According to traditional theory, the need for sleep accumulates during wakefulness and dissipates during sleep. Despite decades of research, it is still uncertain precisely which biological variables form the substrate of sleep need, what characteristics of wake challenge their stability, how information about sleep-wake history is integrated over time, and how sleep mediates the restoration of homeostasis. Daily sleep amount varies greatly across the animal kingdom, yet individuals generally perform a relatively constant and species-specific amount of sleep each day. An even more fundamental question therefore remains whether homeostatic sleep regulation reflects an active process, dynamically shaping daily sleep architecture in response to a physiological need for the homeostatic regulation of specific variables, or whether it corresponds instead to an unknown innate time-keeping process which ensures only that a certain daily quota of sleep is obtained.

The earliest theories of sleep homeostasis supposed the existence of a single variable, termed Process S, which describes sleep drive at the global level (Borbély, 1982). This variable is assumed to always increase during wakefulness, independently of its content, and to decline during sleep. It is widely acknowledged that homeostatic sleep pressure is reflected in the levels of slow wave activity (SWA, 0.5–4 Hz spectral power) observable during NREM sleep in neurophysiological field potentials, such as electroencephalogram (EEG) or local field potential (LFP). Although preceding sleep-wake history is considered a key determinant of sleep amount and intensity, the circadian process also plays an important role (Daan et al., 1984). It is thought that the master pacemaker of circadian rhythmicity is located in the suprachiasmatic nucleus (SCN) of the hypothalamus, which consists of multiple cell-autonomous oscillators (Hastings et al., 2019). Although individual SCN cells may have widely diverse periods, the spatial averaging of their activities results in an occurrence of a stable circadian rhythm at the level of the whole animal (Liu et al., 1997; Kunz and Achermann, 2003).

Current views on the origin of sleep homeostasis emphasise the importance of a local and activity-dependent component (Krueger and Tononi, 2011; Rattenborg et al., 2012; Tononi and Cirelli, 2014). It was shown that SWA is far from uniform across the brain, and that the behavioural and cognitive content of waking, beyond its mere duration, influences subsequent sleep and cortical region-specific SWA (Kattler et al., 1994; Huber et al., 2004; Vyazovskiy and Tobler, 2008; Rector et al., 2009; Murphy et al., 2011; Fisher et al., 2016). Indeed, many candidate mechanisms for the substrate of sleep homeostasis implicate processes occurring at a cellular or local network level. These include the maintenance of cellular homeostasis (Reimund, 1994; Mackiewicz et al., 2007; Vyazovskiy and Harris, 2013; Bellesi et al., 2016), the replenishment of energy stores (Scharf et al., 2008), the influence of sleep-related signalling molecules such as adenosine or cytokines (Krueger et al., 2008), and the regulation of imbalanced synaptic strengths (Tononi and Cirelli, 2003; Vyazovskiy et al., 2008; Liu et al., 2010). Although the equivalence of these processes with Process S has not been conclusively demonstrated (Frank and Heller, 2019), the existing evidence supports the relevance of sleep-wake-dependent differences in neuronal activity for understanding the regulation of sleep.

Cortical neuronal activity is generally higher during waking compared to sleep (Vyazovskiy et al., 2009; McKillop et al., 2018), although this may depend on the cortical region (Hengen et al., 2013). Lower spike firing generally typical of sleep is thought to be due, at least in part, to regular periods of widespread synchronous network silence, termed off periods, intruding on ongoing activity (Steriade et al., 1993b; Steriade et al., 2001; Sanchez-Vives and Mattia, 2014). Importantly, off periods in neural populations are thought to underpin slow wave dynamics at the level of the field potential (Steriade et al., 1993a; Massimini et al., 2004; Buzsáki et al., 2012), and it was shown that their properties reflect homeostatic sleep need (Vyazovskiy et al., 2009; McKillop et al., 2018; Saberi-Moghadam et al., 2018).

Neuronal firing rates typically fluctuate around a homeostatic set point, which is characteristic for individual cells and variable across the population (Turrigiano, 2011; Hengen et al., 2013; O'Leary et al., 2014; Styr et al., 2019). The homeostatic regulation of firing rates may depend on processes occurring specifically in sleep and wakefulness (Grosmark et al., 2012; Hengen et al., 2016) and evidence suggests that firing rates change as a function of time spent awake, conditional on the behaviour (Vyazovskiy et al., 2009; Fisher et al., 2016). Additionally, the magnitude and direction of state-dependent changes in firing rate differs between neurons, depending on the brain region and their individual firing rate set point (Hengen et al., 2013; Miyawaki and Diba, 2016; Watson et al., 2016; Miyawaki et al., 2019). Firing rate homeostasis may occur above the individual cell level (Slomowitz et al., 2015), perhaps through synaptic plasticity induced by the sleep slow oscillation which may act to narrow the population firing rate distribution (Levenstein et al., 2017).

