Some Analytical Criteria for Local Activity of Two-Port CNN with Three or Four State Variables: Analysis and Applications

The local activity principle of the Cellular Nonlinear Network (CNN) introduced by Chua [1997] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. This paper presents some analytical criteria for the local activity of two-port CNN cells with three or four state variables. As a first application, a coupled excitable cell model (ECM) CNN is introduced, which has cells defined by the Chay equations representing ionic events in excitable membranes in terms of a Hodgkin–Huxley type formalism. The bifurcation diagram of the ECM CNN supplies a possible explanation for the mechanism of arrhythmia (from normal to abnormal until stopping) of excitable cells: the cell parameter is changed from an active unstable domain to an edge of chaos. The member potentials along fibers are simulated numerically, where oscillatory patterns, chaotic patterns as well as convergent patterns are observed. As a second application, a smoothed Chua’s circuit (SCC) CNN with two ports is presented, whose prototype has been introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN in order to obtain Turing patterns. The bifurcation diagrams of the SCC CNN are the same as those with one port, which have only active unstable domains and edges of chaos. Numerical simulations show that in the active unstable parameter domains, the evolutions of the patterns of the state variables of the SCC CNNs can exhibit divergence, periodicity and chaos, where, in the parameter domains located in the edge of chaos, periodic patterns and divergent patterns are observed. These results demonstrate once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex patterns of CNNs.