TRANSFER FUNCTION OF THE HEART RATE CONTROL

SYSTEM WITH RESPIRATORY INPUT

The Classical Engineering Approach

Evgeny G. Vaschillo and Bronya Vaschillo

Center of Alcohol Studies, Rutgers, The State University of New Jersey, 607 Allison Road, Piscataway, NJ 08854, U.S.A.

Keywords: HR control system, Transfer function, Fourier filtration, HRV biofeedback, Resonance frequency.

Abstract: The classic “control system theory” approach was used to find the transfer function (TF) of the HR control

system with respiratory input. Eight healthy subjects, ages 19–40 participated in the study. Paced breathing

at seven frequencies in 0.5– 0.04 Hz range was used as sine-wave stimuli to assess the HR control system.

The sine-wave HR oscillation in response to each stimulus as the system’s output was recorded. Amplitude

and phase TFs were calculated for each frequency separately. The Fourier filtration procedure was used for

TF calculation. Experimentally obtained TFs revealed the same important features in all participants: 1) the

amplitude TF had its peak in a narrow frequency range around 0.1 Hz; 2) the phase TF successively

changed from positive to negative when breathing frequency increased, and passed “0” at frequency where

the amplitude TF peaked; 3) the peak frequency and magnitude were unique for each participant. These

features are evidence for the resonance property of the HR control system at a frequency around 0.1 Hz.

This study suggests that accurate identification of an individual’s resonance frequency can be found using

the TF features of HR control system and controlled breathing techniques.

1 INTRODUCTION

Respiratory activity continually perturbs the

cardiovascular system. Respiration modulates the

activity of most sympathetic and vagal efferents

through direct coupling between the respiratory and

autonomic centers and through modulation of central

sensitivity to baroreceptors and other afferent inputs

(Clynes, 1960; Saul et al., 1991). The autonomic

efferents, in turn, modulate peripheral vascular

resistance, heart rate (HR), and other autonomic

functions with respiratory periodicities. Thus,

respiration appears to actively participate in

autonomic and, in particular, HR regulation through

a very complex multiple-loops control system.

Respiration is a voluntary, controllable function and

thus can be exploited by researchers and clinicians

to develop novel approaches for correcting abnormal

autonomic regulation. For example, heart rate

variability (HRV) biofeedback can be used to

normalize autonomic regulation, increase baroreflex

gain and peak expiratory flow (Lehrer et al., 2003),

and treat asthma (Lehrer et al., 2004), major

depression (Karavidas et al., 2007), fibromyalgia

(Hassett et al., 2007), neurosis (Chernigovskaya et

al., 1990), and hypertension (McCrady et al., 2001).

Autonomic variability as well as the frequency

dependence of autonomic reactions to respiration are

critical factors in the development of such methods.

Accordingly, the classic engineering approach may

be useful in furthering our knowledge of this

system’s features.

The goal of this study is to assess the transfer

functions (TF) of the HR control system, using

breathing as the forcing function and heart rate as

the system response.

2 METHOD

The classic “control system theory” approach was

used to examine the transfer function of the HR

control system with respiration input. Sine-wave

stimuli of various frequencies were sent in-series to

the input of the system. The sine-wave HR

oscillation in response to each frequency stimulus

was recorded as the system’s output. Amplitude

(ratio of the amplitudes of the output and input sine-

wave signals) and phase (phase shift between the

233

G. Vaschillo E. and Vaschillo B. (2009).

TRANSFER FUNCTION OF THE HEART RATE CONTROL SYSTEM WITH RESPIRATORY INPUT - The Classical Engineering Approach.

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing, pages 233-238

DOI: 10.5220/0001434002330238

 SciTePress

sine-wave signals) transfer functions were calculated

separately for each frequency.

2.1 Participants and Procedure

Eight healthy subjects (5 males and 3 females), ages

19 – 40 (average 27.5) participated in the study. The

experimental session lasted one hour. Before the

session each participant was taught to breathe slowly

but not too deeply. A sine-wave pacer as well as the

participant’s current respiration curve were

presented on a computer screen. The participant was

instructed to follow the pacer with his/her breathing

such that his/her respiration curve precisely copied

the pacer’s amplitude and frequency. During the

experiment participants performed 7 paced breathing

tasks at 7 randomly presented frequencies (0.5,

0.25, 0.143, 0.11, 0.077, 0.055, and 0.04 Hz,

which corresponded to 30, 15, 5.6, 6.6, 4.6, 3.3,

and 2.4 breaths per minute). Each task included at

least 10 breaths and lasted for 3-5 minutes with 2

minutes break between them. The amplitude of the

pacer’s sine-wave curve was the same at all

frequencies and corresponded to approximately 1050

ml of tidal volume to provide relatively comfortable

breathing without hyperventilation.

