An approach to local diffusion and global propagation in 1-dim. cellular neural networks

Abstract

Summary form only given, as follows. We study the phenomena of local diffusion and global propagation in a one-dimensional CNN described by the space-invariant A-template A = [A/sub -1/ A/sub 0/ A/sub 1/]. Roughly speaking, a CNN behaves in a local diffusion mode when two distant cells do not influence each other if the states of a number r of adjacent cells located between these two cells have reached some value. It behaves in a global propagation mode otherwise, i.e. when one of these two cells can always influence the other one, whatever the value of the state of r adjacent cells located in between these two cells. We can then compute the values of the template parameters for which the CNN has one of these behaviors. The distinction between these two methods of information processing is a radical one that has many practical consequences on: stability; the influence of boundary conditions; the dependence of the number of stable equilibria on the number of cells; the existence of limit cycles; and on the lengths of transients. For example, we can prove that the number of stable equilibria grows exponentially with the number of cells if and only if the CNN has a local diffusion behavior. If it operates in a global propagation mode, this is no longer true, but periodic solutions (one of which can be explicitly computed) are then present for some types of boundary conditions.<>