2.1. The sleep EEG dataset

The present study utilizes a dataset of sleep polysomnography from the Sleep and Cognition Lab at University of California Irvine, led by Dr. Mednick. The dataset is introduced in detail in [22]. The dataset includes full-night sleep EEG data from 22 healthy volunteers (9 females, age 18–34 years) without psychological or neurological issues. EEG were recorded using a 64-channel cap placed according to the international 10–20 System at a 1,000 Hz sampling rate, which was subsequently down-sampled to 128 Hz. Out of the 64 channels, 56 recorded the electrical activity of the brain, while others served as reference, ground, and other biosignal channels (e.g., EOG, EMG). In our analysis, EEG signals were re-referenced to the contralateral mastoids (M1 and M2 channels). Sleep stages (Wake, N1, N2, N3, and REM sleep) were visually scored in 30-second epochs following the R&K manual [23], using the MATLAB toolbox HUME [24]. R&K stages 3 and 4 were combined into a single stage (N3) in accordance with current AASM nomenclature [25], a practice widely used in the literature [26,27]. This study relies on manually sleep-staged data for analysis. Participants all had a good quality sleep (basic sleep outcomes in S1 Table). The dataset used in this study consists solely of spontaneous sleep recordings, and does not include any experimental interventions.

2.2. SO detection

SO detection was performed at each electrode independently, using an algorithm previously utilized in [22,28], closely following the criteria introduced by Massimini et al. [29] and Dang-Vu et al. [30]. The algorithm was applied only to artifact-free data segments, as described below.

2.3. Sleep cycle detection

For sleep cycle detection, we utilized the Feinberg and Floyd algorithm [37]. Briefly, a sleep cycle is defined as the interval from the onset of stage N2 in one NREM period to the onset of stage N2 in the next NREM period, encompassing the intervening REM episode. This algorithm requires a minimum of 5 minutes for the REM period — except for the first REM episode, for which no minimum duration is required — and at least 15 minutes for the NREM period, to prevent brief stage N2 epochs in the REM period from being considered as separate NREM periods. Additionally, the start of a cycle is determined by the start of stage 2. In our study, we consider only sleep cycles that include both REM and NREM periods. Therefore, we excluded the last sleep cycle of participants if they showed no REM period.

2.4. SO percentage measurement

This study investigates patterns in SO counts across successive cycles. To provide a comparable measure across participants, we use SO percentage, a normalized value for each participant. We measure the SO percentage during successive EEG epochs and sleep cycles using the following formulas:

(1)

(2)

SO counts used in these formulas were pooled across all 56 EEG channels, such that each SO event detected at any electrode within a given epoch or cycle contributed to the count. This all-channel aggregation was adopted to capture the overall temporal SO activity across the night. This study focuses on SO percentage during stage N3 and the combined stages N2 and N3 (N2&N3) of the NREM period. We measure SO percentage in 30-second epochs, which we encode as a time series of SO percentage in successive epochs. As we were interested in investigating the pattern of SO percentage across successive cycles and participants, we noted the variability in sleep stages counts in sleep cycles and the resulting different lengths of NREM periods. To achieve comparability, we scaled the length of each NREM period to a fixed 100-point time series. S7 Fig displays the average epoch count across participants in the first four sleep cycles in our dataset. In each NREM period, we utilize a method adapted from our previous work [38] to conduct a temporal normalization. Specifically, the SO percentage of epochs was first calculated using Relationship 1 to yield a time series for each cycle. Because NREM periods vary in duration, these series have a variable number of points (L). Each series was up-sampled via linear interpolation onto a high-resolution grid (e.g., 100 × L points) and subsequently down-sampled by a factor of L. This procedure results in a standardized 100-point time series for every cycle, effectively normalizing out differences in NREM duration.

2.5. Modeling of SO dynamics across sleep cycles

To characterize the evolution of SO occurrence across the night, we modeled the change in SO percentage across successive sleep cycles using parametric functions describing across-cycle decay. We considered two candidate models: a power-law formulation and an exponential formulation. The primary model was a power-law function of the form:

(3)

where y(C) represents the SO percentage in cycle C, is the SO percentage in the first cycle, and T is the decay exponent. As an alternative, we evaluated an anchored exponential model of the form:

(4)

where b is the decay parameter. In both models, was fixed to the first cycle value, such that only a single parameter (T or b) was estimated. Model fitting was performed at both the group level and the individual subject level using least-squares estimation. Model performance was evaluated using goodness-of-fit metrics including the coefficient of determination (R²) and root mean squared error (RMSE).