2.2 Physiological Record

A J&J Engineering (Poulsbo, WA) I-330 DSP-12

physiograph was used to collect electrocardiogram

(ECG) and respiration data. ECG data were

collected from electrodes on the right arm and left

leg (Lead II), digitized at the rate of 1024 Hz. A

respiration strain gauge belt was attached around the

participant's chest. Respiration and pacer sine-wave

curves were digitized at the rate of 4 Hz.

Physiological data were collected during the whole

experiment and divided by tasks for data analysis.

2.3 Data Analysis

Beat-to-beat RR intervals (RRI) of the ECG signal

were measured and recalculated to heart rate [beats

per minute]. Cubic interpolation of the non-

equidistance HR sequence was completed, and the

HR curve was re-sampled at the rate of 4 Hz. The

TFs were calculated separately at each tested

frequency. The Fourier filtration algorithm (Eykhoff,

1974; Vaschillo et al., 2002) was applied for TF

calculation because it allowed more precise and easy

estimation of the amplitude and phase relation

between two sine-wave signals than cross-spectral

Fourier analysis. The algorithm was used to compute

two auxiliary TFs: (1) between the pacer as the input

and respiration as the output [TF(pr-resp)], and (2)

between the pacer as the input and HR as the output

[TF(pr-HR)]. This study targeted the TF of the HR

control system [TF(resp-HR)], which was calculated

by dividing TF(pr-HR) by TF(pr-resp), i.e. by

dividing the TF amplitudes and subtraction of the TF

phases.

3 RESULTS

The paced breathing exercise resulted in the sine-

wave oscillation in the participant’s HR paralleling

the sine-wave respiration curve. The amplitude and

phase relation between the pacer curve, respiration

curve, and sine-wave HR oscillation was strongly

dependent on pacer frequency.

3.1 Transfer Function of the Paced

Breathing

The TF(pr-resp) that reflects the participant’s ability

for precise pacer breathing is illustrated in Figure 1.

Participants were able to accurately maintain the

depth of breathing requested by the pacer, but

always preceded the pacer when asked to breathe at

a rate of less than 18 breaths per minute.

Figure 1: Transfer function of the paced breathing TF(pr-

resp) averaged across 8 participants: input – pacer, output

– respiration. Equality of the pacer and respiration curve

amplitudes correspond to value “1” of the amplitude

TF(pr-resp). Positive phase means that respiration

preceded the pacer.

0.9

0.95

1.05

1.1

1.15

0 0.1 0.2 0.3 0.4 0.5 0.6

Amplitude [arb]

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-5

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Phase [degree]

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3.2 Transfer Function between the

Pacer and Heart Rate (Auxiliary)

Figure 2 presents the auxiliary TF(pr-HR), which

does not have any physiological meaning but is

necessary to calculate the targeted TF of the HR

control system in the Fourier filtration procedure.

Figure 2: Transfer function TF(pr-HR) averaged across 8

participants: input – pacer, output – heart rate.

3.3 Transfer Function of the Heart

Rate Control System

Figure 3 presents TF(resp-HR) and shows how

respiration affected HR. HR response to respiration

was strongest when participants were breathing at

frequencies around 0.1 Hz. Full synchronization of

sine-wave HR oscillation with respiration (phase TF

(resp-HR) is 0˚) occurred at only one frequency,

named as resonance frequency. At this frequency,

the amplitude of the TF(resp-HR) reached its

maximum. The HR curve preceded the respiration

curve at frequencies lower than the resonance

frequency and lagged behind the respiration curve at

frequencies higher than the resonance frequency.

Individual TF(resp-HR) graphs are presented in the

appendix; these figures show that the resonance

frequency and HR oscillation amplitude at resonance

frequency were unique for each participant.

Figure 3: Transfer function of the HR control system

TF(resp-HR) averaged across 8 participants: input –

respiration, and output – heart rate. The amplitude of the

TF(resp-HR) shows the amplitude of sine-wave HR

oscillation elicited by respiration at the corresponding

frequency. The zero phase of the TF(resp-HR)

corresponds to full synchronization of respiration with the

sine-wave HR oscillation; that is, HR goes up with

inhalation and down with exhalation.

4 DISCUSSION

The Amplitude and Phase Transfer Functions of the

HR control system [TF(resp-HR)] that were found in

this study reflect the relationship between respiration

and HR oscillation in 0.04-0.5 Hz frequency range.

As illustrated by the TF(resp-HR), the amplitude of

the respiration-dependent HR oscillations and the

phase between the HR oscillation and respiration

curves both depend on respiration frequency.

