Nonlinear causal influences assessed by mutual compression entropy

Andy Schumann; Berit Fleckenstein; Karl-Jürgen Bär

doi:10.1515/cdbme-2016-0049

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Nonlinear causal influences assessed by mutual compression entropy

Andy Schumann , Berit Fleckenstein and Karl-Jürgen Bär

Published/Copyright: September 30, 2016

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From the journal Current Directions in Biomedical Engineering Volume 2 Issue 1

Abstract

Autonomic control of the heart rate was demonstrated to be complex and nonlinear. Respiratory sinus arrhythima plays a crucial role in heart rate vagal modulation. Here we present an approach of assessing nonlinear causal relationships in bivariate time series, called mutual compression entropy (MCE). We applied MCE to cardiorespiratory data of 29 patients with acute schizophrenia and 29 matched controls. The method is based on data compression and estimates to which extend a (target) time series can be compressed regarding another time series (driver). In schizophrenia an elevated sympathetic and reduced parasympathetic heart rate modulation was found. The nonlinear influence of respiration on heart rate variability was demonstrated by a highly significant reduction of MCE (0.816 vs. 0.808, p¡0.01). In healthy subjects MCE was mainly related to sympathovagal balance. We conclude, that this index has the potential to uncover physiological information beyond linear measures.

Keywords: complexity; heart rate variability; respiratory sinus arrhythmia; symbolic dynamics; schizophrenia

1 Introduction

Exploring nonlinear relationships of complex physiological systems is a major aspect in modern biomedical research. A number of chemo-, pressure- or volume- sensitive regulatory reflexes contribute to a highly complex and nonlinear control of cardiac activity via different compounds of the autonomic nervous system. Regarding the cardiorespiratory system, respiratory sinus arrhythmia is a powerful mechanism of vagal heart rate modulation. [1] In acute schizophrenia pathological changes of cardiorespiratory function were shown in several studies [2], [3], [4], [5]. It was suggested that the breathing pattern characterized by rapid and shallow ventilation determines cardiac autonomic imbalance [6]. The altered heart rate modulation most probably contributes to a higher vulnerability to arrhythmias and severe cardiac events resulting in an overall elevated cardiac mortality [7], [8]. For these reasons, analyzing the influence of breathing on cardiac functioning is of great importance understanding pathophysiological changes of the autonomic nervous system in schizophrenia. Probably the most fundamental assumption about causal relationships, that a cause must temporally precede its effect, was stated by Granger [9]. Here we present mutual compression entropy (MCE) adapting an information-theoretic technique based on symbol transformation to the principle idea of Granger. We extended the approach of Ziv and Lempel, that was already applied to heart rate variability analysis (compression entropy), to bivariate datasets, assuming information transfer to be reflected by symbol sequences of the source series reoccurring in the target series, leading to higher compressibility of the target considering the source [10]. In this study mutual compression entropy was applied to investigate the influence of respiration on heart rate variability.

2 Methods

2.1 Subjects and data acquisition

Twenty nine unmedicated patients with schizophrenia (18 males, 11 females, age: 35.1 ± 9.07 years, BMI: 24,51 ± 4,65) and 29 matched controls (18 males, 11 females, age: 37.2 ± 10.47 years, BMI: 24,74 ± 4,27) were enrolled in this study. Patients were tested before taking medication immediately after stationary admission. Paranoid schizophrenia was established by a staff psychiatrist when patients fulfilled DSM-IV criteria (Diagnostic and statistical manual of mental disorders, 4th edition, published by the American Psychiatric Association) [7]. Healthy control subjects had no present or past history of psychiatric, neurological or other clinically significant disorders. All participants gave their informed written consent in accordance with the protocol approved by the Ethics Committee of Jena University.

Electrocardiogram (ECG) and respiratory excursion of the chest were recorded for 20 min in supine position simultaneously by the MP150 system (BIOPAC Systems Inc, Goleta, CA, USA) and digitized at 1000 Hz. ECG was band pass filtered between 0.05 Hz and 35 Hz and respiration between 0.05 and 5 Hz. Heart beat intervals (BBI) were extracted from the ECG and checked by manual inspection. Respiratory signal was sampled with each heart beat (BBI/2) building a bivariate data set (see Figure 1 I).

Figure 1:

Schematic illustration of MCE calculation. Step I. Extraction of BBI and RSP time series from recorded signals. Step II. Transfer to symbol series according to equation 1. Step III. Compression of symbols in the target window by looking for redundant substrings in the source window (mutual compression).

2.2 Mutual compression entropy (MCE)

At first, input time series x(i) were transformed into sequences of symbols X(i), that reflect the magnitude of temporal fluctuations of the input. Threshold l was defined as a multiple of standard deviation sd (l_x = 2sd_x).

