INTRODUCTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

ANALYSIS OF HEART RATE VARIABILITY (HRV) has received much attention in the recent physiological literature as a method for extracting information on neural mechanisms regulating the cardiovascular system (4, 30, 31, 36). It is generally assumed that fluctuations in cardiovascular parameters originate from the interaction between the sympathetic and parasympathetic neural branches and other low-frequency (LF) sources of noise (such as those caused by humoral, thermal, or vasomotor control). In particular, two rhythms are generally observed in the heart period spectrum: a respiratory or high-frequency (HF) rhythm (located around 0.2-0.25 Hz in humans), which is considered a marker of vagal activity, and a LF rhythm (around 0.1 Hz in humans), which is thought to be a marker of sympathetic activity (28) or may reflect both vagal and sympathetic influences (3). Various maneuvers, which activate the cardiovascular control system (such as posture change or tilting) affect the heart period spectrum, leading to an increase in the LF component compared with the HF component.

Because of its potential physiological and clinical impact, analysis of cardiovascular variability has been the subject of extensive research in the past decades, starting with the pioneering works by Guyton and Harris (18). Fundamental contributions in the field are the works by Akselrod et al. (4) and Koepchen (24). Recent studies include both modern signal processing techniques (7, 33) and mathematical models (11, 15, 23, 39, 43). However, the etiology of HRV in health and disease is still a matter of debate among physiologists. In particular, the role played by the various feedback regulatory loops and of sympathetic versus parasympathetic neural branches is still insufficiently clarified.

Mathematical models may play an important role in clarifying the genesis of cardiovascular parameter variability and testing existing theories in accurate quantitative terms. Indeed, a few models have been presented in the literature with different purposes (11, 15, 23, 39, 43). However, all existing models are based on an oversimplified description of cardiovascular control; in particular, numerical parameter values, heart and vessel dynamics, and/or regulatory mechanism actions do not properly reflect present physiological knowledge. The aim of this study is to use a new mathematical model of short-term cardiovascular regulation to investigate the possible mechanisms leading to heart period fluctuations in reliable physiological terms. The model includes several major aspects that were not addressed in previous theoretical studies devoted to HRV analysis. In particular, the model incorporates a separate distinction of systemic and pulmonary circulation, sympathetic feedback control loops working on systemic resistance, unstressed volume and heart contractility, sympathovagal control of heart period, the mechanical effect of respiration on venous return and a very low-frequency (VLF) vasomotor term. Moreover, all these aspects are simulated on the basis of existing physiological data.

In the present study, as in our previous models, we assumed that heart period is the regulated quantity instead of heart rate (HR). In fact, as pointed out by Hainsworth (20), HR is not the appropriate quantity to quantify autonomic effects, because of the gross nonlinearity of the relationship linking HR to the efferent vagal and sympathetic activities [see Levy and Zieske (27)]. Conversely, these nonlinear relationships are naturally converted to linear if pulse interval is used instead of HR (see Ref. 48 for more details).

Through a sensitivity analysis of model parameters, the present work aspires to clarify the conditions that may lead to clear LF oscillations in the heart period spectrum and to analyze the relationships between the LF and HF rhythms.

	QUALITATIVE MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

The present model differs from a former model, described in Ref. 48, as to the following points.

Simplifications

Improvements

In the following description only the main aspects of the model are presented in qualitative terms. Model equations are given in the APPENDIX.

Heart and Vessels

The vascular system comprehends a separate description of the pulmonary and the systemic circulation. The latter, in turn, contains the parallel arrangement of the splanchnic circulation and of the other, extrasplanchnic systemic vessels. Hemodynamics in each district (pulmonary, splanchnic, and systemic extrasplanchnic) is reproduced by means of resistive, capacitive, and inertial terms, as described previously (48) (see Fig. 1). In particular, blood volume stored in capacitive terms is the sum of an unstressed volume and a stressed volume, the latter being computed as the product of compliance and transmural pressure. A further compartment, not included in the previous work, represents the large systemic veins inside the thorax, carrying venous return to the right heart (see subscript tv in Fig. 1). A separate description of thoracic veins has been adopted to achieve a more accurate description of the effect of respiratory changes on venous return and cardiac output.

