Estimation of the Circadian Phase Difference in Weekend Sleep and Further Evidence for Our Failure to Sleep More on Weekends to Catch Up on Lost Sleep

In the in silico study, we used the same set of model parameters (Table 1, bottom part) to calculate three pairs of time courses of sleep and wakefulness on weekdays and weekends characterized by the identical weekend risetimes (Figure 1 and Table 1, upper part), weekday risetimes (Figure 2), and gaps between weekend and weekday risetimes (Figure 3). The model predicts that, irrespective of the gap between weekend and weekday risetimes (4.0 h and 2.0 h in Figure 1 and Figure 2), the same timing and duration of sleep are predicted to be restored during weekdays (Table 1, upper part). In other words, the prediction is that the endogenously determined sleep duration must be fully reestablished during two days of the weekend after any shift in weekday risetime relative to weekend risetime (e.g., either 4.0 h or 2.0 h in Figure 1 and Figure 2). As illustrated in Figure 1 and Figure 2, the model does not predict that the wake phase of the sleep–wake cycle on weekdays can be delayed after a shorter sleep duration on the previous weekdays. The figures show that the wake phase of the sleep–wake cycle on weekdays is always terminated at the upper threshold of deviation from the setpoint allowed by the sleep–wake regulator to initiate the transition to the sleep phase instead of rising further to accumulate "sleep debt" (i.e., see the upper sine function in Figure 1 and Figure 2). This implies that a shorter weekday sleep duration (i.e., a larger weekday sleep loss or, in other terms, a larger "accumulated sleep debt") cannot cause a larger weekend sleep duration (i.e., in other terms, "compensation of sleep loss" or "paying sleep debt back" or "catching up on lost sleep", etc.).

Moreover, the model also predicts that since a shorter weekday sleep duration cannot cause a larger weekend sleep duration (Figure 1), the difference between earlier and later weekday risers in weekend sleep timing (2.0 h in Figure 1 and Figure 2) is equal to the difference in the circadian phase of sleep (2.0 h in Figure 1 and Figure 2). If this is true, the difference between evening and morning types in the circadian phase of sleep can be estimated as the difference in weekend risetime (or in weekend bedtime). However, these in silico calculations do not take into account that a larger shift in weekday risetime relative to weekend risetime in later risers compared with earlier risers (4.0 h vs. 2.0 h in Figure 1 and Figure 2) can lead to a larger advancing shift in the 24 h cycle of their light exposure, which in turn can lead to a greater advance of the circadian clock phase, and consequently a larger advance of the wake and sleep phases of the sleep–wake cycle on weekdays and even on the following weekend in later risers. Therefore, a non-zero difference between earlier and later weekday risers in weekend risetime is expected instead of a zero difference shown in Figure 1 (i.e., an advance must be larger after an earlier weekday risetime due to a larger gap between weekend and weekday risetimes). Similarly, the calculations shown in Figure 2 do not take into account that such a larger shift in weekday risetime relative to weekend risetime (4.0 h vs. 2.0 h) can lead to a larger advancing shift in light exposure, and so on (see above). Therefore, the difference between earlier and later weekday risers in weekend risetime is expected to be smaller than their difference in the circadian phase (e.g., somewhat smaller than the 2.0 h difference shown in Figure 2).

The question arises: what might be a condition when this advancing shift in the 24 h cycle of light exposure is identical in earlier and later risers? The answer is: this is a condition when, in addition to the identity of durations of weekend sleep, the weekend–weekday gap in risetime is also identical in these risers. In contrast to the results of calculations shown in Figure 1 and Figure 2, the calculations shown in Figure 3 illustrate this condition of the identical shifts in weekday risetimes relative to weekend risetimes (2.0 h vs. 2.0 h in Figure 3 instead of 4.0 h vs. 2.0 h in Figure 1 and Figure 2). Therefore, when the advancing shifts in the 24 h cycle of light exposure are expected to be identical in earlier and later risers, the difference between them in weekend sleep times (either bed- or risetime) is equal to the difference between them in the circadian phase of sleep. Consequently, the difference between chronotypes in the circadian phase of sleep can be calculated from the sleep times reported for weekdays and weekends as the difference in weekend sleep timing in the condition of similarity of weekend–weekday gaps in risetime and weekend sleep duration (Figure 3).

