Global consistency of decisions and convergence of competitive cellular neural networks

In a series of papers published in the seventies, Grossberg had developed a geometric approach for analyzing the global dynamical behavior and convergence properties of a class of competitive dynamical systems. The approach is based on the property that it is possible to associate a decision scheme with each competitive system in that class, and that global consistency of the decision scheme implies convergence of each solution toward some stationary state. In this paper, the Grossberg approach is extended to the class of competitive standard Cellular Neural Networks (CNNs), and it is used to investigate convergence under the hypothesis that the competitive CNN has a globally consistent decision scheme. The extension is nonobvious and requires to deal with the set-valued vector field describing the dynamics of the CNN output solutions. It is also stressed that the extended approach does not require the existence of a Lyapunov function, hence it is applicable to address convergence in the general case where the CNN neuron interconnections are not necessarily symmetric. By means of the extended approach, a number of classes of third-order nonsymmetric competitive CNNs are discovered, which have a globally consistent decision scheme and are convergent. Moreover, global consistency and convergence hold for interconnection parameters belonging to sets with non-empty interior, and thus they represent physically robust properties. The paper also shows that when the dimension is higher than three, there are fundamental differences between the convergence properties of competitive CNNs implied by a globally consistent decision scheme, and those of the class of competitive dynamical systems considered by Grossberg. These differences lead to the need to introduce a stronger notion of global consistency of decisions, with respect to that proposed by Grossberg, in order to guarantee convergence of competitive CNNs with more than three neurons.