Overall, the evidence suggests that the homeostatic regulation of sleep and of neuronal firing rates may be intrinsically related, however, this functional link remains incompletely defined. Mathematical modelling approaches present an opportunity to address this problem. To this end, we developed novel quantitative models of Process S and demonstrated that its temporal dynamics, on both a local and global level, can be derived entirely from local neuronal activity, without any reference to the animal’s sleep and wake states. We found that the magnitude of the deviation of multi-unit firing rate from a locally specified set point carries sufficient information to account for empirically derived patterns of SWA. We then introduced the total time spent in off periods as an alternative measure for Process S with more local origins than SWA, and showed how Process S dynamics may then be described in terms of two opponent processes, dependent on spiking rate and off period occurrence. Our data show that the global dynamics of Process S can be derived from a spatiotemporal integration of local neuronal activities. This suggests that a possible functional role for sleep homeostasis is to provide a precise intrinsic time-keeping mechanism.

We analysed chronic recordings of the frontal electroencephalogram (EEG) alongside local field potential (LFP) and multi-unit activity (MUA) spiking from the primary motor cortex of six mice. These recordings were made continuously over 48 h, starting at light onset, while the mice were freely behaving within their cage and exposed to a standard 12 h - 12 h light-dark cycle. At light onset of the second day a sleep deprivation protocol was enforced for 6 h, involving the presentation of novel objects. Outside of this sleep deprivation period, mice were undisturbed and able to sleep at will.

Process S dynamics can be described as a function of vigilance state history

NREM sleep can be defined and distinguished from waking by the presence of high amplitude slow (0.5–4 Hz) waves in the EEG. The average EEG spectral power in the slow wave range (termed slow wave activity; SWA) in NREM sleep varies as a function of the animal’s recent sleep-wake history, and this relationship has been captured in a classical quantitative theory using the concept of the homeostatic ‘Process S’ (Borbély, 1982; Daan et al., 1984). Process S describes a variable whose magnitude can be estimated from the level of SWA during NREM sleep, reflecting the intensity of sleep, and which is interpreted as corresponding to the homeostatic component of sleep drive (Franken et al., 1991a; Achermann et al., 1993; Huber et al., 2000a; Vyazovskiy et al., 2007; Guillaumin et al., 2018). Theoretically, Process S follows simple dynamics: during wake it increases according to a saturating exponential function towards an upper asymptote, while during NREM sleep it decays exponentially towards a lower asymptote. There are many published variants of the precise equations for this model, but, crucially, all these variants use the sleep-wake state history of the animal as the key predictive variable for Process S. Here, the specific equations used are:

${\displaystyle \begin{array}{rl}\text{Wake / REM}\text{sleep}& :\frac{dS}{dt}=\alpha (S_{max}-S(t))\\ \\ \text{NREM}\text{sleep}& :\frac{dS}{dt}=-\beta (S(t)-S_{min})\end{array}}$

Where S(t) represents the level of Process S as a function of time, S_max and S_min are upper and lower asymptotes, and α and β are rate parameters. The first equation is applied when the animal is scored to be awake or in REM sleep and the second equation is applied when it is scored to be in NREM sleep. Here, we use the typical approach of applying the wake dynamics equation to REM sleep (Franken et al., 1991a; Huber et al., 2000a; Vyazovskiy et al., 2007), but for simplicity do not set separate parameters for wake vs. REM sleep. For convenience, S is expressed in units equivalent to those of SWA. This classical formalism is later abbreviated as model Cl-SWA. Process S as described by these simple dynamics accounts for the time course of empirical SWA with high accuracy. We applied this model to SWA derived from the frontal EEG of all animals, and as expected obtained a high quality fit (Figure 1A).

Figure 1

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Slow wave activity, firing rate and vigilance state time series data.

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Process S dynamics can be described as a function of temporally and spatially integrated local neuronal activity

We first sought to determine whether a model expressing Process S dynamics solely as a function of local multi-unit firing rates might describe the levels of sleep SWA in the corresponding LFP with comparable accuracy to the classical model which is dependent on vigilance states defined at the global level (Figure 2). The classical Process S model was used as a starting point for the development of a novel firing-rate-dependent alternative. To do this, the equations of the classical model were adapted in two ways. Firstly, an instantaneous firing rate threshold (F_θ) was introduced as a new model parameter to replace the wake vs. sleep criterion, assuming an increase in Process S when the threshold is exceeded and a decrease when firing is below. Conceptually, this firing rate threshold resembles a set point: a target firing rate at which the dynamics are stable. Secondly, we assumed that the rate of change of Process S is proportional to the difference between firing rate and this threshold. Introducing this change to the equations ensures that the rate of change of S is equal to zero exactly at the set point, and a continuous function of firing rate around this value. This version of the model is later abbreviated as Fr-SWA. The equations are (Figure 3A):

