Homeostatic forces underlying the daily pattern of sleep propensity

Abstract

The mismatch between rising sleep need and the fluctuating ability to fall asleep underlies insomnia—the most common sleep disorder—yet remains poorly understood. While sleep need increases steadily with time awake, sleep propensity—the likelihood of transitioning from wake to sleep—follows a bimodal pattern, peaking in the mid-afternoon, dipping in the evening, and rising again near bedtime. Building on our previously developed wave model of sleep dynamics, we extend this homeostatic framework to the waking period and show that it predicts the observed bimodal sleep propensity curve. This pattern emerges from two interacting factors: wake-state instability, which increases exponentially across the day, and interaction strength between states, which follows a biphasic trajectory. Together, they produce a daily profile of sleep propensity that closely aligns with experimental data. Notably, the empirical curve demonstrates a deeper evening dip than the model alone predicts—reflecting the known circadian modulation of sleep propensity. The model reveals that the mid-afternoon peak reflects maximal interaction at the homeostatic equilibrium threshold, while the evening dip results from minimal coupling between sleep and wake states, counteracting high instability. A late-day rise in both factors facilitates sleep onset at bedtime and beyond. Experimental data from sleep deprivation further support these predictions. This work provides a mechanistic foundation for understanding daily sleep propensity and may inform strategies to improve sleep and performance in both health and disease.

Statement of Significance

Monotonic increases in sleep need across the day contrast sharply with the bimodal pattern of sleep propensity—the ability to fall asleep—which peaks in the mid-afternoon and again near bedtime. This phenomenon, inherent to normal sleep physiology, may also underlie key forms of insomnia: the inability to fall asleep despite high sleep pressure. Here, we apply the wave model of sleep dynamics to show how similar bimodal sleep propensity pattern can emerge purely from nonlinear, homeostatic changes in the interaction strength between wake and sleep states.

Introduction

Animals alternate between two primary physiological states: wake and sleep. These states follow predictable, species-specific patterns and are regulated by both homeostatic [1] and circadian [2] mechanisms. In humans, the likelihood of transitioning from wake to sleep—referred to as sleep propensity—typically follows a distinct bimodal pattern (Figure 1A) [3, 4]. Experimentally, sleep propensity is assessed by measuring sleep onset latency after varying durations of prior wakefulness or time spent asleep during brief sleep opportunities. Such short sleep protocols show [4] that, after morning awakening, sleep propensity initially declines for 1–2 hours, then gradually increases to a mid-afternoon peak before dipping in the early evening. It rises again shortly before habitual bedtime.

Figure 1.

Difference in daily patterns of sleep need and sleep propensity in humans. (A) Schematics of the bimodal pattern of sleep propensity in the absence of nighttime sleep, based on ultra-short sleep–wake cycle data [4]. (B) Schematics of exponential rise in sleep need, based on the intensity of the initial non-REM (NREM) sleep episode [1]. Both curves are normalized to their respective maxima for illustration purposes.

This empirical profile contrasts with the monotonic rise in sleep need across the day, typically quantified either by the initial intensity of sleep, as measured via slow-wave activity [1], or by behavioral indicators of wake-state instability [5] (Figure 1B). This apparent mismatch between rising sleep need and fluctuating sleep propensity has long puzzled researchers. Most explanations of the bimodal sleep propensity curve attribute it to circadian modulation [6–14].

In humans and other diurnal animals, the circadian system promotes wakefulness during the day and suppresses sleep initiation, thereby counteracting accumulating homeostatic sleep pressure. Forced desynchrony studies [7, 14]—in which participants live for several weeks under highly controlled conditions while their sleep period is delayed by 4 hours each day, increasing the sleep pressure—have demonstrated robust phase-dependent modulation of both percent wakefulness during sleep period (Figure 2A) and sleep latency (Figure 2B). These effects were asymmetric, particularly for sleep latency, and closely corresponded to the evening dip in sleep propensity seen in ultra-short sleep protocols [4] (Figure 1A). This stands in marked contrast to the largely symmetrical circadian profiles of canonical phase markers: melatonin (Figure 2C) and core body temperature (Figure 2D).

Figure 2.

Asymmetric patterns of circadian modulation of wakefulness and sleep latency under forced desynchrony conditions contrast with the symmetrical rhythms of melatonin and core body temperature. (A) Wakefulness within scheduled sleep episodes (% of total recording time). (B) Latency to sleep onset (minutes). (C) Plasma melatonin levels (z-scores). (D) Core body temperature (deviation from mean, °C). All curves are digitized from the double-plotted data presented in figs 3 and 4 of Dijk et al. [14].

Together, these findings suggest that the mid-afternoon peak in sleep propensity arises despite ongoing circadian wake promotion, while the early evening dip coincides with the peak of circadian wake drive in the late evening. As bedtime approaches, the circadian wake signal wanes and sleep-promoting influences increase, reaching a maximum near the core body temperature nadir (Figure 2A and B).

Disentangling homeostatic and circadian influences is essential to understanding how these distinct yet interdependent systems shape the temporal structure of sleep propensity. While circadian signals are often framed as imposing a rhythm on underlying homeostatic drives, an integrative perspective suggests they may instead modulate and amplify homeostatically driven tendencies. This modulation may not only promote internal physiological coherence but also align it with external environmental cycles. In this view, many features attributed to circadian control may reflect the enhancement and temporal refinement of patterns generated by homeostatic dynamics alone.