View larger version (31K): [in this window] [in a new window]   Fig. 1.   Hydraulic analog of the cardiovascular system. P, pressure; F, flow; R, resistance; L, inertance; C, compliance; sa, systemic arteries; sp and sv, splanchnic peripheral and splanchnic venous circulation; ep and ev, extrasplanchnic peripheral and extrasplanchnic venous circulation; tv, systemic thoracic veins; ra and rv, right atrium and right ventricle; pa, pulmonary arteries; pp and pv, pulmonary peripheral and pulmonary venous circulation; la and lv, left atrium and left ventricle; Pmax,rv and Pmax,lv, right and left ventricle pressure in isometric conditions; Pthor, intrathoracic pressure, i.e., the extravascular pressure for the compartments located inside the thorax (surrounded by a dashed line); Pabd, abdominal pressure, which is the extravascular pressure for the splanchnic circulation (delimited by a dash-dotted line); Fo,r and Fo,l, blood flow from the right and left ventricle, respectively.

The mechanical effect of respiration on cardiovascular quantities has been simulated by using time-varying expressions for intrathoracic pressure (P_thor, which is extravascular pressure for all compartments inside the thorax; Fig. 1) and for abdominal pressure (P_abd, which is extravascular pressure for the splanchnic circulation). The expressions for P_thor and P_abd during the respiratory cycle have been given according to the patterns reported in Ref. 35 (see APPENDIX for more details). In the present study we assumed that the subject breathes with a constant respiratory cycle 5 s long.

Finally, respiratory volume (which is an input for lung stretch receptors) has been computed as a linear function of P_thor. Parameters of this pressure-volume relationship were chosen to attain physiological values of tidal volume, minute ventilation, and end-expiration volume in humans (34, 49).

Regulation Mechanisms

View larger version (6K): [in this window] [in a new window]   Fig. 2.   Sympathetic regulation mechanisms acting on the generic effector .  may represent peripheral resistance in the splanchnic or extrasplanchnic systemic vascular beds (Rsp or Rep), venous unstressed volume in the splanchnic or extrasplanchnic systemic vascular bed (Vusv or Vuev), or end-systolic elastance in right or left ventricle (Emax,rv or Emax,rv). Psa and Psan are instantaneous systemic arterial pressure and its normal mean value, respectively; VL indicates lung volume, and VLn is its value at the end of expiration. Psa  Psan and VL  VLn are the error perceived by arterial baroreceptors and pulmonary stretch receptors, respectively; Ga and Gp are the maximal gains of the 2 groups of receptors on effector . The 2 pieces of information from the receptors are summed up and passed through a static sigmoidal relationship (second block). The 3rd and 4th blocks represent a pure delay, D, and a low-pass first-order dynamic with time constant .

The heart period control is more complex, involving a balance between vagal and sympathetic activities. In particular, dynamics of the vagal and sympathetic mechanisms are different: the vagal control is characterized by a rapid response, which is completed within two or three cardiac beats, whereas sympathetic control requires many seconds (20, 22, 32). Accordingly, heart period regulation in the model is described through the more complex diagram in Fig. 3. Worth noting is the existence of different gains and different dynamics for the vagal and sympathetic paths, whereas the sigmoidal relationship still describes the upper and lower limits of the effector response. Of course, some of the gains in Figs. 2 and 3 can be equal to zero (see Assignment of Model Parameters) if the corresponding mechanism plays a negligible role on the specific effector response.

View larger version (13K): [in this window] [in a new window]   Fig. 3.   Feedback regulation of heart period, T. Control of heart period is different from those of the other effectors (see Fig. 2) because it involves a balance between vagal activity (top branch, subscript v) and sympathetic activity (bottom branch, subscript s). The meaning of the symbols is analogous to Fig. 2.

Finally, the existence of other LF sources of noise (such as those caused by humoral and thermal control or vasomotion) has been accounted for in an empirical manner, by superimposing a LF (<0.12 Hz) uniformly distributed noise (zero mean value and assigned variance) on the expression of peripheral extrasplanchnic systemic resistance. The power spectrum of noise used in the simulations decreases quite linearly between 0 and 0.12 Hz.

Assignment of Model Parameters

First, a value for the gains of the lung stretch receptor mechanism on HR and resistance has been given on the basis of experimental data obtained in the dog (14, 19). These authors measured the changes in HR and in total systemic resistance during steady-state variations in pulmonary inflation pressure and/or pulmonary volume (see Fig. 4). Moreover, according to Ref. 1, we assumed that the effect of lung stretch receptors on HR is exclusively mediated by the vagus. Finally, we are not aware of any effect of lung stretch receptors on heart contractility and on venous unstressed volumes; hence, the corresponding mechanism gains were set to zero.

