In this paper, we present an analytical design approach for the class of bipolar cellular neural networks (CNN's) which yields optimally robust template parameters. We give a rigorous definition of absolute and relative robustness and show that all well-defined CNN tasks are characterized by a finite set of linear and homogeneous inequalities. This system of inequalities can be analytically solved for the most robust template by simple matrix algebra. For the relative robustness of a task, a theoretical upper bound exists and is easily derived, whereas the absolute robustness can be arbitrarily increased by template scaling. A series of examples demonstrates the simplicity and broad applicability of the proposed method.

A cellular neural network (CNN) is a recurrent neural network model. Like other models of this kind, the complete stability issue remains an open question. Leaving aside the unstable cyclic output case, the existence of stable outputs itself, known as the partial stability problem, ends up being a quite reliable guarantee for complete stability. Yet, no necessary and sufficient condition for partial stability has been established either. As a workaround, the past ten years provided several sufficient conditions. Some were ported from other neural network models, whereas others came out of various mathematical properties of CNNs. Consequently, the available criteria are disparate and hence, do not help finding any broader criterion. Based on a new viewpoint of the neighborhood consistency condition [1], this paper introduces a design principle of partial stability criteria for CNNs. Every of the currently established partial stability criteria, are then shown to be quite simple derivations of this principle, so opening a new way towards the complete stability problem. KEY WORDS Recurrent neural networks, cellular neural networks, stable state analysis, neighborhood-consistency condition. 1 The Cellular Neural Network Model Introduced in 1988 by professor L. O. Chua, a cellular neural network (a CNN) is a lattice of identical neurons, also called cells [2]. Each of them interact with their nearest neighbors thanks to weighted feedback and feedforward connections, so making a CNN a recurrent neural network model (cf. figure 1). Although there is no theoritical limitation to the extent of such a neighborhood, typical settings restrain it to the directly adjacent cells. The internal dynamics of a cell are described by the following equation: dx (t) = −x(t) + A ∗ N(y(t)) + B ∗ N(u) + I, (1) dt where x(t) is the internal state and u the input value of the cell. The output y(t) equals x(t) in the domain] − 1; +1[,