${\displaystyle \begin{array}{rl}F(t)>F_{\theta }& :\frac{dS}{dt}=\alpha (S_{max}-S(t))(F(t)-F_{\theta })\\ \\ F(t)<F_{\theta }& :\frac{dS}{dt}=-\beta (S(t)-S_{min})(F_{\theta }-F(t))\\ \\ F(t)=F_{\theta }& :\frac{dS}{dt}=0\end{array}}$

Figure 2

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Process S time series and parameters based on SWA for classic and novel models.

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We applied this novel model, alongside the classical model, to describe SWA dynamics at the LFP level, by finding parameter values that would minimise the difference between empirical SWA and modelled Process S. The two models were fit to the SWA from each LFP channel separately, using the multi-unit firing rate from the same channel for the firing-rate-dependent model. The models were also fit to the SWA obtained by averaging the LFP (and firing rate) over the whole population of channels within the same mouse.

Figure 2 shows two examples of the fit of both models, first to SWA derived from the average LFP, and also to SWA derived from a single LFP channel (these examples are from the same animal as in Figure 1A). The overall pattern of Process S dynamics was similar at both LFP and EEG recording levels and was well described by both the classic and novel firing-rate-based model. Throughout the data, this purely firing-rate-dependent model described the overall dynamics of LFP SWA during NREM sleep to a comparable accuracy as the classical model, as reflected in the median percent error deviation between modelled and empirical SWA (Figure 3B). Both the model type and animal had a highly significant effect on the median percent error of the model fit to individual LFP channels, although there is no significant interaction (Model: F_(1,144)=25.9, p=1.1×10⁻⁶; Animal: F_(5,144)=30.2, p=6.3×10⁻²¹; Model x Animal: F_(5,144)=1.65, p=0.15; two-way ANOVA unequal groups). Errors were higher for the novel model, importantly however, the differences in fit quality due to the model type are small relative to the effect of the particular animal and channel, on which fit quality depends much more strongly (Model: η² = 0.079; Animal: η² = 0.459; Model x Animal: η² = 0.025; Channel (residuals): η² = 0.437). The errors of the model fit on the averaged LFP and on the EEG (classical model only) are also shown in Figure 3B, and are more explicitly compared in Figure 3C. We did not find any significant effect of the field potential level (EEG, average LFP, single LFP channels) on the model error (F_(2,171) = 0.2, p=0.82, one-way ANOVA).

The distributions of final optimised rate parameters, ${\displaystyle \mathit{\alpha }}$ and ${\displaystyle \mathit{\beta }}$, are shown in Figure 3D–G for the classic model and Figure 3H–K for the firing-rate-based model. In all cases, it was observed that the optimised rate parameters for Process S derived from EEG and the mean LFP consistently fall within the range of corresponding parameters derived from single LFP channels, suggesting that Process S calculated at spatially higher levels may reflect an averaging of Processes S which exist more locally. Indeed, for both rate parameters in both models, the mean of parameter values over single-channel-derived Process S was not significantly different from the equivalent parameter values obtained from the mean LFP or EEG (Cl-SWA ${\displaystyle \mathit{\alpha }}$: F_(2,10) = 1.0, p=0.39; Cl-SWA ${\displaystyle \mathit{\beta }}$: F_(2,10) = 0.7, p=0.51; two-way ANOVA, factors spatial level and animal; Fr-SWA ${\displaystyle \mathit{\alpha }}$: p=0.079; Fr-SWA ${\displaystyle \mathit{\beta }}$: p=0.24; paired t-test). The final optimised values of the firing-rate threshold parameter, F_θ, in the activity-dependent model are shown in Figure 3L, and the values of S_max and S_min in both models are shown in Figure 3—figure supplement 1. Most parameters in both models were significantly different between animals, with the exception of S_min in the firing-rate-dependent model (Cl-SWA ${\displaystyle \mathit{\alpha }}$: F_(5,77) = 4.03, p=2.8×10⁻³; Cl-SWA ${\displaystyle \mathit{\beta }}$: F_(5,77) = 8.65, p=1.9×10⁻⁶; Cl-SWA S_max: F_(5,77) = 14.35, p=9.8×10⁻¹⁰; Cl-SWA S_min: F_(5,77) = 12.42, p=1.1×10⁻⁸; Fr-SWA ${\displaystyle \mathit{\alpha }}$: F_(5,77) = 8.33, p=3.0×10⁻⁶; Fr-SWA ${\displaystyle \mathit{\beta }}$: F_(5,77) = 9.27, p=7.5×10⁻⁷; Fr-SWA F_θ: F_(5,77) = 14.3, p=1.0×10⁻⁹; Fr-SWA S_max: F_(5,77) = 6.64, p=3.9×10⁻⁵; Fr-SWA S_min: F_(5,77) = 2.16, p=0.07; one way ANOVA).