2.6. Modeling of inter-arrival time of SOs

This study models the timing of SO occurrence within and across cycles. First, we describe our modeling approach for the timing of SO occurrence within cycles. Then, by modeling the parameters of the model across cycles, we introduce a probabilistic modeling of SO occurrence using a point process [39]. Point process models are commonly used in neuroscience to describe the statistical structure of temporally discrete events, such as neuronal spikes or oscillatory bursts [40]. The first step in modeling such events involves identifying a probability distribution function (PDF) that fits the empirical inter-event intervals. After determining the PDF for event occurrence, the inversion method, which is a commonly used technique for simulating random variates, can be employed to simulate event occurrences. The inversion method samples from a uniform distribution and then uses the inverse of the cumulative distribution function (CDF) to transform these uniform samples into samples from the desired distribution [41]. Since the CDF describes the probability that a random variable takes on a value less than or equal to a specified value [42], the inverse CDF describes the value of the random variable corresponding to a given probability. By generating random samples from a uniform distribution for the inverse CDF, we can simulate the occurrence of events ⁴¹.

This study employed an inverse Gaussian (IG) function as a PDF to model the probability distribution of SO occurrence inter-arrival times. Among the candidate distributions evaluated (S4 Table), the IG showed the highest overall similarity across sleep cycles when averaged across the dataset. Notably, in the first sleep cycle—whose parameter estimates were used to extrapolate model behavior across subsequent cycles—the IG provided a better fit than the others. In addition to this empirical performance, the IG distribution is well suited for modeling inter-arrival times of discrete events because it is defined only for positive values and captures the skewed structure commonly observed in neural event interval distributions [43]. Furthermore, the IG distribution has been widely used in neuroscience to model stochastic neural timing processes such as spike trains and event intervals [44–46]. For these reasons, we selected the IG distribution to model SO inter-arrival times. The IG model consists of two parameters: mean (μ) and lambda (λ), defined according to the following formula [47]:

(5)

where is the random variable. The optimal parameters of the IG model were determined by fitting an IG model to the distribution of the inter-arrival times of SOs within each sleep cycle. The CDF of the IG can be described by the following formula [48]:

(6)

where is CDF of the standard normal distribution, as described by the formula below [49]:

(7)

The CDF of normal distribution and its inverse are not available in closed form, requiring the use of numerical procedures [50]. Consequently, the inverse CDF of IG is also not available in a closed form. Therefore, we employed MATLAB to estimate the inverse CDF of the IG distribution through numerical procedures, leveraging the error function [51] for estimating inverse CDF [52].

By fitting the IG model to the inter-arrival times of SOs within each sleep cycle, we obtained μ and λ parameters for every cycle. Parameter estimation was performed independently for each cycle, at each electrode and for each subject, resulting in four independent sets of cycle-specific parameter estimates. In a separate analysis, parameters derived from the first sleep cycle were used to simulate SO timing in Cycles 2–4, enabling evaluation of the extent to which early-night dynamics predict later-cycle behavior. This process was repeated independently across all 56 EEG channels for each of the 22 participants, resulting in a 56-by-22 matrix of μ and λ values per cycle. That is, SO inter-arrival times were computed from the trough timestamps of SOs detected at each individual electrode separately, and the IG model was fitted to each channel’s time series independently. The reported μ and λ values therefore reflect averages of these channel-level parameter estimates across subjects and electrodes. Parameter estimation was performed independently for each electrode and each participant using the empirical inter-arrival times at that specific channel-subject-cycle combination. The reported group-level values therefore represent averages of individual channel × subject estimates, computed separately per cycle, rather than estimates derived from a pooled or averaged time course. To characterize how SO timing dynamics evolve across the night, we analyzed the variation of these parameters across successive cycles. Specifically, we examined whether μ and λ values estimated from early cycles could be used to describe their progression in later cycles. For each cycle, we averaged μ and λ across electrodes and participants to obtain representative values ( and ). The functional forms used to describe the cross-cycle evolution of μ and λ were selected empirically on the basis of goodness-of-fit and biophysical interpretability. For μ, a power-law model was chosen because the trajectory was well-linearized in log–log space, and because this form is consistent with the power-law model applied to SO counts across cycles (Section 2.5), providing a unified modeling framework. For λ, the trajectory was better linearized in log-linear space, motivating an exponential function analogous to the anchored exponential model in Eq. 4. Both models are anchored to first-cycle estimates, so that later-cycle behavior is predicted from early-night dynamics. Prior to averaging, sleep cycles with a low SO count (below a cycle-specific threshold) were excluded, where the threshold was defined as the median of the lower quartile of SO counts across channels and subjects. Eight electrodes (TP7, TP8, P1, P2, P5, P6, PO3, and PO4) were excluded entirely from this modeling due to insufficient SO counts. For modeling the inter-arrival time of SO occurrence, we focused exclusively on N3.

