Exact number of mosaic patterns in cellular neural networks

This work investigates mosaic patterns for the one-dimensional cellular neu ral networks with various boundary conditions. These patterns can be formed by combining the basic patterns. The parameter space is partitioned so tha t the existence of basic patterns can be determined for each parameter regi on. The mosaic patterns can then be completely characterized through formul ating suitable transition matrices and boundary-pattern matrices. These mat rices generate the patterns for the interior cells from the basic patterns and indicate the feasible patterns for the boundary cells. As an illustrati on, we elaborate on the cellular neural networks with a general 1 x 3 templ ate. The exact number of mosaic patterns will be computed for the system wi th the Dirichlet, Neumann and periodic boundary conditions respectively. Th e idea in this study can be extended to other one-dimensional lattice syste ms with finite-range interaction.