Characterizing Nonlinear Heartbeat Dynamics within a Point Process Framework

Abstract

Heartbeat intervals are known to have nonlinear and non-stationary dynamics. In this paper, we propose a nonlinear Volterra-Wiener expansion modeling of human heartbeat dynamics within a point process framework. Inclusion of second-order nonlinearity allows us to estimate dynamic bispectrum. The proposed probabilistic model was examined with two recorded heartbeat interval data sets. Preliminary results show that our model is beneficial to characterize the inherent nonlinearity of the heartbeat dynamics.

I. Introduction

Heart rate (HR) and heart rate variability (HRV) are important quantitative markers of cardiovascular control, as regulated by the autonomic nervous system [1]. It has long been understood that the healthy heart is influenced by multiple neural and hormonal inputs that result in variations of duration in the interbeat intervals. Specifically, various nonlinear neural interactions and integrations occur at the neuron and receptor levels, and underlie the complex output of structures such as the sino-atrial (SA) node in response to changing levels of sympathetic and vagal activities.

In characterizing the nonlinear heartbeat dynamics, both linear and nonlinear system identification methods have been applied to R-R interval series [8], [9], [14]. Examples of higher order characterization for cardiovascular signals include nonlinear autoregressive (AR) model, Volterra-Wiener series expansion, and Volterra-Laguerre model [12]. However, none of these nonlinear models included nonlinear elements in the framework of a statistical definition of heart beat generation. In this paper, we apply nonlinear modeling to heartbeat dynamics with the point process paradigm. The point process theory is a natural statistical tool to characterize the probabilistic generative mechanism of the heart beat, while allowing us to model the instantaneous HR and HRV [3], [4]. Furthermore, by inclusion of the second order nonlinear term our model offers an opportunity to monitor dynamic higher-order spectra [13].

II. Volterra Series for Nonlinear System Identification

Rooted in the Volterra Theorem, the Volterra series expansion is a general method for nonlinear system modeling and identification [12]. In functional analysis, a Volterra series denotes a functional expansion of a dynamic, nonlinear, time-invariant function. The Volterra series can represent a wide range of nonlinear systems. Because of its generality, Volterra series expansion was widely used in nonlinear modeling in engineering and physiology [12]. For instance, computational procedures based on a comparison of the prediction power of linear and nonlinear models of the Volterra-Wiener form have been applied to measure the chaotic dynamics of heartbeats [2].

Consider a nonlinear single-input and single-output system y = g(x). According to the Volterra series theory, the nonlinear system can be expanded by a (finite or infinite) set of kernel expansion terms:

$$y(t)=k_0+\sum _{m=0}^{M-1}{k}_1(m)x(t-m)+\sum _{m=0}^{M-1}\sum _{n=0}^{M-1}{k}_2(m,n)x(t-m)x(t-n)+\dots$$ (2)

where M is the memory of the nonlinear system. The Volterra kernels {k₀, k₁, k₂, …,} describe the dynamics of the system. Estimation of the Volterra coefficients is generally performed by computing the coefficients of an orthogonalized series, and then recomputing the coefficients of the original Volterra series; a common method is based on the least squares [12]. In this paper, we will apply adaptive filtering method to recursively estimate the time-varying Volterra coefficients.

III. Heartbeat Interval Point Process Model

A. Heartbeat Interval

Suppose we are given a set of R-wave events ${\left\{u_j\right\}}_{j=1}^J$ detected from the ECG, let RR_j = u_j − u_j−1 > 0 denote the j^th R-R interval, or equivalently, the waiting time until the next R-wave event. By treating the R-wave as discrete events, we may develop a probabilistic point process model in the continuous-time domain [4].

