Covert sleep-related biological processes are revealed by probabilistic analysis in Drosophila

Conditional Wake and Doze Probability Determine Drosophila Sleep.

Measurement of the amount of sleep in Drosophila is based on movement data, typically taken in bins of ≤1 min, where a sleep episode is defined as a certain period of time without movement, usually 5 min (17, 18). The locomotor data are intrinsically binary since only two activity states (active and inactive) can be directly discriminated. Between each individual observation, the fly may choose to either remain in its current activity state or switch. The sequence of activity states reveals not only the amount of time spent in each state, but also the conditional probability of switching states, e.g., from active to inactive. We call the conditional probability P(Movement_{time = i} = 0 | Movement_{time = i-1} > 0) P(Doze) because it is the probability that the fly will switch from the active state to the inactive state. We call the conditional probability P(Movement_{time = i} > 0 | Movement_{time = i−1} = 0) P(Wake) because it is the probability that the fly will switch from the inactive state to the active state (Fig. 1A). A detailed explanation of the algorithm for calculating these probabilities and the rationale for using activity/inactivity as the input rather than sleep defined by the 5-min threshold is described in more detail in Methods and SI Appendix.

These probabilities should change in response to changes in sleep drive and arousal. In order to assess the pattern of activity-state transition probability, we analyzed behavioral data acquired with both the well-characterized Drosophila Activity Monitor (DAM) system (Canton Special [wild-type, WT] female flies; n = 60; Fig. 1) and with the more recently developed FlyBox (WT female flies; n = 55; SI Appendix, Fig. S1) (19). Both P(Wake) and P(Doze) change across the day, but do not exactly replicate activity, sleep, or one another regardless of the method of data acquisition (Fig. 1 B and C and SI Appendix, Fig. S1C). Both P(Wake) and P(Doze) have strong circadian influences, with P(Wake) showing a similarity to locomotor activity. P(Doze) cycles more weakly and circumscribes sleep.

In order to assess the degree of interdependence among these variables quantitatively, we measured the between-animal Pearson correlation of the measures. P(Wake) and total sleep are strongly anticorrelated (R = −0.92, P < 0.0001), while P(Doze) and total sleep are correlated, but less strongly (R = 0.26, P < 0.0001). Interestingly, P(Wake) and P(Doze) are only moderately anticorrelated with one another (R = −0.18, P < 0.0001). These relationships, and the shape of the curves in Fig. 1B, suggest that the transition probabilities are not simply each other’s inverse, but are instead likely to be driven by distinct sets of biological mechanisms.

The three-way relationship between P(Wake), P(Doze), and sleep (calculated using the 5-min inactivity definition) can be illustrated using a heatmap of total sleep generated at each combination of probabilities (Fig. 1D). In order to interrogate the causal relationships between these variables, we used a Markov-chain model to generate in silico behavior. The Markov-chain model is intentionally simplified: it has no circadian rhythms, environmental inputs, etc., making it an ideal environment for determining how changes of only one parameter affect behavior. For the probability bins in which we have in vivo data, there is a surprisingly good match between the in vivo and simplified in silico data (root mean square error = 5.8% Time Asleep, R = 0.99, P < 0.0001). The lack of unexplained variance in the in vivo data with respect to the in silico data suggests strongly that the combination of P(Wake) and P(Doze) causally determines the amount of total sleep for each individual fly. In addition to the total sleep, sleep episode duration is strongly regulated by P(Wake) (SI Appendix, Fig. S2). Previous research has found that each fly has an idiosyncratic preference for a range of behaviors, including total daily sleep (20, 21). We find that P(Wake) and P(Doze) are similarly idiosyncratic, but the population mean is a reasonable estimate of the central tendency of the P(Wake) and P(Doze) distributions for WT flies (SI Appendix, Fig. S3).

