## Subjects

Prior to the experiment, participants were instructed to abstain from alcohol for at least 24 hours, and from tea, coffee, nicotine, cocoa, and any food and beverages containing methylxanthine for at least 12 hours. Intense exercise training was prohibited for at least 6 hours prior to testing, and all participants were asked to empty their bladder within 30 minutes of testing (Ellingson et al., 2022). Before each procedure, participants rested for 10 minutes in a comfortable chair in a quiet room.

*Patient group*: The patient group consisting of 16 patients with stable ischemic HF (documented by prior myocardial infarction, percutaneous coronary intervention, or coronary artery by-pass grafting) and LVEF ≤ 50%. Additional inclusion criteria included sinus rhythm, optimal pharmacological therapy, stable clinical condition for at least 3 months before enrollment, and without significant features of hypervolemia at the moment of enrollment. The patients were excluded if they were younger than 18 years old, had a history of sustained ventricular arrhythmia (ventricular tachycardia or ventricular fibrillation) or cardiac arrest, were in New York Heart Association functional class IV, permanent atrial fibrillation or flutter, permanent second- or third-degree atrioventricular block, had an implanted pacemaker, had clinical features of coronary instability at the moment of enrolment, a revascularization (coronary angioplasty or/and surgery by-pass) within 3 months prior to the study, or incomplete coronary revascularization status (scheduled control coronarography, coronary angioplasty or surgery by-pass), clinical evidence of autonomic neuropathy, concomitant terminal disease and non-cardiologic comorbidities with a potentially unfavorable effect on survival. Baseline characteristics of the patients group were as follow: 15 out of 16 patients were men with a mean age of 62 ± 6 years and BMI of 29.1 ± 3.8 kg/m2. Most of the patients underwent myocardial infarction (88%), had hypertension (75%), hypercholesterolemia (69%) and were active smokers (75%). Diabetes and atrial paroxysmal fibrillation were present in 25%. Regarding pharmacotherapy 94% of patients received beta-blockers, 88% had angiotensin-converting enzyme inhibitors/angiotensin receptor blockers, 94% patients were treated with statins, 88% with antiplatelet drugs, 50% with mineralocorticoid receptor blockers and 25% with diuretics and anticoagulants. The study was conducted in accordance with the Helsinki recommendations and was approved by the Ethics Committee of Medical University of Gdansk (NKBBN/864/2022–2023).

**Control group**

Ten control participants (9 males and 1 female, age 28.5 ± 9.1 years, BMI = 24.1 ± 1.2 kg/m2), who were healthy, non-smokers, and over 18 years of age, participated in the study. The Ethics Committee of University of Regina (REB#2017-013) approved this study and the experimental protocol. All control participants provided signed informed consent forms.

### Experimental design

For both groups (control and patient) testing was carried out between 08.00 a.m. and 1.00 p.m. in a quiet, comfortable room set to a temperature of 18–20°C with minimal ambient lighting. Once all the necessary medical research equipment was attached to the volunteer participant (see "Measurements" section below), they were directed to lie down on a bed with a headrest for 15 minutes of the stabilization period (provided with a blanket if necessary).

## Measurements

During the experiment the researchers used a Finometer (Finapres 2300, Ohmeda) to measure the participants' blood pressure (BP). The Finometer employs a finger-cuff to monitor beat-to-beat blood pressure from the left middle finger. The heart rate (HR) was determined from an electrocardiogram (ECG) signal, with the ECG ground electrode placed on the left anterior superior iliac spine and the two primary leads located under the middle portion of each clavicle (Gruszecka et al. 2020).

The BP and ECG signals were obtained at a sampling rate of 300 Hz and then processed for analysis. Pre-processing steps involved detrending and normalization of the signals by subtracting their mean and dividing by their standard deviation (Gruszecki et al. 2018). Additionally, the signals were downsampled to a lower frequency of 20 Hz to reduce the amount of data while retaining important biological waveform features of the ECG and BP signals.

