shows the group-average profiles for the fitted homeostatic and circadian processes during each of the 36-h total sleep deprivation periods (left panel). It also shows the overall two-process model fit (right panel), as well as the group-average data and their standard deviations across time awake (collapsed over the sleep deprivation periods). The figure shows that the model captured the group-average data well. The size of the standard deviations (over participants) was proportional to the magnitude of group-average performance impairment (Pearson’s r = 0.92). This pattern suggests that individual differences were exposed by increased homeostatic and/or circadian influences across the sleep deprivation period (Doran et al., 2001
Fig. 2 Two-process model fit for neurobehavioral performance impairment across repeated 36-h periods of total sleep deprivation. Left: Best-fitting curves for the homeostatic (black) and circadian (gray) processes across the repeated sleep deprivation periods. (more ...)
(top) shows the parameter estimates and their standard errors for the full model (i.e., with all three random effects). The error term standard deviation was fairly large, and indicated that at the level of individuals, the model did not capture the data as well as in the collapsed, group-average representation of (right). This may be due to known order effects (i.e., gradually increasing vulnerability) across the repeated sleep deprivations (Tucker et al., 2007
; Van Dongen et al., 2004
), which are not accounted for in the present modeling. Nonetheless, the model captured 64.2% of the variance in the data set.
Parameter estimates and standard errors (S.E.), and parameter correlation matrix.
Of primary interest are the random effects for β, γ and κ. Likelihood ratio tests revealed that there was a significant contribution to the model goodness-of-fit from the random effect for β (χ12 = 91.2, P < .001), and a trend for γ (χ12 = 3.7, P = .054). However, there was no significant contribution from the random effect for κ (χ12 < 0.01, P > .99).
The abstract mathematical processes S and C have no physically meaningful absolute scales. The magnitude of the individual differences attributed to each process separately was therefore quantified by expressing the standard deviation of the random effect relative to the corresponding scaling factor. For process S, this yielded σβ/β = 56.0% ± 17.8% (mean ± standard error). For process C, this yielded σγ/γ = 62.7% ± 31.5%. Thus, the relative magnitudes of individual variability were very similar for the two processes S and C, with no significant difference (t6 = 0.19, P = 0.86).
Over the time points considered during each of the sleep deprivation periods (from 2 h until 34 h since scheduled awakening), the range of the modeled effect of the homeostatic process on PVT lapses was 2.02 times as big as that of the circadian process (see , left). Thus, given that the relative magnitudes of the individual differences were found to be similar for the two processes, overall the impact of the individual differences in the homeostatic process was effectively about twice as large as the impact of the individual differences in the circadian process in this study.
Before interpreting these results, it is important to review the parameter correlation matrix shown in (bottom). There were no pairwise correlations exceeding 0.5, and most were much smaller. It follows that there were no major linear interdependencies (colinearities) among the parameters. This means that each of the model parameters was well estimable, and as such the processes S and C and their individual variabilities were satisfactorily dissociable in this study.
The absence of individual differences in the intercept κ
is noteworthy - in earlier estimates of individual variability in the two-process model used to predict neurobehavioral performance (Van Dongen et al., 2007
), individual differences in the intercept were substantial (σκ
= 6.02). However, the earlier parameterization of the two-process model was different, and not focused specifically on assessing the individual difference contributions of the dissociated homeostatic and circadian processes. Furthermore, the earlier modeling effort did not include individualization of the initial homeostatic state S0
as in the present work. That said, on a scale theoretically ranging from 0 to 1, the initial homeostatic state S0
varied only modestly among individuals in our data set (between 0.05 and 0.11). In the two-process model, small variations in the value of S0
wash out within one or two sleep/wake cycles, and thus there should have been no noticeable carry-over to the sleep deprivation periods. Indeed, rerunning the model with the value of S0
fixed at the average over participants did not increase the estimate for individual differences in the intercept.
Two modeling assumptions we made deserve some discussion. First, we assumed that the random effects were normally distributed over participants, even though there have been no studies assessing whether or not individual differences in the homeostatic and circadian processes are normally distributed. However, it can be demonstrated mathematically (Olofsen et al., 2004
) that the results of model fitting do not depend critically on the assumed distributions of the random effects when more than a handful of data points are available per subject (as was the case). Second, we used an additive model to predict neurobehavioral performance based on the processes S
. This form has been shown previously to describe neurobehavioral function during total sleep deprivation adequately (Achermann and Borbély, 1994
; Daan et al., 1984
). Nevertheless, a forced desynchrony study has revealed that there is a nonlinear interaction in the contributions of the two processes to neurobehavioral performance (Dijk et al., 1992
). This interaction has also been confirmed under conditions of total sleep deprivation (Van Dongen and Dinges, 2003
). Yet, in the first 36 h of sleep deprivation the interaction effect was negligible, and it was therefore justifiably ignored here.
Finally, there are some important caveats. The expression of trait individual differences in vulnerability to sleep deprivation depends on the kind of neurobehavioral performance considered (Van Dongen et al., 2004
), and possibly on the specific cognitive processes involved (Ratcliff and Van Dongen, 2009
). As such, the results of our study could vary upon examining data from other performance tasks in the neurobehavioral test battery. Furthermore, the present results were obtained using data from a (repeated) total sleep deprivation experiment. Whether or not the relative contributions of individual differences in the homeostatic and circadian processes to trait vulnerability to sleep loss are equivalent in other sleep loss and/or circadian disruption paradigms (e.g., Darwent et al., 2010
) remains to be determined. Lastly, the present findings may be specific to the population of healthy men and women aged 22 - 40 from which the study sample was drawn.