In this study, we addressed whether the homeostatic regulation of sleep might contribute to the observed bimodal pattern of sleep propensity in humans. At this time, the model specifically focuses on the homeostatic regulation, drawing on physical principles that transitions require not only rising state instability but also sufficient interaction between states. Even a highly unstable wake state can remain metastable—that is, it can persist temporarily near a transition point—if its interaction with the sleep state is too weak to trigger the sleep onset. We hypothesize that the daily bimodal pattern of transitions between wake and sleep arises from a nonlinear homeostatic relationship between these states, driven by two dynamic factors: (i) the relative instability of the current state and (ii) the strength of interaction between states.

We test this hypothesis using the previously introduced wave model of sleep dynamics [15], in which sleep and wake states are represented as interacting probability waves, each evolving within its own Morse potential: U_S for sleep and U_W for wake (Figure 3A). These potentials define the system’s energy landscape and govern transitions between states. Transitions are driven by a regulatory variable x, which integrates destabilizing influences from homeostatic, circadian, and environmental sources. As x increases, instability energy ε rises exponentially during wakefulness and stepwise declines during sleep cycles (Figure 3B). Though not defined here as a direct physiological signal, ε quantifies systemic instability, consistent with physical models where higher energy implies lower stability.

Figure 3.

The schematics of the wave model of sleep dynamics. (A) In the wake state (U_W, red potential), the driving force (horizontal red arrow) of homeostatic, circadian, or environmental nature increases the value of the regulating parameter x (horizontal axis) beyond the sleep–wake homeostatic equilibrium setpoint x_c (vertical cyan line) and the x_c region of efficient interaction of two states (gray area). This rises the energy of state instability (vertical red arrow and vertical axis) above the homeostatic energy threshold (Ux_c, black dot; the crossing point of the two potential curves). Sleep (U_S, purple potential) is initiated at x_o (red dot) of the initial energy level J_in (first sleep cycle). Relaxation of x and occurs in the form of sleep wavepacket propagating along energy levels (blue arrows)—NREM episodes, where x_in—Maximal deviation achieved during the first sleep cycle. At the end of each cycle, around x_c, a portion of the wavepacket energy (yellow arrows) is released—REM sleep. Stars—Maximal stability of the corresponding state. The schematics serves to only illustrate the concept. For actual position of Ux_c and J_in within U_S, see Figure 4A. (B) Top panel: Time-dependent changes in the regulating parameter of state stability x (vertical axis) over wake (white area) and sleep (purple area) states. In sleep, the period of x oscillations corresponds to the duration of consecutive NREM episodes, while a decline in their amplitude squared correlates with the drop in NREM intensity. Each cycle ends around x_c (horizontal cyan line), the region of sleep–wake equilibrium where REM episodes occur (multicolor blocks). F—The driving force (red arrow) that increases x during wakefulness. Bottom panel: Time-dependent increase in the energy of state instability (vertical axis) over the wake state (white area) and stepwise decline over consecutive sleep cycles (block width—Duration; NREM—Cyan; REM—Multicolor). Small reduction in at the start of wake period corresponds to sleep inertia. Maximal energy level reached () defines initial NREM intensity (SWA). Linearly increasing portions of energy released correspond to REM intensity (block height). Horizontal cyan line—Potential energy of homeostatic threshold, Ux_c. Vertical cyan line—Timing of Ux_c, with daytime timing around mid-afternoon peak in sleep propensity. Horizontal axis—Time since awakening (hours). The figure is taken from [15], with permission.

A second key variable, interaction strength Z, reflects the dynamic proximity of the sleep and wake potentials. At the homeostatic equilibrium point Ux_c, where the potentials intersect, Z is maximal, and the system has equal probability of occupying either state. Above this threshold, interaction strength varies nonlinearly with x, producing a biphasic profile over the waking day. Together, ε and Z define a time-varying transition probability, i.e. sleep propensity, generated from internal system dynamics.

To operationalize these dynamics, the model draws on empirical sleep architecture. Specifically, non-rapid-eye-movement (NREM) and rapid-eye-movement sleep (REM) episode durations are used to define the shape of the sleep potential and to estimate the energy level at equilibrium, while also serving as benchmarks for model validation. The model uses changes in NREM duration—treated as oscillation periods—to estimate the potential’s width (σ) and the energy level J_in at which wake-to-sleep transition occurs. The REM duration profile—modeled as the lifetime of coherent superposition between the sleep and wake waves—further defines the equilibrium energy (Ux_c). These parameters predict NREM intensity (proportional to amplitude squared), REM intensity (proportional to energy gaps Δε), and the number of quantized levels in U_S(J_max), which correlates positively with σ.

Once this architecture is established, the model can simulate sleep–wake transition dynamics, such as sleep propensity. Validation against experimental data has confirmed the model’s ability to reproduce core features of human sleep dynamics with mathematical precision, including its response to sleep deprivation and sleep extension, and to predict invariant relationship between NREM and REM [15].

Here, we show that the wave model predicts a bimodal daily pattern of sleep propensity based solely on homeostatic dynamics. This pattern emerges from a bimodal profile of interaction strength across the instability curve. We further demonstrate that this pattern is modulated under sleep deficit conditions, and that these changes align closely with empirical data supporting a mechanistic, homeostatic basis for the daily structure of sleep propensity, further modulated by the circadian clock.

Methods

Documenting normal sleep architecture

Modeling of the normal daily sleep propensity profile was based on the analysis of sleep dynamics in a group of young, healthy volunteers, as described previously [15]. Representative group data for regular sleep were obtained as part of our larger study on the circadian regulation of sleep and hormonal functions (“Multimodal Circadian Rhythm Evaluation”; PI: I.V.Z., funded by Pfizer Inc.). The study was conducted in accordance with the Declaration of Helsinki on Ethical Principles for Medical Research Involving Human Subjects and was approved by the Boston University Institutional Review Board (protocol code H-33035 and date of approval is November 15, 2015). All participants provided written informed consent.