Thus, the results of the in silico study indicated that sleep times can be used for direct estimations of the differences in the circadian sleep phase between chronotypes with similar weekend sleep duration and similar weekend–weekday gap in risetime (Figure 3). Therefore, data from the empirical study can be used to exemplify the application of this approach to the calculation of the difference between chronotypes in the circadian phase of sleep as the difference between them in weekend risetime (Table 2 and Tables S9–S13).

Moreover, data from an empirical study can be used to perform the estimations of circadian phase difference using the whole dataset, i.e., individuals with different weekday gaps in risetime. The calculations shown in the notes to Table 3 suggest that the difference in circadian phase of sleep (Table 3, lower part) can be directly calculated as the sum of the difference between earlier and late weekday risers (Table 3, middle part) and the difference between morning and evening types (Table 3, upper part). This implies that the whole set of data on two compared chronotypes can be used to calculate a difference between these chronotypes in the circadian phase of sleep (Table 3, Table S14 and S16, and Figure 4B).

The mean ages ± standard deviations in subsamples of 1271 male and 3075 female university students (age 17–25 years) were 19.3 ± 2.1 and 19.4 ± 1.7 years, respectively, and the mean ages ± standard deviations in subsamples of 138 male and 456 female lecturers and other staff members (age > 25 years) were 41.0 ± 13.7 and 42.5 ± 12.2 years, respectively.

As shown in, the survey participants from two age subsamples were further divided into earlier and later weekday risers, based on whether or not their risetime was earlier or later than 7:00, and into seven chronotypes according to their answer to the SIC (Single-Item Chronotyping) question: "Self-assess your chronotype by choosing one of six patterns of daily change in alertness level". The chronotypes are abbreviated as LIVEMAN types: Lethargic, Inconclusive, Vigilant, Evening, Morning, Afternoon, and Napping. The significant differences between these chronotypes were confirmed by assessing chronobiological characteristics such as weekend sleep times (), and sleepiness, sleepability, and wakeability in the morning, daytime and evening/night hours (). Table S1 Table S2 and Figures S1–S3 Table S17 and Figures S4–S7

contains the results of ANOVAs of sleep times in the subdivisions of the whole sample applied for testing the significance of four independent factors (seven chronotypes, two ages, two sexes, and two weekday risetimes).contain the estimates of mean sleep times in the whole sample and two age subsamples (they are also illustrated in). These ANOVAs () with the following post hoc pairwise comparisons () confirmed a model-based prediction of independence of weekend time in bed from the weekend–weekday gap in risetime (i.e., a shorter weekday time in bed caused by a larger gap does not lead to a longer weekend time in bed). Since this gap was significantly larger in evening than morning types, this excludes the possibility of simple comparison of the difference between all morning types and all evening types in the circadian phase of sleep using the difference in their weekend sleep timing (the latter difference is smaller due to a larger advance of the circadian phase in evening types due to a larger advance of the 24 h cycle of light exposure in response to early weekday wakeups). Table S2 Tables S3 and S4 Figures S1–S3 Table S2 Tables S5 and S13

Post hoc pairwise comparisons of weekend time in bed in chronotypes reported in Table 2 suggested that, despite the independence of this time from weekend–weekday gap in risetime predicted by the model (Figure 1), it was, nevertheless, shorter in one of six chronotypes (vigilant type) compared to lethargic, napping, and afternoon types. Indeed, such a shorter time in bed is expected for this vigilant type due to its chronobiological characteristics illustrated in Figures S4–S7, e.g., low sleepiness levels, high wakeability, and low sleepability both in the morning and evening hours. Since weekend time in bed in the vigilant type was significantly different from that in some of the other types, this type was not included in the comparison of the circadian phase difference between all chronotypes presented in Tables S6–S13.

