Regularity of Cardiac Rhythm as a Marker of Sleepiness in Sleep Disordered Breathing

Aim The present study aimed to analyse the autonomic nervous system activity using heart rate variability (HRV) to detect sleep disordered breathing (SDB) patients with and without excessive daytime sleepiness (EDS) before sleep onset. Methods Two groups of 20 patients with different levels of daytime sleepiness -sleepy group, SG; alert group, AG- were selected consecutively from a Maintenance of Wakefulness Test (MWT) and Multiple Sleep Latency Test (MSLT) research protocol. The first waking 3-min window of RR signal at the beginning of each nap test was considered for the analysis. HRV was measured with traditional linear measures and with time-frequency representations. Non-linear measures -correntropy, CORR; auto-mutual-information function, AMIF- were used to describe the regularity of the RR rhythm. Statistical analysis was performed with non-parametric tests. Results Non-linear dynamic of the RR rhythm was more regular in the SG than in the AG during the first wakefulness period of MSLT, but not during MWT. AMIF (in high-frequency and in Total band) and CORR (in Total band) yielded sensitivity > 70%, specificity >75% and an area under ROC curve > 0.80 in classifying SG and AG patients. Conclusion The regularity of the RR rhythm measured at the beginning of the MSLT could be used to detect SDB patients with and without EDS before the appearance of sleep onset.


Time Frequency Representation
Time Frequency Representation based on Choi-Williams distribution (CWD) (1) is calculated by convoluting the Wigner distribution (WD) (2) and the Choi-Williams (CW) exponential (3) [1]. CWD removes the interference terms present in the Wigner distribution, and is thus used here to clearly characterize the dominant frequency components. Moreover, it has a very good resolution in both the time and frequency domains. (1) where  c was set to 0.005 using the estimation criterion proposed in a previous study [2]. Using this value of  c the interferences produced in the time-frequency plane by using sinusoidal functions, located in the RR frequency bands of LF and HF are eliminated. In order to minimize the boundary effect, each segment was multiplied by a Hanning window with length 256 samples.

Auto-mutual information function
AMIF is a metric to estimate both linear and nonlinear dependences between two time series [3,4], x t and x t+τ . It can be regarded as a nonlinear equivalent of the correlation function. AMIF is calculated by the distribution of the probability amplitudes of x t =RR(t) and x t+τ =RR(t+τ), and the joint probability of these time series, based on Shannon entropy.
Probabilities and joint probabilities were computed on the basis of a quantization in 5 bits. This function describes how the information of a signal (AMIF value at τ=0) decreases over a prediction time intervals (AMIF values at τ>0). Increasing information loss is related to decreasing predictability, and increasing complexity of the signal [5].
AMIF was also defined from Rényi entropy as in (5).
In equation 5, the largest probabilities most influence the AMIF_Re q when q>1 and the smallest probabilities most influence the values of AMIF_Re q when 0<q<1. The AMIF_Re q converge to the Shannon AMIF when q 1. In this work, AMIF was calculated using a discrete time delay 0≤τ≤100 samples for different values of the control parameter of Rényi entropy: q = {0.1, 0.2, 0.5, 2, 3, 5, 10, 30, 50, 100} and q→1 using (4). AMIF was normalized by its maximum value that corresponds to τ=0.

Correntropy function
CORR is a similarity measure of signals that generalizes the autocorrelation function to nonlinear spaces.
is a symmetric positive kernel function, defined by the Gaussian kernel: where σ is the size of the kernel determined in this study by Silverman's rule [6] of density estimation: In this rule, A is the smaller value between the standard deviation of the data samples and data interquartile range scaled by 1.34 and N is the number of samples. A powerful advantage of the CORR function is its robustness against impulsive noise. This advantage is due to the fact that when an outlier is present, the inner product in the feature space computed via the Gaussian kernel tends to be zero.
In this work, CORR was calculated using a discrete time delay 0≤τ≤100 samples.

Defined measures from AMIF and CORR
AMIF and CORR were applied to RR(t) time series (TB) and to RR ( In the following table, the notation used in the main text is associated with the calculated measures of AMIF and CORR: