Turing Patterns in CNNs-Part 11: Equations and Behaviors

The general state equations describing two-grid cou- pled CNNs based on the reduced Chua's circuit are derived, and the analysis of 'hring pattern formation is approached from a specific point of view: spatial-eigenfunction based equation decoupling. Discrete spatial eigenfunctions for two common types of boundary conditions are presented, and the way the dynamics is influenced by the shape and position of the dispersion curve (i-1 i) N PART I of this paper (l), it was shown that two-grid I coupled CNNs based on the reduced Chua's circuit have the potential of producing reaction-diffusion type (Turing) patterns.' In the above paper the general principle of pattern formation has been described, the conditions concerning the individual cell have been analyzed, a qualitative explanation of the pattern formation mechanism in a 1-D c"J has been given, and the multiple equilibria property has been exemplified. The key mechanism of pattern formation is based on two kinds of conditions. The first kind refers to requirements for the isolated cells to have a unique, stable equilibrium point. These conditions, in terms of the nonlinear characteristic and of he Jacobian matrix of the isolated cell at the equilibrium point, have been derived in (l). For the whole interconnected array, the above equilibrium is still an Fig. 1. The (i,j) node of the two-grid coupled CNN.