Head-direction cells are neurons that signal a rat's directional heading in the horizontal plane. Head-direction cells in the anterior thalamus are anticipatory, so that their firing rate is better correlated with the rat's future head direction than with the present or past head direction. We recorded single-unit activity from head-direction cells in the anterior thalamus of freely moving rats. We measured the time interval by which each individual cell anticipated the rat's future head direction, which we refer to as the cell's anticipatory time interval (ATI). Head-direction cells in the anterior thalamus anticipated the rat's future head direction by an average ATI of approximately 17 ms. However, different anterior thalamic cells consistently anticipated the future head direction by different ATIs ranging between 0 and 50 ms. We found that the ATI of an anterior thalamic head-direction cell was correlated with several parameters of the cell's directional tuning function. First, cells with long ATIs sometimes appeared to have two peaks in their directional tuning function, whereas cells with short ATIs always had only one peak. Second, the ATI of a cell was negatively correlated with the cell's peak firing rate, so that cells with longer ATIs fired at a slower rate than cells with shorter ATIs. Third, a cell's ATI was correlated with the width of its directional tuning function, so that cells with longer ATIs had broader tuning widths than cells with shorter ATIs. These relationships between a cell's ATI and its directional tuning parameters could not be accounted for by artifactual broadening of the tuning function, which occurs for cells that fire in correlation with the future (rather than present) head direction. We found that when the rat's head is turning, the shape of an anterior thalamic head-direction cell's tuning function changes in a systematic way, becoming taller, narrower, and skewed. This systematic change in the shape of the tuning function may be what causes anterior thalamic cells to effectively anticipate the rat's future head direction. We propose a neural circuit mechanism to account for the firing behavior we have observed in our experiments, and we discuss how this circuit might serve as a functional component of a neural system for path integration of the rat's directional heading.