It is well known that, all things being equal, the accuracy of mammalian timing and memory decays gradually with the passage of time. The gradual decay of temporal accuracy is also observed in single-unit neural recordings. Here we review recent modeling work describing a specific mechanism for timing and memory and relevant neural data. The model describes a neural mechanism that can give rise to a logarithmically compressed representation of the recent past. We examine the specific predictions of the model, in particular that the elapse of time is represented by sequentially activated cells which fire for a circumscribed period of time. Such cells, called time cells, have been observed in neural recordings from several brain regions in multiple species. As predicted by the model, the cells show accuracy that decreases with time.