In recent years, spiking neural networks (SNNs) have demonstrated great success in completing various machine learning tasks. We introduce a method for learning image features with locally connected layers in SNNs using a spike-timing-dependent plasticity (STDP) rule. In our approach, sub-networks compete via inhibitory interactions to learn features from different locations of the input space. These locally-connected SNNs (LC-SNNs) manifest key topological features of the spatial interaction of biological neurons. We explore a biologically inspired n-gram classification approach allowing parallel processing over various patches of the image space. We report the classification accuracy of simple two-layer LC-SNNs on two image datasets, which respectively match state-of-art performance and are the first results to date. LC-SNNs have the advantage of fast convergence to a dataset representation, and they require fewer learnable parameters than other SNN approaches with unsupervised learning. Robustness tests demonstrate that LC-SNNs exhibit graceful degradation of performance despite the random deletion of large numbers of synapses and neurons. Our results have been obtained using the BindsNET library, which allows efficient machine learning implementations of spiking neural networks.