Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 7960467, 11 pages

https://doi.org/10.1155/2017/7960467

## Measuring Coupling of Rhythmical Time Series Using Cross Sample Entropy and Cross Recurrence Quantification Analysis

^{1}MORE Foundation, 18444 N. 25th Ave, Suite 110, Phoenix, AZ 85023, USA^{2}Center for Research in Human Movement Variability, University of Nebraska Omaha, 6160 University Drive, Omaha, NE 68182-0860, USA^{3}College of Public Health, University of Nebraska Medical Center, 984355 Medical Center, Omaha, NE 68198-4355, USA

Correspondence should be addressed to Jennifer M. Yentes; ude.ahamonu@setneyj

Received 14 April 2017; Revised 11 July 2017; Accepted 20 August 2017; Published 22 October 2017

Academic Editor: Christopher Rhea

Copyright © 2017 John McCamley et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this investigation was to compare and contrast the use of cross sample entropy (xSE) and cross recurrence quantification analysis (cRQA) measures for the assessment of coupling of rhythmical patterns. Measures were assessed using simulated signals with regular, chaotic, and random fluctuations in frequency, amplitude, and a combination of both. Biological data were studied as models of normal and abnormal locomotor-respiratory coupling. Nine signal types were generated for seven frequency ratios. Fifteen patients with COPD (abnormal coupling) and twenty-one healthy controls (normal coupling) walked on a treadmill at three speeds while breathing and walking were recorded. xSE and the cRQA measures of percent determinism, maximum line, mean line, and entropy were quantified for both the simulated and experimental data. In the simulated data, xSE, percent determinism, and entropy were influenced by the frequency manipulation. The 1 : 1 frequency ratio was different than other frequency ratios for almost all measures and/or manipulations. The patients with COPD used a 2 : 3 ratio more often and xSE, percent determinism, maximum line, mean line, and cRQA entropy were able to discriminate between the groups. Analysis of the effects of walking speed indicated that all measures were able to discriminate between speeds.

#### 1. Introduction

It has long been suggested that various biological rhythms, such as walking and breathing rhythms, have some degree of synchrony. Hey et al. [1] noted in a 1966 paper that breathing frequency was often a submultiple of stepping frequency. Subsequent studies observed this synchrony of breathing and walking or running rhythms, and it has become commonly referred to as locomotor-respiratory coupling [2–5]. The synchrony of breathing and stepping rhythms was considered to be present when the interval between heel strike and the beginning of breath inspiration or expiration was constant for a series of breaths [2]. Synchrony of breathing rhythms was observed for 8 of 15 participants while walking on a treadmill [3]. Stepping frequency influenced breathing frequency even when no rhythmical coupling existed [4]. These findings and those of many others [5–7] have soundly established that locomotor-respiratory coupling exists during walking in humans.

Previously, locomotor-respiratory coupling has been investigated using standard deviation of the interval between inspiration and heel strike [2], the percentage of inspirations starting at the same stage of the walking cycle [8], creating a cross-correlogram of breathing period versus step period [3], and discrete relative phase and return maps [6]. These tools, while providing useful information concerning the relationships between measured breathing and walking rhythms, may not be appropriate for biological systems that are nonstationary and noisy [9]. It has been suggested that biological rhythms may be considered nonlinear in nature [10]. It is appropriate, therefore, that the coupling between biological rhythms be investigated using measures that can assess the nonlinear nature of the rhythms under observation. Two such measures are cross sample entropy [11] and cross recurrence quantification analysis [12].

The concept of entropy as a means to describe the randomness of a finite sequence was described by Kolmogorov and Uspenskii [13]. Pincus [14] introduced approximate entropy to quantify regularity using relatively few data points. Pincus and Singer [15] developed cross approximate entropy as an extended form of approximate entropy to quantify asynchrony or conditional irregularity in interconnected networks [16]. Sample entropy was subsequently introduced to overcome a bias in approximate entropy caused by the counting of self matches [11]. It was noted that cross approximate entropy, while not affected by self matches, lacked relative consistency [11]. In a similar manner to cross approximate entropy, cross sample entropy (xSE) was developed from sample entropy to allow the measurement of asynchrony with relative consistency. xSE provides a method of determining if patterns that are similar within one data series are also similar in another data series [11]. It has been used to examine relationships in situations as diverse as stock markets [17], voice disorders [18], and renal sympathetic nerve activity in rats [19].

