Spectral decomposition of heart rate variability using generalized harmonic analysis
Introduction
The autonomic nervous system plays an important role in monitoring conditions and making appropriate changes in the internal environment, such as in the heart rate, digestion, respiratory rate, salivation, perspiration, pupillary dilation, micturition (urination), and sexual arousal. The autonomic nervous system has two subsystems, the sympathetic nervous system and the parasympathetic nervous system, which differ in their anatomical characteristics and functional properties. The sympathetic nervous system is activated by either norepinephrine (noradrenaline) or epinephrine (adrenaline) neurotransmitters and operates when an individual is in an excited state, such as during exercise. The parasympathetic nervous system utilizes mainly acetylcholine and operates when an individual is relaxed—for instance, when eating or sleeping [1].
Fluctuations in the heart rate (heart rate variability, or HRV) in electrocardiogram (ECG) signals reveal the autonomic nervous function. There are two major approaches to measuring HRV. The RR mean interval, variance, and related statistics are used in time domain-based methods. Spectral analysis of the RR interval time series is used in frequency domain-based methods. In the frequency domain approach, the power within each frequency band is calculated. The bands are typically low-frequency (LF) bands, ranging from 0.04 to 0.15 Hz, which corresponds to the variation in blood pressure and high-frequency (HF) bands ranging from 0.15 to 0.4 Hz, which are derived from fluctuation in respiration [2], [3], [4], [5], [6], [7], [8].
To calculate the power of the RR interval variation within each frequency band (LF or HF), the fast Fourier transform (FFT) or an autoregressive (AR) model can be used. In both the FFT and AR methods, the power of an RR interval variation is calculated by integrating the area under the power spectrum curve within the frequency band. However, the frequency resolution of the FFT is determined by the data length, and its spectrum is affected by the time window, which is used to reduce the prediction error of the spectrum. Moreover, integrating the area under the power spectrum curve includes the power of undesired signals such as noise. In contrast, in the AR approach, the spectrum can be factorized into separate components. This method often yields negative power values [9], [10], [11]. Furthermore, AR parameters are determined using simple criteria such as the final prediction error (FPE) [12], Akaike’s information criterion (AIC) [13], criterion autoregressive transfer (CAT) function [14], or minimum prediction length (MDL) [15]. These criteria, however, seem to underestimate the optimum AR model order [9], [16].
Generalized harmonic analysis (GHA) [17] is a simple method that searches for frequency components in the signal, one by one. Specifically, a sinusoidal signal that minimizes the residual is found in the observed data section. The sinusoid is subtracted from the original signal. The second sinusoid is then found in the new original signal and these steps are repeated. The power of the extracted sinusoid can also be calculated. Unlike FFT, the frequency resolution of the GHA is not restricted to the inverse of the data time length and can be freely selected. GHA is widely used in acoustics research for tasks such as audio coding [18], [19] or reducing scratch noise in 78 rpm records [20].
The Pisarenko harmonic decomposition (PHD) [21] is widely used in various signal processing fields. The PHD method uses the eigenvector associated with the minimum eigenvalue to estimate the frequencies of the source signal.
The Lomb-Scargle (LS) periodogram [22], [23] is a well-defined technique for calculating the power spectrum density of an unevenly sampled signal. But, this method is also equivalent to FFT and AR in the equally-spaced RR interval time series [24]. The LS technique is not suitable for decomposing spectra and the GHA method is not adequate for calculating power spectral density of an unequidistantly sampled signal.
In this paper, the GHA method, which extracts a sinusoid and calculates its power from the RR interval time series signals, has been applied to the spectral analysis of HRV using the integral pulse frequency modulation (IPFM) model. The theoretical and measured power values and frequency calculated by GHA and PHD are compared. Besides, power value in the LF and HF bands from one real ECG signal with change in position by the three methods are compared. Furthermore, to evaluate the interchangeability of GHA and LS methods, Bland-Altman (B-A) [25] analysis is applied to power value in the LF and HF bands from MIT-BIH normal sinus rhythm database.
Section snippets
Integral pulse frequency modulation (IPFM) model
The IPFM model was proposed for exploring the properties of nerve spike trains [26], and subsequently has been applied to the RR interval time series of HRV [27], [28]. In the IPFM model, spikes are generated when the integrated value of the input signal p(t) = 1 + m(t) reaches a predefined threshold Im. Each time the integrated value reaches the threshold, a spike is generated and the integrated value y(t) is reset [27], [29], [30]. When m(t) = 0, output x(t) consists of a spike series with a
IPFM simulation signals
Fig. 3 displays the HRV spectral profiles of the simulated IPFM signal, which underwent spectral analysis using the GHA, PHD, and AR methods (Fig. 3(a) to 3(c), respectively). The vertical axes for the GHA and PHD methods are in power units of ms2. Thus, the power of the sinusoids is proportional to the vertical value. The vertical axes for the AR method are in conventional power spectral density units of ms2/Hz. Note that the vertical axes values in Fig. 3(a) are constant, but vary in Fig. 3
Discussion
The aim of the present paper is to propose a new method for decomposing sinusoids in the spectrum of HRV using the GHA method.
The GHA and PHD methods can compute the power of extracted sinusoids. The power computed using GHA does not depend on the number of extracted sinusoids, but the frequency and power of the sinusoids extracted by PHD vary with respect to the number of extracted sinusoids. Fig. 3(b) illustrates spectral profiles obtained for 1 to 48 sinusoids extracted by PHD using the
Conclusion
In this paper, a new technique for analyzing HRV spectra based on GHA was introduced. According to this method, the power of a sinusoid within LF or HF bandwidth can be obtained. GHA can also obtain measurements that are obtained by FFT-like methods such as the integrated area under a spectral curve. LF and HF band power value calculated by the GHA are interchangeable with ones by the Lomb-Scargle transform using public-domain RR interval database from short term recordings of 30 min. This
CRediT authorship contribution statement
Noriaki Satoh: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The author would like to thank Professor Emeritus Yutaka Tomita, Keio University, Japan and Professor Masahiro Aruga, Saitama Institute of Technology, Japan for their useful comments at Japanese domestic conferences about ergonomics. We thank Kimberly Moravec, PhD, from Edanz Group (https://en-author-services.edanzgroup.com/) for editing a draft of this manuscript.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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