STATE EVALUATION FUNCTIONS AND LYAPUNOV FUNCTIONS FOR NEURAL NETWORKS

For a network of binary state elements, consider functions from the set of state configurations into the set of real numbers. We first characterize the existence of such state evaluation functions through the properties on their difference functions. A method to restore the original state evaluation function from their difference functions is also shown. A state evaluation function is a Lyapunov function for some network if the function value decreases as the system undergoes state changes. Then we apply the results for networks of McCulloch-Pitts type model neurons to see when there can be Lyapunov functions. In the simplest linear analysis, the weight matrix W of the network has non-negative diagonal elements and must be quasi-symmetric with respect to a positive diagonal matrix C, that is, CW must be symmetric. We have also derived, as another example, more complicated conditions under which neural networks have Lyapunov functions.