A sleep model and an observer using the Lotka–Volsterra equation for real-time estimation of sleep

Abstract

This paper describes a novel method that estimates sleep behaviors in a real-time manner. A Luenberger-type observer was employed, which requires a dynamic model of the sleep as a priori information about the sleep stages. A reasonable sleep model is essential in this approach. The classic Lotka–Voltera equation with interpretations associated with the sleep was built and an observer was developed. The observer using heartbeat rhythm and body movement as input signals estimated and compensated the sleep depths, which shows better results than the original inputs in the standpoint of the percentage of δ-waves in total brainwaves, used as a reference. The observer compensated for phase-shift errors and non-cyclic errors of the sleep cycle. The correlation between the reference and the compensated sleep cycle behavior was 0.79, whereas that between the reference and the measurement itself was 0.65. The application of the observer improved the accuracy of the sleep cycle measurement.

Appendix 1: Index characterizing wakefulness and non-REM sleep

With regard to SDI(k), the sleep depth index is defined using the heartbeat and body movement signals. Wakefulness and non-REM sleep have the characteristics as shown in Table 2. The sleep depth index is defined based on No. 3 “When sleep deepens, body movement becomes smaller and less frequent.” The discrete time for every 1 min is defined as k, and the value changes from k = 1, 2, …, to TIB (total measurement time (min) with the subject lying on a bed). Let $A_k^{\text{body}}$ and $A_k^{\text{heart}}$ be the mean amplitude of the body movement and heartbeat signal at the discrete time k, respectively. The value of $A_k^{\text{body}}$ becomes large when the sleep is shallow, as the body movement is frequent. On the other hand, when the sleep is deep, the value of $A_k^{\text{body}}$ becomes rather small. The amplitude of the body movement and heartbeat signal vary with the differences among subjects. To standardize these deviations, $A_k^{\text{body}}$ is divided by $A_k^{\text{heart}} + A_k^{\text{body}}$. $A_k^{\text{body}}$ shows large fluctuation when body movement is detected and when not detected at all. $A_k^{\text{body}}$ shows subtle fluctuation when a few body movements are detected. To reduce the large fluctuations and to magnify the subtle ones, the logarithm of $A_k^{\text{body}} /(A_k^{\text{heart}} + A_k^{\text{body}} )$ is calculated, and the sleep depth index SDI(k) becomes:

$$ {\text{SDI}}(k) = \log_2\left ({\frac{{A_k^{\text{body}} }}{{A_k^{\text{heart}} + A_k^{\text{body}} }}}\right) $$

(3)

Table 2 Characteristics of non-REM sleep

The value of the index SDI(k) becomes small when the sleep is deep; and the shallower the sleep, the larger the value.

1.1 Appendix 2: Index characterizing REM sleep

With regard to RSI(k), the REM sleep index is defined using the heartbeat signal. REM sleep has the characteristics shown in Table 3.

Table 3 Characteristics of REM sleep

The R-K Method, in determining the REM sleep stage, focuses on brainwaves, eye movement, and the EMG of jaw muscle, as shown in characteristics No. 1 to No. 4 in Table 3. We focus on the fact that the heartbeat becomes less rhythmical during REM sleep (Characteristic No. 5 in Table 3). Let $H_k^{\text{former}}$ be the heart rate of the former 30 s of the 1 min of discrete time k. Similarly, let $H_k^{\text{latter}}$ be the heart rate of the latter 30 s of discrete time k. The change in heart rate between the former and latter halves of the 1-min discrete time k is thus obtained from $| {H_k^{\text{former}} - H_k^{\text{latter}} } |$. The same procedures are performed for each of k = 1,2, … TIB, and index RSI(k) to show the REM sleep stages as in Eq. (4) is obtained through the moving average of the data amount of q, which is equally dispersed before and after k.

$$ {\text{RSI}}(k) = \frac{1}{2q + 1}\sum_{i = - 10}^{10} {| {H_{k + i}^{\text{former}} - H_{k + i}^{\text{latter}} } |} $$

(4)

The order of moving average of RSI(k) in Eq. (4) was set to q = 10. The values of index RSI(k) become large during REM sleep due to the aforementioned characteristic No. 5 of REM sleep in Table 3.