The daily behavioral cycle of flies can be visualized as trajectories through this probability space (Fig. 1 F–I). In wild-type animals, the presence of a light cycle generates a double “U”-shaped trajectory (Fig. 1F). Removing the light cycle results in a purely linear trajectory through probability space while clock mutant animals maintain the U-shaped trajectory (Fig. 1 G and H), indicating that the divergence from linearity is caused by light and does not require an intact clock. In the absence of both light cues and a circadian clock, flies simply find their behavioral set point and remain stationary in the probability space (Fig. 1I). Given the apparent importance of P(Wake) and P(Doze) in setting the level of total sleep, we sought to relate each to a biological process.

Conditional Wake Probability Is a Measure of Sleep Depth.

During sleep, movement and responses to external stimuli are suppressed. The more strongly these behaviors are suppressed, the “deeper” the sleep. Because the likelihood of waking up is logically linked to the depth of sleep, we hypothesized that P(Wake) is a measure of sleep depth. Sleep depth is most commonly measured in flies using arousal threshold (17, 18, 22–25). To examine the relationship between transition probabilities and sensitivity to stimuli, we measured P(Wake), P(Doze), and arousal threshold at two temperatures: a standard temperature (25 °C) and an elevated temperature (29 to 30 °C). Since elevated temperature increases sleep during the day and decreases it at night (26), we hypothesized that sleep depth should also be affected by the change in temperature. As predicted, temperature affects both P(Wake) and P(Doze). P(Wake) is significantly decreased during the daytime (P < 0.0001) and elevated during the nighttime (P < 0.0001) at 29 °C (Fig. 2B). Daytime P(Doze) is significantly elevated by heat (P < 0.0001), but nighttime P(Doze) is unchanged (P = 0.057).

Because sensitivity to sensory stimuli is also regulated by time of day (24, 27, 28), we measured mechanical arousal threshold with gentle tapping at 3-h intervals across the day using FlyBox (Zeitgeber time [ZT] 1, 4, 7, 10, 13, 16, 19, and 22; n = 89 WT female flies). Tapping the flies produced an excess of awakenings compared to chance at time points across the daily cycle at both baseline and high temperatures (Fig. 2C). Heat produces significantly anticorrelated effects on P(Wake) and P(Doze) (R = −0.82, P = 0.012), so the method of partial correlations was used to determine their independent associations with arousal threshold. The effect of heat on the response to tapping across time points is significantly correlated with the effect of heat on P(Wake) (R = 0.84, P = 0.016) but not the effect of heat on P(Doze) (P = 0.77; Fig. 2D). When we normalized P(Wake) and P(Doze) between time points, individual flies that responded to the tap had a significantly higher P(Wake) and significantly lower P(Doze) than the nonresponsive flies (P = 0.024 and P < 0.0001, respectively; Fig. 2E). We conclude from this experiment that P(Wake) and arousal threshold are regulated identically by heat, in line with our prediction. As an independent probe of the association of transition probabilities with sleep depth, we measured light-mediated arousal threshold with flashes of red light across the day using the DAM system (n = 117 WT female flies; SI Appendix, Fig. S4). The effect of heat on light arousal is highly correlated with the effect of heat on P(Wake) (R = 0.92, P = 0.02) but not well-correlated with the effect of heat on P(Doze) (P = 0.06). Within each time point, individual flies that responded to the light had a significantly higher P(Wake) than the nonresponsive flies (P < 0.0001), but there was no difference in P(Doze) (P = 0.15).

In summary, the effect of heat on arousal threshold is tightly coupled with the effect of heat on P(Wake) regardless of the sensory modality or monitoring system used. Coupling between mechanical arousal threshold and P(Doze) was seen only when normalizing P(Doze) between time points in the comparison of responding vs. nonresponding individuals (Fig. 2E). We posit that the association in this experiment is due to the latent correlation between P(Wake) and P(Doze) noted previously and does not reflect a major role for P(Doze) in regulating arousal threshold. This latent correlation is removed by the partial linear correlation method (used to analyze data in Fig. 2D where there is no P(Doze)/arousal threshold association) but is not removed in the comparison of responders and nonresponders in Fig. 2E. We therefore conclude that P(Wake) and arousal threshold, but not P(Doze), measure the same underlying biological process, namely, sleep depth.

