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The neural dynamics of the vertebrate olfactory system have been simulated with integrodifferential equations, in the search for insight into the nature of a primitive but still highly successful pattern recognition mechanism. The basic neural circuitry consists of a mass of interactive neurons having the capability of alternating between a stable equilibrium state and a stable limit cycle state. The spatial amplitude modulation of the limit cycle seen as a carrier appears to reflect information that is stored in the olfactory bulb from previous exposure to odors. This store is conceived to exist in the form of patterns of modified synapses constituting templates. Experiments have been conducted with two types of template, one associated with attention and desired input, the other with habituation and unwanted input. As constituted the model has certain desirable features, but it fails to account for the physiological result that the limit cycle patterns observed in the bulb appear to conform more to expectations of input than to actual input.