The relationship between the optimal firing set point and state specific firing rates is shown in Figure 3M. Firing rate threshold was z-normalised (subtract the mean firing rate over a given state and divide by the standard deviation) separately with respect to the distribution of firing rates in wake, NREM and REM sleep. This shows, as expected, that the set point is typically below mean firing in wake (−1.0 ± 0.6; mean ± sd), well above mean firing in NREM sleep (1.6 ± 1.1), and slightly above mean firing in REM sleep (0.7 ± 1.1). These were all significantly different from zero (Wake: p=1.6×10⁻¹³, NREM: p=1.9×10⁻¹⁴, REM: p=3.4×10⁻⁷, two-sided Wilcoxon signed rank test). Figure 3N further shows the state-dependent distribution of average firing rate, expressed as a percentage of the chosen firing set point parameter. Mean firing was 182 ± 57.6% of the firing rate set point during wake, 61.8 ± 17.6% during NREM sleep, and 84.2 ± 31.8% during REM sleep. Again, these distributions were all significantly different from 100% (Wake: p=6.7×10⁻¹⁴, NREM: p=2.0×10⁻¹⁴, REM: p=5.4×10⁻⁵). Note that, in a few channels, the optimal firing rate threshold is actually above the mean firing rate during waking. This occurs when the waking firing rate distribution overlaps substantially with the NREM sleep firing rate distribution but has a heavier tail. REM sleep, in this data, is typically associated with firing below the set point, and therefore Process S decreases, albeit at a reduced rate compared to NREM sleep (Figure 3O). Interestingly, there are no significant correlations between the model fit error and the threshold normalisation relative to any one vigilance state (Wake: p=0.06, r = 0.21; NREM: p=0.35, r = −0.11; REM: p=0.94, r = −0.01), suggesting that there is no clear relationship between the firing rate set point and firing rate distribution of any one particular state.

Off occupancy as a novel metric for local sleep-wake related neuronal dynamics

When considering the relationship between multi-unit firing rates and LFP slow waves, a conceptual complication arises due to the different origins of the LFP and MUA from within the same channel. While the MUA firing rate represents the activity of only a few individual neurons, factors such as volume conduction result in spatial smoothing of the LFP signal, and as such it can be influenced by the activities of neurons covering a cortical area potentially on the order of several millimetres (Kajikawa and Schroeder, 2011). Subthreshold currents also influence the LFP (Reimann et al., 2013; Ness et al., 2016). This means that when a slow wave is detected in the LFP, it is not guaranteed that all local neurons contributing to the MUA are necessarily silent (Todorova and Zugaro, 2019). Similarly, not every long interspike interval occurs during a slow wave. An estimation of the occurrence of off periods may be obtained by combining LFP and spiking data (Figure 4A). Slow wave detection was performed on each LFP over the whole 48 h, including all vigilance states (0.5–6 Hz filter followed by an amplitude threshold, values shown in Figure 4C). All multi-unit inter-spike intervals (ISIs) which coincide with the peak of detected slow waves were identified. The distribution of the duration of these ISIs was often (64 out of 75 channels) unambiguously bimodal (Figure 4B). This could be interpreted as evidence of the existence of two spiking states occurring locally during the more widespread network slow oscillation: high frequency spiking (on period) and extended silence (off period). Note that these two distributions do not simply correspond exactly to sleep vs. wake conditions, because bimodality is often evident in the distribution of slow wave coincident ISIs from NREM sleep only, or even from REM sleep only (Figure 4B). The distribution of slow wave coincident ISIs (over all vigilance states) was used to define an ISI duration threshold for the detection of off periods, separately for each channel. The values used for this threshold are shown in Figure 4D. The average multi-unit firing rate aligned to the peak of slow waves in detected off periods reveals a clear suppression of firing, consistent with expectations (Figure 4E). A rebound increase in average firing is visible in this example channel immediately after the off period, as has been previously documented in some cortical neuronal populations (Chauvette et al., 2010). The total fraction of time each channel spends in off periods was calculated over all epochs of 4 s, and termed the ‘off occupancy’. Off occupancy defined in this way is high in NREM sleep and low in both wake and REM sleep (Figure 4F). The existence of a non-zero frequency of local cortical off periods has been previously reported during wakefulness (Vyazovskiy et al., 2011; Vyazovskiy et al., 2014; Fisher et al., 2016), and REM sleep (Funk et al., 2016). The off occupancy measure displays similar temporal dynamics to LFP (and EEG) SWA over both sleep deprivation and spontaneous sleep and wake (Figure 4G). These results together demonstrate that the detected off periods were likely to be physiologically meaningful and that, for instance, the possible inclusion of interneurons in the MUA which may spike during off periods (Zucca et al., 2019) did not represent a significant problem.

