In this work, we present an application of the local activity theory by demonstrating the emergence of complex patterns in a Memristor Cellular Nonlinear Network (M-CNN) structure. The proposed M-CNN structure consists of identical memristive cells, which are resistively coupled to each other in a two-dimensional (2-D) grid form. Each cell contains a locally active NbOx memristor, a DC voltage source, a bias resistor, and a capacitor. Firstly, the locally active memristor together with its AC equivalent circuit is introduced. Secondly, the stability analysis of the single cell is performed. Then, the opportune parameter space, associated with local activity, edge-of-chaos, and sharp-edge-of-chaos domains, is determined in terms of cell characteristics, namely, the DC operating point, the capacitor, and the coupling resistor. Precisely, all the derivations are performed parametrically and a simplified generic memristor model is employed to enhance the simulation speed. Simulation results successfully show that complexity can be observed in resistively coupled M-CNNs utilizing locally active memristors.