The most general framework of continuous Hopfield neural networks

A generalization of the energy function of the classical continuous Hopfield neural network is presented, the stationary points of which coincide with the complete set of equilibrium conditions of the network. Instead of applying statistical mechanical arguments, a direct proof is given. An energy expression of a Hopfield network having built-in constraints, namely of the so-called Potts glasses type, is presented. By performing a far-reaching generalization, the most general framework of continuous Hopfield networks is created, where almost arbitrary energy functions can be chosen and where constraints of all kind can be incorporated in the neural net. The analysis includes the presentation of several stability theorems concerning various sets of differential equations. Finally, a discussion on the possibilities to apply the presented theoretical results as well as an outlook on future topics of research is included.