Spiking neural networks (SNNs) with a lattice architecture are introduced in this work, combining several desirable properties of SNNs and self-organized maps (SOMs). Networks are trained with biologically motivated, unsupervised learning rules to obtain a self-organized grid of filters via cooperative and competitive excitatory-inhibitory interactions. Several inhibition strategies are developed and tested, such as (i) incrementally increasing inhibition level over the course of network training, and (ii) switching the inhibition level from low to high (two-level) after an initial training segment. During the labeling phase, the spiking activity generated by data with known labels is used to assign neurons to categories of data, which are then used to evaluate the network's classification ability on a held-out set of test data. Several biologically plausible evaluation rules are proposed and compared, including a population-level confidence rating, and an $n$-gram inspired method. The effectiveness of the proposed self-organized learning mechanism is tested using the MNIST benchmark dataset, as well as using images produced by playing the Atari Breakout game.