The brain builds dynamic models of the body and the outside world to predict the consequences of actions and stimuli. A well-known example is the oculomotor integrator, which anticipates the position-dependent elasticity forces acting on the eye ball by mathematically integrating over time oculomotor velocity commands. Many models of neural integration have been proposed, based on feedback excitation, lateral inhibition or intrinsic neuronal nonlinearities. We report here that a computational model of the cerebellar cortex, a structure thought to implement dynamic models, reveals a hitherto unrecognized integrator circuit. In this model, comprising Purkinje cells, molecular layer interneurons and parallel fibres, Purkinje cells were able to generate responses lasting more than 10 s, to which both neuronal and network mechanisms contributed. Activation of the somatic fast sodium current by subthreshold voltage fluctuations was able to maintain pulse-evoked graded persistent activity, whereas lateral inhibition among Purkinje cells via recurrent axon collaterals further prolonged the responses to step and sine wave stimulation. The responses of Purkinje cells decayed with a time-constant whose value depended on their baseline spike rate, with integration vanishing at low (< 1 per s) and high rates (> 30 per s). The model predicts that the apparently fast circuit of the cerebellar cortex may control the timing of slow processes without having to rely on sensory feedback. Thus, the cerebellar cortex may contain an adaptive temporal integrator, with the sensitivity of integration to the baseline spike rate offering a potential mechanism of plasticity of the response time-constant.