A three states sleep–waking model

A three states sleep–waking model

Chaos, Solitons and Fractals 29 (2006) 808–815 www.elsevier.com/locate/chaos A three states sleep–waking model J.C. Comte b a,* , M. Schatzman b, P...

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Chaos, Solitons and Fractals 29 (2006) 808–815 www.elsevier.com/locate/chaos

A three states sleep–waking model J.C. Comte b


, M. Schatzman b, P. Ravassard a, P.H. Luppi a, P.A. Salin


a Laboratoire de Physiopathologie des Re´seaux Neuronaux du Cycle Veille-Sommeil, UMR 5167 (CNRS/Universite´ Claude Bernard Lyon1), Faculte´ de Me´decine RTH Laennec 7, Rue Guillaume Paradin 69372 Lyon Cedex 08, France MAPLY (Laboratoire de Mathe´matiques applique´es de Lyon), UMR5585 (CNRS/Universite´ Claude Bernard Lyon1) 21, Avenue Claude Bernard, 69622 Villeurbanne Cedex, France

Accepted 30 March 2005

Abstract The mechanisms underlying the sleep-states periodicity in animals are a mystery of biology. Recent studies identified a new neuronal population activated during the slow wave sleep (SWS) in the ventral lateral preoptic area of the hypothalamus. Interactions between this neuronal population and the others populations implicated in the vigilance states (paradoxical sleep (PS) and wake (W)) dynamics are not determined. Thus, we propose here a sleep–waking theoretical model that depicts the potential interactions between the neuronal populations responsible for the three vigilance states. First, we pooled data from previous papers regarding the neuronal populations firing rate time course and characterized statistically the experimental hypnograms. Then, we constructed a nonlinear differential equations system describing the neuronal populations activity time course. A simple rule playing the firing threshold role applied to the model allows to construct a theoretical hypnogram. A random modulation of the neuronal activity, shows that theoretical hypnograms present a dynamics close to the experimental observations. Furthermore, we show that the wake promoting neurons activity can predict the next SWS episode duration. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction The slow waves sleep and paradoxical sleep (or rapid eyes movement sleep) alternation is a basic feature of sleep in almost all mammals. The relationship between two sleep states remains ones of the major mystery of sleep. It is well admitted that sleep–waking dynamics results from interactions between several neuronal populations [1–5]. Actors of this dynamics located in the brainstem are, switch on during the paradoxical sleep (PS) such as pontine reticular neurons (PS-on neurons) or switch off during PS such as locus coeruleus and dorsal raphe neurons (PS-off neurons). Classical electrophysiological and pharmacological studies have demonstrated that these brainstem populations are implicated in promoting and in maintaining PS or wake (W) and are in a mutual interaction. Recently, it has also been shown that several specific regions in the hypothalamus are activated during W, PS and slow wave sleep (SWS) reviewed


Corresponding author. E-mail address: [email protected] (J.C. Comte).

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.054

J.C. Comte et al. / Chaos, Solitons and Fractals 29 (2006) 808–815


by [6]). Notably, the ventrolateral preoptic anterior hypothalamic (VLPO) neurons, that are active during SWS [7], have been proposed to play a key role in sleep induction [8,9]. Furthermore, sleep-promoting VLPO neurons may interact with wake active neurons in the brainstem in mutual inhibition [1]. Thus, the aim of the present paper is to propose a theoretical model able to describe the interactions between these neuronal populations of brainstem and hypothalamus and to reproduce the vigilance states dynamics during the diurnal period (the period of sleep for the rat). The PS sleep oscillator model developed by McCarley and Hobson (MH model) implying the two interactive brainstem neuronal populations (PS-on and PS-off) cited above [3,10] presents the activity of the two neuronal populations as a prey–predator interaction described by a Lotka–Volterra (LV) system. However, the MH model that describes the dynamics of two coupled neuronal populations does not allow to model the vigilance state dynamics. Indeed, this one does not take in account the SWS promoting neurons, while in order to represent the alternation of W, SWS, and PS, at least three neuronal populations are required. Furthermore, the neuronal populations underlying the vigilance states dynamics do not display the same pattern of activity. As underlined by Saper et al. [1], the VLPO neurons present a large temporal spread of activity which is fuzzier than those of putative PSon neurons [5,11,12] and Won neurons of the brainstem and the posterior hypothalamus [2,13]. Thus, in order to model the three vigilance states alternation without destroy the MH hypotheses, we introduce the sleep-promoting VLPO neurons activity with a third differential equation. The nonlinear differential equations system proposed here to describe the neuronal populations dynamics allows to produce a cyclic behavior as observed experimentally in average. Our model leans on experimental results such as the neuronal populations firing rate dynamics to determine, on the one hand, the interactions during the transition states and on the other hand, their function in induction of vigilance states. The results of the present paper indicate that our nonlinear differential equations systems based on three first-order ordinary equations models accurately the neuronal population dynamics. A simple rule: ‘‘the winner takes all’’ (WTA) applied at each time step allows us to construct the theoretical hypnogram resulting from the neuronal populations activities and reproduces the experimental alternation of the vigilance states. Futhermore, our results reveal that the wake promoting neurons firing rate can predict the following SWS episode duration. Finally, one suggests that the sleep architecture is strongly linked to the firing rate stochasticity of each population involved in vigilance states alternation. The paper is organized as follow. First we present Section 2 the data an their statistical analysis, then we introduce Section 3 our theoretical model, while Section 4 is devoted to a discussion and finally Section 5 is relative to some concluding remarks.