Twenty-four young, healthy male volunteers (Mean ± SEM: 24.5 ± 4.4 years; range: 19–34 years) were selected based on self-reported criteria: 7–9 hours of habitual nighttime sleep, minimal (<1.5 hours) changes in sleep duration on weekends, no sleep complaints, no history of chronic disorders or regular medications, no recent trans-meridian travel, no drug use, no smoking, and habitual coffee consumption not exceeding three cups per day.

Over the 2 weeks preceding the inpatient phase, sleep–wake cycles were documented using activity monitors (Philips Inc.), placed on four extremities and torso, and sleep logs. Subjects then spent three consecutive nights in the General Clinical Research Center at Boston University School of Medicine. Bedtimes were scheduled individually based on habitual sleep patterns, and subjects were allowed 9 consecutive hours in bed. Sleep was recorded using polysomnography (Nihon Kohden PSG system) following standard techniques, with sleep stages visually scored in consecutive 30-s epochs. To be included in the regular sleep dataset, each sleep night had to have a sleep efficiency of at least 85% and show no evidence of sleep apnea or other sleep disorders (n = 39 nights total).

NREM–REM cycles were defined by the succession of a NREM episode (minimum duration: 10.5 minutes) followed by a REM episode (minimum duration: 3 minutes). No minimum criterion for REM duration was applied for the final cycle. A NREM episode was defined as the interval between the first two epochs of stage 2 and the first occurrence of REM within a cycle, whereas a REM episode was defined as the interval between two consecutive NREM episodes or as the interval between the last NREMS episode and final awakening. Intermittent awakenings (up to 5 minutes in this population of normal sleepers) were not included in the NREM or REM periods and were instead quantified as a percentage of wakefulness per sleep cycle or per time unit.

Documenting sleep propensity following mild and severe sleep deficits

Sleep propensity was analyzed in 18 male cadets (age: 18–22 years) from the Novosibirsk military school. The experiments conformed to the ethical standards of the Declaration of Helsinki and were approved by the Ethics Committee of the Siberian Branch of the Russian Academy of Medical Sciences. Informed written consent was obtained from all participants.

Prior to the experimental period, cadets followed a military schedule providing regular 7-hour nighttime sleep opportunities with no daytime naps. Given that the recommended average sleep duration for young males is 8 hours [16], this schedule represented a state of chronic mild sleep restriction (mild sleep deficit). Group 1 (mild sleep deficit, n = 9) maintained their regular sleep schedule, while group 2 (severe sleep deficit, n = 9) was kept awake for the entire night preceding the experimental procedures. Both groups were studied in parallel.

The detailed experimental protocol and results for sleep measures other than sleep propensity were reported previously [17–19]. Briefly, during a 24-hour experimental period starting at 06:00 hours, subjects underwent testing under sleep laboratory conditions using a short sleep paradigm consisting of 20-minute sleep opportunities interspersed with 100-minute wake periods. Polysomnographic data were collected using a Medicor polygraph (EEG8S, Micromed, Hungary), and sleep stages were scored in 30-second epochs. Sleep initiation was defined as the occurrence of two consecutive epochs of stage 2. Sleep propensity was quantified based on either the total sleep duration during each 20-minute sleep opportunity or the latency to stage 2 sleep.

Mathematical Modeling

In accordance with the wave model of sleep dynamics [15], the daily sleep–wake cycle is viewed as the interplay between two interacting probability waves representing the probability of sleep (S) and wake (W) states. This model accurately describes four key polysomnographic measures, including the durations and intensities of consecutive NREM and REM episodes [15]. Through quantitative analysis of overnight sleep architecture, the model identifies parameters influencing state transitions—namely, state instability and the strength of interaction between the two states—which together determine the probability of transitions between S and W states.

Both S and W states arise from the coherent interplay of numerous underlying homeostatic processes. Within the wave model, these underlying processes are represented by S and W wavefunctions, and , respectively, where ξ denotes the essential set of internal parameters characterizing these processes. The wave functions are normalized to unity, i.e. for i = s, w. The orthogonality of the S and W wavefunctions emphasizes their fundamental internal differences because the inner product of the wavefunctions is a measure of overlap of different physical states. The functions can be represented as eigenfunctions that include only real parts.

In our modeling, the single regulating parameter of state stability x is employed, and its value defines the specific forms of the two wavefunctions. This parameter is a one-dimensional reduction of the overall physiological state that reflects the system's capacity to maintain stability. Dynamic changes of x, accompanied by the resulting alteration of the internal parameters of S and W waves, can be initiated by homeostatic and other factors, such as the circadian clock or environmental perturbations. The probabilistic nature of both S and W waves allows us to employ the well-developed mathematical framework of quantum mechanics to quantitatively describe the transitions between these two states. While we refer to a coherent superposition of S and W waves in REM sleep, this is a formal modeling construct and does not imply that sleep or wake states possess quantum physical properties.

Our prior investigations yielded successful results through this approach, providing precise quantitative descriptions of normal human sleep architecture by solving the wave equation for the x-variable [15].

For the examination of the 24-hour dynamics of S–W transitions in this study, we adopted a simpler semi-classical approach [20, 21]. This method relies on the classical-like motion of the center of a wave packet corresponding to the S or W wave. Within this approximation, the regulating parameter x is treated as a predetermined function of time, denoted as x = x(t), and the evolution and transitions take place due to the dynamic adjustments in x.

Semi-classical evolution of the two-state system

Within the framework of the wave model, the time-evolution of the two-state system can be described by the superposition of sleep and wake waves:

(1)

where and are amplitudes of the S and W waves and

(2)

where is the phase of S or W wave accumulated since wakeup time at t = 0. The constant is introduced for the transformation of the unitless time t to hours. The potential energies and modify the propagation of S and W waves, respectively, or the motion of their wavepacket, and reflect the degree of state instability.

