Sleep slow oscillation emergence on the scalp as a renewal point process

Sleep slow oscillations (SOs), characteristic of NREM sleep, are causally tied to cognitive outcomes and the health-promoting homeostatic functions of sleep. Characterization of SO organization during a night of sleep is an active area of research, with most existing work focused on individual SO events rather than the temporal dynamics across sleep cycles or channels. Hence, the probabilistic structure governing the timing and distribution of SOs in one individual across the sleep night remains underexplored. To address this gap, we introduce a computational model characterizing SO emergence over time as a function of sleep cycle and electrode location. SOs were detected in a dataset of nighttime sleep from 22 subjects (9 females), acquired with polysomnography including 64 EEG channels. Modeling of SO occurrence was performed separately for SOs detected during stage N3, and during a combination of stages N2 and N3 (N2&N3). We analyzed SO emergence at two temporal scales. First, we modeled cumulative SO occurrences across successive sleep cycles using a power law fit (across-cycles model). Second, we characterized SO timing within each cycle using a renewal point process (within-cycle model), fitting an inverse Gaussian distribution to the inter-event intervals of SOs and estimating its parameters μ (mean) and λ (shape) for each sleep cycle and channel. Both models were fit to individuals and to a generic idealized 'average' SO emergence behavior, describing both general and individualized patterns. The decay rate of SO count per cycle was 1.70 for N3 and 1.14 for N2&N3, with participant-level variance of 1.00 and 0.53, respectively. Within-cycle modeling showed consistent increases in μ (0.83 ± 0.14) and λ (4.59 ± 0.66) across cycles. This probabilistic framework captures structured SO timing and supports descriptive modeling of large-scale SO dynamics across the night, offering a basis for future investigations of variability in sleep organization.