Analyzing Global Dynamics of a Neural Field Model

We study global dynamics of the neural field, or a neural network model that represents densely distributed cortical neurons as a spatially continuous field. By analyzing the Lyapunov functional for the neural field with finite and infinite domains, we show that the state in the finite field necessarily converges to a steady solution and that the infinite field cannot have a limit cycle attractor. We also show that the Lyapunov functional of the neural field model can be considered to be a natural extension of the Lyapunov function of the Hopfield model to the continuous field. The result suggests that the two neural systems have, generally, common global dynamics characterized by the intimately related Lyapunov functional/function.