The numerical parameters of the and potentials were obtained from the detailed analysis of sleep architecture in data collected by our group (see above) and data reported by Barbato and Wehr [22]. The results of the analysis are described in detail in [15]. The established potential curves and provided the basis for the present study, allowing us to determine how the energy of state instability changes depending on the 24-hour variation in the regulating parameter x(t).

The time-dependent amplitudes and of S and W waves provide the time-dependent probabilities of being in S or W state:

(3)

The equations governing the two-state dynamics with time-dependent parameters have been the subject of thorough exploration in quantum physics. Examples include analysis of populations of excited states in atomic collisions [20], the experimental examination of the state transitions in ultracold gases [23] and the investigation of atomic state populations under the influence of time-dependent electric fields [24]. The equations corresponding to the amplitudes of S and W states were derived from the wave equation governing the total wave function . This was accomplished through the conventional process of projecting the time-dependent wave equation onto the basis functions and [21]:

(4)

(5)

where is the matrix element describing the potential of interaction between S and W states. In our model, the value of includes an explicit dependence on x only, therefore the time variation of changes S–W interaction. The value of (x) peaks in the vicinity of the crossing point and we use the Gaussian shape of this peak to simplify the calculations: [], where parameters and reflect the maximal value of the interaction matrix element and the width of the interaction area, respectively. Under real conditions, may also include an explicit dependence on time t due to circadian and environmental factors,

The knowledge of the time dependence of the regulating parameter x = x(t) allows us to rewrite Equations (4) and (5) in the form convenient for numerical solutions:

(6)

(7)

where is the velocity associated with the transformation of the regulating parameter during the 24-hour S–W cycle. The difference of S and W phases in Equations (6) and (7) depends on the energy separation and the velocity of x, :

(8)

where must be calculated for the known trajectory x = x(t) or for the velocity which is defined as the function of x. The appearance of v(x) in the denominator reflects the change of variables applied during the quasi-classical transformation from the time domain to position-based coordinates. Specifically, v(x) acts as the Jacobian term, representing the effective “velocity” of the wavepacket along the x-trajectory. Thus, the dt value in the left side of Equation (8) is the time interval required for the displacement of the wave packet on the distance dx, i.e. . This “technical” change of the integration variable ensures that time evolution is expressed correctly in the new coordinate frame.

In Equations (6) and (7), the matrix element includes the contributions of two possible mechanisms of interaction between S and W waves:

(9)

Analysis of sleep–wake interaction and sleep propensity

Equations (6) and (7) describe the dynamics of the amplitudes corresponding to the S and W waves. As a result, they offer insight into the time-dependent probability of remaining awake or transitioning into sleep. These dynamics of also reflect the effectiveness of the S–W interaction across various regions of the regulating parameter x. The mathematical characteristics of Equations (6) and (7) allow to predict the region where efficient transitions between S and W can take place.

The rate of S ↔ W transitions can be notably elevated within regions characterized by slow alterations of the phase difference ∆φ(t) or within domains exhibiting elevated values of . In alignment with the principles of the theory of non-adiabatic transitions [21], these criteria can be succinctly expressed through Massey’s parameter Z:

(10)

where is the characteristic time of the S–W interaction at certain x. All values in Equation (10) are expressed in dimensionless units. The typical dependence of Z on the regulating parameter x is shown in Figure 5B. We assume a constant τ to simplify the analytical expression for Z and focus on a qualitative role of instability and interaction strength in shaping state transition probability. Numerical solutions of the system in Equations (6) and (7) required knowledge of interaction between S and W states and their potential curves. Physiologically, τ represents the characteristic timescale of resolution for state instability. Under stable sleep–wake conditions, this timescale is not expected to vary significantly. While this assumption can be relaxed in future extensions of the model (e.g. under pathological or pharmacological perturbations), it serves as a reasonable first-order approximation here. Maximal interaction strength corresponds to the region of the crossing of the and potential curves, where as illustrated in Figure 4.

Figure 5.

Daily variation in the homeostatic pattern of state instability and probability of sleep–wake state transitions. (A) Modeled transition probability (homeostatic-only, i.e. no circadian input is included) or sleep propensity in the absence of overnight sleep, as based on system instability (ε) and state interaction strength. Sleep propensity zones (pink numbers) as in Figure 4. (B) The likelihood of sleep onset (sleep propensity) during the day (white area) and across an 8-hour nighttime sleep period (purple area). Daytime sleep propensity is assessed as described in panel A. Nighttime sleep propensity is shown using three complementary approaches: (i) black curve—Model prediction based on the ratio of REM to NREM episode durations (see Methods); (ii) cyan curve—Empirical estimate from our dataset (n = 39 nights, healthy young males) based on the inverse of % wakefulness per hour of regular nighttime sleep; (iii) red dashed curve—Same method applied to previously published data on hourly wake % (inverted to represent sleep propensity) from a forced desynchrony protocol [14], Figure 3A). Horizontal axis: Hours since awakening. (C) Comparison of model-derived (black) and experimental (cyan) sleep propensity curves over the wake period. To emphasize differences in dip depth, both curves are scaled and aligned at their respective peak values. The experimental curve (cyan) shows a markedly deeper evening dip, consistent with additional circadian suppression of sleep propensity not captured by the homeostatic-only model. Note: Vertical axes are plotted on separate baselines to reflect differing units and capture the difference in the evening dip (left: Transition probability; right: Minutes asleep). While normalization of both curves would allow direct axis alignment, it would also attenuate the apparent circadian effect we aim to highlight—Specifically, the deeper evening dip in the empirical data.

Figure 4.

