
