Conditional Doze Probability Is a Measure of Sleep Pressure.

Sleep deprivation in flies, like in other animals, leads to increased subsequent sleep (17, 18). A common model for the generation of increased sleep after release from sleep deprivation posits that there is an increase in sleep “pressure” which increases sleep drive (29, 30). Because sleep drive increases the tendency to fall asleep, we hypothesized that P(Doze) is related to sleep pressure.

To increase sleep pressure and examine its relationship to P(Doze), we performed a daytime (ZT 0 to 12) sleep deprivation (SD) experiment (WT female flies; n = 116 SD, 118 controls; Fig. 3A). Because the expression of recovery sleep is regulated by time of day (31), we hypothesized that the sleep homeostat would maintain a “memory trace” of sleep debt during times of day when recovery sleep is not normally expressed. Flies acquire sleep debt during shaking, P(Wake) is up-regulated, and P(Doze) is down-regulated (Fig. 3 B and C). Following release from sleep deprivation at ZT 12, P(Doze) is up-regulated, but sleep debt is only minimally discharged during the night, with most of the recovery sleep occurring during the next light period (Fig. 3B). P(Doze) remains elevated for 24 h until sleep debt is fully discharged (P < 0.0005; Fig. 3D). In contrast, P(Wake) is significantly suppressed only during the times of day when recovery sleep is actually being performed, i.e., in the early evening (ZT 12 to 18; P = 0.0005) and in the morning presiesta (ZT 0 to 6; P < 0.0001; Fig. 3E). We also performed nighttime SD by shaking flies between ZT 12 and 24 (WT female flies; n = 120 SD, 117 controls; SI Appendix, Fig. S5). As expected, sleep debt was discharged immediately following release from nighttime SD, so it did not allow either P(Wake) or P(Doze) to be excluded as a measure of sleep pressure.

These experiments demonstrate that P(Doze) has properties consistent with a measure of sleep pressure: it is elevated by sleep deprivation and returns to baseline only after sleep debt is discharged. In contrast, suppression of P(Wake) is necessary for the expression of rebound sleep but does not have the same memory trace properties as P(Doze). Consistent with P(Wake) driving the expression of rebound sleep, the total amount of rebound is more strongly correlated with change in P(Wake) (R = −0.97, P < 0.0001) than change in P(Doze) (R = 0.19, P = 0.06), and the effect size of the change in P(Wake) is larger than the effect size of P(Doze) (Cohen’s d = −1.46 and 0.81, respectively) (32).

One implication of this result is that the magnitude of rebound sleep is a flawed measure of sleep homeostasis, consistent with recent reports (21). Interestingly, P(Doze) also appears to be regulated by “wake pressure” in addition to sleep pressure. This is most obvious during the shaking period, when P(Doze) is down-regulated because disruptive mechanosensory inputs provide an environmentally driven wake pressure (Fig. 3C). There are also likely effects of internally generated wake pressure. These can also be inferred from the decrease in P(Doze) during the anticipatory evening activity peak before ZT 12 in unperturbed flies (Fig. 1B). We conclude that P(Doze) is a measure of sleep pressure, with the caveat that strongly wake-maintaining processes (sensory, circadian, or otherwise) can mask our ability to detect sleep pressure via behavioral outputs.

Dopaminergic Tone Regulates Sleep Depth.

Sleep is regulated by a multitude of molecular processes, and many null and hypomorphic mutations disrupt sleep in flies (33). Dopamine in particular has a role in regulating sleep quantity by increasing arousal (13, 14), and acute activation of dopaminergic neurons has been shown to decrease total sleep (16, 34–36). In order to validate the proposed biological significance of P(Wake) and P(Doze) described above, we measured the effect of manipulating dopaminergic tone on activity-state transition probability by performing a 2-d thermogenetic activation (37) of TH-Gal4–positive dopamine neurons (Fig. 4A). In order to separate the effect of dopamine neuron activation from the direct effect of temperature on sleep (Fig. 2), we compared the effect of our heat manipulation on the experimental line (TH-Gal4/UAS-dTrpA female flies, n = 31) with the effect on parental strain controls (TH-Gal4/+ female flies, n = 31; UAS-dTrpA/+ female flies, n = 61). An effect on behavior was attributed to dopamine neuron activation only when the experimental line was significantly different from both control genotypes.