Assuming history dependence, the waiting time t − u_t until the next R-Wave event follows an inverse Gaussian model:

$$p(t)={\left(\frac{\theta }{2\pi t^3}\right)}^{\frac{1}{2}}exp\left(-\frac{\theta {(t-u_t-{\mu }_t)}^2}{2{{\mu }_t}^2(t-u_t)}\right)(t>u_t),$$

where u_t denotes the previous R-wave event occurred before time t, θ > 0 denotes the shape parameter, and μ_t denotes the instantaneous R-R mean that can be modeled as a generic function of the past (finite) R-R values μ_t = g(RR_t−1, RR_t−2,…, RR_t−h)where RR_t−j denotes the previous jth R-R interval occurred prior to the present time t. In our previous work [3], [6], the history dependence is defined by expressing the instantaneous mean μ_RR(t) as a linear combination of present and past R-R intervals (in terms of an AR model), i.e., function g is linear. Here, we propose to include the nonlinear terms of past R-R intervals in an attempt to improve the model fit. Specifically, the mean μ_t is defined as

$${\mu }_t\equiv {\mu }_{RR}(t)=a_0+\sum _{i=1}^p{a}_{i}R{R}_{t-i}+\sum _{i=1}^q\sum _{i=1}^q{b}_{ij}(R{R}_{t-i}-\langle RR\rangle )(R{R}_{t-j}-\langle RR\rangle )$$ (2)

where $\langle RR\rangle =\frac{1}{h}{\sum }_{k=1}^{h}R{R}_{t-k}$. Equation (2) can be interpreted as a discrete Volterra-Wiener series with degree of nonlinearity d = 2 and memory h = max{p, q}[2]. Given the proposed parametric model, the nonlinear indices of the HR and HRV will be defined as a time-varying function of the parameters θ = [a₀, a₁,…, a_p, b₁₁, …, b_qq, θ].

B. Instantaneous Indices of HR and HRV

Heart rate is defined as the reciprocal of the R-R intervals. For RR measured in seconds, r = c(t − u_t)⁻¹ (where c = 60 s/min) is a physiological measurement in beats per minute (bpm). By the change-of-variables formula, the HR probability p(r) = p(c(t − u_t)⁻¹) is given by

$$p(r)=\mid \frac{dt}{dr}\mid p(t),$$ (3)

and the mean and the standard deviation of heart rate r can be derived [3],[4]. Essentially, the instantaneous indices of HR and HRV are characterized by the mean μ_HR and standard deviation σ_HR, respectively.

It is known from the point process theory [3],[4] that the conditional intensity function (CIF) λ(t) is related to the inter-event probability p(t) by a one-to-one transformation:

$$\lambda (t)=\frac{p(t)}{1‐{\int }_{u_t}^{t}p(\tau )d\tau }.$$ (4)

The estimated CIF can be used to evaluate the goodness-of-fit of the probabilistic model for the heartbeat dynamics.

C. Adaptive Point Process Filtering

Let θ denote the unknown parameters in the parametric probabilistic model, we can recursively estimate them via adaptive point process filtering [3]:

$${\mathbf{\theta }}_{k\mid k-1}={\mathbf{\theta }}_{k-1\mid k-1}$$ (5)

$$P_{k\mid k-1}=P_{k-1\mid k-1}+W$$ (6)

$${\mathbf{\theta }}_{k\mid k}={\mathbf{\theta }}_{k\mid k-1}+P_{k\mid k-1}(\nabla log{\lambda }_k)[n_k-{\lambda }_k\mathrm{\Delta }]$$ (7)

$$P_{k\mid k}={\left[P_{k\mid k-1}^{-1}+\nabla {\lambda }_k\nabla {{\lambda }_k}^T\frac{\mathrm{\Delta }}{{\lambda }_k}-{\nabla }^{2}log{\lambda }_k[n_k-{\lambda }_k\mathrm{\Delta }]\right]}^{-1}$$ (8)

where P and W denote the parameter and noise covariance matrices, respectively; Δ=5ms denotes the time bin size; $\mathrm{\Delta }{\lambda }_k=\frac{\partial {\lambda }_k}{\partial {\xi }_k}$ and ${\mathrm{\Delta }}^2{\lambda }_k=\frac{{\partial }^2{\lambda }_k}{\partial {\xi }_k\partial {\xi }_k^T}$ denotes the first- and second-order partial derivatives of the CIF w.r.t. θ at time t = kΔ, respectively. The indicator variable n_k =1 if a heart beat occurs in time ((k − 1)Δ, kΔ] and 0 otherwise.

