To determine the changes in alveolar PCO₂ (P_ACO₂) and PO₂ (P_AO₂) during periodic and regular breathing epochs in the same infants, we studied 11 preterm infants during quiet sleep (birth weight 1,630 ± 94 g; gestational age 31 ± 1 weeks; postnatal age 32 ± 3 days). A total of 94 breathing/apneic cycles were analyzed and compared with regular periods. During periodic and regular breathing epochs, there were negative correlations of P_AO₂ on P_ACO₂. Short (5 s) and long ( > 5 s) apneas for individual infants occurred along the regression line for that infant. There was not a single overall critical P_ACO₂ below which apnea occurred, but for individual infants the P_ACO₂ and the P_ACO₂ of the breath preceding apnea varied within a limited range. Apneas occurred in clusters of P_ACO₂ and P_AO₂ along the average regression line of P_AO₂ on P_ACO₂. Analysis of the data showed that apnea occurred at the lowest P_ACO₂ and highest P_AO₂ levels if allowance was made for circulation time. During apnea, ‘the best fit’ for the increase in P_ACO₂ and the decrease in P_AO₂ was linear, rather than logarithmic. The findings suggest the following. (1) There is not a single overall critical level of P_ACO₂ for apnea to occur, but in a given infant this level varies within a limited range. This indicates that these infants are likely breathing near the apnea threshold. (2) Short and long apneas appear to occur randomly along the regression of P_AO₂ on P_ACO₂ for a particular infant. (3) The changes in alveolar gases are linear during apnea. The data are consistent with the idea that each infant has an optimum narrow range of P_ACO₂ and P_AO₂ values in which apnea occurs.