While the present amplitude TF findings are

consistent with the results of other studies (Angelone

and Coulter, 1963), (Cook et al., 1998), (Hirsch and

Bishop, 1981), (Saul et al., 1991), the phase TF

findings reported herein are only partially supported

by prior research. These results are in close

agreement with the sympathetically-controlled TFs

reported by Saul et al. (1991). However, they differ

from Angelone and Coulter (1963), who found that

respiration and HR oscillation were in phase at

frequencies lower than 0.05 Hz and, that at ~0.1 Hz,

the phase angle was about -90º. They also differ

from Eckberg (1983) who reported the 0º phase

between respiration and HR oscillation occurred at

the ~0.2 Hz frequency.

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Ampitude [arb]

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Amplitude [arb]

TRANSFER FUNCTION OF THE HEART RATE CONTROL SYSTEM WITH RESPIRATORY INPUT - The Classical

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These discrepancies can be explained in part by

the findings of Saul et al. (1991) who noted a

dramatic difference between the TF(resp-HR) that is

only under sympathetic control (parasympathetic

blockade) and the TF that is under vagal control.

Indeed, most prior studies, including Angelone and

Coulter (1963) and Eckberg (1983), did not control

sympathetic and vagal activity in their studies. In the

present study, the paced breathing procedure was

challenging enough to activate sympathetic and

depress parasympathetic systems. Moreover,

discrepancies may exist due to the use of cross-

spectral Fourier analysis in these earlier studies,

which may not have provided accurate phase

estimation. The present study used more precise

Fourier filtration procedures to estimate phase

relations, and was thus able to obtain individual TFs

that were stable for all participants.

The TF(resp-HR)s reported in our study

consistently demonstrated that: 1) the amplitude TF

has its peak in a narrow frequency range around 0.1

Hz; 2) the phase TF successively changed from

positive to negative when breathing frequency

increased, and passed “0º” at same frequency where

the amplitude TF peaked; 3) the frequency and

magnitude of the peak were unique for each

participant. These findings lend support for the

identification of the resonance property of the HR

control system at a frequency of approximately 0.1

Hz.

It is known that HR baroreflex closed-loop

provides the ~0.1 Hz resonance properties in the HR

control system (Cevese et al., 2001), (Vaschillo,

Vaschillo, & Lehrer, 2006). Earlier studies showed

that HR resonance can be triggered not only by

respiration, but by other rhythmical stimuli, such as

rhythmical emotional stimulation (Vaschillo et al.,

2008) or rhythmical muscle tension (Vaschillo et al.,

2007). This suggests that the resonance property of

the HR control system does not depend on

respiration and the TF(resp-HR) reflects properties

of the HR baroreflex.

The studies also revealed that the ~0.1 Hz

resonance oscillation in HR usually is accompanied

by the same frequency high amplitude oscillation in

other cardiovascular functions (e.g., in arterial blood

pressure and in vessel tone (

Cooke et al., 1998; Lehrer

et al., 2003, 2004).

HRV biofeedback is used to train participants to

control breathing such that they can harness the 0.1

Hz resonance in the cardiovascular system. The

therapeutic effects of HRV biofeedback occur as the

result of generalized high-amplitude oscillations in

autonomic functions elicited by the biofeedback

procedure (Chernigovskaya et al., 1990; Lehrer et

al., 2003, 2004). These oscillations train autonomic

reflexes, and systematic training of autonomic

reflexes normalizes and improves autonomic

regulation. To maximize oscillations and,

accordingly, the therapeutic effects of biofeedback,

the patient’s precise resonance frequency should be

determined; however this historically has been

difficult to assess (Vaschillo, et al., 2002, 2006).

This study suggests that knowledge of the TF(resp-

HR) features and the use of controlled breathing

techniques may allow more accurate identification

of an individual’s resonance frequency.

5 CONCLUSIONS

Classical control system theory applied to the

investigation of physiological systems can be a

useful tool for the medical practice. Physiological

systems function via closed-loop reflexes within a

very narrow frequency range, thereby suggesting

that application of sine-wave stimuli with Fourier

filtration procedure may be more effective for

testing such systems than multi-frequency stimuli

with the cross-spectral Fourier analysis.

ACKNOWLEDGEMENTS

This research was supported by grants from the

National Institute of Alcohol Abuse and Alcoholism

(R01 AA015248 and K02 AA00325) and the

National Institute of Drug Abuse (P20 DA017552).

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APPENDIX

Individual Transfer Functions for 8

Participants

Participant A

Participant B

Participant C

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Participant D

Participant E

Participant F

Participant G

Participant H

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Phase (Degree)

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