Some theorems concerning the free energy of (Un) constrained stochastic hopfield neural networks

Abstract

General stochastic binary Hopfield models are viewed from the angle of statistical mechanics. Both the general unconstrained binary stochastic Hopfield model and a certain constrained one are analyzed yielding explicit expressions of the free energy. Moreover, conditions are given for which some of these free energy expressions are Lyapunov functions of the corresponding differential equations. In mean field approximation, either stochastic model appears to coincide with a specific continuous model. Physically, the models are related to spin and to Potts glass models. By means of an alternative derivation, an expression of a 'complementary free energy is presented. Some surveying computational results are reported and an alternative use of the discussed models in resolving constrained optimization problems is discussed.