To simulate the inter-arrival time of SOs, we generated random numbers from a uniform distribution. We then used the inverse CDF to convert those values to simulated inter-arrival times of SOs. Only inter-arrival times exceeding a minimum inter-event interval for the given sleep cycle and electrode were accepted into the simulation. This minimum inter-event interval was defined empirically as the smallest observed inter-trough interval at each electrode for each subject and sleep cycle — that is, the minimum value of the pairwise differences between successive SO trough timestamps. For use in the simulation, this value was averaged across subjects per electrode and cycle. This threshold serves as a data-driven lower bound on the inter-arrival time distribution, ensuring that simulated SO sequences respect the minimum spacing observed in the recorded data. Physiologically, it approximates the minimum time required for the thalamocortical network to complete one SO event before a new SO trough can be detected, and therefore reflects the data-derived time duration in between two subsequent SO troughs. The simulation was built on an iterative process, including continued generation of simulated inter-arrival time values until the last inter-arrival time exceeded the cumulative summation of all inter-arrival times of SOs.

2.7. Model evaluation

The primary goal of this study is to model the timing of SO events using probabilistic distributions, rather than to classify individual events or generate pointwise predictions. Accordingly, we evaluated model performance using goodness-of-fit metrics that quantify how well the simulated inter-arrival distributions match observed data. This approach emphasizes probabilistic structure over detection accuracy, and avoids reliance on sensitivity or specificity, which are suited to classification tasks.

We used three primary metrics to quantify model fit: absolute difference error, the Kolmogorov–Smirnov (KS) test, and Kullback–Leibler (KL) divergence, each applied using 100 bootstrapped simulations per cycle. Absolute difference error measures the average relative deviation between model-generated and empirical SO timings:

(8)

KL divergence is a measure to quantify the dissimilarity between two probability distributions by explaining how one probability distribution diverges from another [53]. It can be defined using the following formula:

(9)

where P and Q are probability distributions on the same sample space 𝜒, and x is the bin number in the histogram. p(x) and q(x) are probability density function of P and Q respectively. Values of this measure closer to zero indicate lower dissimilarity.

We used the KS-test to assess whether the distributions of our data and simulations could be derived from the same probability distribution. To obtain a cumulative value across multiple simulations, we defined a “distribution similarity” metric, as the fraction of 100 simulations which passed the KS-test (p-value > 0.05) when compared to a sample of real data. We interpret this measure as an indicator of similarity, ranging between 0 and 1, with values closer to 1 indicating higher similarity. It should be noted that, as a statistic derived from the KS test, this metric is sensitive to the number of inter-arrival intervals available in each cycle. Because SO counts decline across successive cycles, the KS test has lower statistical power to detect distributional differences in later cycles, which introduces a sample-size-dependent component when comparing distribution similarity values across cycles. For this reason, this metric should be interpreted alongside the absolute difference error and KL divergence, which provide complementary and more sample-size-stable assessments of distributional correspondence.

In addition to the three mentioned measures, we utilize a probability-probability (P-P) plot to visually assess the performance of the model by examining how closely a dataset aligns with a particular model. This method entails plotting the two CDFs against each other. If they closely resemble each other, the data will appear as a nearly straight line [54]. Confidence intervals represent the lower and upper bounds of the range within which we expect the CDF of the data to lie with a certain level of confidence, 95% in this study. If the points in the P-P plot closely adhere to the diagonal line (y = x), it indicates a good fit between the distribution of the data and the model. Conversely, if a significant number of points in the P-P plot fall outside the confidence intervals, it may suggest a lack of fit between the distributions of the data and the model.