View larger version (22K): [in this window] [in a new window]   Fig. 4.   Response of heart rate (HR; A) and systemic vascular resistance (B) to lung-stretch receptor activation. Continuous lines are model results. Simulations were performed in the absence of arterial baroreflex (i.e., only lung stretch receptors are operative). Hence, all the baroreflex gains were set at zero. , , , results of 3 experiments reported in Hainsworth (19); * , experimental data from Daly et al. (14). To facilitate comparison, we maintained the same input stimuli as used in in vivo experiments, i.e., inflation pressure or inflation volume. Increase in inflation volume has been expressed in % of tidal volume (VT).

Second, a preliminary value for the gains of all sympathetic mechanisms activated by the arterial baroreflex [i.e., parameter G_a in Fig. 2] and the upper and lower limits of the sigmoidal relationships (with the exception of the relationship for heart period, which is described below) have been given to simulate the results of open-loop experiments performed in vagotomized dogs (5, 9, 12, 13, 38, 42, 44). In these experiments the carotid sinuses are isolated from the rest of the circulation and their pressure is changed in steps. However, because the vagus is cut, neither the efferent vagal activity nor the afferent activities from aortic baroreceptors and lung stretch receptors concur with the observed responses. Hence, all vagal gains and lung stretch receptor gains on peripheral resistances were set to zero during these simulations. The results are shown in Fig. 5.

View larger version (36K): [in this window] [in a new window]   Fig. 5.   Summary of open-loop responses of arterial baroreflex in vagotomized conditions. A: systemic arterial pressure. B: cardiac output. C: heart period changes. D: systemic resistance. E: splanchnic unstressed volume. F: extrasplanchnic unstressed volume. Continuous thick lines are model simulation results. To simulate open-loop conditions, pressure at baroreceptors was given a constant nonpulsating value different from arterial pressure, changed in steps from 35 to 155 mmHg. To simulate vagotomy, pulmonary receptor gains (Gp in Figs. 2 and 3) and baroreflex vagal gain (GaTv in Fig. 3) were set at zero. Model results are compared with experimental data in the vagotomized dog in open-loop conditions. To compare model data with data in the dog, the corresponding quantities were normalized or expressed per unit weight. Experimental points are from Refs. 5, 9, 12, 13, 38, 42, and 44. Only in the case of heart period changes (C) has the simulation been repeated (dashed line) by including the baroreflex vagal gain as well as pulmonary receptor gains, to reproduce results of phenylephrine infusion in young human volunteers (; Ref. 25). Moreover, during the last simulation, baroreflex sympathetic gain on heart period (GaTs in Fig. 3) and on peripheral resistances (GaRep and GaRsp in Fig. 2) have been enhanced to account for the role of aortic baroreceptors (17, 40, 41). The 3 experimental lines in E and F (unstressed volume control) are means ± SD.

Third, to complete the calculation of the arterial baroreflex, we must assign the gain of the vagal control on HR. Moreover, the possible role of the extracarotid (mainly aortic) arterial baroreceptors must be assessed. In this regard, some authors have observed that these baroreceptors play a major role in the control of HR in humans and that they significantly contribute to the increase in sympathetic activity to resistance vessels (17, 40, 41). Accordingly, the vagal and sympathetic gains that characterize heart period control by the arterial baroreceptors have been reassigned to reproduce the changes in heart period observed in young, healthy men (25) during pharmacological changes in arterial pressure (see Fig. 5). To account for the possible role of extracarotid baroreceptors, the gain of the sympathetic control of HR exhibits a higher value compared with the value inferred from experiments in vagotomized dogs. Similarly, the gain of the sympathetic control on systemic arterial resistances has been increased, to account for the possible effect of extracarotid baroreceptors. We assumed a ratio of extracarotid versus carotid sympathetic control on resistance and heart period as large as 2.5, which approximately agrees with data by Ferguson et al. (17).

Mechanism dynamics. All dynamic parameters in the control loop have been assigned on the basis of dog experiments. The time constant and time delay of the control of contractility (elastance; E_max) have been given to reproduce the frequency dependence of the open-loop transfer function from carotid sinus pressure to end-systolic elastance (26). The time constant and pure delay of the resistance control have been given according to data reported previously (16). The time constant and pure delay of the venous unstressed volume control are higher than those of the resistance control: in fact, ~1 min is required before the accomplishment of complete active venoconstriction (45). The time constants and pure delays of the HR control have been assigned considering that the effect of vagus stimulation on heart period is completed within two or three beats, whereas the sympathetic control is characterized by slower dynamics (a few seconds) (20, 22, 32).