Temporal relationship between the wake-state instability and strength of interaction between the sleep and wake states. (A) The dynamics of the energy of state instability, (magenta line) depends on the regulating parameter of state stability x (horizontal axis). Sleep propensity zones: The numbers (1–6 in pink) and corresponding thin vertical lines indicate points of comparison between and the proximity (blue lines) between sleep (U_S, purple) and wake (U_W, magenta) potential curves (as in Figure 3). Following morning awakening (1), the energy of state instability continues to decline (2, sleep inertia), then rises towards the homeostatic equilibrium (3, x_c and Ux_c, blue dot) and beyond it (4–6) during consolidated wakefulness. The point x₀ (5)—habitual sleep onset at the initial energy level J_in, corresponding to the first sleep cycle. Point 6—Increase in state instability and the proximity of two states during sleep deprivation. Yellow lines illustrate gradual increase in energy gaps of the Morse potential towards low energy levels. As detailed in [15], on average, regular sleep starts around level 10 and Ux_c is at around level 7 in young adults with normal sleep. (B) the strength of interaction (Z, blue line) between sleep and wake states positively correlates with x and proximity between the two states, thus negatively correlates with the distance between the potential curves (blue lines in Figure 4A). Following morning awakening (point 1), Z declines (point 2, sleep inertia) and then increases towards the crossing of the two potential curves x_c, where the proximity is maximal, while the two states are at equilibrium (point 3). Further rise in state instability is associated with large distance or minimal proximity between the two states (point 4), hence, with weak strength of their interaction, which then gradually increases towards the point of habitual sleep onset x₀ (point 5) and further (point 6). If sleep is prevented (point 6, after the whole-night sleep deprivation), Z continues to rise due to the rise in state instability and growing proximity between the potential curves at high x.

The primary objective of this study was to formulate accurate theoretical predictions for the daily variations in sleep propensity. In the context of our model, sleep propensity refers to the probability of transitioning from wake to sleep during a short time interval, Δt = 6 minutes (0.1 hour), at a given point along the trajectory. If this probability is divided by the constant Δt, it yields the physical rate of W → S transitions per unit time. Both the probability and the rate represent sleep propensity equivalently, as long as Δt remains fixed across experimental or theoretical evaluations.

We computed for the time span  hours, where t = 0 corresponds to the habitual morning awakening time. The temporal trajectory of the regulating parameter x = x(t) was determined using a series of reference points established in our previous study. These model points were inferred from an extensive quantitative analysis of human sleep architecture within specific experimental groups with regular and extended sleep [15] (Figure 4A). The temporal trajectory refers to the modeled time course of sleep propensity over the waking period. It was determined through iterative numerical calculations of the system described in Equations (6) and (7), by simulating the dynamic increase in instability energy (ε) and aligning with characteristic time points observed in experimental sleep propensity data. The non-monotonic profile of interaction strength (Z), which also shapes the time-dependent probability of transitioning from wake to sleep, was incorporated to establish an initial zero-approximation trajectory.

Probability of state transitions in the absence of sleep

First, we carried out calculations to determine the probability of state transitions over a 24-hour interval in the absence of sleep. Utilizing the established time-dependent value of the regulating parameter x(t), we were able to derive the time-evolution of the instability energy ε(t) as follows: ε(t) = Uw(x(t)). The outcomes of these calculations are depicted in Figure 5A, illustrating the variations of ε(t) attributed to the variable rate of progression of the regulating parameter x. The initial conditions required for the determination of the 24-hour sleep propensity are: and for any . The calculated rate of transition in is depicted in Figure 5B.

Probability of state transitions in the presence of overnight sleep

The analysis of the probability of state transitions within a 24-hour timeframe, which encompassed an 8-hour overnight sleep, consisted of two distinct stages (Figure 5C). The initial step involved the computation of (t,) throughout the 16-hour period of wakefulness, where . The superscript index “16” indicates the duration of the continuous wakefulness period. This calculation employed the same x(t) trajectory and parameters as detailed earlier, in the absence of sleep, with (t,) (Figure 5).

During the 8-hour sleep interval, encompassing the time period , the evaluation of sleep propensity followed different initial conditions, with . These conditions corresponded to the transition speed from S to W state, resulting in the calculation of (t,  ). The straightforward relationship between the probability of staying asleep and the probability of waking up during sleep, , allows for the theoretical evaluation of the sleep propensity during 8-hour sleep.

The quantitative description of sleep architecture through the wave model [15] and the determination of the durations of consecutive NREM and REM episodes facilitated the computation of (t,  ) using a theoretical formula for the probability of remaining asleep after undergoing “n” sleep cycles:

(11)

where and are the durations of REM and NREM episodes in the k-th sleep cycle. The formula in Equation (11) has been derived with the assumption of Landau–Zener’s type of interaction [24] between S and W states in the vicinity of the region where the two potential curves cross (Figures 3 and 5A). Since the model accurately predicts and [15], in Equation (11) we used either experimental or theoretical data to calculate , with similar results. The speed of W–S transition for the two time intervals, 16 and 8 hours, was matched at t = 16 hours (Figure 5C, black line).

In experimental settings, alterations in wake propensity during the sleep period can be approximated by directly measuring the percentage of time spent awake [7]. Analogously to the previously discussed method, this parameter can be transformed into an estimation of sleep propensity using the straightforward relationship . The outcomes of employing this approach are depicted in Figure 5C (red dashed line) and exhibit similarities with those yielded by Equation (11).