To represent the performance of the model in a P-P plot across all subjects, we followed the method outlined in reference [38] to standardize the CDF lengths of both the data and model to 250 points for each participant. In this method for standardizing the lengths of different signals, each signal is first interpolated to a fixed length N (in this study, 250 points) by generating additional data points between successive samples using a uniform interpolation method. The interpolated signal is then down-sampled at a rate proportional to its original length, resulting in a final signal scaled to the desired length N. Subsequently, we computed the mean CDF across all participants for both the data and the model. Plotting these mean CDFs against each other for each cycle produced the P-P plot. This resulting plot visually illustrates the average performance of the model across all participants in our study.

This study employs evaluation methods to assess how well the model replicates the empirical distribution of SO inter-arrival times. We use quantitative metrics—including absolute difference error, KL divergence, and the KS test—alongside visual tools such as P–P plots to evaluate the alignment between model-generated and observed data. These comparisons help characterize how accurately the model captures the probabilistic structure of SO timing across the night.

3.1. SO percentage across cycles

We were firstly interested in estimating and modeling the cumulative trend of SO count decay across subsequent sleep cycles. For this analysis, we focused on the first four sleep cycles for each participant, since most participants had 4 or fewer cycles (Fig 1A) and cycles after the fourth had a very small amount of N3, which is the target of this modeling investigation. Consistent with known trends of SWA, we observed a decrease in the SO percentage during successive sleep cycles, both in each participant (Fig 1B shows one example) and in a normalized averaged representation of cycle-by-cycle SO emergence across all participants (Fig 1C). When considering the cumulative amount of SOs found within each cycle, Fig 1D shows again the decreasing trend across them, which we modeled with a power law relationship:

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Fig 1. SO percentage across successive cycles.

A) cycle count in different participants of our dataset. The x-axis shows participant number in the dataset. Most participants have 4 or fewer cycle. The average cycle count in our study is 4.2. B) SO percentage in successive epochs across cycles during N3 of the first participant in our dataset. The x-axis shows time per hour from sleep onset to end of fourth cycle. C) SO percentage across successive cycles during N3 by averaging across all participants. The x-axis shows normalized time, with cycle length equalized to 100 points. D) Total SO percentage during N3 across successive cycles obtained by averaging SO percentage across all participants. The across-cycle trend was modeled using a power-law function , with T = −1.701. The x-axis shows sleep cycle, and error bars indicate standard error. In panels B and C, each dot represents the SO percentage at a single time point (a 30-second epoch in B; a normalized time point in C), and the brown line shows the smoothed SO percentage obtained by a running median across 10 consecutive time points.

https://doi.org/10.1371/journal.pcbi.1014572.g001

(10)

where represents the SO percentage in a given cycle (a cumulative count across all cycle divided by the total count of SOs in the night), is the SO percentage in the first cycle, C denotes the cycle number (), and T, the rate of decay, is equal to -1.701. The RMSE of the model equaled 0.0085. We modeled the across-cycle trend for each participant with a power law separately and found that the average decay rate across participants was -1.6641.002. Thus, the SO percentage in cycles 2–4 declined systematically relative to the first cycle, allowing cross-cycle dynamics to be described based on early-night measurements. To assess whether an exponential decay formulation—commonly used in sleep homeostasis models—provides a better description of the across-cycle SO decline, we compared the power-law model with an anchored exponential model using a single fitted parameter (S8 Fig). At the group level, the exponential model provided a slightly better fit (R² = 0.992 vs. 0.988; RMSE = 1.92 vs. 2.29), with a fitted decay rate of R = −1.151 (Equation 4). However, at the individual subject level, both models performed similarly (power-law: R² = 0.758 ± 0.370, RMSE = 7.74 ± 5.41; exponential: R² = 0.762 ± 0.364, RMSE = 7.97 ± 5.25), with the power-law yielding slightly lower subject-level RMSE and no significant differences in R² and RMSE between models (p > 0.6). Given this comparable performance, we retained the power-law formulation for its interpretability and its linearization in log–log space, which permits direct visual inspection of the decay exponent. Repeating this analysis for the combined N2&N3 stage revealed a qualitatively similar decreasing trend in SO percentage across cycles (S9 Fig), with a group-level decay rate of −1.14 (RMSE = 0.0064) and a mean per-subject decay rate of −1.21 ± 0.73. A paired comparison of individual decay rates between N3 and N2&N3 showed that the N3 decay was significantly steeper (t(21) = −3.96, p = 0.0008, 95% CI [−1.08, −0.34]; S10 Fig), consistent with the expectation that including N2 epochs — which carry fewer SOs — attenuates the apparent rate of decline. The inter-subject variance in decay rates was substantially smaller for N2&N3 (variance = 0.28) than for N3 (variance = 1.12), indicating greater cross-subject consistency in the combined-stage analysis.