	RESULTS

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

Figure 6A shows a segment of the heart period time pattern, simulated with the model using the basal parameter values as in Table 1. Figure 6B reports the power spectral density computed from a longer simulated signal (1,000 s) with the Welch averaged periodogram method (2). The sequence was first detrended to eliminate frequencies close to zero; then several overlapping sections (60-s duration each, windowed by means of the Hanning window) were used to compute the averaged periodogram. The results clearly show the presence of two peaks in the power spectral density. The HF peak (0.2 Hz) reflects the respiratory activity, transmitted to the cardiovascular system both via the extravascular thoracic and abdominal pressures and via action of the lung stretch receptors. The LF peak represents a resonance of the control loops, located around 0.1 Hz. Quite evident in the spectrum also is the effect of the additional noise at very low frequencies.

View larger version (16K): [in this window] [in a new window]   Fig. 6.   A: 200-s segment of heart period (HP) temporal pattern simulated with the model by using the basal parameter values (Table 1). B: power spectral density (PSD) computed on a longer simulated signal (1,000 s). The spectrum was obtained by breaking the signal into several sections (60-s duration each; detrended to eliminate the frequencies close to zero and then windowed by means of the Hanning window and zero-padded) and by averaging the periodograms of these sections (Welch's averaged modified periodogram method).

To clarify the role of the vagal and sympathetic mechanisms in the genesis of the power spectrum, we performed a sensitivity analysis on the strength of the sympathetic and vagal branches. This is summarized in Fig. 7. Figure 7, A and B, shows the effect of increasing or decreasing the gains of all sympathetic mechanisms (working on heart period, peripheral resistance, venous unstressed volume, and heart contractility) by ±20% of their basal value. Simulations show that even a moderate increase in the sympathetic gains causes a significant increase in the LF component, whereas the HF component exhibits just a mild decrease (Fig. 7A). By contrast, a moderate reduction in the sympathetic gains strongly attenuates the LF component (which becomes almost indistinguishable from noise) with a small increase in the HF component.

View larger version (28K): [in this window] [in a new window]   Fig. 7.   Results of sensitivity analysis on sympathetic and vagal gains. A and B: effect on the PSD of increasing (A) and decreasing (B) the gains of all sympathetic mechanisms (working on peripheral resistances, venous unstressed volumes, heart period, and heart contractility) by 20% of their basal values. C and D: effect of increasing (C) and decreasing (D) vagal gains (i.e., the vagal gains on heart period from arterial baroreceptors and the lung stretch receptors) by 20% of their basal values. E: combined effect of increasing sympathetic gains by 20% and decreasing vagal gains by 20% (i.e., the simultaneous occurrence of the two situations depicted in A and D); F: effect of decreasing the amplitude of the very low-frequency random noise by 50%. For comparison, the basal power spectrum is reported in each panel (dashed lines).

Figure 7, C and D, shows the effect of a moderate change (±20%) in the vagal gains (that is, the vagal gains on heart period from both arterial baroreceptors and lung stretch receptors). Increasing the vagal gains induces a large increase in the HF component, whereas the LF component is reduced (although the sympathetic gains were still set at their basal value). The opposite effect (significant decrease in the HF band with an increase in the LF band) occurs when the vagal gains are reduced.

The previous results are summarized in Fig. 7E, where the sympathetic gains were increased by 20% and the vagal gains simultaneously decreased (20%). The effect is a dramatic rise in the LF component of the spectrum, with a reduction of the HF peak.

Furthermore, to analyze the effect of VLF noise (such as that induced by thermal or hormonal regulation) on the spectrum, the simulation was repeated, with basal parameter values, by reducing the amplitude of the random noise by 50% of the initial value. The results, shown in Fig. 7F, attest that the presence of this VLF noise is important in the production of a clear LF component. The resonance in the control loops amplifies the existing noise at ~0.1 Hz, thus resulting in a more evident peak compared with the case with almost no noise.

In conclusion, sensitivity analyses in Fig. 7 point out that the HF peak mainly reflects the strength of vagal control (in fact, it is only mildly affected by the sympathetic gains), whereas the LF peak exhibits a more complex dependence on sympathetic and vagal mechanisms and on additional noise. In fact, this peak is affected both by a change in sympathetic strength and by a change in the vagal component. Moreover, the LF peak is also influenced by the level of superimposed noise.