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Off occupancy time series and off period detection parameters.

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Process S dynamics, defined using off periods, can be described as a function of vigilance states or neuronal firing rates

In order to investigate whether the off occupancy measure reflects Process S, we applied the classical state-based model to the time course of off occupancy, exactly as was done with single channel SWA. The classical model was applied with its equations unchanged, and is abbreviated as Cl-Off. Figure 5A shows an example of the resulting Process S time course obtained in this way with a high quality of fit. In contrast, some changes were introduced to the firing-rate-based model in order to describe off occupancy dynamics. We considered that the different dynamics above vs. below a particular firing rate set point may be due to two opponent processes simultaneously active in dynamic opposition but with differential magnitude in wake vs. NREM sleep. Specifically, we consider a Process S increasing term which is proportional to instantaneous firing rate, and a Process S decreasing term which is proportional to the time spent in off periods (off occupancy). The equation used is:

${\displaystyle \frac{dS}{dt}=\alpha F(t)(S_{max}-S(t))-\beta X(t)(S(t)-S_{min})}$

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Process S time series and parameters based on off occupancy for classic and novel models.

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In this model (Figure 5B), one term drives S towards an upper asymptote (S_max) in proportion to firing rate (F), while the other drives S towards a lower asymptote (S_min) in proportion to the off occupancy (X). Again, two rate parameters α and β are required. This model behaves similarly to previous models because firing is high in wake and low in NREM sleep, whereas off occupancy is high in NREM sleep and low in wake. This variant of the model is abbreviated as Fr-Off. Figure 5A also includes the fit from this model, demonstrating a high level of agreement between the two models and an accurate fit to the data.

The distribution of the median percent errors for the fits from both models over all animals and channels is shown in Figure 5C. As before, the model type and animal has a significant effect (Model: F_(1,138) = 8.06, p=5.2×10⁻³; Animal: F_(5,138) = 17.2, p=3.2×10⁻¹³; Model x Animal: F_(5,138) = 0.52, p=0.76; two-way ANOVA with unequal groups), and the classic model achieved a slightly lower median percent error. However, this effect was again very weak compared with the variation in fit quality between animals and channels (Model: η² = 0.034; Animal: η² = 0.367; Model x Animal: η² = 0.011; Channel (residuals): η² = 0.588). Figure 5D shows the distribution of values for the change in modelled Process S from one simulated time step to the next resulting from the firing-rate-based model (Fr-Off), in wake, NREM and REM sleep, pooling all animals, channels and time. Unlike in the previous firing-rate-based model, REM sleep is now typically associated with an increase in Process S. The distributions of final optimised rate parameters, ${\displaystyle \mathit{\alpha }}$ and ${\displaystyle \mathit{\beta }}$, are shown in Figure 5E–F for the classic model and Figure 5G–H for the firing rate and off occupancy model. Values for S_max and S_min are shown in Figure 5—figure supplement 1. Most parameters in both models were different between animals (Cl-Off ${\displaystyle \mathit{\alpha }}$: F_(5,74) = 2.4, p=0.047; Cl-Off ${\displaystyle \mathit{\beta }}$: F_(5,74) = 2.29, p=0.055; Cl-Off S_max: F_(5,74) = 9.49, p=6.4×10⁻⁷; Cl-Off S_min: F_(5,74) = 16.79, p=8.0×10⁻¹¹; Fr-Off ${\displaystyle \mathit{\alpha }}$: F_(5,74) = 9.86, p=3.9×10⁻⁷; Fr-Off ${\displaystyle \mathit{\beta }}$: F_(5,74) = 5.14, p=4.6×10⁻⁴; Fr-Off S_max: F_(5,74) = 10.08, p=2.8×10⁻⁷; Fr-Off S_min: F_(5,74) = 10.25, p=2.3×10⁻⁷; one way ANOVA). Notably, the weakest evidence for inter-animal differences were for ${\displaystyle \mathit{\alpha }}$ and ${\displaystyle \mathit{\beta }}$ in the classic model.