D. Goodness-of-fit Tests

The goodness-of-fit of the model is based on the Kolmogorov-Smirnov (KS) test. Given a point process specified by J discrete events: 0 < u₁ <···< u_J < T define the random variable $z_j={\int }_{u_{j-1}}^{u_j}\lambda (\tau )d\tau$ for j = 1, 2,…, J − 1. If the model is correct, then the variables v_j = 1− exp (−z_j) are independent, uniformly distributed within the region [0, 1], and g_j = Φ⁻¹(v_j) (where Φ(′) denotes the cumulative density function (cdf) of the standard Gaussian distribution), are independent standard Gaussian random variables. To compute the KS test, the v_j s are sorted from the smallest to the largest value, and plotted against the cdf of the uniform density defined as $\frac{j-0.5}{J}$. If the model is correct, the points should lie on the 45 degree line. In Cartesian plot of the empirical cdf as the y-coordinate versus the uniform cdf as the x-coordinate, the 95% confidence interval lines are $y=x\pm \frac{1.36}{{(J-1)}^{1/2}}$. The KS distance, defined as the maximum distance between the KS plot and the 45° line, is used to measure the lack-of-fit between the model and the data. We also compute the autocorrelation function of g_j s: $\mathit{ACF}(m)=\frac{1}{J-m}{\sum }_{j=1}^{J-m}{g}_j{g}_{j+m}$. If the g_j s are independent, ACF(m) shall be small (around 0 and within the 95% confidence interval $\frac{1.96}{{(J-1)}^{1/2}}$) for all values of m.

IV. Instantaneous Higher-Order Spectral Analysis

Given the second-order Volterra-Wiener expansion for the instantaneous R-R interval mean {μ_RR(t)}, we may compute the linear time-varying parametric autospectrum:

$$Q(f,t)=\frac{{\sigma }_{RR}^2(t)}{{\mid 1-{\sum }_{k=1}^p{a}_k(t)e^{-j2\pi f}\mid }^2},$$ (9)

Note that the frequency unit is in Hz instead of beat/sample. By integrating (9) in each frequency band, we may compute the index within the VLF (0.01–0.05 Hz), LF (0.05– 0.15 Hz), or HF (0.15–0.5 Hz) ranges. In addition, let $B(f_1,f_2,t)={\sum }_{k=1}^q{\sum }_{l=1}^q{b}_{kl}(t)e^{-j2k\pi f_1}{e}^{-j2k\pi f_2}$ denote the Fourier transform of the second-order kernel coefficients {b_kl(t)}. From (2) we obtain [13]

$$B(-f_1,-f_2,t)=\frac{C(f_1,f_2,t)}{2Q(f_1,t)Q(f_2,t)},$$ (10)

where C(f₁, f₂, t)denotes the bispectrum (Fourier transform of the third-order moment). Let b(t) denote a vector that contains all of coefficients {b_kl (t)}; in light of (10), we may compute an index that quantifies the fractional contribution of the linear terms on the total power as

$$\rho (t)=\frac{\mid Q(f,t)\mid }{\mid Q(f,t)\mid +\mid C(f_1,f_2,t)\mid }=\frac{1}{1+2\mid \mathbf{b}(t)\mid `\mid Q(f,t)\mid (t)},$$

which can be viewed as a dynamic counterpart of the following static power ratio (with assumed stationarity)

$$\mathit{ratio}=\frac{\mathit{power}\mathit{spectrum}}{\mathit{power}\mathit{spectrum}+\mathit{bispectrum}}.$$ (11)

V. Experimental Data and Results

The first hearbeat data set was recorded under the protocol of “autonomic blockade assessment of the sympatho-vagal balance and RSA”. Detailed description of the experimental data was given in [16]. The recorded R-R interval time series last about 5 minutes. Here we only applied a subset of the data set which has been tested previously using the linear predictive model [6], [7]. The second heartbeat data set, which was retrieved from a public resource: Phyisonet (http://www.physionet.org/) [10], consists of R-R time series recorded from 7 congestive heart failure (CHF) patients (from BIDMC-CHF Database) and 10 healthy subjects (from MIT-BIH Normal Sinus Rhythm Database). Each R-R time series was artifact-free and lasted about 50 minutes (small segments of the original over 20-hrs recordings). Since these recordings are longer, they are suitable for studying the heart beat intervals that have complex dynamics.

