EDGE OF CHAOS AND LOCAL ACTIVITY DOMAIN OF FITZHUGH-NAGUMO EQUATION

The local activity theory [Chua, 97] offers a constructive analytical tool for predicting whether a nonlinear system composed of coupled cel ls, such as reaction-diffusion. and lattice dynamical systems, can exh ibit complexity. The fundamental result of the local activity theory a sserts that a system cannot exhibit emergence and complexity unless it s cells are locally active. This paper gives the first in-depth applic ation of this new theory to a specific Cellular Nonlinear Network (CNN ) with cells described by the FitzHugh-Nagumo Equation. Explicit inequ alities which define uniquely the local activity parameter domain for the FiteHugh-Nagumo Equation are presented. It is shown that when the cell parameters are chosen within a subset of the local activity param eter domain, where at least one of the equilibrium state of the decoup led cells is stable, the probability of the emergence of complex nonho mogenous static as well as dynamic patterns is greatly enhanced regard less of the coupling parameters. This precisely-defined parameter doma in is called the ''edge of chaos'', a terminology previously used loos ely in the literature to define a related but much more ambiguous conc ept. Numerical simulations of the CNN dynamics corresponding to a larg e variety of cell parameters chosen on, or nearby, the ''edge of chaos '' confirmed the existence of a wide spectrum of complex behaviors, ma ny of them with computational potentials in image processing and other applications. Several examples are presented to demonstrate the poten tial of the local activity theory as a novel tool in nonlinear dynamic s not only from the perspective of understanding the genesis and emerg ence of complexity, but also as an efficient tool for choosing cell pa rameters in such a way that the resulting CNN is endowed with a brain- like information processing capability.