As has been previously shown, activating dopamine neurons significantly reduces total sleep throughout the activation period (P < 0.0001; Fig. 4B). This is accompanied by a very substantial increase in P(Wake) throughout the activation period (P < 0.0001; Fig. 4D) with only a modest daytime increase in P(Doze) (P < 0.0001; Fig. 4C). Sleep episode duration is also significantly reduced during the daytime by dopamine neuron activation (SI Appendix, Fig. S6). Following the cessation of activation, the experimental line had significantly increased sleep (P < 0.0018), but P(Wake) and P(Doze) were not significantly altered compared to controls. Close examination of the behavioral traces (Fig. 4A) suggests that there may be transient differences in P(Wake) and P(Doze) immediately after temperature is restored to baseline, but the change is not maintained long enough to reach statistical significance when averaged across the entire lights-on period. Because temperature has a strong effect on sleep independent of dopaminergic activation, we also investigated chronic genetic perturbations of the dopamine system (SI Appendix, Fig. S7). Both mutants have increased P(Wake) but have opposite changes in P(Doze) (SI Appendix, Fig. S7 C and D).

The overall effect of dopamine on P(Doze) is variable and contingent: only some genetic contexts, time windows, and environmental conditions produce perturbations. On the other hand, acute activation of dopamine neurons and chronic increases of dopaminergic tone increase P(Wake) across all experimental conditions, supporting the idea that P(Wake) is a measure of sleep depth. The conditional changes in P(Doze) may reflect indirect processes or network level linkage.

High-Sucrose Diet Promotes Sleep Consolidation by Decreasing Sleep Pressure.

Both sleep deprivation and manipulations of the dopamine system produce dramatic changes in total sleep (Figs. 3 and 4). In order to test the relationship of P(Wake) and P(Doze) to sleep structure, we measured behavior of WT flies fed either a low (2.5%) or a high (30%) concentration of sucrose, a manipulation which has been shown to regulate sleep fragmentation independent of total sleep (23). In agreement with these findings, we do not observe dramatic changes in the daily sleep pattern between low- and high-sucrose diet (WT female flies; n = 90 low sucrose, 94 high sucrose; Fig. 5A). The effects on the amount of sleep are modest: no effect of diet on daytime sleep and a small increase in nighttime sleep (P = 0.0001; Fig. 5B).

There is, however, a large decrease in the number of sleep episodes both during the day and at night in the high-sucrose group (P < 0.0001; Fig. 5C). Interestingly, we find that P(Wake) is unaffected by diet during the daytime but significantly decreased at night (P < 0.0001; Fig. 5D), in concordance with the effect of sucrose on total sleep. P(Doze) is decreased by high sucrose during both daytime and nighttime, mirroring the effect of sucrose on sleep structure (P < 0.009; Fig. 5E). In sum, the effect of sucrose on P(Wake) is variable and contingent—it is perturbed only at night—while the effect of sucrose on P(Doze) and sleep structure is consistent.

This was an unexpected finding because the initial report of a sucrose effect on sleep structure attributed the effect to sleep depth (23). This discrepancy is likely due to this prior study measuring sleep depth (via arousal threshold) only at night, a time at which we also find a small difference in P(Wake) (Fig. 5D). With our improved analysis tools, sleep depth and pressure can be estimated throughout the day, allowing us to separate the different dimensions of diet/environmental interactions and conclude that the change in sleep structure is more likely due to P(Doze), a surprising and interesting finding since fragmentation is usually thought to be due to changes in arousal (Discussion).

Changes in Sleep with Age Are due to Both Sleep Pressure and Depth.

