The local activity criteria for "difference-equation" CNN

In this paper we use an exponential conformal mapping and a z-transform to "translate" the local activity criteria for continuous-time reaction-diffusion cellular nonlinear networks (CNN) to those for difference-equation CNNs. A difference-equation CNN is modeled by a set of difference equations with a constant sampling interval deltat > 0. Since a difference-equation CNN tends to a continuous-time CNN when deltat --> 0, we can view the Laplace transform of a continuous-time CNN as the limit of the conformal-mapping z-transform of a corresponding difference-equation CNN. Based on the relation between Laplace transform and our conformal-mapping z-transform, we extend the local activity criteria from a continuous-time CNN to a difference-equation CNN. We have proved the rather surprising result that the class of all reaction-diffusion difference-equation CNNs with two state variables and one diffusion coefficient is locally active everywhere, i.e. its local passive region is empty. In particular, as deltat --> 0, the local-passive region of a continuous-time CNN cell transforms into the "edge-of-chaos" region of a corresponding difference-equation CNN cell with deltat > 0. Remarkably, as deltat --> 0 the focally active edge-of-chaos region degenerates into a locally passive region as the difference equation tends to a differential equation. These results highlight a fundamental difference between the qualitative properties of systems of nonlinear differential- and difference-equations.