Adult neurogenesis has long been documented in the vertebrate brain and recently even in humans. Although it has been conjectured for many years that its functional role is related to the renewing of memories, no clear mechanism as to how this can be achieved has been proposed. Using the mammalian olfactory bulb as a paradigm, we present a scheme in which incorporation of new neurons proceeds at a constant rate, while their survival is activity-dependent and thus contingent on new neurons establishing suitable connections. We show that a simple mathematical model following these rules organizes its activity so as to maximize the difference between its responses and can adapt to changing environmental conditions in unsupervised fashion, in agreement with current neurophysiological data.