The use of recurrence plots to analyze experimental time series was originally proposed by Eckmann et al. [20]. A recurrence plot is a tool that reduces a potentially high-dimensional, nonlinear, dynamical system into a two-dimensional representation of points, revealing recurring patterns or trajectories within the system. While a visual inspection of such plots exposes many interesting qualitative features, further quantitative analysis was proposed by Zbilut and Webber [21]. For example, quantifying the recurrence rate of these plots will measure how often a system revisits a state it already visited. Cross recurrence quantification analysis (cRQA) is an adaptation of recurrence quantification. Cross recurrences between two signals are found by calculating the distances between all points in one series with all points of another series, rather than within one system to itself [12, 22]. The cRQA plot provides a visual representation of the coupling of two different time series on one time scale. This visualization and the measures which are subsequently extracted do not provide a direct measure of the strength of coupling between two signals. When the cRQA measures are viewed in combination they do give useful information concerning how the two signals relate to one another over time, from which information about coupling may be deduced. cRQA has previously been used to investigate interpersonal coordination [22], to identify cover songs [23], and for intra- and interpersonal interlimb coordination [24].

The purpose of this paper was to compare and contrast the use of xSE and cRQA measures, for the coupling assessment of rhythmical patterns. The measures were first assessed using simulated signals with three known fluctuations in frequency, amplitude, and a combination of both. Further, locomotor-respiratory coupling for two experimental groups was studied as models of normal and abnormal coupling. The same measures were used to assess the coupling of breathing and walking rhythms for older healthy subjects and patients with chronic obstructive pulmonary disease (COPD). COPD affects the breathing rhythm [25] and walking rhythms [26] of patients. However, it is not well understood whether the observed changes in walking rhythms are coupled to the altered breathing rhythms of patients with COPD.

#### 2. Methods

##### 2.1. Simulated Data Analysis

Fluctuations in biological signals may occur in a variety of ways. For example, changes in walking speed may be achieved through altered frequency (e.g., step time), amplitude (e.g., step length), or a combination of both. The frequency of the signal (e.g., step time) may change over time, such as the continual variations that may occur in stride frequency. An alternative way to change walking speed is to change the amplitude, the step length. In reality, both will generally change. It is important to understand the effect that alterations in amplitude, frequency, or both have on the ability of analysis methods to determine the nature of coupling between signals. To understand the ability of different measures to differentiate between signals, it is necessary to assess the differences in measures calculated from simulated signals with known levels of complexity.

Nine different types of synthetic oscillating, quasi-sinusoidal signals were generated using custom Matlab codes (MathWorks, Inc., Natick, MA), forming a matrix. The first dimension of the matrix represented the fluctuations present in the signal (periodic, chaotic, or random) and the second dimension represented the parameter(s) which varied in the signal (frequency, amplitude, or both). For each of these nine signal types, different frequency ratios ( = 1 : 1, 2 : 3, 1 : 2, 2 : 5, 1 : 3, 2 : 7, and 1 : 4) were generated. Ten signals of each ratio were generated for random and chaotic fluctuations. As the periodic signals do not vary, one signal of each ratio was generated for analysis (Figure 1). The signal with the higher frequency was constructed to have a similar frequency as walking, which for the biological data recorded averaged approximately 0.75 Hz. Thus, for each of the “5-minute” long trials at 30 Hz (9000 data points), 225 cycles would occur. The lower frequency signals varied in the number of cycles based on the frequency ratio. For example, a 2 : 3 ratio would have 225 cycles for one signal and 150 cycles for the second and a 1 : 2 ratio would have 225 cycles for one signal and 112.5 cycles for the second.