Results

Bimodal pattern of the strength of interaction between sleep and wake states

Figure 4 illustrates the x-dependent dynamics of ε in parallel with Z, the strength of interaction between the two states. When the U_S and U_W potential curves are close to one another, the interaction between the two states becomes stronger, leading to an increased likelihood of state transitions. Notably, the potential curves are maximally close to each other near the homeostatic equilibrium point (Ux_c). Below Ux_c, the proximity between the curves decreases, whereas above Ux_c, the proximity of the two states demonstrates a nonlinear relationship with x. Specifically, the proximity between the two states is first declining and then increasing again over the period of wakefulness (Figure 4A). This gives rise to a bimodal Z pattern (Figure 4B).

The mathematical analysis of these two parameters, ε and Z, follows a continuous function (see Methods). In Figure 4, we illustrate their dynamics using six key points, designated as sleep propensity zones for further analysis of the probability of state transitions. Point 1 represents awakening or the onset of the daily wake period, followed by a relaxation of state instability due to the inertia of the damping forces (Figure 4A). This minor decrease in state instability is associated with a reduction in the proximity of the two states (point 2). Subsequently, the driving forces of homeostatic, circadian, and environmental nature favor the wake state and elevate state instability towards the homeostatic equilibrium threshold (x_c, point 3). At this threshold, the potential energies of the two states are equalized (Ux_c), their proximity reaches its maximum and Z reaches its peak (Figure 4B).

Beyond the x_c equilibrium region, the rise in state instability favors the transition from the increasingly unstable wake state to the more stable sleep state (Figure 4A). However, this is accompanied by a significant reduction in the proximity of the potential curves representing these two states, reaching its maximum distance and thus minimal strength of interaction Z at point 4 (Figure 4B), which makes transitioning from wake to sleep difficult. Once point 4 is surpassed, the further increase in the energy of state instability brings the two potential curves closer together (points 5 and 6), thus increasing Z. Consequently, both critical parameters—state instability and strength of interaction—now favor the transition from wake to sleep.

Wave model predicts the bimodal pattern of sleep propensity

Using the wave model fit on the experimental observations of NREM and REM episode durations in young, healthy males, we then examined its predictions of how the probability of transitioning into sleep state depends on the time since awakening. The 24-hour dynamics of x and , as predicted by the model (Figure 3B), along with the established correspondence between x,  , and Z (Figure 4), enabled us to reconstruct the daily pattern for the probability of state transition (P) (see Methods).

The overall bimodal profile and the timing of its specific peaks and dips in the absence of sleep (Figure 5A) closely resembled the profile of sleep propensity documented experimentally using an ultra-short sleep protocol [4] (Figure 1A). Specifically, the morning dip in sleep propensity corresponded to minimal wake-state instability following the period of sleep inertia (point 2). On the other hand, the mid-afternoon “siesta” peak aligned with the homeostatic equilibrium and maximal proximity of the two states at Ux_c (point 3).

The evening dip in sleep propensity was within the region of low interaction between the two waves (point 4). The surge in sleep propensity at the habitual sleep onset time, known as the sleep gate [4], corresponded to the rapid increase in P at point 5 due to the surge in both the states’ interaction and energy of state instability. In the absence of overnight sleep (Figure 5A), further rise in the probability P of the wake-to-sleep transition corresponded to the continued exponential increase in ε and Z around point 6 (Figure 4A).

Figure 5B compares experimental and model-based estimates of sleep propensity during nighttime sleep. Experimentally, sleep propensity is often assessed by the inverse of wake time, i.e. the percentage of time spent awake per hour of recording or per sleep cycle [7]. We applied this method to our dataset of healthy young males and also included an inverted published wake propensity curve from a forced desynchrony protocol [7]. Both empirical curves show a similar trajectory: a sharp increase in sleep propensity following sleep onset, peaking at the end of the first sleep cycle, and then declining monotonically across the night.

The model-derived curve (Figure 5B, black) closely mirrors this empirical pattern, despite being based on a completely different mechanism. In the wave model, sleep propensity during sleep is estimated from the ratio of REM to NREM episode durations (within the same sleep cycle), reflecting the reduction in instability energy (ε) after each REM episode (Figures 3 and 4). This mechanistic approach produces a declining curve that parallels empirical results, supporting the model’s ability to capture the intrinsic homeostatic structure of overnight sleep transitions. Importantly, this straightforward estimation of sleep propensity decline further supports the model’s central premise: that the energy of instability is released primarily during REM sleep. Note that the initial rise in sleep propensity following sleep onset corresponds to the wavepacket’s movement from the transition point x₀ to x_in, the right turning point of the initial energy level J_in (Figure 3B, top panel).

Documenting sleep propensity requires providing subjects with repeated sleep opportunities, during which they spend some time asleep [4]. The model predicted that daytime sleep would delay the onset of the mid-afternoon peak and evening dip in sleep propensity, as the energy of instability (or sleep need) would not accumulate during sleep periods. Accordingly, the model’s peaks and dips occurred slightly earlier than those observed experimentally (Figure 5C).

Notably, while the overall profiles of the model-derived and empirical curves are aligned, the experimental data exhibit a substantially deeper dip in sleep propensity during the early evening hours. This divergence is consistent with the influence of the circadian system, which is absent from our homeostatic-only model. The circadian wake-promoting drive is known to peak in the late evening, suppressing sleep propensity despite rising homeostatic pressure [7, 14]. This pattern is clearly seen in the short sleep protocol data [4], further emphasizing that while our model captures the intrinsic homeostatic architecture of sleep propensity, the full empirical profile includes an additional, temporally specific circadian modulation.

Effects of sleep deficit on the sleep propensity profile

Previously, the wave model accurately predicted the effects of acute sleep deprivation on sleep architecture by accounting for the increase in wake-state instability [15]. Sleep initiation at a higher energy level of the U_S curve (Figure 3A) explained not only the characteristic higher intensity of NREM sleep but also the reduction in both the duration and intensity of the initial REM sleep episodes following sleep deprivation. By incorporating an additional parameter—the strength of interaction between the two states (Figure 4B)—we now addressed the changes in sleep propensity following sleep deprivation.