## Wavelet transform

Wavelet analysis was employed to identify and analyze the physiological mechanisms underlying the oscillations in the cardiovascular system. The wavelet transform is a powerful mathematical tool used in signal processing that enables the transformation of a signal from the time domain to the time-frequency domain. Unlike traditional Fourier analysis, which provides information about the frequency content of a signal but not its temporal evolution, wavelet analysis allows us to examine how the frequency content of a signal changes over time. The definition of the wavelet transform is:

$$W\left(s,t\right)=\frac{1}{\sqrt{s}}\underset{-\infty }{\overset{+\infty }{\int }}\phi \left(\frac{u-t}{s}\right)g\left(u\right)du$$

,

where \(W\left(s,t\right)\) is the wavelet coefficient, \(g\left(u\right)\) is the time series and \(\phi\) is the Morlet mother wavelet, scaled by factor \(s\) and translated in time by \(t\). The Morlet mother wavelet is defined by the equation:

$$\phi \left(u\right)=\frac{1}{\sqrt[4]{\pi }}\text{e}\text{x}\text{p}(-i2\pi u)\text{e}\text{x}\text{p}(-0.5{u}^{2})$$

,

where \(i=\sqrt{-1}\). The Morlet wavelet is a commonly used wavelet in signal processing due to its Gaussian shape, which allows for good localization of events in time and frequency domains (Bernjak et al. 2012). This means that the Morlet wavelet can accurately capture changes in frequency content over time, making it a valuable tool for analyzing complex signals with non-stationary oscillations, such as those seen in physiological systems. The Gaussian shape of the Morlet wavelet also helps to reduce spectral leakage, which can occur when analyzing signals with sharp, sudden changes in frequency content. When the Morlet wavelet is used for wavelet analysis, the resulting wavelet coefficients are complex numbers in the time-frequency plane:

$$X\left({\omega }_{k},{t}_{n}\right)={X}_{k,n}={a}_{k,n}+i{b}_{k,n}$$

.

They define the instantaneous relative phase,

$${\theta }_{k,n}=\text{a}\text{r}\text{c}\text{t}\text{a}\text{n}\left(\frac{{b}_{k,n}}{{a}_{k,n}}\right)$$

,

and the absolute amplitude,

$$\left|{X}_{k,n}\right|=\sqrt{{a}_{k,n}^{2}+{b}_{k,n}^{2}}$$

,

for each frequency and time.

During the measurement, heart may create phase modulations. A mathematical tool to find the relationship between the phases of two signals is the wavelet phase coherence (WPCO). WPCO enables us to determine whether the oscillations detected are significantly correlated over time and can be a valuable tool for investigating the complex dynamics of heart signals and their relationship to other physiological signals. To estimate the WPCO we used the following expression (Lachaux et al. 2002):

$${C}_{\theta }\left({f}_{k}\right)=\frac{1}{n}\left|\sum _{t=1}^{n}\text{e}\text{x}\text{p}\left[i\left({\theta }_{2k,n}-{\theta }_{1k,n}\right)\right]\right|$$

,

where \({\theta }_{k,n}=\text{a}\text{r}\text{c}\text{t}\text{a}\text{n}\left(\frac{{b}_{k,n}}{{a}_{k,n}}\right)\) is an instantaneous measure of phases at each time \({\text{t}}_{\text{n}}\) and frequency \({f}_{k}\) for both signals. WPCO approaching zero when two oscillations are unrelated and their phase difference continuously changes with time. When two oscillations are related, their phase difference remains constant with time, and the wavelet phase coherence (WPCO) approaches 1.

Additionally, we can obtain information about the phase lag of one oscillator relative to the other by calculating the phase difference defined as:

## Statistical analysis

Nonparametric statistical tests were used to avoid assumptions of normality. Specifically, the Wilcoxon rank sum test was used to assess the significance of differences in median values between the control and patient groups. Significance was set at p < 0.05.

To determine the statistical significance of the estimated values of phase coherence, we employed the surrogate data testing method (Lancaster et al. 2018). Since fewer cycles of oscillations naturally occur at lower frequencies, there can be artificially increased wavelet phase coherence, even in cases where there is none. The surrogate analysis enables us to identify a significance level above which the phase coherence can be deemed physically meaningful. For this purpose, we used intersubject surrogates (Sun et al. 2012), which assume that the signals collected from different subjects are independent but have similar characteristics. The threshold value of phase coherence obtained at each frequency was then compared to the surrogate threshold, and coherence values above the threshold were considered statistically significant.