Variation in local process S dynamics is averaged away at higher scales and is stable across conditions

Although we demonstrated that the rate parameters obtained at the local level appear to correspond closely to those obtained for ‘global’ EEG, the question remains how local activity-dependent Processes S might relate to the global Process S which reflects sleep-wake history. To enable the direct comparison of their time courses, each Process S was normalised in a range from zero to one, determined by the corresponding fitted values of S_max and S_min. We observed that the mean over individual Processes S derived from applying activity-dependent models in single channels matches the time course of Process S derived from the EEG with the classical model. This held for both local SWA (Figure 6A) and off occupancy (Figure 6B) models. Note that the Cl-SWA model is considered in both cases here, as this is the most ‘global’ level at which the model can be applied, and off periods are not defined for the EEG. Interestingly, over all animals the global level Process S was typically in between the average of local Processes S derived from the two activity-dependent models (Figure 6C). This result suggests that global Process S calculated at a higher spatial scale might reflect an averaging across a manifold of local Processes S that exist on a finer spatial level.

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The modelling approaches described here all identify variability in Process S between recording channels, and indicate that a local component determines its temporal dynamics, which, in turn, is spatially integrated to give rise to the global dynamics of Process S. One important caveat that remains is that, even within the same global behavioural state, neuronal activity recorded from the same local cortical region may differ markedly between recording conditions, depending, for example, on the time of day or specific wake behaviours. Therefore, if the dynamics of global Process S represent spatial integration of local neuronal activities, it is important to determine how stable the locally-derived rate parameters are, and how sensitive our overall modelling approach is, to factors which are expected to influence local neuronal firing rates.

To address this question, we performed parameter estimation for each model separately for baseline day, when the animals are awake spontaneously for most of the dark phase, and for the day when sleep deprivation was conducted, when the animals are engaged in novel object exploration and spent time awake for an extended period of time during the light phase, when nocturnal mice are habitually asleep. Invariably, we observed that both the rate parameters and the median error were close to those derived from the whole 48 h, with parameters clustering around 100% of their original values and error differences around zero (Figure 6—figure supplement 1). Statistical analysis (Figure 6—figure supplement 1) suggested that the day, baseline or sleep deprivation, did have a significant effect on all parameters and error differences, suggesting some influence on Process S which is unaccounted for both by global sleep-wake history and cortical activity. The model type, either classical (state-based) or novel (activity-based), was generally only significant for Process S models based on SWA, not off periods. Importantly, the interaction between day and model was not significant in most cases, supporting that the activity-dependent models describe day-dependent variation in Process S generally similarly to the classical sleep-wake state model. Indeed, in the one case in which a significant interaction effect was found (β in off periods models), day-dependent differences were greater for the classical model than activity-dependent model (Figure 6—figure supplement 1E). The effect of animal was variably significant, although the interaction between animal and day was significant in all cases. This further suggests that variation in Process S errors may be associated with inter-individual variability arising from technical factors, such as precise location of the recording electrode, or systematic differences in the animals’ behaviour, rather than due to the modelling approach.

Here, we demonstrated that Process S dynamics can be derived entirely from multi-unit neuronal spiking activity, without reference to global sleep-wake states. A model was outlined whereby the integrated history of local multi-unit firing rates, relative to a locally-defined firing rate set point, predicted the temporal dynamics of LFP SWA. The accuracy of this model was demonstrated in data recorded from mouse frontal cortex over 48 h, including both voluntary sleep and wake, and sleep deprivation. A novel metric for Process S was then presented, termed off occupancy, which measures the fraction of time a neural population spends in off periods, defined by the coincidence of LFP slow waves and multi-unit spiking silence. The modelling approach was combined with the off occupancy metric and tested on the same data to present a quantitative framework for understanding dynamics of sleep pressure at a highly local level in terms of neural spiking and off periods. Importantly, it is concluded that Process S at the global level reflects sleep-wake history through an integration of local cortical neuronal activity levels over time and also across spatially distributed networks.

Central to this modelling perspective is the assumption that the generation of spikes by a neural network is in some way correlated with an increase in sleep pressure, and that this manifests in the subsequent expression of slow waves and off periods, reflecting Process S. The energetic cost of spiking is high (Attwell and Laughlin, 2001), the regulation of a neuron’s firing rate set point is linked to cellular energetics (Styr et al., 2019; Vergara et al., 2019), and neurons are susceptible to cellular stresses that can result from sustained metabolic load, such as oxidative stress (Wang and Michaelis, 2010; Cobley et al., 2018; Kempf et al., 2019; Cirelli et al., 2006). Furthermore, spiking activity may be mechanistically associated with synaptic plasticity, and it was suggested that firing rates are an important determinant of overall changes in synaptic strength, with higher spiking leading to greater changes (Graupner et al., 2016; Lappalainen et al., 2019). Compensatory processes exist within neurons to oppose cellular stress and related homeostatic challenges (Kültz, 2005) and off periods could provide the opportunity for neurons to prioritise such processes, therefore mediating the restorative benefits of sleep (Vyazovskiy and Harris, 2013). Similarly, off periods are associated with distinct synaptic plasticity rules (González-Rueda et al., 2018), and so their prevalence and patterning is likely also related to whatever regulation of synaptic strength occurs during sleep (Tononi and Cirelli, 2014; Timofeev and Chauvette, 2017; Seibt and Frank, 2019).

