The heart rate is analyzed in the time domain, frequency domain and by using nonlinear parameters.

The time-and frequency-domain measures of HR variability were analyzed by the methods recommended by the Task Force of the European Society of Cardiology 30. The standard deviation of all normal R-R intervals (SDNN) and the difference between the maximum hourly HR variability (circadian rhythm) were computed as standard time-domain measures of HR variability. Spectral power was quantified both by Fast Fourier Transform analysis and by autoregressive analysis in 4 frequency bands 11:, ≤ 0.003 Hz(ultralow frequency [ULF]), 0.003 to 0.04 Hz(very-low frequency [VLF]), 0.04 to 0.15 Hz (low frequency [LF]), and 0.15 to 0.40 Hz (high frequency [HF]). ULF and VLF spectral components were computed over the entire recording interval by the fast Fourier method 11. LF and HF components were computed from segments of 512 R-R intervals by the autoregressive method.

The Poincare plot, a technique taken from nonlinear dynamics, portrays the nature of R-R interval fluctuations. It is a plot in which each R-R interval is plotted as a function of the previous R-R interval. Poincare plot analysis is an emerging quantitative-visual technique whereby the shape of the plot is categorized into functional classes that indicate the degree of the heart failure in a subject 36 The plot provides summary information as well as detailed beat-to-beat information on the behavior of the heart 13.

The geometry of the Poincare plot is essential and can be described by fitting an ellipse to the graph. The ellipse is fitted onto the so called line-of-identity at 45° to the normal axis. The standard deviation of the points perpendicular to the line-of-identity denoted by SD1 describes short-term variability which is mainly caused by respiratory sinus arrhythmia (RSA). The standard deviation along the line-of-identity denoted by SD2 describes long-term variability.

Statistically, the plot displays the correlation between consecutive intervals in a graphical manner. Nonlinear dynamics considers the Poincare plot as the two dimensional (2-D) reconstructed R-R interval phase-spaces, which is a projection of the reconstructed attractor describing the dynamics of the cardiac system. The R-R interval Poincare plot typically appears as an elongated cloud of points oriented along the line-of-identity. The dispersion of points perpendicular to the line-of-identity reflects the level of short term variability. The dispersion of points along the line-of-identity is thought to indicate the level of long-term variability.

The Poincare plot may be analyzed quantitatively by calculating the standard deviations of the distances of the R-R(i) to the lines y = x and y = -x+2*R-R_m, where R-R_m is the mean of all R-R(i) 33. The standard deviations are referred to as SD1 and SD2, respectively. SD1 related to the fast beat-to-beat variability in the data, while SD2 describes the longer-term variability of R-R(i) 33. Figure 1 shows the Poincare plot of a normal young subject.

Fig. 1 Poincare plot of a normal young subject

Approximate entropy quantifies the regularity of time series. ApEn is also called a "regularity statistic". It is represented as a simple index for the overall "complexity" and "predictability" of each time series. In our study ApEn quantifies the regularity of the R-R interval. The more regular and predictable the R-R interval series, the lower will be the value of ApEn. On the other hand more the randomness in the R-R interval series, the higher will be the value of ApEn. Therefore, this method quantify the unpredictability of fluctuations in a time series such as an instantaneous R-R interval time series, R-R(i).

To compute ApEn of each data set, m-dimensional vector sequences [x[n]] were constructed from the R-R interval time series

x(n) = [R-R(n),.....R-R(N+m-1)]

Where the index n can take on values ranging from 1 to N-m+1 and N is the total number of data points in the R-R interval time series. If the distance between two vectors x(i) and x(j) is defined as d [x(i), x(j)], then we have

Where,

N, is a sequence of 1000 consecutive instantaneous R-R intervals.

m, specifies the pattern length which is 2 in this study.

r defines the criterion of similarity which has been set at 15% of the standard deviation of 1000 R-R intervals and (r) = { Number of x(j) such that d [x(i), x(j)] ≤ }/(N-m+1).

ApEn (N,r,m) is then defined as follows in equation (3)

ApEn = [ApEn_m(N,r,m) - ApEn_m+1(N,r,m + 1)]     (3)

The criterion of similarity, r, was chosen such that it was larger than most of the noise but at the same time not so large that detailed information about the system dynamics would be lost. On the basis of the work of Pincus et al 242526, for m = 2, N = 1000 and r = 15% of the standard deviation of the data would produce reasonable statistical validity of ApEn.

In our study we use data set of 1000 adjacent R-R intervals. We divide the data set into smaller sets of length m = 2. This amounts to 500 smaller sub-sets. The next step is to determine the number of sub-sets that are within the criterion of similarity r = 15% of deviation of 1000 R-R intervals. Then we repeat the same process for the second sub-set till each sub-set is compared with the rest of the data set. This process computes the part of equation (2) and N-m+1 = 1000-2+1 = 999. We repeat the same process for m = 3. Approximate entropy is then calculated using equation (3).

