MATHEMATICAL ENGINEERING TECHNICAL REPORTS Existence of Lyapunov Functional for Neural Field Equation as an Extension of Lyapunov Function for Hopfield Model

We show that there is a Lyapunov functional for the neural field equation, a neural network model which represents highly dense cortical neurons as a spatially continuous field, and that the system necessarily converges to an equilibrium point as far as the length of the field is finite. We also show that the Lyapunov functional is a natural extension of the Lyapunov function of the Hopfield model. The results suggest that the two models have generally common global dynamics characterized by the intimately related Lyapunov functional/function.