2. Experimental data analysis overview In order to construct a realistic model of the vigilance states alternation, we analyzed hypnograms during the same diurnal period for six rats. The statistical analysis that consists to quantify the episode duration and the inter-episode intervals has been extracted from polygraphic recordings analyzed at a 10 s time scale. Unlike it is currently thought, the vigilance states durations do not follow a Gaussian distribution. Indeed, as shown in Fig. 1 the statistical distribution of the vigilance states duration follow power laws as observed in critical system such as avalanches or seismological [14]. One observes Fig. 1 that both distributions of W and PS have a minimal bouts duration of 15 s and a width to half height of 10 s, while the third distribution regarding the SWS presents a minimal bouts duration equal to 15 s and a spread width to half height of 100 s. Using a Kolmogorov–Smirnov test, we verified that each distribution obtained from the different animals can be pooled. The inter-episodes intervals distributions for W and PS (Fig. 1) present a large spread in comparison with the SWS. The inter-episodes intervals half height width is equal to 50 s for the SWS, and to 115 s and 320 s for W and PS, respectively. One note the strong reciprocal relation between the two representation spaces. Indeed, narrow time duration distributions lead to wide inter-episode interval distributions and reciprocally, wide time duration distributions lead to narrow inter-episode interval distributions. Also, the sharp SWS inter-episode interval distribution indicates its strong periodicity, while the wide PS inter-episode interval reveals a strongly variable apparition frequency of this state. The W inter-episode interval following a bell shape right side indicates a slight modulation of the emergence frequency. These remarks regarding the time duration and inter-episode interval are confirmed by the transitions states statistics represented Fig. 2(a). Those ones reveal a high transition probability between W and SWS and reciprocally. Transitions from SWS to PS, and from PS to W, corresponding to mode A, and represented with a single arrow Fig. 2(b) are less frequent than the transitions corresponding to mode B represented by a double arrows in Fig. 2(b). Moreover, the results show that the transition probability from PS to SWS is very low in comparison with the transition probability from PS to W, and express a cyclic behavior following the mode A trajectory. Furthermore, this remark is enforced by the transitions probabilities from SWS to PS, and from PS to W that are almost equal.


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Fig. 1. Duration epsidoes and inter-episodes intervals distributions during the diurnal period. (a) Episode duration probability density for the three vigilance states (W: Wake, PS: paradoxical sleep, and SWS: slow wave sleep). Data pooled from six adult tracks recorded during 8 h (same diurnal period). The probability densities sum is equal to one for each vigilance. Note that, the white spread SWS distribution indicates a strong episodes duration variability. (b) Interval inter-episodes probability density for the three vigilance states (same data as in (a)) displaying a narrower shape for SWS in comparison with PS and W.

3. Theoretical model In order to integrate in our model the principal characteristics described in the previous section and extracted from the literature, we constructed the following nonlinear differential equations system: 8 dw > > ¼ a0 w þ b0 wp > > > dt > < dp ð1Þ ¼ a1 p  b1 wp > dt > > > > > : dsw ¼ a2 ðsw  sw0 Þ3  b2 wp dt where, w, p, and sw are the neuronal populations activities (firing rate) corresponding to the wake, paradoxical sleep 1 1 and slow waves sleep states, respectively. Time constants a1 0 ; a1 , and a2 define the own dynamics of each neuronal population. The wp product, represents the interaction between the PS and W neuronal populations as it has been proposed by McCarley and Hobson [3,10]. Although the MH model allows to reproduce properly the W and PS neuronal populations activity alternation, this model does not allow to describe the sleep dynamics. Thus, in order to fill this hiatus, we propose to add to the MH model a third equation relative to the SWS promoting neurons. As we said above, this population presents a certain fuzziness (no specificity) regarding its activity level which presents a no null minimum value during the three vigilance states. It is the reason why we introduced the coefficient sw0, which defined

