We consider a Cellular Neural Network (CNN) with a bias term z in the integer lattice on the plane . We impose a symmetric coupling between nearest neighbors, and also between next-nearest neighbors. Two parameters, a and , are used to describe the weights between such interacting cells. We study patterns that can exist as stable equilibria. In particular, the relationship between mosaic patterns and the parameter space (z, a; ) can be completely characterized. This, in turn, addresses that so called "Learning Problem" in CNNs. For z = 0, a > 0 and > 0, we obtain various types of feasible local-defect patterns. Moreover, we derives some "Exclusion Principles" and "Coexistence Principles" to determine whether two feasible local-defect patterns can be glued together. Using such principles, we are able to give a rather complete characterization of global-defect patterns. The complexities of mosaic and defect patterns are also addressed. In particular, we obtain a region in (z, a; )-space for which the corresponding mosaic patterns have zero spatial entropy, while the associated defect patterns have non-zero spatial entropy.