According to this view of sleep homeostasis, in principle, regulation can occur entirely at the local level, within cortical networks or perhaps even within single neurons. Indeed, this is reflected in our results through the differences between channels both with respect to the accuracy of the model and in the values of optimised parameters. Why then is global sleep preferred over a hypothetical state including asynchronous local off periods, which presumably could be used to sustain longer periods of behavioural wakefulness? It has been argued previously that it is ecologically optimal to synchronise off periods and undergo dedicated periods of total behavioural shutdown, because local off periods during waking impair behaviour (Vyazovskiy et al., 2011; Rattenborg et al., 2012). Mechanistically, the occurrence of off periods might be obstructed by strong synaptic coupling and shared neuromodulatory tone. Indeed, it has been observed that the degree of coupling between an individual neuron’s firing and the population firing rate, is variable between cells but characteristic to an individual neuron and likely reflects total synaptic strength with its neighbours (Okun et al., 2015). Some neurons may therefore be less able to express asynchronous off periods than others.

The preference for global sleep, despite its fundamentally local mechanisms, may be evidence that sleep homeostasis ultimately does not serve a single specific local function. Instead, the recent history of neuronal activity levels may regulate the local tendency to generate a signal (reflected in Process S) which is integrated over larger neuronal populations through intrinsic network mechanisms in order to produce a global sleep propensity signal that estimates the total time spent awake with great accuracy. This mechanism would enable the brain to enforce a daily quota of sleep, which could have many benefits at the physiological and ecological level, rather than to initiate sleep in response to the homeostatic need of one specific regulated variable (Vyazovskiy, 2015). Importantly, in this mechanism, the variables which change as a function of time spent awake or asleep do not themselves need to be directly regulated by sleep. Conversely, variables which remain stable during continuous wake or sleep could still make essential contributions to the time-keeping mechanism. Note that this property is present in our model, since the levels of neural activity themselves are not directly proportional to the levels of Process S. The approximately equivalent overall accuracy of both firing-rate-based and vigilance-state-based models supports this possibility, as does the evidence that homeostatically regulated cellular variables can actually be stable during extended wakefulness, and that maintenance processes in sleep may be ultimately prophylactic (Vyazovskiy and Harris, 2013). In order to maximise the efficiency of global sleep, single neuronal activities may be modulated such that sleep pressure accumulates, on average, as uniformly as possible. This is consistent with recent reports that sleep regulates the population level firing rate distribution (Watson et al., 2016; Levenstein et al., 2017; Miyawaki et al., 2019) and could account for sleep’s link with synaptic and firing rate homeostasis, reconciling the local origins of sleep pressure accumulation with the global level of its dissipation.

Here, spike firing rate is used to represent the level of neural activity because it is convenient to record, locally variable, and directly linked to neuronal functionality. However, no strong claim is made that firing rates are necessarily causally responsible for the accumulation of sleep pressure. Indeed, it would likely be possible to obtain a reasonable quantitative account of sleep homeostasis using any physiological variable (or set of variables) that are consistently higher in either the wake or sleep state. For example, a model assuming that Process S increases in proportion to local cortical temperature, which, in laboratory rodents, drops by ~2°C when falling asleep (Franken et al., 1991b; Tobler et al., 1997) and which has been mechanistically implicated in sleep regulation (Hoekstra et al., 2019), might also provide a plausible description of Process S dynamics. However, our results demonstrate that firing rates are a useful measurable correlate of the processes that directly underpin Process S, and therefore firing rate variance resulting from differences in experience and behaviour may well account for the variance in Process S accumulation in normal individuals, between waking periods and between cortical regions.

It should be noted that the firing-rate-based models used here typically slightly under-perform, relative to the classic model, in terms of minimising the error between simulation and empirical data. A limit on the accuracy of the model might be expected, related to technical restrictions rather than conceptual ones. In this approach, neuronal activity was considered at the multi-unit level, which comprises only a few randomly sampled nearby neurons. Certainly, individual neurons vary in their firing properties (Watson et al., 2016) and grouping these as a single measure of local network activity is somewhat arbitrary. However, since it remains possible that firing rate homeostasis acts on the population level (Levenstein et al., 2017) and also that there exist substantial local spatial correlations in cortical spiking (Okun et al., 2015), this choice is justifiable. Therefore, it may be significant that local Process S reflects global sleep-wake history more closely than the history of local activity, supporting the idea of a spatial integration of Process S. It remains possible that these models might be applicable to firing derived from well isolated single units, although a reduction in fit quality would be predicted, owing to the proposed intrinsic spatial integration of Process S.