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Fig. 2. Vigilance states transitions probabilities. (a) Transitions probabilities map of the vigilance states quantified for six rats (same data as in Fig. 1). The sum of the transitions states probabilities is equal to 100%. (b) Transitions trajectories diagram extracted from (a). Transitions between vigilance states can be depicted in two different modes. Mode A (single arrow), the trajectory follows SWS, PS, W and SWS. Mode B (double arrows), the trajectory is restricted to SWS and W alternation. (c) Hypnogram extracted (beginning at 7H00 a.m.) from one rat during the diurnal period (sleep period for the rat) showing the wide SWS duration variability, and both mode A and B described in (a) and (b).

the minimum activity level of this population. In order to maintain the sw population activity during a bigger time period than for the two others, we introduce the cubic term (sw  sw0)3. Indeed, this ones allows to converge quickly to a plateau reflecting the neuronal population characteristics observed experimentally [7]. An exponent 2 does not model a plateau while values superior to 3 do not lead to a best fitting. Then, in order to reproduce the populations activity time course, we have performed numerical simulations of the nonlinear differential system [1] with a C++ Runge–Kutta 45 algorithm and an integration time step equal to dt = 102 warranting the solution stability (see Fig. 3). Parameters a0 = 4, a1 = 2, a2 = 1, b0 = 1, b1 = 4, b2 = 1, and SW0 = 5.5, have been chosen to match (after a time scale change) with the neuronal activities (Fig. 3b) of each neuronal populations obtained experimentally (not represented in a sake clarity) by [5,11,15] for PSon neurons, by [7] for the SWS-promoting neurons, and by [13,16,17] for the Won neurons. The theoretical hypnogram (Fig. 3a) reproducing the vigilance states alternation is obtained by applying at each time step the WTA rule, that is in a first approximation, the most active neuronal population is responsible for the current vigilance state activation. Thus, the vigilance state output variable H writes formally as follow: If ðw > swÞ&ðw > psÞ ) H ¼ W ; If ðsw > wÞ&ðsw > psÞ ) H ¼ SWS; If ðps > swÞ&ðps > wÞ ) H ¼ PS;


where upper-case variables represent a given vigilance state, while the lowercase variables are relative to the instantaneous firing rate of the neuronal populations. Note that, the special cases corresponding to equalities between neuronal activities are not considered here. As seen Fig. 3(a), hypnogram presents a large SWS duration and a PS duration bigger than W. However, the statistics of Fig. 1(a) indicate a slightly superior W duration in comparison with PS, and suggest that the raw activities comparison is not suitable. Indeed, the direct neuronal activities comparison does not take into account the unknown efficiency of this ones to flip in one vigilance stage or another one. Thus, in order to compare the neuronal activities without a priori we normalized the neuronal activities before comparison as represented Fig. 3(d). This process allows to construct the hypnogram of Fig 3(c) from the WTA rule and satisfies in average the statistics of experimental hypnograms. However, although our model allows to produce the mean cycle of vigilance stages alternation, nevertheless experimental hypnograms are characterized by a large degree of variability (see Fig. 1). In order to


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Fig. 3. Theoretical hypnograms and neuronal populations dynamics. Hypnograms (a) and (c) are obtained from the rule ‘‘The winner takes all’’ applied to the neuronal populations dynamics (b) and (d), respectively. (b) Neuronal populations (w: blue, sw:green, and p: red) dynamics obtained from numerical simulations with parameters a0 = 4, a1 = 2, a2 = 1, b0 = 1, b1 = 4, b2 = 1, and SW0 = 5.5 allowing to reproduce accurately the neuronal population time course pulled from literature. (d) Neuronal populations dynamics represented in normalized amplitude. (For the interpretation of colors in the figure reader is referred to the web version of this article.)

reproduce qualitatively the same behavior as observed Fig. 4(e), we chose the same parameters value as above and we modulated randomly the wake activity (w) and the slow wave sleep (sw) activity with a multiplicative white Gaussian ‘‘noise’’ of null mean and standard deviation equal to rw = 0.2, and rsw = 0.05 for w and sw, respectively, since those ones present the most duration variability. The p population activity has not been perturbed because this one presents statistically a small duration variability in comparison with w and sw populations as observed experimentally [19,7]. Furthermore, the p population is slightly perturbed by induction with the pw product (equation two of system [1]). As observed, Fig. 4(a), we recover the wide episodes duration variability for the three vigilance stages like in Fig. 2(c). The statistical analysis (not represented here) of the numerical results (hypnograms) for a sleep period of eight hours follow the same power law described in Section 2 for the three vigilance states. The phase plan Fig. 4(b) displays the cyclic activity of the three neuronal populations and shows the cycle period modulation by the neuronal activities amplitude. This property common to the nonlinear oscillators allows to demonstrate that the next stage duration is strongly correlated to the neuronal activity of the population underlying the current stage. In the present case, the relation between the SWS stage duration versus the normalized firing rate of the population underlying the W stage is linear and presents a correlation coefficient equal to q = 0.99. Consequently our model based on three coupled differential equations and including only, one interaction between the w and p populations and a small perturbation of the w and sw populations, allows to reproduce accurately the sleep–wake dynamics, and suggests that the vigilance states episode duration is intimately linked to the neuronal populations activity variability.