To test the utility of our analysis tools in understanding a complex, longitudinal biological process, we compared the behavior of flies of different ages. The total amount and structure of sleep produced by an individual is strongly regulated by age in many animals, including flies (15, 16). Sleep was measured in cohorts of WT flies during weeks 1, 3, 5, and 7/8 post eclosion (n = 48, 30, 30, and 39, respectively; weeks 7 and 8 were combined due to a low rate of survival to this age). In agreement with prior results, the daily profile of sleep is altered in aged flies, most dramatically during the day (Fig. 6A). In these cohorts of flies, old flies slept significantly longer than young flies (P < 0.0002; Fig. 6B). Perhaps surprisingly, given our results with groups of young flies (Figs. 1–3) where the amount of sleep was determined largely by changes in P(Wake), nighttime P(Wake) does not vary significantly as flies age, and daytime P(Wake) is significantly decreased only in the oldest group of flies (P < 0.0001; Fig. 6E). In contrast, P(Doze) increases with age during both the day and the night (P < 0.0001; Fig. 6D).

Plotting the changes in P(Wake) and P(Doze) as a trajectory through probability space allows us to assess how aging-related changes in behavioral transition probability are related to total sleep (Fig. 6F). Nighttime behavior follows a linear path, the extent of which falls within a rather uniformly high-sleep area of the probability space, consistent with the relatively modest changes in sleep quantity at night (Fig. 6 A and B). Daytime behavior follows a more complex path that moves from a low-sleep area of the space toward a high-sleep area. In this aging dataset, changes in behavior are generated by alterations in both P(Wake) and P(Doze). The effect of aging on nighttime P(Doze) is much stronger than the effect on the amount of sleep. Our method of analysis reveals an “invisible” sleep/aging interaction that would not be apparent with currently used sleep metrics. Determining the biological basis of this shift in transition probability will provide insight into the aging process.

A Four-State Hidden Markov Model of Sleep Pressure and Depth.

P(Doze) and P(Wake) provide a continuous integrated view of the activity of all potential wake- or sleep-promoting systems. But sleep-regulating centers likely do not operate completely independently: electrophysiological data in mammals has shown that coordination of activity in particular sleep circuits can produce sleep substates with distinct properties. Human sleep is composed of Rapid-Eye Movement (REM) sleep and either three or four non-REM stages (38, 39). Rodent sleep is usually quantified more coarsely using only two substates (REM and non-REM) (40), but the existence of additional substates has been proposed (41). Distinct sleep stages have been proposed in flies (24), but there has not been a method for detecting these stages in free-behaving animals. Building upon our validation of P(Wake) and P(Doze) as markers of sleep depth and pressure, we developed an HMM that classifies fly behavior into one of four hidden states (Fig. 7).

The four-state model is composed of deep sleep, light sleep, early wake, and full wake (Fig. 7A). The model definition requires that transitions from wake to sleep must begin in light sleep and that transitions from sleep to wake must begin in early wake (a “sleep inertia”-like property). The permitted hidden state transition probabilities were fitted to the behavior of WT flies in a 12:12 light:dark cycle and per⁰¹ flies in constant darkness. Despite the substantial differences in sleep phenotype between WT and per⁰¹ flies, the inferred hidden state transition probabilities and emission probabilities were extremely similar (SI Appendix, Table S1), indicating that the existence and properties of these hidden states are robust and both clock and light independent. The fitted HMM can be used to find the posterior probability of each hidden state in a behavioral sequence of an individual animal, thereby decoding its behavior into one of four states (Fig. 7B). The average occupancy of each hidden state varies over the daily cycle, with deep sleep occurring predominantly at night and full wake occurring primarily during the morning and evening activity peaks (Fig. 7 C and D). Interestingly, the total sleep (light and deep) inferred by the four-state HMM is identical to the total sleep as measured using the 5-min threshold (17, 18), even though the sleep definitions are independently derived (Fig. 7E).