A. Nonlinearity Test

A time-domain nonlinearity test [5], which uses the bootstrap technique based upon the third-order moment and bispectrum, was applied to the R-R time series for testing the presence of nonlinearity in the heart beat intervals. Null hypothesis assumes that the given time series is linear and stationary. The result of the hypothesis test is either H=0 (which indicates that the null hypothesis is accepted and p > 0.05) or H=1 (which indicates that the null hypothesis is rejected with 95% confidence). In the experiments here, we restricted the test to short term dependence by setting the number of laps M=8, and a total of 500 bootstrap replications were simulated for every test.

In the first data set, the level of nonlinearity varies from different postures and pharmacological conditions. For instance, most R-R time series in upright posture failed to reach significance in the nonlinearity test, and similar observations were also found in the PROP condition. In the second data set, 1 out of 10 R-R time series from the healthy subjects showed significant nonlinearity (p < 0.05), whereas in the CHF group, 3 out of 7 R-R time series failed to reach significance in the nonlinearity test. This result confirms that to some degree the heartbeat dynamics from healthy subjects are more nonlinear [15].

B. Performance Comparison

The performance comparison between the proposed nonlinear Volterra modeling and the standard linear modeling was measured by the KS distance. Specifically, the smaller the KS distance, the better is the model fit. In order to keep the number of unknown parameters approximately the same for fair comparison, we set p=8 in linear modeling and set p=4~6 and q=2 in nonlinear modeling. The initial parameters were optimized by first solving a Yule-Walker equation for the linear part using the first 0.5 or 1 min of recordings, and then fitting the residual error with the nonlinear part (via least squares). The comparative results are summarized in Tables I and II. As seen from the tables, nonlinear volterra modeling generally improves the model fit, especially when the R-R time series exhibit more nonlinearity (with small p-value in the nonlinearity test).

TABLE I.

RESULTS FROM SELECTED 4 SUBJECTS IN THE FIRST DATA SET (p-VALUES ARE OBTAINED FROM THE NONLINEARITY TESTS. THE NUMBERS IN BOLD FONT INDICATE THE KS FIT IS WITHIN THE 95% CONFIDENCE BOUND. THE TICK INDICATES AN IMPROVEMENT OF FIT BY INCLUSION OF NONLINEARITY.)

subject	epoch	p-value	KS dist. (linear→nonlinear)
11	control, supine	0.020	0.1220→0.1093 ✓
11	control, upright	0.166	0.0424→0.0517
11	ATR, supine	0.004	0.1085→0.1004 ✓
11	ATR. upright	0.002	0.1202→0.1135 ✓
11	DB, supine	0.018	0.1442→0.0805 ✓
11	DB, upright	0.156	0.1684→0.1374 ✓
15	control, supine	0.090	0.0590→0.0709
15	control, upright	0.132	0.0765→0.0927
15	ATR, supine	0.138	0.1277→0.0914 ✓
15	ATR, upright	0.128	0.1339→0.1191 ✓
15	DB, supine	0.288	0.1584→0.1087 ✓
15	DB, upright	0.230	0.1540→0.0940 ✓
20	control, supine	0.012	0.0416→0.0406 ✓
20	control, upright	0.136	0.0599→0.0354 ✓
20	PROP, supine	0.230	0.0994→0.0793 ✓
20	PROP, upright	0.142	0.0994→0.0570 ✓
20	DB, supine	<1e-8	0.1026→0.0684 ✓
20	DB, upright	0.004	0.1010→0.1269
21	control, supine	0.178	0.1343→0.1161 ✓
21	control, upright	0.208	0.0713→0.0578 ✓
21	PROP, supine	0.232	0.0506→0.0279 ✓
21	PROP, upright	0.012	0.1337→0.1227 ✓
21	DB, supine	0.094	0.1130→0.1063 ✓
21	DB, upright	0.430	0.1087→0.1017 ✓

TABLE II.