We analyzed the effects of sleep deprivation resulting in varying degrees of sleep deficit and residual wake-state instability, as reflected by greater deviations in ε and x at the onset of sleep propensity assessment (Figure 6A). The model predicts that more severe sleep deficits produce more pronounced changes in the daily sleep propensity profile. Mild sleep deficits result in wakefulness initiated near the homeostatic equilibrium threshold, Ux_c (points 1–3) (Figure 6A). This masks the typical mid-afternoon peak, which instead manifests as a modest initial increase in sleep propensity due to higher and Z. However, the evening dip in sleep propensity remains evident, as the gradual rise in ε moves through the region around x = 0, characterized by minimal interaction between sleep and wake states (point 4).

Figure 6.

Modifications of sleep propensity profile due to different degree of sleep deficit. (A) Under normal sleep conditions, relaxation of the energy of state instability ε during sleep reaches point 2 (pink number). Mild sleep deficit (SD) results in lesser relaxation of ε (points 1–3), masking the mid-afternoon peak, though preserving the evening dip in sleep propensity around point 4. Moderate SD negates the mid-afternoon peak but can also modify the evening dip, since involves the points 3–5 interval of weak interaction between the sleep and wake states. Severe SD negates the entire bimodal pattern of sleep propensity due to a combination of relatively high ε and strength of interaction between states beyond point 5. Sleep propensity zone 1–6, as in Figure 4. (B) Sleep propensity (minutes asleep during 20-minute sleep opportunities provided at 100-minute intervals) over 24-hour period in subjects with mild SD (red) and severe SD (blue), n = 9 per group, ^*p<.01.

The model further predicts that a moderate sleep deficit (i.e. sleep propensity measurements begin at x values near point 4 of weak interaction between the two states; Figure 6A), will result in reduced initial sleep propensity despite elevated ε due to low initial Z. Subsequently, sleep propensity will continue to rise as it typically does after the evening dip, since both ε and Z gradually increase.

In contrast, severe sleep deficits—such as those resulting from overnight sleep deprivation—sustain x at levels beyond those associated with typical evening sleep onset (point 5). In this case, sleep propensity assessment begins around point 6, where both state interaction and ε are elevated (Figure 6A). This combination significantly amplifies sleep propensity and suppresses its normal biphasic variations, as the entire evaluation process takes place within the x interval characterized by concurrent increases in state instability ε, regulating parameter x and the strength of interaction Z.

To test these predictions, we analyzed sleep propensity data from healthy young male cadets who adhered to a military schedule that provided regular 7-hour nighttime sleep opportunities with no daytime naps [17–19]. Given a recommended sleep duration of around 8 hours per night for young males [16], this schedule represented a condition of chronic mild sleep restriction. Two groups were studied during a 24-hour short sleep protocol, consisting of twelve 120-minute cycles of alternation of 100-minute wake period with 20-minute polysomnographically monitored sleep opportunity. Prior to the experiment, one group maintained regular 7-hour nighttime sleep (mild sleep deficit), while the other group experienced 24 hours of total sleep deprivation (severe sleep deficit).

Figure 5B illustrates that the daily sleep propensity profiles for both groups aligned with the wave model's predictions. Consistent with the wave model, the mild sleep deficit group exhibited no discernible mid-afternoon peak, while their evening dip remained prominent. In contrast, the severe sleep deficit group displayed high sleep propensity with no significant variations during the mid-afternoon or evening periods, as was predicted by the model.

Discussion

Precise mathematical description of a physiological process is more than just a theoretical construct—it serves as a framework that reveals the underlying principles governing that process. It has long been recognized that sleep need does not always correspond to the ease of falling asleep. In his 1992 review [25], Lavie, whose lab played a pivotal role in characterizing the daily bimodal profile of sleep propensity [4, 26–30], addressed this issue and noted that a comprehensive model of sleep–wake regulation must integrate at least three aspects of sleep behavior: the speed of sleep onset, sleep duration, and sleep composition. The wave model of sleep dynamics offers such a framework by linking the durations of consecutive NREM and REM sleep episodes with the probability of state transitions, embodying the concept of dynamic homeostasis.

Our analysis shows that wake-state instability and the strength of wake–sleep interaction can be quantified by analyzing changes in overnight NREM and REM sleep episode durations. Previously [15], we found that fitting the wave model to these experimental parameters—readily available from any polysomnographic record—enables a mathematically accurate representation of normal sleep architecture and yields experimentally validated predictions regarding its modification in response to acute sleep deprivation or sleep abundance [15]. Importantly, the model revealed an invariant relationship between NREM and REM sleep. This study now demonstrates that such experimental sleep data are also sufficient to uncover the homeostatic regulation of daily sleep propensity profile. This is analogous to predicting the ease of ascending a hill based on the experience of descending it.

Experimental studies have demonstrated how sleep need [1], reflected in the intensity of the initial NREM sleep episode, accumulates over the wake period, necessitating a transition from wake to sleep. A complementary concept from cognitive performance studies—wake-state instability—described the increasing variability in attention and reaction time during sleep deprivation, reflecting a growing physiological difficulty in maintaining stable wakefulness [5]. While distinct in origin, this behavioral concept captures a similar principle to what the wave model formalizes more broadly: a rising systemic instability as the organism moves away from homeostatic equilibrium.

However, while increased instability of one state relative to another is a necessary condition for transition, it is not sufficient on its own. Low interaction strength or a high activation energy barrier can result in a metastable state, reducing the probability of transition. Using our wave model, we demonstrate how a nonlinear change in the strength of interaction between wake and sleep states can occur and how it can account for the bimodal sleep propensity pattern—even while sleep need increases steadily. The model’s predictions align closely with experimental observations on sleep propensity under habitual sleep–wake conditions, overnight wakefulness, and following both mild and severe sleep restriction.

