Bifurcations and Oscillatory Behavior in a Class of Competitive Cellular Neural Networks

Abstract

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some stationary state. In this paper it is shown that the interconnection symmetry, though ensuring complete stability, is not in the general case sufficient to guarantee that complete stability is robust with respect to sufficiently small perturbations of the interconnections. To this end, a class of third-order CNN's with competitive (inhibitory) interconnections between distinct neurons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This result is of obvious relevance in view of CNN's implementation, since perfect interconnection symmetry in unattainable in hardware (e.g. VLSI) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leonard model of the voting paradox, a special Volterra-Lotka model of three competing species. Finally, it is shown that the results in this paper can also be viewed as an extension of previous results by Zou and Nossek for a two-cell CNN with opposite-sign interconnections between distinct neurons. Such an extension has a significant interpretation in the framework of a general theorem by Smale for competitive dynamical systems. *

A study on WTA cellular neural networks

“…Section III in Reference [5], where the allowed parameter variations preserving a correct WTA functionality is shown to be about 17 per cent). ¶ It is worth mentioning that there are even pathological situations where the domain of attraction of the unstable equilibrium points is not of zero measure, see References [16,17] for some examples in the CNN framework.…”

“…This is di erent from the ignition mechanism highlighted in this paper. For example it is seen in Figure 2 that atThen, it is certainly useful to normalize x w according to Equation (17), before starting the computation of the WTA CNN. Such a normalization does not impose signiÿcant limitations in practice.…”

Robustness of convergence in finite time for linear programming neural networks

“…Indeed, recent work has shown that there are classes of nominally symmetric additive neural networks for which convergence is not robust with respect to perturbations due to tolerances [4,5]. Namely, even arbitrarily small perturbations cause the birth of large-size non-vanishing oscillations in the long-run behaviour of the trajectories, an highly undesirable situation which makes the networks not useful for solving signal processing tasks.…”

Existence and characterization of limit cycles in nearly symmetric neural networks

“…A first answer to the fundamental question raised in [11], [12] is given in [13]. In that paper, a special class of third-order standard cellular neural networks (CNNs) [5] with inhibitory (competitive) interconnections between distinct neurons has been introduced.…”