# Assessment of Cardiorespiratory Interactions during Apneic Events in Sleep via Fuzzy Kernel Measures of Information Dynamics

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Database

#### 2.2. Preprocessing and Extraction of Time Series

#### 2.3. Cardiorespiratory Information Dynamics

#### 2.4. Fuzzy Kernel Entropy Estimation

#### 2.5. Statistical Analysis

## 3. Results

## 4. Discussion

_{X->Y}quantifies the part of the information carried by the present of RR that can be predicted exclusively by the past of RESP, and this behavior is also expected given the presumed unidirectional nature of cardiorespiratory interactions [24,53]. The CE has been already demonstrated to represent a proper measure of the overall directed interactions from driver to target when their nature is unidirectional [24], and thus can be regarded as a simpler and more effective approach in this context, returning more significant values (compare Figure 6d vs. Figure 6b).

_{X->Y}) in any other respiratory sleep disorder. Such results may add clinical relevance to RERA events, which has been debated in the literature [58], since the effects of RERA episodes are surely not evident as OSA (being not associated to concomitant oxygen desaturation), but are longer-lasting in terms of variation of cardiorespiratory interactions. This suggests that RERA events may be indicative of the development of more serious apneic events in a later stage if unnoticed and inadequately addressed [29,30] and may reinforce the still not definitely confirmed findings which relate RERA events with the development of cardiovascular comorbidities, which have already been suggested in [59] reporting repetitive increases in blood pressure concomitant with RERA episodes during sleep.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Exemplary windows of the respiratory airflow signal recorded for different breathing effort events during sleep: (

**a**) normal respiration; (

**b**) RERA; (

**c**) hypopnea; (

**d**) obstructive sleep apnea; (

**e**) central apnea. The signal parts in blue correspond to the labelled events.

**Figure 2.**Schematic representation of the methodological processing steps used for forming observation matrices: (

**a**) analyzed segments, which included the whole duration of the event (DURING) and 20 s epochs immediately before (PRE) and after (POST) it; (

**b**) ECG and airflow signals, respectively acquired from the cardiac and the respiratory system (R-peaks and the corresponding AF value are highlighted in red and green, respectively), from which RR and RESP time series are extracted; (

**c**) extraction of an instance of synchronous segments consisting of RR intervals and AF amplitude values; (

**d**) repetition of more instances for each given experimental condition (PRE, DURING, POST) to form 300-point time series; vertical lines separate different segments. Note that the segments are joined in (

**d**) for visualization purposes only, since the realizations of the RR and RESP patterns used for information analysis are extracted separately from each segment (see Section 2.4).

**Figure 3.**Graphical representation of the information decomposition of the predictive information ${P}_{Y}$ as (

**a**) information storage ${S}_{Y}$ and information transfer ${T}_{X\to Y}$ and (

**b**) as internal information ${S}_{Y|X}$ and cross information ${C}_{X\to Y}$.

**Figure 4.**Exemplary representation of the fuzzy probability estimation procedure: (

**a**) representative segments of generic time series Y and X of N = 7 samples, with the samples forming the second observation of the present value of Y, ${Y}_{n}$, and of the past values of X and Y, ${X}_{n}^{-}$ and ${Y}_{n}^{-}$, encircled in green and red; (

**b**) observation matrix formed placing in the ith row the samples of the ith observation of ${Y}_{n},{X}_{n}^{-},{X}_{n}^{-}$ (Chebyshev distances between the second observation and the other observations are indicated below the matrix); (

**c**) Gaussian kernel function and probability estimate for the second observation calculated as the average of the kernel function applied to the Chebyshev distances between the second and all other observations.

**Figure 5.**(

**a**) Predictive information and its decomposition in terms of (

**b**) information storage and (

**c**) information transfer, and in terms of (

**d**) internal information and (

**e**) cross information, for different breathing disorder events and at different phases (CONTROL, PRE, DURING, POST). Statistical significance (p < 0.05) between the time segments is indicated by a bridging line, whereas statistical significance (p < 0.05) in pairwise comparisons between each segment and CONTROL is indicated by *.

**Figure 6.**Statistical significance measured using a one-sided surrogate test for: (

**a**) information storage, (

**b**) information transfer, (

**c**) internal information and (

**d**) cross information, for different breathing disorder events and at different phases (CONTROL, PRE, DURING, POST).

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**MDPI and ACS Style**

Lazic, I.; Pernice, R.; Loncar-Turukalo, T.; Mijatovic, G.; Faes, L. Assessment of Cardiorespiratory Interactions during Apneic Events in Sleep via Fuzzy Kernel Measures of Information Dynamics. *Entropy* **2021**, *23*, 698.
https://doi.org/10.3390/e23060698

**AMA Style**

Lazic I, Pernice R, Loncar-Turukalo T, Mijatovic G, Faes L. Assessment of Cardiorespiratory Interactions during Apneic Events in Sleep via Fuzzy Kernel Measures of Information Dynamics. *Entropy*. 2021; 23(6):698.
https://doi.org/10.3390/e23060698

**Chicago/Turabian Style**

Lazic, Ivan, Riccardo Pernice, Tatjana Loncar-Turukalo, Gorana Mijatovic, and Luca Faes. 2021. "Assessment of Cardiorespiratory Interactions during Apneic Events in Sleep via Fuzzy Kernel Measures of Information Dynamics" *Entropy* 23, no. 6: 698.
https://doi.org/10.3390/e23060698