On stability and equilibria of the M-lattice

Both the analog Hopfield network and the cellular neural network are special cases of the M-lattice system, recently introduced to the signal processing community. We prove that a subclass of the M-lattice is totally stable, This result also applies to the original cellular neural network as a rigorous proof of its total stability. By analyzing the stability of fixed points, we derive the conditions for driving the equilibrium outputs of another subclass of the M-lattice to binary values. For the cellular neural network, this analysis is a precise formulation of an earlier argument based on circuit diagrams. And for certain special cases of the analog Hopfield network, this analysis explains why the output variables converge to binary values even with nonzero neuron auto-connections. This behavior, observed in computer simulation by researchers for quite some time, is explained for the first time here.