We tested the relevance of HMM decoding by testing the sensitivity of flies to an arousing stimulus in each hidden state (Fig. 7F). Flies have significantly greater arousal probability in light sleep than in deep sleep, and greater arousal probability in full wake than light wake (P < 0.0001). Because the emission probabilities of deep and light sleep are both zero, the primary difference between these hidden states is their respective probability of transitioning to wake (SI Appendix, Table S1). Likewise, the primary difference between early and full wake is their respective probability to transition to sleep. We hypothesized that the fraction of sleep in deep sleep and the fraction of wake in full wake would be the HMM correlates of P(Wake) and P(Doze), respectively. We tested this hypothesis by examining the sleep mutants fumin (14), and insomniac (42) (SI Appendix, Fig. S7) using the HMM. The % deep sleep and % light wake differ significantly between w¹¹¹⁸, fumin, and insomniac flies (P < 0.0001; Fig. 7G). In these disparate groups of flies, P(Wake) is correlated with % light sleep (R = 0.84, P < 0.0001) while P(Doze) is correlated with % light wake (R = 0.90, P < 0.0001). Overall, the HMM approach recapitulates the results of the P(Wake)- and P(Doze)-based analysis with the benefit of allowing decoding of behavior into discrete categories that may be tied to different circuit configurations or balances between circuits.

Probability of Wake Is a Measure of Sleep Depth.

The depth of sleep is clinically important and tied to its restorative power (3). In humans, sleep depth is measured using vital signs, EEG, and behavior, which together provide a holistic view of the process (43). Sleep depth in flies is more difficult to characterize. Total local field potential power in flies is significantly reduced immediately after the initiation of sleep, but does not scale to indicate increased depth of sleep even as the arousal threshold changes (24, 44). Sleep depth is therefore most commonly measured in flies using arousal threshold in response to an applied sensory stimulus (17, 18, 22–25).

Unfortunately, reliance on sensory arousal thresholds incurs two significant confounds for the experimenter: the regulation of sensory systems by time of day and the disruptive nature of the measurement. Like other animals, the sensory systems of Drosophila are regulated by the circadian clock. This has been most robustly demonstrated in the response to light (27, 45, 46) and to odors (28, 47). The major confounding feature of sensory arousal threshold, however, is that it unavoidably disrupts sleep. Rousing the flies is likely to disrupt the deeper-sleep substates found in longer-sleep episodes (24). Repeated rousing may activate homeostatic mechanisms that alter sleep depth (18), and fragmentation of sleep without sleep loss can even trigger sleep rebound (48). Finally, the arousing stimulus may act as a Zeitgeber to the circadian system (49).

In light of these confounds to the sensory arousal threshold, a noninvasive measure of sleep depth would be extremely helpful in understanding the function and regulation of sleep in Drosophila. P(Wake) is an attractive alternative measure, and several lines of evidence reinforce its validity as a measure of sleep depth. Most directly, our arousal threshold experiments demonstrate that P(Wake) correlates very significantly with ability to be aroused by either a mechanical or light stimulus (Fig. 2 and SI Appendix, Fig. S4). We also find that the arousal-associated molecule dopamine increases P(Wake) in both chronic and acute activation experiments (Fig. 4 and SI Appendix, Fig. S7). This is consistent with observations that the rate of spontaneous self-awakening is lower during deeper sleep stages in humans (50). Considering this evidence, we propose that P(Wake) is a behavioral correlate of sleep depth in flies the magnitude of which is closely aligned with dopaminergic tone.

Probability of Doze Is a Measure of Sleep Pressure.

After long periods of wake, the desire to sleep also increases. This homeostasis-promoting desire to sleep is described as “sleep pressure,” or process S, in the influential Borbély two-factor model of sleep (30). Sleep pressure is difficult to measure directly because it is an internal drive rather than an external behavior. The most common measure of sleep pressure is indirect—a change in the amount of total sleep from baseline, i.e., rebound sleep after deprivation (25). However, this metric is inevitably confounded by sleep depth, which increases in rebound sleep and which we show is the major driver of the amount of sleep across conditions (Figs. 1, 3, and 4). Another common measure of sleep pressure is the latency to sleep after an environmental cue, such as lights turning off (23, 51). Sleep latency is not confounded by sleep depth, but it can only be measured relative to a predefined marker and only once per fly per event, limiting both its temporal resolution and statistical power.