4. Discussion Recent works [18] has suggested that classical LV systems such as proposed by MH are not able to present the sleep– wake cycle experimental stochastic properties. Indeed, first, the sleep–wake dynamics involves three vigilance states,

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Fig. 4. Vigilance states alternation simulation emerging from noisy neuronal populations. (a) Theoretical hypnogram obtained from the WTA rule applied to the noisy normalized neuronal populations. (b) Simulation parameters are the same than Fig. 3, while the multiplicative Gaussian white noise with null mean value applied to the w and sw neuronal populations present a standard deviation equal 0.2 and 0.05, respectively. (c) Neuronal populations phase plan displaying an amplitude and period modulated cycle owing to the noise applied to the sw and w populations. (d) Relation between the w firing rate amplitude and the next SWS duration presenting a linear correlation equal to q = 0.99.

that is at least three underlying neuronal populations. Second, the neuronal populations activity presents a certain degree of variability that is a stochastic component. In order to include these characteristics in our model, we added to the MH model a third equation relative to the slow-waves sleep neuronal population. Then, we perturbed the slow-waves and wake neuronal populations firing rate pattern from a Gaussian white ‘‘noise’’ of null mean and standard deviation equal to rw = 0.2, rsw = 0.05, respectively. This process allows us to produce neuronal population activ-


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ities close to the experiment. Also, others studies [2,5] suggested that neuronal populations (PS_on and W_on) responsible for the vigilance stage are similar to flip–flop alternating devices. However, this hypothesis based on a bistable behavior of the W and PS promoting neurons needs to be completed by the SWS promoting neurons since this one inhibits strongly the Won neurons and reciprocally [1]. Thus, this last observation raises the question of the inhibitory or excitatory neuronal populations efficiency, since these ones do not present a same firing rate amplitude. Also, in order to integrate the preceding remarks, we decided to compare the three normalized neuronal populations firing rate through the rule ‘‘the winner takes all’’. In others words we consider in first approximation that the efficiency balances the firing rate amplitude. Like this, we introduce the flip–flop concept between the three neuronal populations. One note that a neuroelectronic flip–flop ‘‘timer’’ system imply analog (not binary or digital) time course and threshold, like met in electronic (monostable, bistable, astable), where the analog part is an integration or accumulation circuit while the second one is a threshold comparator. Consequently, the model proposed here allows to merge the neuronal populations dynamics with the vigilance states alternation and confirms that the firing rate time course is important to describe the hypnogram patterns richness, since the stochastic part come from the neuronal populations noise. Narcolepsy is a pathology characterized by a strongly distributed wake–sleep alternation. Normally, in humans SWS always precedes a PS episode. However, in the narcolepsy pathological case the PS state emerge spontaneously into wakefulness. Our model avoid for the chosen parameters the W to PS transition corresponding to the narcolepsy pathological case and confirms theoretically recent findings [19] regarding the relation between the current state duration and the firing rate of the neuronal population associated to the preceding vigilance stage.

5. Conclusion In summary, we have first statistically characterized the vigilance states episodes duration and their periodicity. It emerges from this analysis that the sleep–waking dynamics presents a strong cyclic component following in first approximation the trajectory W, SWS, PS, W, . . . Then, in order to reproduce accurately the neuronal populations activity underlying the vigilance states alternation obtained experimentally, we have constructed a nonlinear differential equations system. The WTA rule applied to the normalized numerical solutions of our model provides the searched cyclic alternation. Actual hypnograms presenting a certain stochasticity degree, we have also superimposed a multiplicative white Gaussian ‘‘noise’’ to the w and sw populations activity, since those ones present statistically the most duration variability. The previously cited rule, applied to the normalized neuronal populations activity in the noisy conditions allows to produce hypnograms close to actual ones and suggests that the that the sleep–waking pattern richness is due to the stochastic component of the w and sw neuronal populations activity. The three neuronal populations activity phase plan analysis reveals as expected an amplitude dependent period of the three cyclic neuronal populations and indicates that the neuronal population activity amplitude underlying the current vigilance stage predicts the duration of the next one. A linear and strongly correlated relation between the SWS stage duration and the firing rate amplitude of the neuronal population underlying the W stage confirms this property. Finally, one note that the W to PS transition relative to the narcolepsy case does not appear with the proposed parameters corresponding to the physiological case. Thus, without being to speculative one says that our model may help to characterize further sleep changes mechanisms during and after learning as strongly studied recently [20,21].

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