The results suggesting non-significant differences in weekend time in bed and weekend–weekday gap in risetime in some of the chronotypes (e.g., evening vs. napping and lethargic vs. afternoon;) provided the possibility to interpret the differences between these types in weekend risetime as estimates of the differences between them in their circadian phase of sleep (). The results suggested that the circadian phase of the evening type was significantly delayed relative to the phase of the napping type, but this difference did not exceed one h (at least, 0.5 h), and the lethargic type did not significantly differ from the afternoon type in this phase (). Tables S6 and S13 Table S6 Table S6

Moreover, since weekend time in bed in morning types was not significantly different from time in bed in vigilant and other types, this provided the possibility to compare the circadian sleep phase of this type with circadian sleep phases of all the other types by means of separate ANOVAs of samples including only morning type and one of the other types (Table 3 and Tables S13–S16 and Figure 4 and Figure 5).

Figure 4A illustrates the similarity of chronotypes on the advancing shifts in weekend bed- and risetimes in response to the advancing shift in weekday risetime. This similarity is confirmed by non-significant interactions between the factors chronotype and weekday risetime (Table S2, upper part). However, similar advancing shifts in morning and evening types also suggested that, irrespective of the shift, there was a profound difference between these types in the weekday–weekend gap in risetime (i.e., this weekend–weekday gap in risetime was larger in evening earlier weekday risers than in morning earlier weekday risers and it was also larger in evening later weekday risers than in morning later weekday risers; Tables S7 and S8). In contrast, this difference in the gap between these chronotypes was found to be non-significant in the comparison of earlier weekday risers of morning type with weekday risers of evening type (Table S8, upper part). This non-significant difference provided the possibility to interpret the difference between these sub-subsamples of morning and evening types in weekend risetime as the estimate of the difference between them in the circadian phase of sleep (Tables S8–S13). Such comparison of morning and evening types indicated that this difference exceeded two hours (Table S8, upper part, Tables S12 and S13). The difference between earlier risers of morning type with later risers of three other types (lethargic, napping, and afternoon) was smaller by at least 0.5 h, i.e., the circadian phase position of lethargic, napping, and afternoon types was between the positions of morning and evening types. Similar estimates of circadian phase differences between morning and other chronotypes were obtained by using another approach to this estimation when ANOVAs were performed on datasets consisting of morning type and one of the other types (Table 3, Tables S14 and S16). Figure 4B illustrates the differences between chronotypes obtained by comparison of earlier risers of morning type (the earliest bed- and risetimes for this type) with later risers of evening, lethargic, napping, and afternoon types (the later bed- and risetimes for these types).

The results illustrated in Figure 5A,B suggested that, as expected, the two age subsamples drastically differed in bed- and risetimes. These times on weekends in older age were advanced relative to the times in younger age (Table S4, Figure 5). Despite this, two ages showed practically identical differences between chronotypes in the circadian phase of their sleep, estimated as the difference between early risers of morning type and later risers of evening type in weekend risetime (Figure 5, and Tables S8 and S12, upper parts, and Table S15). Other compared pairs of chronotypes also demonstrated similarity of older and younger ages in the difference between chronotypes in weekend risetime (Table S12, upper part, and Table S15) despite the similarly large differences between ages in weekend sleep timing (Table S4).

