The local activity theory [Chua, 97] offers a constructive analytical tool for predicting whether a nonlinear system composed of coupled cells, such as reaction-diffusion and lattice dynamical systems, can exhibit complexity. The fundamental result of the local activity theory asserts that a system cannot exhibit emergence and complexity unless its cells are locally active. This paper gives the first in-depth application of this new theory to a specific Cellular Nonlinear Network (CNN) with cells described by the FitzHugh–Nagumo Equation. Explicit inequalities which define uniquely the local activity parameter domain for the FitzHugh–Nagumo Equation are presented. It is shown that when the cell parameters are chosen within a subset of the local activity parameter domain, where at least one of the equilibrium state of the decoupled cells is stable, the probability of the emergence of complex nonhomogenous static as well as dynamic patterns is greatly enhanced regardless of the coupling parameters. This precisely-defined parameter domain is called the "edge of chaos", a terminology previously used loosely in the literature to define a related but much more ambiguous concept. Numerical simulations of the CNN dynamics corresponding to a large variety of cell parameters chosen on, or nearby, the "edge of chaos" confirmed the existence of a wide spectrum of complex behaviors, many of them with computational potentials in image processing and other applications. Several examples are presented to demonstrate the potential of the local activity theory as a novel tool in nonlinear dynamics not only from the perspective of understanding the genesis and emergence of complexity, but also as an efficient tool for choosing cell parameters in such a way that the resulting CNN is endowed with a brain-like information processing capability.

A cellular neural/nonlinear network (CNN) [Chua, 1998] is a biologically inspired system where computation emerges from a collection of simple nonlinear locally coupled cells. This paper reviews our recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions. The key element in this evolutionary process is the replacement of the linear discriminant (offset) function w(σ) = σ in the "standard" CNN cell in [Chua, 1998] by a piecewise-linear function defined in terms of only absolute value functions. As in the case of the standard CNN cells, the excitation σ evaluates the correlation between a given input vector u formed by the outputs of the neighboring cells, and a template vector b, which is interpreted in this paper as an orientation vector. Using the theory of canonical piecewise-linear functions [Chua & Kang, 1977], the discriminant function w(σ) = z + z 0 σ − s m k=1 (−1) k |σ − z k | is found to guarantee universality and its parameters can be easily determined. In this case, the number of additional parameters and absolute value functions m is bounded by m < 2 n − 1, where n is the number of all inputs (n = 9 for a 3 × 3 template). An even more compact representation where m < n − 1 is also presented which is based on a special form of a piecewise-linear function; namely, a multi-nested discriminant: w(σ) = s(z m +|z m−1 +· · · |z 1 +|z 0 +σ|||). Using this formula, the "benchmark" Parity function with an arbitrary number of inputs n is found to have an analytical solution with a complexity of only m = O(log 2 (n)). Int. J. Bifurcation Chaos 1999.09:1-48. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/02/15. For personal use only. Int. J. Bifurcation Chaos 1999.09:1-48. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/02/15. For personal use only.