Our results suggest that, during wakefulness, while state instability increases according to an inverse exponential function, the strength of interaction follows a more complex, bimodal pattern (Figure 4). It peaks at the homeostatic equilibrium where the two states are in maximal proximity (Ux_c), corresponding to the mid-afternoon peak in sleep propensity. The interaction then declines sharply near the minimal energy on the sleep potential curve (U_S), which coincides with the evening dip. Later, as the proximity of the two potential curves increases again, interaction strength rises, leading to a surge in sleep propensity during extended wakefulness. Under prolonged sleep deprivation, when state instability is very high, the interaction strength also remains consistently elevated, and the bimodal pattern of sleep propensity disappears.

During each sleep cycle, the strength of interaction fluctuates, being maximal in the x_c region during REM sleep, defining its duration [15], and lower during NREM sleep. This may explain the higher probability of state transition (awakening) in REM sleep. Over the entire sleep period, state instability decreases with each sleep cycle due to the drop in the energy of state instability. This results in a gradual reduction in sleep propensity, the magnitude of which can be accurately predicted by the model based on the ratio of REM/NREM episode durations of each cycle. The prediction is consistent with our experimental results and the earlier findings on the percent awake during sleep based on the forced desynchrony studies [7].

Beyond the homeostatic process, additional factors—such as circadian, environmental, or pharmacological influences—can also affect the dynamics of sleep propensity. It is therefore essential to determine which factors predominantly influence state instability and which primarily modulate the strength of interaction. The model provides clear predictions that can distinguish these effects based on which specific experimental parameters change.

From a broader perspective, the circadian system may not simply oppose homeostatic pressures to fit sleep into a fixed daily schedule. Rather, it may act as a synergistic modulator that reinforces and temporally refines underlying homeostatic tendencies. In this view, the circadian system helps amplify the internal coherence of physiological processes—including sleep—while coordinating them with environmental cycles. This highlights the value of isolating homeostatic contributions, as done here, to provide a foundation for future integration with circadian mechanisms.

For instance, the circadian clock plays a pivotal role in sleep regulation. The wave model predicts that if the clock’s effect targets state instability, then the principal measures of sleep architecture (the intensities and durations of NREM and REM episodes, plus sleep onset latency) would vary with circadian phase. In contrast, if the clock modulates the strength of interaction between sleep and wake states, it will primarily affect two measures: REM sleep episode duration and state transitions (i.e. sleep onset latency or sleep propensity). This is because, according to the model, these two sleep measures, but not the other three, depend on the strength of interaction between states [15]. Experimental observations, particularly those in which homeostatic and circadian influences are desynchronized, support this latter scenario, with the clock primarily affecting REM episode duration and sleep propensity [2, 6, 7, 14]. Furthermore, a synergistic homeostatic and circadian effect that reduces interaction strength in the early evening is consistent with the more pronounced evening dip in sleep propensity observed under regular entrained conditions, compared to model-generated curves based solely on homeostatic regulation (Figure 5C).

We therefore propose that the circadian clock exerts its daytime effect by reducing the strength of interaction between sleep and wake states without directly affecting state instability. Toward the end of the day, as clock activity diminishes, disinhibition of state interactions facilitates a surge in sleep propensity and, along with the homeostatic effect on interaction strength, ultimately promotes sleep onset. This is not to be confused with the indirect effect of the clock, which is by attenuating state transitions during the day and early evening prolongs and consolidates wakefulness, thereby indirectly increasing overall wake-state instability by the end of the day.

These analytical capabilities of the wave model can be extended to evaluate how specific pharmacological agents or other interventions (e.g. light, exercise, meals) differentially affect state instability and interaction strength. This is particularly relevant for insomnias that manifest as an impaired ability to initiate or maintain sleep, despite objective signs of increased sleep deficit and wake-state instability [31]. While we do not directly map modeled sleep propensity to absolute hours of prior sleep loss in this study—aside from distinguishing overnight sleep deprivation from chronic restriction in the datasets presented—we anticipate that future work could calibrate homeostatic states more precisely using experimental markers such as REM episode structure following graded sleep restriction.

Additionally, the empirical data used for model validation were collected from male participants. The wave model describes fundamental homeostatic dynamics that generalize across populations; however, sex-related differences in sleep architecture, such as potential width, rate of energy accumulation, or transition stability, may influence model parameters and deserve further investigation.

In conclusion, mathematical modeling is particularly powerful in fields with abundant experimental data yet unclear regulatory logic. Sleep research has amassed extensive molecular and physiological findings—from neuronal activity patterns to genetic influences—but integrating them into a coherent framework remains challenging. Our model helps organize these findings by distinguishing factors that destabilize wake state (state instability) from those that modulate state transitions. This distinction enables systematic mapping of molecular and physiological changes onto sleep dynamics. By providing a precise mathematical depiction, our model can enhance understanding of sleep regulation, inform targeted treatments for sleep disturbances, and improve overall sleep health.

Author contributions

Vasili Kharchenko (Conceptualization [equal], Formal analysis [lead], Funding acquisition [equal], Methodology [lead], Software [lead], Supervision [equal], Writing—original draft [equal], Writing—review & editing [equal]), Arcady A. Putilov (Investigation [equal]), Michael G Rozman (Formal analysis [supporting], Software [supporting]), Irina V. Zhdanova (Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Writing—original draft [equal]).

Disclosure statement

Financial disclosure: none.

Non-financial disclosure: none.

Data availability

Preprint repositories: bioRxiv. Polysomnographic data are available upon request.