In higher animals, complex and robust behaviors are produced by the microscopic details of large structured ensembles of neurons. I describe how the emergent computational dynamics of a biologically based neural network generates a robust natural solution to the problem of categorizing time-varying stimulus patterns such as spoken words or animal stereotypical behaviors. The recognition of these patterns is made difficult by their substantial variation in cadence and duration. The neural circuit behaviors used are similar to those associated with brain neural integrators. In the larger context described here, this kind of circuit becomes a building block of an entirely different computational algorithm for solving complex problems. While the network behavior is simulated in detail, a collective view is essential to understanding the results. A closed equation of motion for the collective variable describes an algorithm that quantitatively accounts for many aspects of the emergent network computation. The feedback connections and ongoing activity in the network shape the collective dynamics onto a reduced dimensionality manifold of activity space, which defines the algorithm and computation actually performed. The external inputs are weak and are not the dominant drivers of network activity.