SOME PROPERTIES OF THE ATTRACTORS OF DISCRETE-TIME CELLULAR NEURAL NETWORKS

Some theoretical properties concerning discrete-time (discrete-state) cellular neural networks (DTCNNs) are presented which illustrate the role of self-feedback and shed some light on the sizes and shapes of the attraction basins. The main results proved in the paper are the following. (1) With no self-feedback, two stares cannot both be fixed points of the DTCNN dynamics if they differ or coincide in isolated grid positions. (2) With no self-feedback, every state which differs from a fixed point v (coincides with v) only in isolated grid positions is attracted to v(-v) along some orbit with block-sequential dynamics. (3) For a k-attractor of a DTCNN we have k < (2r + 1)(2)/2, where r is the neighbourhood radius. (4) The attraction domain of a k-attractor v includes all the states which differ from v in al most k positions per neighbourhood. The properties proved herein are a consequence exclusively of the local connectivity of DTCNNs.