Table 4 and Figure 6 exemplify the way in which a model-based simulation (Figure 6) can be applied to confirm the results of in silico (Figure 3) and empirical studies (Figure 4B). The simulation supported the results of direct estimation of the differences between chronotypes in the circadian phase of weekend sleep. In this example, the same set of model parameters, with the exception of the circadian phase parameter, was used to simulate rise- and bedtimes reported by earlier risers of morning type and later risers of evening type for weekdays and weekends. In particular, the results of this simulation (Table 4 and Figure 6) supported the results of the empirical study, indicating a more than 2.0 h difference between these two chronotypes in the circadian phase of sleep. They also supported the predicted (Figure 1, Figure 2 and Figure 3) and observed similarity (Table 2, Table 3, Tables S8, S13, S14 and S16) of weekend sleep duration in these two chronotypes.

For the present in silico and simulation studies, we used one of the variants [48] of the two-process model of sleep–wake regulation [27] proposing that any parameter of the homeostatic process in the classical variant of the two-process model [27] is modulated by the circadian clock. This modulation [48] was included in the model as the counterpart of the circadian process in the classical variant of the model [27]. Mathematically, this circadian term is the simplest periodic (sine) function with the 24 h period (the period was assigned to 24.00 h because, in natural settings, the circadian clocks always remain under control of (i.e., are entrained to) the external light–dark cycle with the 24 h period [48]). If t1 and t2 are the initial times for the buildup and decay phases of this process of sleep–wake regulation St, it can be calculated using the following equations:(1a)St=[Su+C(t)]−{[Xu+C(t)]−Sb }∗e−(t−t1)/[Tb − k ∗ C(t)],(1b)St=[Sl+C(t)]−{Sd−[Sl+C(t)]}∗e−(t−t2)/[Td−k ∗ C(t)] where

(2)C(t)=A∗sin(2π∗t/24+φ0) is a sine function with a 24 h period (Table 1 and Table 4).

The parameters of the model [48] were initially derived from data on (1) the durations of recovery sleep after 6 gradually increasing intervals of extended wakefulness [49] and (2) the levels of slow-wave activity (SWA) in 10 naps [50] and two recovery sleep episodes [51,52]. The simulation of such experimental data provided the possibility of measuring St as relative slow-wave activity (rSWA), i.e., rSWA = 1 in the baseline sleep. The initial parameters of the model were slightly modified in the previous (e.g., [28,29]) and present computations and simulations (Figure 1, Figure 2, Figure 3 and Figure 6, and Table 1 and Table 4) to account for the difference between the initially simulated experimental sleep durations [49] and times in bed calculated from self-reported bed- and risetimes on weekdays and weekends [28,29]. Detailed description of the model, its physical counterpart, and its parameters can be found in [28].

All calculations and simulations were performed using macros in Excel software. In the in silico study, the model of the sleep–wake regulating process [48] was used to calculate the sleep–wake cycles of two hypothetical earlier and later risers, and the results of the calculations are illustrated in Figure 1, Figure 2 and Figure 3. It was proposed that the process of sleep–wake regulation St is identical in these earlier and later risers (lower part of Table 1). It was additionally proposed that these risers have either identical weekend risetimes and circadian phases (Figure 1) or identical weekday risetimes but different circadian phases (Figure 2) or identical gaps between weekend and weekday risetimes but different circadian phases (Figure 3). The 24 h sleep–wake cycle in the model of this process St is represented by daily alternations of an inverse exponential buildup phase (1a) and an exponential decay phase (1b), i.e., the wake and sleep phases of the cycle. The parameters of St are additionally modulated by the circadian clock represented by a sine-form function with a 24 h period, Ct (2). In these figures, the modulation is illustrated by the 24 h oscillations of Sbt and Sdt that are the highest buildup and the lowest decay of St, i.e., bedtime and risetime, respectively, determined by the endogenous sleep–wake regulating process St (1,2) measured in rSWA (see the model parameters in Table 1). These Sbt and Sdt are, in fact, the thresholds to which the sleep–wake regulator allows the deviations of the regulated process from the setpoint.