The simulation results in Fig. 7 summarize the role of the sympathetic and vagal efferent branches, and of noise, on heart period variability. However, the strength of oscillations not only depends on efferent activity but also on any other component within the control loops. In particular, any change in the sensitivity of the different effectors to efferent activity may cause a change in the oscillation strength. This point is usually ignored in the physiological literature, whereas the amplitude of the LF and HF peaks is ascribed merely to efferent activity, by neglecting the role of the other components in the loop.

To analyze this aspect, we performed a sensitivity analysis on each effector response separately from the others. To this end, we selectively changed the central slope of the sigmoidal relationship for each effector (systemic peripheral resistance, venous unstressed volume, heart contractility, and heart period) by maintaining the sensitivity of all other effectors at the basal value. The slope of each effector was modified by ±50% compared with the normal level, and the effect on the LF and HF peaks was evaluated.

The results are shown in Fig. 8, with reference to a modification of the response of peripheral resistances (Fig. 8, A and B) and heart period (Fig. 8, C and D). The results concerning a modification of the response for venous unstressed volumes and contractility are not shown because a change in these effectors did not cause any appreciable change in the strength and position of the LF and HF peaks.

View larger version (28K): [in this window] [in a new window]   Fig. 8.   Results of sensitivity analysis on effector responses. A and B: effect on the PSD of increasing (A) and decreasing (B) the slope of the sigmoidal curve of the peripheral resistances (both splanchnic and extrasplanchnic) by 50% of its basal value. C and D: effect on the PSD of increasing (C) and decreasing (D) the slope of the sigmoidal curve of the heart period by 50% of its basal value. Changes in the slope of the sigmoidal curve of the other 2 effectors (heart contractility and venous unstressed volume) do not cause any appreciable alterations in the power spectrum; hence the corresponding data are omitted.

The results in Fig. 8, A and B, show that the response of arterioles (which are the effectors for the resistance control) plays a pivotal role in the genesis of the LF wave. In fact, if the resistance response is depressed, the LF peak almost completely disappears, whereas a strong resistance response manifestly increases the LF peak. Moreover, as well expected, the resistance sensitivity has no effect on the HF peak.

The results in Fig. 8, C and D, reveal that a strong sensitivity of the cardiac pacemaker (and hence of heart period response) to efferent activity results in a higher HF peak, but this sensitivity has a modest effect on the amplitude of the LF peak. As well expected, a scarce sensitivity of the sinus node causes a depression of both LF and HF peaks.

These results stress that the amplitudes of the HF and LF peaks represent not only the variations in efferent activity but also the sensitivity of the effectors (mainly arterioles and sinus node). In particular, the LF peak is especially affected by the strength of the resistance control (which depends both on variations in sympathetic efferent activity and on the reactivity of resistance vessels) whereas the HF peak is especially expressive of the vagal control on heart period and of sinus node reactivity.

	DISCUSSION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

The present work investigates the role of some short-term regulatory mechanisms on the genesis of HRV with the use of mathematical modeling and computer simulation techniques. Although various models have been presented in recent years for the study of HRV, the present model introduces several new aspects and overcomes previous limitations.

The idea that the arterial pressure control loop can cause oscillations in cardiovascular quantities has been portrayed in the physiological literature for many decades [see Guyton and Harris (18); see also Koepchen (24) for an ample review]. Kitney (23) first proposed a nonlinear model that involves a negative feedback loop, a pure time delay, and a switching element to theoretically analyze the oscillatory behavior of the blood pressure control system. A popular model of HRV was proposed by deBoer et al. (15) in the late 1980s. It consists of a set of difference equations describing the baroreflex control, the input impedance of the systemic arterial tree, the contractile properties of the myocardium, and the mechanical effect of respiration. With that model, the authors were able to explain the presence of a LF peak in the HR spectrum, ascribing it to a resonance of the baroreflex control loop, mainly due to the sympathetic time delay. However, these models are based on a very simplistic description of the cardiovascular system and of the heart, and they include just a few features of the pressure control loop. Hence, their value lies more in the analysis of the mathematical properties of the equations than in physiological soundness.

A simple model of HRV, which may exhibit chaotic behavior, was recently presented by Cavalcanti and Belardinelli (11). In this case, too, however, the emphasis is more on the mathematical route leading to chaos than on physiological reliability. "Reduced" models, oriented to the analysis of a simple feedback loop, have also been presented (10, 39).

