A three states sleep–waking model
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], [2], [3], [4], [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 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.
Section snippets
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
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:where, w, p, and sw are the neuronal populations activities (firing rate) corresponding to the wake, paradoxical sleep and slow waves sleep states, respectively. Time constants , and define the own dynamics of each neuronal
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, 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
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
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