In Figure 1, the difference in risetime on weekends is equal to zero because, despite two different 2 h or 4 h gaps in weekend–weekday risetime, the circadian phases of sleep are identical (i.e., rise- and bedtimes on Sunday are equal to 9:00 and 24:00, respectively). The model predicts that, due to the circadian modulation Ct of the parameters of St, the endogenously determined bed- and risetimes (shown for Sunday) are restored during just one day with ad lib sleep (between Friday and Saturday). This restoration is not affected by the advance of weekday risetime relative to weekend risetime (e.g., either 4.0 h or 2.0 h in earlier and later risers, respectively). Therefore, weekend rise- and bedtimes on Sunday are equal to 9:00 and 24:00, respectively, i.e., they are expected to be identical after 5 days of earlier and later weekday risetime at 5.0 a.m. and 7.0 a.m., respectively.

In Figure 2, not only are two different weekend–weekday gaps in risetime (2.0 h or 4.0 h) hypothesized, but also a 2.0 h difference in the circadian phase of sleep is hypothesized. This makes identical the weekday risetimes of earlier and later risers, and, therefore, the model predicts that, due to the combination of the 2.0 h difference in this phase with the identical 2.0 h difference in bed- and risetimes on weekends, the difference in weekday risetime is equal to zero.

In Figure 3, two identical (2.0 h) weekend–weekday gaps in risetime and the 2.0 h difference in the circadian phase of sleep are hypothesized. Since there is no difference in this gap, the difference in weekday risetime is equal to the difference in weekend risetime (and weekend bedtime either). Consequently, the results of model-based calculations shown in Figure 1, Figure 2 and Figure 3 suggest the following variants of the equation: difference in weekday risetime = difference in weekend risetime plus weekend–weekday gap in risetime. 1. The difference in weekday risetime is equal to the gap in risetime (Figure 1). 2. The difference in weekend risetime is equal to the gap in risetime (Figure 2). 3. The difference in weekday risetime is equal to the difference in weekend risetime (Figure 3).

The calculations shown in Figure 1 and Figure 2 do not take into account a larger advancing shift in the 24 h light exposure expected from a larger advancing shift in weekday risetime, while Figure 3 suggests that these shifts are expected to be identical because the difference in weekday risetime is equal to the difference in weekend risetime (i.e., the shifts in the 24 h light exposure are identical in earlier and later risers illustrated in Figure 3). Consequently, the difference in weekend sleep times is expected to be equal to the difference in the circadian phase of sleep due to a zero gap between weekend and weekday risetimes (Figure 3).

In the winter season, lecturers invited students and staff members of older age to respond to the questions asking about their sleep–wake habits and behavior (). No exclusion criteria were applied for participation in the survey. All students attended classes/office in the university buildings, which excludes the possibility that they were in unhealthy physical and mental condition at the time of participation in the survey. The responses were collected from smartphones using the Google Forms questionnaire. Informed consent was obtained from each participant in the form of a response of "Agree" to the first statement of the questionnaire: "I give informed consent to anonymously and voluntarily participate in this online survey of sleep–wake behavior and habits". Table S1

The survey participants were asked to report clock hours for bed- and risetimes on weekdays and weekends (Figures S1–S3, and Tables S3 and S4). Single-Item Chronotyping (SIC) [42] was used for the subdivision of the whole sample into chronotypes (Table S1). Additionally, survey participants were asked to complete two multi-item chronotypological questionnaires (Figures S3–S7): the 19-item Visuo-verbal Judgment Task (VJT) [53], for reporting expected levels of sleepiness on a 1.5-day interval of permanent wakefulness, and the reduced version of the 60-item Sleep–Wake Adaptability Test (SWAT) for self-assessment of abilities to sleep or wake on demand at different times of the day [54,55].