A direct experimental test of the conclusion that global Process S derives from a spatial and temporal integration of local neuronal activities would require neuronal activity to be recorded and manipulated simultaneously across many brain areas, including cortical and subcortical regions. A recent study, which locally activated a specific cortical area during sleep with optogenetics, did not find an increase in SWA or off periods in subsequent NREM sleep (Rodriguez et al., 2016). The failure of such a localised manipulation of activity to alter measurable Process S is consistent with our conclusion of spatial integration, which implies that local deviations would indeed tend to be averaged away. More global manipulations of neuronal activity, using a pharmacological approach, unfortunately come with other interpretative difficulties. For example, another study found that systemic atropine administered during behavioural wakefulness produces slow wave activity and reduces neuronal activity, yet increases the duration of subsequent NREM sleep (Qiu et al., 2015). While this was interpreted as evidence that spiking activity is not related to sleep pressure accumulation, crucially, the direct effects of atropine on Process S and the functional significance of the resulting induced slow oscillation are unclear. Manipulations at an intermediate scale, for example which consistently perturbed a whole cortical region, might be effective and could, in theory, exert effects beyond simple local compensatory responses, impacting also global sleep-wake states. This might not be easy to achieve methodologically, as existing approaches are often not sufficiently selective and may lead to unspecific compensatory effects.

It is reasonable to assume that sleep homeostasis unfolds over multiple time scales and that Process S as defined by these models describes a relatively fast one, approaching its upper asymptote after continuous wakefulness on a time scale of hours. The inclusion of processes acting over longer or shorter time scales might explain discrepancies in all these models, however, the challenge remains to identify what these could be. Brain processes which unfold over multiple scales have previously been explained using the theory of criticality, which implies self-similarity across many spatial and temporal scales. It is possible that the magnitude of the deviation of cortical activity from the critical point might reflect the levels of Process S, as has been previously suggested (Pearlmutter and Houghton, 2009; Meisel et al., 2013). Alternatively, since waking dynamics may be slightly subcritical, and NREM sleep slightly supercritical (Priesemann et al., 2013), the deviation from criticality could, in principle, contribute to the dynamics of Process S. Crucially, it remains unclear to what extent criticality and sleep homeostasis are linked and by what mechanisms, although firing rate homeostasis is likely also relevant (Hsu and Beggs, 2006). Addressing this question would require datasets comprising a large number of widely spatially distributed recording sites, and well isolated individual neurons.

The role of REM sleep in Process S dynamics has not been explicitly addressed or considered in the construction of these models. Depending on the model variant, REM sleep is associated either with a small increase or small decrease in Process S, because firing rates are low in REM sleep (closer to NREM sleep than waking) and yet off period occupancy is also low (closer to wake levels than NREM sleep). It is possible that REM sleep might represent a homeostatically neutral state, in which the level of Process S changes minimally, or not at all (Vyazovskiy and Delogu, 2014). Another important factor which remains unaddressed is the influence of the circadian rhythm on Process S and also on neuronal activity. It may be possible, for example, that the time-keeping mechanism derives information also from circadian signals, possibly running faster or slower at different phases of the light-dark cycle or that the neuronal firing rate set-point experiences circadian regulation.

In summary, our data suggest that Process S is reflected at multiple scales in the brain, from EEG and LFP slow wave activity to the occurrence of local multi-unit off periods. Its dynamics across all scales can be described quantitatively using information derived only from local neuronal activity with comparable accuracy to the classical model of Process S which depends on global sleep-wake history. We postulate that, despite its local origins, Process S may be integrated across networks for the purpose of tracking time spent awake and used to enforce a daily quota of global sleep, rather than serving the regulation of specific physiological variables.

Data analysed during this study are included in the manuscript and supporting files. Source data files have been provided for Figures 1, 3, 4 and 5.

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Author details

1. Christopher W Thomas

   Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, United Kingdom

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5371-7257

2. Mathilde CC Guillaumin

   Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, United Kingdom

   Contribution

   Conceptualization, Formal analysis, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-8611-3852

3. Laura E McKillop

   Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, United Kingdom

   Contribution

   Resources, Investigation, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3085-1175

4. Peter Achermann

   1. Institute of Pharmacology and Toxicology, University of Zurich, Zurich, Switzerland
   2. The KEY Institute for Brain-Mind Research, Department of Psychiatry, Psychotherapy and Psychosomatics, University Hospital of Psychiatry, Zurich, Switzerland

   Contribution

   Conceptualization, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0208-3511

5. Vladyslav V Vyazovskiy

   Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, United Kingdom

   Contribution

   Conceptualization, Supervision, Funding acquisition, Methodology, Project administration, Writing - review and editing

   For correspondence

   vladyslav.vyazovskiy@dpag.ox.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-4336-6681

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