Analysis for Numerical Implementation of Continuous Time Hopfield Network Model

The choices of discrete time step for Euler method and trapezoidal method and terminating condition of iteration in trapezoidal method are discussed in this paper for numerical implementation of continuous time Hopfield network. The decreasing conditions of an energy function are investigated by the use of convex function. By utilization of the primal convex function, the conditions are analyzed under which its conjugate function minus a quadratic function is also convex. Based on the analysis of the proof for convergence of the continuous time Hopfield network model, a generalized model is proposed. For the common Euler and trapezoidal methods, the choice of their discrete time step is discussed for numerical implementation of the continuous time Hopfield network. As the trapezoidal method is an implicit scheme, its realization needs an iterated procedure. The conditions to terminate the iterated procedure are analyzed. According to the special forms of the continuous time Hopfield network model, an improved iterated algorithm for trapezoidal method is proposed and analyzed. The numerical results show that choosing a suitably large discrete time step will be helpful not only to accelerate the numerical implementation but also to improve the optimization performance.