The original English version of the SIC was proposed by O. Mairesse and A. Putilov for distinguishing between six rather than just two morning and evening chronotypes [42]. The English names for the seven response options to the SIC's question ("Self-assess your chronotype by choosing one of six patterns of daily change in alertness level") were abbreviated as "LIVEMAN" (Lethargic, Inconclusive, Vigilant, Evening, Morning, Afternoon, Napping type). The illustrations and short descriptions of levels of alertness in the morning, afternoon, and evening were assigned to the graphs and names of these chronotypes (see the SIC in the Supplementary Materials to [42]): (1) Lethargic type ("Moderately energetic type" in Russian): low level in the morning, afternoon, and evening; (2) Vigilant type ("Highly energetic type" in Russian): high level in the morning, afternoon, and evening; (3) Evening type: low level in the morning, middle in the afternoon, high in the evening; (4) Morning type: high level in the morning, middle in the afternoon, low in the evening; (5) Afternoon type ("Daytime type" in Russian): low level in the morning, high in the afternoon, middle in the evening; (6) Napping type ("Daytime sleepy type" in Russian): high level in the morning, low in the afternoon, middle in the evening; and (7) Inconclusive type ("None of the above").

The VJT [53] was designed by N. Marcoen and O. Mairesse to evaluate how sleepy survey participants thought they would be performing sleepiness-neutral activity (sitting and reading) at different randomly presented times after having habitual night sleep. Such sleep is expected to be terminated at approximately 7:30, either by a waking-up signal or due to a spontaneous awakening. The time cues from 8 a.m. to midday and from 8 p.m. to midnight are presented with one-hour intervals, while time cues between midday and 8 p.m. and after midnight are presented with two-hour intervals (Figure S4). Moreover, the participants would see a visual aid on the screen that consisted of clock times along the scale illustrating the daily variation in the outdoor illumination level and indicating the duration of the waking period (see [53] for these illustrations and other details). The randomly collected sleepiness self-ratings are then subsequently ordered for constructing a curve of expected sleepiness (Figure S4) for each survey participant and for averaging sleepiness within the morning, daytime, and evening/nighttime subintervals (Figures S5 and S7C).

The SWAT [54,55] was constructed by A. Putilov for the assessment of the ability to sleep or wake at either appropriate or inappropriate sleep and waking times. The SWAT includes 10-item scales for assessing Morning Sleepability (MS), Nighttime Wakeability (NW), Daytime Wakeability (DW), and Daytime Sleepability (DS). A positive/negative score on each of these scales indicates ability/inability to sleep or wake on demand in the morning (MS; Figure S6A), in the evening/at night (NW; Figure S6B), and in the daytime (DW and DS; Figure S7A,B).

All these questionnaire tools for chronobiological assessment were previously cross-validated (e.g., in [42,55]).

We used the SPSS₂₆ statistical software package (IBM, Armonk, NY, USA) for statistical analysis. Two-, three- and four-way ANOVAs of sleep times were run to test significance of the differences between the subdivisions (Table S1) of the whole sample (Tables S2 and S13), and post hoc pairwise Bonferroni comparisons were then used to confirm significance of differences between subsamples of different chronotypes and/or weekday risetimes (Table 2 and Tables S5–S13). Additionally, such ANOVAs were applied for comparison of morning chronotype with each of the other chronotypes (Table 3 and Tables S14–S16). The differences between chronotypes are illustrated in Figure 4, Figure 5 and Figures S1–S3. Finally, three-way ANOVAs of scores on questionnaire scales designed to assess sleepiness, sleepability, and wakeability in the morning, afternoon, and evening/night were used to confirm the differences between self-chosen chronotypes (Table S17). The differences between chronotypes in these chronobiological characteristics are illustrated in Figures S4–S7.

The least-squares method was used to simulate sleep times in morning and evening types (Figure 6 and Table 4). In the process of simulation, the empirical and simulated sleep times and the empirical and simulated weekday–weekend gaps in sleep times were compared several times until a drop in the difference between simulated and empirical value below 0.15 h (9 min) for each of these nine estimates (see the middle part of Table 4).

Generative artificial intelligence (GenAI) has not been used in this paper.