Probability of ceasing activity, P(Doze), is conceptually similar to sleep latency. The primary difference between the two measures is that sleep latency is measured against a discrete external time marker, while P(Doze) is measured using fly-initiated activity, providing a continuous readout. Due to this similarity, we expected that P(Doze) would measure a similar biological process to sleep latency, namely sleep pressure. Empirically, we find that P(Doze) behaves as we would expect a measure of sleep pressure to behave. First, P(Doze) increases precede sleep increases and P(Doze) decreases lag behind sleep as debt is discharged (Fig. 3C). Second, P(Doze) is not a major determinant of total sleep in unmanipulated flies, consistent with findings showing that inhibition of the sleep homeostat has only a minimal effect on baseline sleep (52, 53). Third, and perhaps most importantly, we show that P(Doze) has “memory trace” properties: after sleep deprivation it remains elevated until the completion of rebound sleep (Fig. 3).

While the similarities to process S are striking, the daily pattern of P(Doze) does not have the same shape as Borbély’s conceptualization (30). Process S is noncircadian and increases steadily during wakefulness, while P(Doze) is fairly flat with troughs during the morning and evening peaks of activity (Fig. 1) (54). This could be due to P(Doze) not being a “pure” sleep pressure signal: P(Doze) is clearly influenced by the clock and by light (Fig. 1), tying P(Doze) to the animals’ internal and external states. This is consistent with the idea that the homeostat has a relatively minor role in the specification of sleep drive in unperturbed animals and is engaged only in the case of deviation from the norm. Notably, in our sleep deprivation experiments the trajectory of P(Doze) becomes much more similar to S as sleep debt builds up (Fig. 3C). Alternatively, it may be that the two-process model simply does not reflect regulation of sleep pressure in Drosophila. Recent evidence suggests that sleep pressure in flies is primarily regulated by the circadian clock rather than by sleep debt (21). The lack of agreement between model and empirical results suggests the need for a revised and more broadly based model of sleep drives.

Counterintuitively for a measure of sleep pressure, P(Doze) is correlated with increased sleep fragmentation (Fig. 5). P(Doze) is the probability of initiating sleep episodes, and since the median sleep episode is shorter than the mean episode in the typical distribution of episode durations for normal animals (55, 56), it follows that increasing P(Doze) will usually result in shorter mean sleep episode duration. The important implication of this finding is that sleep fragmentation is not, on its own, evidence of a wake-promoting process. Our results indicate that fragmented sleep can be the result of decreased sleep depth [i.e., high P(Wake)], increased sleep pressure [i.e., high P(Doze)] or a combination of the two. In line with this, we have recently shown that activation of serotonergic neurons that fragment sleep without changing its total amount primarily increases P(Doze) (48). Biologically, this may make sense. Allowing fragmentation of sleep (which can have advantages under some conditions, e.g., starvation) to be generated by a process untethered to arousal threshold or dopaminergic tone gives the brain more degrees of freedom in determining sleep patterns.

Integration of P(Wake) and P(Doze).

Our transition-probability analytical method defines two parameters that can assess the core drives that determine sleep and wake. Importantly, these parameters are only moderately correlated with one another at baseline and are not always affected in concert by perturbations. Some perturbations appear to act primarily on P(Wake) and some primarily on P(Doze). For example, acute increases in dopaminergic tone have strong and consistent effects on P(Wake) and variable effects on P(Doze). In contrast, other perturbations of the system (e.g., aging and mechanical sleep deprivation) alter both P(Doze) and P(Wake). During sleep deprivation, P(Doze) measures sleep need, but decreased P(Wake) is the primary driver of the amount of rebound sleep (Fig. 3), indicating that the two processes interact. These interactions may give additional insight into the privileged roles of some neural circuits in increasing total sleep after sleep deprivation (34) and into the ability of some forms of sleep deprivation to bypass engaging the homeostat (57).

A Hidden Markov Model for Drosophila Sleep Substates.