(1)sx={3;x⁢(i)−x⁢(i−1)>lx2;|x⁢(i)−x⁢(i−1)|≤lx1;x⁢(i)−x⁢(i−1)<lx

The original compression procedure by Ziv and Lempel is extensively described elsewhere [10]. In the following we briefly outline the idea modified in order to analyze physiological signals [7], [8], [11], [12].

The encoding process is conducted in two adjacent time windows shifted along the input. One filled with already encoded symbols (memory) and one covering the current data point and subsequent symbols (buffer). If the current substring, that is going to be encoded (stored in the buffer), appears on the memory, it can be skipped storing only the start and length of the redundant information on the memory. So the number of iterations needed to encode the input can be reduced. Thus, self-reproducing input strings can be compressed without losing input information.

In the bivariate approach the two windows are distributed to two time series in order to encode the target series Y(i) (target window) regarding the symbols of the source series X(i) (source window). Subsequences of the input series (here X(i)) from element n to m are denoted as X^m_n = [x_n; x_{n + 1};…;x_m]. The basic steps to calculate MCE are illustrated in Figure 1. The target window covers N_T symbols Xpp+NT starting at the current data point X_p. These target symbols are encoded using the symbols of the source window Xp−NS−1p−1 with length N_S. In Figure 1 both windows have a length of three samples. In the source window the longest subseries Xp−NS+vp+NS+v+n−1, lasting n symbols starting at element v, matching the target sequence Xpp+n−1, is extracted. Instead of encoding the whole target string, only the starting point v of its equivalent in the source and its length n is stored. So n target symbols can be passed and the next symbol to be encoded is X_{p + n} .

We defined MCE(Y—X) as proportion of iterations that can be saved compressing Y(i) by X(i) with respect to the original length of Y(i). Thus, MCE rises with increasing information of the source series recurring in the future of the target series. Assuming the input Y(i) of length L is compressed by X(i) in M iterations, MCE is calculated by the equation below.

(2)MCE(Y|X)=1−ML

MCE is dependent on the length of the two windows N_S (source window) and N_T (target window). Additionally, we introduced a shift of the target window back in time, overlap τ, to allow temporal overlapping sequences in both windows and immediate interaction (delay d = 0). It is possible to consider several coupling delays within MCE analysis defined by the presets of its calculation. All interaction delays d with 1−τ≤d<NS+NT−τ contribute to MCE estimation.

2.3 Standard indices of heart rate variability and blood pressure

Mean heart rate (HR) and breathing rate (BR) were estimated by averaging the respective time series. Standard deviation of heart beat intervals (sdNN), root mean square of successive interval differences (RMSSD) and ratio of low to high frequency power (LF/HF) were chosen to assess global, short-term vagal variability of heart rhythm and sympathovagal balance [13]. Univariate compression entropy CE was estimated following the original algorithm described by Baumert with recommended window lengths w=7, b=3 [14], [15]. Respiratory sinus arrhythmia RSA was calculated by peak-valley-method [1], [4].

2.4 Statistical analysis

To compare patients with schizophrenia and healthy controls, a two-sample t-Test was conducted. Normal distribution of results was proofed by Shapiro-Wilks-test. Pearson correlation coefficient was used to assess bivariate linear dependencies of the parameters. Levels of statistical significance were defined according to common conventions [p¡0.05 (*), p¡0.01 (**)].

3 Results

In patients with acute schizophrenia heart rate was elevated by more than 13 min⁻¹ (see Table 1). Global heart rate variability (sdNN: 53.06 vs. 41.93 ms, p < 0.05), respiratory sinus arrhythmia (RSA: 68.85 vs. 45.74 ms, p < 0.05) and short term modulation (RMSSD: 39.28 vs. 25.45 ms, p < 0.05) were significantly decreased in patients. These changes indicate a diminished vagal influence on heart rate in patients with schizophrenia. The ratio of low and high frequency heart rate modulation was not changed (2.28 vs. 3.02, p > 0.05).

Table 1:

Cardiorespiratory indices in healthy controls and patients with schizophrenia.

Parameter	Controls	Patients	Significance
HR [min⁻¹]	63.79 ± 8.49	77.05 ± 9.91	**
sdNN [ms]	53.06 ± 21.17	41.93 ± 14.13	*
RMSSD [ms]	39.28 ± 26.7	25.45 ± 16.89	*
CE	0.74 ± 0.08	0.7 ± 0.06	n.s.
LF/HF	2.28 ± 1.68	3.02 ± 2.18	n.s.
RSA [ms]	68.85 ± 48.09	45.74 ± 21.69	*
BR [min⁻¹]	15.16 ± 3.01	17.08 ± 3.58	*
T_in [s]	1.82 ± 0.47	1.65 ± 0.38	n.s.
T_ex [s]	2.35 ± 0.96	2.02 ± 0.63	n.s.
T_in/T_ex	0.86 ± 0.29	0.88 ± 0.25	n.s.

