The neuronal transition probability (NTP) model for the dynamic progression of non-REM sleep EEG: the role of the suprachiasmatic nucleus

Abstract

Little attention has gone into linking to its neuronal substrates the dynamic structure of non-rapid-eye-movement (NREM) sleep, defined as the pattern of time-course power in all frequency bands across an entire episode. Using the spectral power time-courses in the sleep electroencephalogram (EEG), we showed in the typical first episode, several moves towards-and-away from deep sleep, each having an identical pattern linking the major frequency bands beta, sigma and delta. The neuronal transition probability model (NTP)--in fitting the data well--successfully explained the pattern as resulting from stochastic transitions of the firing-rates of the thalamically-projecting brainstem-activating neurons, alternating between two steady dynamic-states (towards-and-away from deep sleep) each initiated by a so-far unidentified flip-flop. The aims here are to identify this flip-flop and to demonstrate that the model fits well all NREM episodes, not just the first. Using published data on suprachiasmatic nucleus (SCN) activity we show that the SCN has the information required to provide a threshold-triggered flip-flop for TIMING the towards-and-away alternations, information provided by sleep-relevant feedback to the SCN. NTP then determines the PATTERN of spectral power within each dynamic-state. NTP was fitted to individual NREM episodes 1-4, using data from 30 healthy subjects aged 20-30 years, and the quality of fit for each NREM measured. We show that the model fits well all NREM episodes and the best-fit probability-set is found to be effectively the same in fitting all subject data. The significant model-data agreement, the constant probability parameter and the proposed role of the SCN add considerable strength to the model. With it we link for the first time findings at cellular level and detailed time-course data at EEG level, to give a coherent picture of NREM dynamics over the entire night and over hierarchic brain levels all the way from the SCN to the EEG.

Figure 1. Sleep structure across NREM-REM sleep cycles 1 to 4 for a typical subject.

Spectral power time-course data in the major frequency bands: beta (β) 18–25 Hz; sigma (σ) 12–15 Hz and delta (δ) 1–4 Hz, together with the corresponding hypnogram in the uppermost panel. W = wake; REM = REM sleep (shaded blue); 1, 2, 3, 4 = NREM sleep stages. The total duration of the time-courses is 6 h 50 min with the hours indicated by vertical dashed lines. Note the repeated alternations between going towards deep sleep (delta rising) and going away from deep sleep (delta falling) in each NREM episode. This is best visualised on the delta panel. Towards-and-away cycles (indicated by a red bracket) constitute the basic building blocks of the NREM episode.

Figure 2. Power time-courses for beta, sigma and delta in NREM episode 1.

Data are averaged over six healthy subjects having a single delta peak in NREM 1; error bars represent the standard error on the mean (adapted from figures 3a and 3b in Merica and Fortune [2]). The relation between the shapes of the time-course curves in the move towards deep sleep displays the distinct pattern on which the NTP model is based: beta power drops exponentially, delta power rises in an S-curve and sigma power reaches its maximum while delta is still rising. The number of neurons (N) in each mode beta, sigma and delta, is expressed as a percentage of the number of neurons in beta mode (N₀) at the start of the NREM episode. N₀ is the fixed-size of the generating population of the NTP model. The model fits the data well: overall goodness of fit as measured by the coefficient of determination R² (%) = 92.6 (R² delta = 97.8, R² sigma = 82.0, R² beta = 90.3).

Figure 3. Three-stage cascade processes involving stochastic transitions in the hard sciences.

(a) Physics: a 3-element cascade radioactive decay (adapted from Fermi Lectures: Chicago University press; 1950). This process starts at time = 0 with a fixed size population of radioactive atoms all in an identical initial state, where each atom has the same probability of transitioning to an intermediate radioactive state. Atoms arriving in that intermediate state are immediately subject to a probability of transitioning to the final stable state. The intermediate state reaches a maximum, when the number of atoms/unit time entering it, is the same as the number leaving it. Using the 2 probability parameters one can calculate the time courses of the relative number of atoms in each state. An important factor which simplifies the mathematics is that individual transitions are of very short duration relative to the duration of the population transition. (b) Chemistry: a 3-stage consecutive reaction of solutes within a solvent (adapted from Chemical Engineers Handbook (1973) with kind permission of the publisher). Gives the concentration-time profile for consecutive reaction A→B→D.

Figure 4. Vigilance states, slow-wave activity and SCN neuronal activity: 24 hour recording.

