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We simulate and formally analyze the emergent operations from the specific anatomical layout and physiological activation patterns of a particular local excitatory-inhibitory circuit architecture that occurs throughout superficial layers of cortex. The circuit carries out two effective procedures on its inputs, depending on the strength of its local feedback inhibitory cells. Both procedures can be formally characterized in terms of well-studied statistical operations: clustering, and component analyses, under high-feedback-inhibition and low-feedback-inhibition conditions, respectively. The detailed nature of these clustering and component procedures is studied in the context of extensive related literatures in statistics, machine learning, and computational neuroscience. The two operations (clustering and component analysis) have not previously been shown to contain deep connections, let alone to each be derivable from a single overarching algorithmic precursor. The identification of this deep formal mathematical connection, which arose from analysis of a detailed biological circuit, represents a rare instance of novel mathematical relations arising from biological analyses. ¹