Results are given in mean ± standard deviation. HR: mean heart rate, sdNN: standard deviation of BBI, RMSSD: root mean square of successive BBI, CE: compression entropy of BBI, LF/HF: ratio of low and high frequency HR components, RSA: respiratory sinus arrhythmia, MCE: mutual compression entropy, BR: breathing rate, T_in: inspiration time, T_ex: expiration time, T_in/T_ex: ratio of inspiration to expiration time.

Table 2:

Correlation of MCE and cardiorespiratory indices in healthy controls

Parameter	Pearson correlation	Significance
HR	−0.105	n.s.
sdNN	0.187	n.s.
RMSSD	0.372	*
CE	0.105	n.s.
LF/HF	−0.557	**
RSA	0.366	*
BR	−0.061	n.s.
T_in	0.057	n.s.
T_ex	−0.049	n.s.
T_in/T_ex	0.060	n.s.

Pearson correlation coefficients are given with p-values symbolized by asterisks (*p < 0.05, **p < 0.01). HR: mean heart rate, sdNN: standard deviation of BBI, RMSSD: root mean square of successive BBI, CE: compression entropy of BBI, LF/HF: ratio of low- and high frequency HRV power, RSA: respiratory sinus arrhythmia, BR: breathing rate, T_in: inspiration time, T_ex: expiration time, T_in/T_ex: ratio of inspiration to expiration time.

Breathing rates were higher in patients (15.2 vs. 17.1 min⁻¹, p < 0.05), with an similar ratio of inspiration and expiration periods (0.86 vs. 0.88, n.s.). MCE estimating respiratory influence on the heart rate was significantly lower in patients (0.816 ± 0.005 vs. 0.808 ± 0.007, p < 0.01). In Figure 2 results of both groups are illustrated by boxplots.

Figure 2:

Boxplots of MCE in controls and patients with schizophrenia.

In healthy controls MCE correlated positively to vagal heart rate modulation RMSSD and RSA (r=0.372 and r=0.366 , both p < 0.05) and negatively to LF/HF ratio (0.557, p < 0.01).

4 Discussion

In this work we presented mutual compression entropy (MCE) for the assessment of complex nonlinear directed interaction combining the information-theoretic procedure of data compression with the central idea of causality [9]. MCE was used to investigate cardiorespiratory coupling in schizophrenia and revealed a loss of causal influence of breathing on heart rate modulation.

In patients suffering from schizophrenia an increased heart rate and short term heart rate variability was found in several studies [2], [3], [7], [12]. An autonomic imbalance expressed by an elevated sympathetic and reduced parasympathetic activity was suggested to cause these cardioregulatory changes resulting in an increased cardiac mortality in schizophrenia [2], [7], [12]. Breathing rate was increased in patients by two breaths per minute. A shallow and accelerated breathing in acute schizophrenia was already reported [4], [6]. Additionally, the modulation of heart rate due to respiration was less pronounced.

The causal influence of respiration on heart rate was demonstrated by RSA and MCE. The nonlinear relationship seemed to be more effected, with a highly significant reduction of MCE (0.816 vs. 0.808, p < 0.01). Schulz et al. investigated nonlinear cardiorespiratory coupling in terms of joint symbolic dynamics and demonstrated this attenuation as well [2], [3]. It was suggested that rapid and shallow breathing might be the reason for impairment of short term heart rate variation. Cognitive load evoked by pathophysiological paranoia might lead to a persistently elevated level of arousal expressed as vagal inhibition and dominating sympathetic activity [7], [6]. High negative correlation of MCE to LF/HF (r = −0.56**) indicated the dependency on sympathovagal balance. Additionally MCE was less strongly correlated to RMSSD and RSA (both r = 0.37*). MCE estimates information transfer and reveals a directed and weighted interaction index. Symbolization is robust against outliers, does not require stationarity and is very customizable. MCE integrates interactions with a variety of delays within a range defined by calculation setting.

In conclusion, we introduced MCE as a measure of complex nonlinear interaction that has some basic methodological advantages. Impairment of respiratory influence on heart rate was demonstrated by highly significant MCE reduction in acute schizophrenia. We are convinced that MCE is augmenting the variety of available tools for coupling analysis.

Author’s Statement

Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Material and methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration.

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Published Online: 2016-9-30

Published in Print: 2016-9-1

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https://doi.org/10.1515/cdbme-2016-0049

Keywords for this article

complexity; heart rate variability; respiratory sinus arrhythmia; symbolic dynamics; schizophrenia

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