To determine whether the observed power-law decay in SO percentage is driven by changes in sleep architecture, we examined the relationship between NREM stage duration and SO expression. N3 duration decreased markedly across cycles, while N2 duration increased (S11 Fig). Cross-sectional analyses (across participants within each cycle) revealed that N3 duration was strongly and consistently associated with SO percentage across all cycles (r = 0.81–0.91, all p < 0.0001; S12 Fig), whereas N2 duration showed weaker and less consistent relationships. Within-subject analyses further demonstrated that declines in N3 duration closely tracked reductions in SO percentage across cycles (mean r = 0.97, p < 0.0001; S13 Fig). To directly test whether N3 duration accounts for the across-cycle decay, we performed a mixed-effects mediation analysis. N3 duration accounted for 64.5% of the cycle-related decline in SO percentage, while a significant direct effect of cycle remained (β = -0.279, p < 0.0001), indicating partial mediation. Given that N3 is characterized by a higher prevalence of SOs, this result reflects a strong statistical association between N3 duration and SO expression.

3.2. Characterizing the temporal structure of SO occurrence

To characterize the temporal structure of SO occurrence, we modeled the inter-arrival times of successive SOs and computed their cumulative sums to approximate SO timing within each cycle. Using the procedure outlined in Section 2.5, we obtained IG parameters (μ and λ) for each of the 56 EEG channels across 22 participants, resulting in a total of parameter pairs per cycle. Fig 2A schematically shows the matrix of μ values for one sleep cycle. Separate matrices were generated for both μ and λ across all four cycles. Fig 2B and 2C illustrate the distribution of μ and λ values across cycles, respectively. The distributions of both parameters are right-skewed, particularly in later cycles where a subset of channel×subject combinations with low SO counts yielded high μ and λ estimates; consequently, the arithmetic mean reported in panels D and E exceeds the modal peak visible in the histograms of panels B and C. An increasing trend in the mean μ values across successive cycles was observed in Fig 2D, which we modeled using a power law.

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Fig 2. Inverse Gaussian (IG) model parameters for SO inter-arrival times across sleep cycles.

A) Matrix of μ values across subjects (22) and EEG channels (56) for a single sleep cycle (totaling 1,232 values per parameter per cycle). B) Histogram of μ values across four sleep cycles. The x-axis shows μ, and the y-axis indicates frequency. A magnified inset is provided for the first cycle. C) Histogram of λ values across four cycles; inset shows the first cycle. D) Average μ across subjects and channels for each cycle. The trend follows a power law. E) Average λ across subjects and channels per cycle, modeled with an exponential fit. For panels D and E, averages were computed over 1,232 values per cycle (56 channels × 22 subjects). Error bars represent the standard error of the mean (SEM). Note that the SEM is substantially smaller than the full spread of values visible in panels B and C, which display the complete distribution of individual channel×subject parameter estimates. This distribution is right-skewed, particularly in later cycles where SO counts are lower, which causes the arithmetic mean in panels D and E to exceed the modal peak of the histograms in panels B and C.

https://doi.org/10.1371/journal.pcbi.1014572.g002

(11)

where represents the value of in the first cycle, C is cycle number, and T, which represents the increasing rate, was estimated as 0.8348. The RMSE of the model was 0.2381. By modeling μ for each participant separately, the average value of μ across participants was 0.831, with a variance of 0.44. Fig 2E shows the trends that emerged when the same procedure was applied to the distributions of λ values. The increasing trend in average λ across cycles was fitted using an exponential relationship as:

(12)

where represents the value of in the first cycle, C represents cycle number, and R represents the increasing rate estimated as 4.5893. The RMSE of the model was 0.0096. To justify the choice of functional form for both parameters, S14 Fig presents the cross-cycle trajectories of μ and λ on both linear and transformed scales, with power-law and exponential fits overlaid. For μ, the power-law fit appears as a straight line in log-log space while the exponential deviates, confirming the power-law as the better-fitting form. For λ, the exponential fit is linearized in log-linear space while the power-law deviates, supporting the exponential choice. This visual comparison directly motivates the functional form selected for each parameter. The parameter μ represents the mean inter-arrival time between successive SOs at a given electrode (in seconds). Its power-law increase across cycles means that SOs arrive progressively less frequently as the night advances — consistent with the well-established decline in slow-wave activity under diminishing homeostatic sleep pressure. The parameter λ governs the shape of the inverse Gaussian distribution: a larger λ corresponds to a narrower distribution of inter-arrival times around their mean, indicating more regular (less variable) SO timing. The exponential increase in λ across cycles therefore reflects a progressive regularization of SO occurrence: although fewer SOs occur in later cycles, those that do occur are more evenly spaced in time. Together, the increases in μ and λ capture a shift across the night from a high-density, highly variable SO regime — characteristic of maximal homeostatic drive in early sleep — toward a lower-density but more rhythmically predictable pattern in later cycles. By modeling individually for each participant, the mean value of across all participants was 8.506, with a variance of 0.113. Relationships 9 and 10 describe how the μ and λ parameters evolve across cycles in a systematic manner based on values observed during the first sleep cycle.

To assess whether the within-cycle temporal structure of SOs is preserved when N2 is included, we estimated the IG parameters μ and λ for the N2&N3 condition using the same renewal process procedure (S5 Table). Both parameters showed a monotonically increasing trend across cycles, consistent with N3 results. N2&N3 values were consistently larger: μ increased from 14.46 ± 0.41 s (Cycle 1) to 59.20 ± 2.25 s (Cycle 4), and λ from 4.10 ± 0.05 (Cycle 1) to 16.04 ± 1.90 (Cycle 4). Cycle-by-cycle paired comparisons revealed significant differences in μ in cycles 1 and 3, and in λ in cycles 1–3, indicating that while the absolute magnitude of SO timing structure is altered by the inclusion of N2, the cross-cycle trajectory is preserved. The larger μ in N2&N3 reflects lower SO density in N2 epochs; the larger λ indicates a more regular inter-arrival time distribution in the combined condition, likely reflecting the more homogeneous and lower-density SO distribution in N2 epochs, which reduces variability in inter-arrival times when combined with N3. To assess whether the observed model parameters reflect spatially meaningful differences in SO dynamics, we performed a regional analysis comparing frontal and posterior electrode groups. Frontal electrodes (FP1, FP2, FPz, F3, F4, Fz, F7, F8) exhibited consistently lower μ values compared to posterior electrodes (P3, P4, P1, P2, P5, P6, Pz, O1, O2, P7, PO3, PO4, POz, P8), indicating shorter inter-SO intervals and higher SO density at frontal sites (S15 Fig). Additionally, λ values were higher in frontal regions during early sleep cycles, suggesting more regular SO timing at frontal electrodes early in the night, with this difference diminishing across later cycles. Despite these regional differences in magnitude, both frontal and posterior groups showed similar across-cycle trajectories, with increasing μ and evolving λ across successive cycles. These results are consistent with the well-established frontal predominance of SOs and indicate that the model parameters capture regionally differentiated dynamics rather than purely global scalp averages.

