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Brain machine interfaces (BMIs) have recently received significant attention from the neuroscience and engineering communities as a result of striking advances in monitoring, processing, and modeling brain function at multiple temporal and spatial resolutions. These advances, however, have also raised significant challenges to both communities that are becoming the focus of numerous ongoing research efforts. Broadly categorized based on their level of invasiveness, BMIs relying on implantable microelectrode arrays (MEAs) have received the most attention. This paper briefly reviews some fundamental concepts underlying the operation of MEA-based BMIs and highlights in particular the signal processing challenges faced by these systems in light of their resource-constrained operation. Finally, we summarize some of our recent progress in this area and suggest some open questions for future research.