Abstract Spectral analysis was performed on ventilation and instantaneous heart rate data recorded in 15 term infants during quiet sleep in the first week after delivery, and in 11 of these infants during active sleep. There was a close relation between the main peaks of the ventilation spectra and the corresponding histograms of the reciprocals of T tot. The spectra for instantaneous heart rate showed power at the rate of breathing (HF) and also at lower frequencies, 0.04–0.2 Hz (LF). During quiet sleep, the relative magnitudes of the HF and LF peaks for heart rate were found to depend on the respiratory rate and the variability of T tot. During active sleep, most of the power in the heart rate spectrum was concentrated in the LF region. Weighted coherences between ventilation and heart rate were higher during quiet than active sleep, both in the HF and LF spectra. LF power was higher during active than quiet sleep in both ventilation and heart rate. The results suggest that the pattern of breathing has a marked effect on the shape of the heart rate spectrum. In most infants, however, there is no fixed phase relationship between oscillations in ventilation and heart rate, at high or low frequencies. These oscillations are affected by sleep state and hence, by implication, by central nervous system rhythm generators.