Continuous-state Hopfield dynamics based on implicit numerical methods

A novel technique is presented that implements continuous-state Hopfield neural networks on a digital computer. Instead of the usual forward Euler rule, the backward method is used. The stability and Lyapunov function of the proposed discrete model axe indirectly guaranteed, even for reasonably large step size. This is possible because discretization by implicit numerical methods inherits the stability of the continuous-time model. On the contrary, the forward Euler method requires a very small step size to guarantee convergence to solutions. The presented technique takes advantage of the extensive research on continuous-time stability, as well as recent results in the field of dynamical analysis of numerical methods. Also, standard numerical methods allow for synchronous activation of neurons, thus leading to performance enhancement. Numerical results axe presented that illustrate the validity of this approach when applied to optimization problems.