Shared anatomical and physiological features of primary, secondary, tertiary, polysensory, and associational neocortical areas are used to formulate a novel extended hypothesis of thalamocortical circuit operation. A simplified anatomically based model of topographically and nontopographically projecting ("core" and "matrix") thalamic nuclei, and their differential connections with superficial, middle, and deep neocortical laminae, is described. Synapses in the model are activated and potentiated according to physiologically based rules. Features incorporated into the models include differential time courses of excitatory versus inhibitory postsynaptic potentials, differential axonal arborization of pyramidal cells versus interneurons, and different laminar afferent and projection patterns. Observation of the model's responses to static and time-varying inputs indicates that topographic "core" circuits operate to organize stored memories into natural similarity-based hierarchies, whereas diffuse "matrix" circuits give rise to efficient storage of time-varying input into retrievable sequence chains. Examination of these operations shows their relationships with well-studied algorithms for related functions, including categorization via hierarchical clustering, and sequential storage via hash- or scatter-storage. Analysis demonstrates that the derived thalamocortical algorithms exhibit desirable efficiency, scaling, and space and time cost characteristics. Implications of the hypotheses for central issues of perceptual reaction times and memory capacity are discussed. It is conjectured that the derived functions are fundamental building blocks recurrent throughout the neocortex, which, through combination, gives rise to powerful perceptual, motor, and cognitive mechanisms.