SWA (EEG power density 1–4 Hz) and SCN neuronal activity are plotted as a percentage of the mean activity during NREM sleep over 24 h (each data point is the mean of 6 10-s epochs). The grey background indicates the subjective night (dark period) characterised by a lower level of SCN activity where the nocturnal rat is mostly active but exhibits short bouts of sleep. SWA shows high values during NREM sleep (N) and low values during REM sleep (R) and waking (W), adapted from Deboer et al. with the kind permission of the publisher. The two green curves we added on the SCN activity figure enclose most of the data points. We also added about half way between these green lines a blue threshold line which separates SCN activity into two zones: one corresponding to a WAKE or REM level where SWA is about zero and the other to a NREM level where SWA is high. These zones correspond to the ‘Away’ and ‘Towards’ phases of the NTP model. The blue colour-coded bar above the SWA figure show the instants of transition of the SCN data across the threshold line, in either the upward or the downward direction. The inserted vertical green alignment lines facilitate the visualisation of simultaneity between transition events in the SCN activity and SWA.

Figure 5. Vigilance states, slow-wave activity and SCN neuronal activity.

The left hand panels show an expansion of the data in Figure 4 (labelled c), representing 2 hours around the circadian peak. Right hand panels are an expansion of a portion of the recordings in the left hand panels (labelled a). Vertical lines on the right hand panels indicate vigilance state transitions, adapted from Deboer et al. with the kind permission of the publisher. We have added a horizontal blue line on the SCN activity figure, above which SWA is falling rapidly to zero and below which SWA is rising slowly. These zones correspond to the ‘Away’ and ‘Towards’ phases of the NTP model. The blue colour-coded bar above the SWA figure shows the start and stop times of each phase. The orange rectangle on this bar indicates the position of an away phase of the SWA where the SCN data do not cross the threshold line as would be expected. The green colour-coded bar added above the SWA figure on the left hand panel, based on the green line threshold at about 160% on the SCN activity panel, show the SCN potential to control NREM-REM state transitions.

Figure 6. SCN to EEG: a graphic summary of the NTP model for NREM sleep structure.

Dual control by the SCN switch and the NTP model equations: the SCN, the acknowledged time-keeper of the brain, changes its level of activity under the influence of feedback from several sources in particular homeostatic drive, it then uses these changes to create a flip-flop (F) timing control that switches downstream action between the two dynamic states going towards and going away from deep sleep i.e. between P₁P₂ and P₃P₄. These instructions are conveyed to the sleep-promoting (S-P) and wake-promoting (W-P) population pair which by reciprocal interaction are locked in anti-phase, like the 2 ends of a see-saw. Here the NTP equations take over and determine, as generating population at the brainstem, the template pattern of firing-rate time-courses of the brainstem activating neurons. The template propagates downstream and modulates thalamic output. Thalamocortical networks then form the complex wave sequences observed on the EEG, while following the power time-course pattern dictated by the brainstem. For simplicity, only the first towards phase of the template pattern is shown in the inset boxes. The box for the S-P population is in grayscale to indicate that the vertical axis is different from that of the colored patterns following: it indicates the fraction of neurons in each equivalent firing-rate mode (β, σ and δ) corresponding to the cascade (relatively silent→moderate firing-rate→fast firing-rate) rather than the cascade (fast firing-rate→moderate firing-rate→relatively silent) in the colored patterns. Thus the mean firing-rate in the S-P population (grayscale) progressively increases with time, while in the W-P population (colour) it decreases.

Figure 7. The NTP model fit to the data of representative NREM episodes 1 to 4.

Data is represented by filled circles: beta (blue), sigma (green) and delta (red). The time scale is given as a percentage of the total duration of the given NREM episode. The number of neurons (N) in each mode beta, sigma and delta, is expressed as a percentage of the number of neurons in beta mode (N₀) at the start of the NREM episode. The model with the fixed probability set fits well all four NREM episodes: NREM 1 with 3 towards-and-away (TA) cycles: overall goodness of fit as measured by R² (%) = 81.2, (R² delta = 88.6, R² sigma = 63.6, R² beta = 84.7). NREM 2: 5 TA cycles; overall R² = 75.5, (R² delta = 89.2, R² sigma = 48.6, R² beta = 76.2). NREM 3: 5 TA cycles; overall R² = 75.7, (R² delta = 90.8, R² sigma = 52.6, R² beta = 70.3). NREM 4: 7 TA cycles; overall R² = 63.1, (R² delta = 86.2, R² sigma = 46.1, R² beta = 42.4).

Figure 8. Overview of SCN sleep timers.

As we see it, the SCN, influenced by feedback from several sources in particular homeostatic drives, has all the information necessary to provide sleep timing flip-flops not only for wake-sleep (W-S) and NREM-REM (NR-R) transitions but also for the towards (T) and away (A) switchovers within NREM. Thus there would appear to be a progressive penetration of SCN control into all layers of the complex behaviour that is sleep: in the outer layer a control of the circadian process (wake-sleep alternation), in the next layer a control of the ultradian process (NREM-REM alternation) and in the next deeper layer a control of the process responsible for NREM sleep structure (towards and away alternation within NREM).