To quantify model performance, we used three evaluation metrics: absolute difference error, distribution similarity (based on the KS test), and KL divergence (Table 1). To assess robustness, each metric was computed over 100 bootstrapped simulations. The analysis was conducted for cycles two through four using model parameters derived from the first cycle. Both absolute difference error and KL divergence increased across cycles, indicating a decline in model-data correspondence as the night progressed. To clarify the interpretation of these results, we emphasize that the model is not intended to provide precise pointwise prediction of SO timing in later cycles, but rather to capture partial temporal continuity in the statistical structure of SO occurrence across the night. In this context, distribution similarity was defined as the fraction of bootstrap simulations in which the KS test failed to reject the null hypothesis, indicating statistical indistinguishability between simulated and observed inter-arrival time distributions. Under the null hypothesis (α = 0.05), a baseline rejection rate of 0.05 is expected by chance. Therefore, distribution similarity values substantially greater than 0.05 indicate that the first-cycle model captures non-random structure in later-cycle SO timing. The observed values (Cycles 2–4: 0.51, 0.70, 0.63) reflect moderate but imperfect distributional correspondence between model-generated and empirical inter-arrival times. These values should be interpreted with the following caveat: because SO counts decline systematically across the night, later cycles provide fewer inter-arrival intervals per channel-subject combination, which reduces the statistical power of the KS test to detect distributional differences. This sample-size dependence means that the absolute values of distribution similarity are not directly comparable across cycles. Nevertheless, all three values substantially exceed the 0.05 chance baseline, confirming that the first-cycle model captures genuine non-random temporal structure in later cycles. The non-monotonic pattern — with similarity peaking in Cycle 3 before declining in Cycle 4 — likely reflects the combined influence of declining SO counts (which reduces KS test sensitivity and mechanically inflates apparent similarity in Cycle 3) and the genuine increase in inter-individual variability by Cycle 4, which degrades distributional correspondence beyond what the sample-size effect alone would predict. Notably, distribution similarity peaked during the third cycle, suggesting relatively stronger alignment between simulated and observed SO timing in that interval. Although distribution similarity was highest in the third cycle, visual comparison of the parameter trends in Fig 2D–2E may suggest a larger apparent discrepancy between the model and empirical data relative to cycle 2. This occurs because the similarity metric in Table 1 evaluates the correspondence between the distributions of simulated and empirical inter-arrival times, whereas Fig 2D–2E illustrate cycle-averaged parameter trends (μ and λ). Consequently, small deviations in the averaged parameter trajectories may appear visually larger even when the underlying inter-arrival time distributions remain closely aligned.

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Table 1. Absolute difference error, distribution similarity and Kullback-Leibler divergence using 100 times bootstrapping.

https://doi.org/10.1371/journal.pcbi.1014572.t001

To visually demonstrate model performance, Fig 3 presents a simulation of SO timing during the third cycle of Participant 4, for a single EEG channel. This example illustrates modeled SO occurrences based on inter-arrival times generated from the fitted distribution. The μ and λ parameters used in the simulation were estimated from the participant’s first cycle and extrapolated to the third cycle using Equations 9 and 10. Simulated SO intervals were generated by transforming uniform random values via the inverse CDF, following the procedure described in Section 2.5. This example reflects the stochastic nature of SO timing and provides a representative case of how the model reproduces observed patterns in spontaneous sleep data.

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Fig 3. Simulated and observed SO timing in EEG channel Fz during the third sleep cycle of one participant.

A) Hypnogram from sleep onset through the end of the third cycle. The x-axis represents time (in hours); wake periods correspond to EEG epochs marked by major body movements. B) Detected SOs in the original EEG recording for this cycle (n = 101). C) Simulated SO occurrences generated using μ = 8.93 and λ = 3.87. D) Comparison of SO timing between the empirical data and one simulation run for the third cycle.

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We also used probability–probability (P–P) plots to evaluate the alignment between the model-generated and empirical distributions of SO inter-arrival times across successive sleep cycles. Following the procedure outlined in Section 2.6, within each cycle, the length of the CDF of both the data and the model was equalized to 250 points and then averaged across all participants. In the resulting P–P plots, closer alignment to the diagonal line indicates greater similarity between the empirical and modeled distributions. As shown in Fig 4, the majority of points fell within the 95% confidence interval, supporting an acceptable goodness of fit between the model and observed data.

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Fig 4. Probability–probability (P–P) plots comparing cumulative distributions of observed and simulated SO inter-arrival times across sleep cycles.

The y-axis shows the empirical cumulative distribution function (CDF); the x-axis shows the model-generated CDF. The red dashed line represents the 95% confidence interval. Panels A–D display results for cycles 1 through 4, respectively. The black diagonal line represents the identity line (y = x), indicating perfect agreement between model and data. In Panel A (green), the IG model was directly fit to the inter-arrival times from the first cycle. In Panels B–D (blue), the model parameters were extrapolated from first-cycle estimates using Equations 9 and 10.

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