RESULTS FROM SELECTED 10 SUBJECTS IN THE SECOND DATA SET (p-VALUES ARE OBTAINED FROM THE NONLINEARITY TESTS).

subject	group	p-value	ρ (0.01–0.15 Hz)	KS dist. (linear→nonlin.)
01	CHF	0.10	0.9669	0.0813→0.0933
05	CHF	0.122	0.9837	0.1176→0.1224
08	CHF	0.012	0.9153	0.1004→0.0954 ✓
10	CHF	<1e-8	0.9510	0.0902→0.1321
11	CHF	0.02	0.9446	0.1043→0.0996 ✓
16265	healthy	<1e-8	0.8012	0.0933→0.0919 ✓
16273	healthy	<1e-8	0.9112	0.0925→0.0822 ✓
16483	healthy	0.004	0.9547	0.0819→0.0798 ✓
16786	healthy	<1e-8	0.8983	0.0938→0.0931 ✓
16795	healthy	<1e-8	0.9052	0.0925→0.0903 ✓

The same observation can also be found in the second dataset, although neither linear nor nonlinear models being tested thus far has passed the KS test (within 95% confidence bound). This may be due to the highly complex dynamics involved in these non-stationary heartbeat time series, or may be due to the insufficiency of our current model. Increasing the model memory might help to improve the goodness-of-fit, but it also increases both model and computational complexity. Determining a good tradeoff between the complexity and performance remains an issue that needs to be decided by practitioners.

C. Quantification of Self-similarity via Scaling Exponent

Heartbeat intervals have been reported to have fractal behavior and chaotic dynamics [11], [15]. Research effort was largely devoted to characterizing such nonlinear behavior at different timescales using relatively short recordings. In time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a non-stationary signal [14]. Based on the original R-R time series and the estimated μ_RR(t), we computed the scaling α-exponent, respectively, for each subject in the second data set. The fine temporal resolution (5 ms) enabled us to reveal the fractal structure of the evenly-spaced continuous-time signals without using any interpolation technique.

To evaluate the statistical differences between two groups for each index, we also conducted a nonparametric rank-sum test to test the null hypothesis that the medians of two sample groups are equal. The results are summarized in Table III and partly illustrated in Fig. 1. As seen in Table III, on average the CHF patients have lower HRV and greater HR. It also appeared that nearly all statistical indices are statistically significant. The insignificance of the α-exponent computed from the R-R time series might be due to the insufficiency of samples; in contrast, the α-exponent estimated from μ_RR(t)seems more accurate to characterize the group difference (using 24-hrs recordings, it was reported in [14] that the scaling exponent statistic computed from R-R time series is 1.24±0.22 for the CHF group and 1.00±0.11 for the healthy group). Hence, the HR, HRV, power ratio, and scaling exponent statistics can serve as useful metrics to distinguish the healthy and pathological conditions given relatively short heartbeat recordings.

TABLE III.

MEAN±STD STATISTICS OF THE SECOND DATA SET (p-VALUES ARE OBTAINED FROM THE RANK-SUM TESTS BETWEEN TWO GROUPS).

statistical index	CHF	Healthy	p-value
R-R (ms)	737±158	889±84	0.030
μHR (bpm)	86.4±18.9	71.9±10.1	0.031
σHR (bpm)	2.1 ± 0.8	5.9±2.2	0.003
ρ (0.01–0.15 Hz)	0.939±0.048	0.887±0.051	0.030
Ratio (0.01–0.15 Hz)	0.998±0.003	0.923±0.079	2 × 10−4
μRR ⇒ α-exponent	1.242±0.057	1.168±0.058	0.033
R-R ⇒ α-exponent	0.992±0.176	0.918±0.090	0.536

Figure 1.

Left panel: scatter plot comparison between 7 CHF (triangle) and 10 healthy (circle) subjects. Right panel: KS plot comparison for one selected subject 16483 (top linear vs. bottom nonlinear).

VI. Discussion

We have presented a method for characterizing nonlinear dynamics of the human heartbeat with the point process paradigm. Unlike other nonlinear modeling methods developed in the literature, our probabilistic model computes the instantaneous HR and HRV statistics. Based on the second order Volterra-Wiener series expansion, we use an adaptive point process filter to track the kernel coefficients and estimate the instantaneous parametric autospectrum and bispectrum, as well as the power ratio. In comparing the healthy and CHF subjects, the heartbeat exhibits less nonlinear dynamics in the pathological condition, which was confirmed by the nonlinearity test and the relative linear/nonlinear power ratio. Preliminary results have shown that our proposed point process model is beneficial to characterize the inherent nonlinearity of the heartbeat dynamics. In the future work, we will conduct a systematic study on more heartbeat data and use the local likelihood method [4] to determine proper orders for p and q. The examination of nonlinear interactions between heartbeat and other covariates (such as respiration and blood pressure) is also currently under investigation.