Lyapunov Exponent (λ) is a quantitative measure of the sensitive dependence on the initial conditions. It defines the average rate of divergence of two neighboring trajectories. An exponential divergence of initially nearby trajectories in phase space coupled with folding of trajectories, to ensure that the solutions will remain finite, is the general mechanism for generating deterministic randomness and unpredictability. Therefore, the existence of a positive λ for almost all initial conditions in a bounded dynamical system is widely used definition of deterministic chaos. To discriminate between chaotic dynamics and periodic signals Lyapunov Exponent (λ) are often used. It is a measure of the rate at which the trajectories separate one from other. The trajectories of chaotic signals in phase space follow typical patterns. Closely spaced trajectories converge and diverge exponentially, relative to each other. For dynamical systems, sensitivity to initial conditions is quantified by the Lyapunov Exponent (λ). They characterize the average rate of divergence of these neighboring trajectories. A negative exponent implies that the orbits approach a common fixed point. A zero exponent means the orbits maintain their relative positions; they are on a stable attractor. Finally, a positive exponent implies the orbits are on a chaotic attractor.

The algorithm proposed by Wolf et al 35 is used to Largest LE (LLE) from EEG data. For a given the time series x(t) for m dimensional phase space with delay coordinate t, that is a point on the attractor is given by

{ x(t), x(t + t),..., x(t + (m-1)t }

We locate nearest neighbor to initial point

{ x(t0), x(t0+ t),..., x(t0 + (m-1)t }

And denote the distance between these two points as L(t0). At a later time t1, initial length will evolve to length L'(t1). The mean exponential rate of divergence of two initially close orbits is characterized by

In implementation of this program, the following set of numerical parameters has to be chosen:

P = { m, t, T, Smax, Smin,thmax}

where m is the embedding dimension, t is delay, T being evaluation time (= t k+1 - t k-1) and Smax, Smin are the maximum and minimum separations of replacement point respectively and thmax is the maximum orientation error. According to Das et al [Das et al, 2002] an embedding dimension between 5 to 20 and a delay of 1 should be chosen when calculating LE for EEG data. In our analysis we have chosen an embedding dimension of 10 and delay of 1.

The concept of fractal is most often associated with geometrical objects satisfying two criteria: Self-similarity means that an object is composed of sub-units on multiple levels that statistically resemble the structure of the whole object. The second criterion for a fractal object is that it has a fractal dimension, also called fractal, that can be defined to be any curve or surface that is independent of scale. This concept of fractal structure can be extended to the analysis of physiological signals.

The DFA was used to quantify the fractal scaling properties of heart rate signals of short interval. Therefore, we apply DFA to the analysis of biological data, which calculates the root-mean-square fluctuation of integrated and detrended time series, permits the detection of intrinsic self-similarity embedded in a non-stationary time series, and also avoids the spurious detection of apparent self-similarity 1. This method has been applied to a wide range of simulated and physiologic time series in recent years 19892122.

To illustrate the DFA algorithm, the total length of the heart rate signal (N) is integrated

Where HR(i) is the i th heart rate signal and HR_avg is the average heart rate of N samples. Next, the integrated time series is divided into boxes of equal length n, a least-squares line is fit to the data (representing the trend in that box). The y-coordinate of the straight line segments is denoted by y_n(k). Then, we detrend the heart rate data y(k), by subtracting the local trend, y_n(k), in each box. The root mean square fluctuation of this integrated and detrended heart rate data is calculated by

This computation is repeated over all the box sizes to provide a relationship between F(n), the average fluctuation as a function of box size n.

In this study, the box size is ranged from 4 to ~300 beats. A box size larger than 300 beats would give a less accurate fluctuation value because of finite length effects of data.

Typically, F(n) will increase with box size. A linear relationship on a double-log graph indicates the presence of scaling i.e. F(n) ≈ n^α. Under such conditions, the fluctuations can be characterized by a scaling exponent α, the slope of the line relating log F(n) to log(n). An α of 0.5 corresponds to white noise, α = 1 represents 1/f noise and α = 1.5 indicates Brownian noise or random walk. A good linear fit of the log F(n) to log(n) plot (DFA plot) indicates F(n) is proportional to n^α, where α is the single exponent describing the correlation properties of the entire range of heart rate data [shown in Fig. 1]. However in some cases, we found that the DFA plot was not strictly linear but rather consisted of two distinct regions of different slopes separated at a break point n_bp. This observation suggests there is a short range scaling exponent, α_s, over periods of 4 to 13 [n_bp] beats, and a long – range exponent, α_l, over long periods [Woo et al, 1992; Kamen et al, 1996].

The purpose of surrogate data is to test for any nonlinearity in the original data 32. To test if the attractor geometry and correlation dimension are truly due to chaotic dynamics one must examine these characteristics for surrogate data sets.

Nonlinear indexes such as ApEn are computed for several surrogate data series. Their values are compared with that assumed by the nonlinear index computed for the original index 32. The demonstration of statistically significant difference in ApEn between the original and surrogate data are in keeping with the presence of nonlinear dynamics in the original data.

Surrogate data have Fourier decomposition with the same amplitudes as the empirical data decomposition but with random phase components. This is obtained from the Chaos Data Analyzer.

To test for a statistical significance of difference in original ApEn and the surrogate data, 10 surrogate data series were generated to match each original signal. Then it is subjected to Student t test distribution. We found that, the surrogate data ApEn and original data ApEn, are distinct and they differ by more than 50%. This rejects the null hypothesis and hence the original data contain nonlinear features.

Table 5 Results of various parameters using surrogate data