P(Wake) and P(Doze) are of only limited utility for classifying flies as being in discrete sleep/wake substates. The P(Wake)/P(Doze)/Sleep landscape (Fig. 1) shows smooth gradients without clear breakpoints or substates. The technical ability to identify and perturb sleep substages in real time has yielded significant insight into the neurophysiology of sleep in mammals (58, 59), so we developed a tool for classifying fly sleep into hidden states using a HMM. The hidden states decoded by the HMM are distinct in their responses to outside stimuli and internally generated movement in a way that is extremely similar to P(Wake) and P(Doze) (Fig. 7).

Despite their similarities, the conditional probability approach and the HMM have different strengths and weaknesses. Conditional probabilities are naturally scalar, agnostic with regard to underlying biology, and can be estimated from relatively small samples of flies. However, the conditional probabilities are sometimes undefined (Methods) and are of limited usefulness in real-time applications. The HMM excels in classifying behavior into distinct hidden states for post hoc analysis or real-time perturbation using the short-time Viterbi algorithm (60). The HMM also requires significant training data if the genetic background or baseline conditions of an experiment preclude reusing a previously trained network model. Importantly, however, we find that the conditional probability and HMM approaches appear to respond to the same underlying drives and are altered in similar manners by experimental perturbations. The complementary strengths of each approach will allow greater flexibility for investigators in experimental design and analysis.

Animals.

Flies were raised on cornmeal–dextrose–yeast food in a 25 °C incubator. The incubator maintained a 12 h:12 h light:dark cycle. Transgenic flies used were the following: per⁰¹(CS)—a gift of Michael Rosbash (Brandeis University, Waltham, MA) —w¹¹¹⁸ and w¹¹¹⁸; dat^fmn—gift of Amita Sehgal (University of Pennsylvania, Philadelphia, PA) —PBac{WH}inc^f00285, w¹¹¹⁸ (inc²; BDSC # 18307), TH-Gal4 (BDSC # 8848), and UAS-dTrpA (37).

Sleep Data Collection.

Data were collected using the DAM system and Flybox as previously reported (19, 71). Unless otherwise specified, experiments were carried out at 25 °C with a 12 h:12 h light:dark cycle. The light-driven arousal threshold in the DAM system was measured in a dimly lit incubator by delivering 3-s pulses of red light. The mechanical arousal threshold in the Flybox was measured by applying a single tap to the side of the 96-well plate using a solenoid.

Generation of In Silico Data.

Simulation data were generated by custom programs written in Matlab (MathWorks, Natick, MA). The simulation was initialized in a random state with equal probability of wake and sleep. At each subsequent time step, the simulator checked the current state of the fly, generated a random number between 0 and 1, and compared the value with P(Wake) or P(Doze) to determine if the in silico fly woke or dozed, as appropriate.

Data Analysis.

DAM system, Flybox, and in silico data were analyzed with a common set of modular programs written in Matlab. P(Wake) was calculated as follows: 1) Every bin of inactivity was counted except the last one: this quantity was the denominator, the total number of state transitions from inactivity; 2) For every bin of inactivity, the number of times the fly was active were counted in the subsequent bin: this quantity was the numerator, the total number of state transitions from inactive to active; 3) The total of state transitions from inactive to active were divided by the total state transitions from inactive. If there were no bins where the fly did not move (i.e., denominator is zero), the transition probability was considered undefined. P(Doze) was calculated identically, but with activity and inactivity reversed. The choice of using 1-min activity bins instead of 5-min sleep/wake as the basis for the analysis was made after considering both methods (SI Appendix, Fig. S8).

Statistical Analysis.

Statistical analysis was performed using the Matlab statistics toolbox. Fitting of HMMs and decoding of hidden states were performed using the Matlab Statistics and Machine Learning Toolkit.

Data Sharing and Code.

Statistical treatments and information are provided for each figure in SI Appendix, Supplemental Statistical Tables. The MATLAB scripts for analysis of P(Wake)/P(Doze) and for HMM functions have been deposited in GitHub and can be accessed at https://github.com/Griffith-Lab/Fly_Sleep_Probability.