Seydnejad and Kitney (43) recently presented a more comprehensive model of cardiovascular regulation, finalized to the study of blood pressure and HRV. The model consists of a set of differential equations and includes several aspects of cardiovascular regulation, such as the afferent baroreflex, the vagal and sympathetic modulation of HR, the sympathetic control of the vasculature, the mechanical effect of respiration, a centrogenic oscillator, and a VLF vasorhythm. However, the nonlinear dependence of blood pressure on HR, on respiration, and on sympathetic excitation (i.e., the overall circulatory response) was identified empirically, through the analysis of Volterra series expansion, without the use of a cardiovascular model. Empirical mathematical models, based on the identification of autoregressive equations, were extensively used by Baselli et al. (see Ref. 7 for a review). Although these last models represent helpful empirical tools to quantitatively analyze spectra and extract specific features from data, they do not provide a physiological underpinning for the mechanisms leading to cardiovascular variability.

The previous compendium underlines the need for a comprehensive mathematical model that carefully embodies physiological knowledge and can elucidate the possible origin of cardiovascular variability without the need for empirical learning procedures. The present model incorporates several aspects that were previously neglected, such as the control of venous unstressed volume and contractility, the lung stretch receptor reflex, heart pulsatility, the effects of intrathoracic and abdominal pressure changes on pulmonary circulation and on venous return, and a VLF vasomotor noise. Moreover, each feedback loop is separately described with its own parameter values. We are not aware of any previous model that summarizes all these aspects into a single theoretical structure. Furthermore, parameters in the model were given on the basis of physiological experiments in the literature (generally performed in open-loop conditions) and not on the basis of the final desired behavior. The pattern of HRV arises ultimately as an emergent property, when all model components are combined, and the model works in its natural closed-loop condition. This aspect strongly differentiates physiological models (like the present) from empirical models.

The primary result of the present study is that a spectrum of heart period variability, similar to that observed in human subjects, emerges spontaneously from model simulations using basal parameter values. Subsequently, the sensitivity analysis of the effect of parameter changes furnished interesting clues on the origin and physiological significance of the HF and LF spectral components.

HF Component

A common viewpoint in the literature is that the HF peak can be considered as an index of vagal activity. The previous description, and the sensitivity analysis shown in Fig. 7, confirm that the HF peak is significantly affected by the vagal gains and it is quite independent of the sympathetic gains. Hence, changes in the HF peak can be considered to be almost entirely caused by vagal control actions. However, care must be taken in considering the HF peak as an index of the strength in vagal control. In fact, the HF peak is modulated by all factors affecting the input to baroreflex and the lung stretch reflex [such as the depth and frequency of breathing (21), venous compliances in the thoracic and abdominal cavity, posture changes (47), etc.]. Moreover, as shown in Fig. 8, C and D, the HF peak depends strongly on the sensitivity of the cardiac pacemaker to efferent activity. Hence, as suggested by Akselrod (3) and Malpas (31), in different subjects and/or under different breathing conditions, the HF spectral component may be largely different even in the presence of an equivalent vagal gain.

LF Peak

However, the sensitivity analysis in Fig. 7 shows that other factors may modulate the LF peak considerably. First, a reduction in the vagal gains induces an appreciable increase in the LF peak, and vice versa. The model suggests that if the vagal mechanisms are strong they can damp the alterations in heart period induced by the sympathetic loop. By contrast, if the vagal gains are reduced, oscillations in heart period induced by the sympathetic resonance can be entirely manifested. In addition, the LF peak is dramatically affected by the presence of vasomotor noise in the LF and VLF range. This noise is amplified by the resonance in the baroreflex loop, placed around 0.1 Hz: the higher the noise, the higher is its amplification and so the higher the peak in the power spectrum. The latter result underlines the difficulty in considering the LF peak merely as a marker of short-term baroreflex regulation, without regard for other factors (such as vasomotion, thermal regulation, or hormonal regulation) that may represent noise terms for the autonomic control loops. This conclusion agrees with the observation that calcium blockade, which attenuates vasomotor fluctuations, also reduces the LF peak (50).

The effect of changes in vagal gains on the LF peak (Fig. 7, C and D) deserves a comment. The result in Fig. 7 seems to contradict the observation by some authors (47), according to whom the LF peak is reduced by atropine, which is a vagal blocking agent. Conversely, in our model a reduction in vagal gain causes an increase in the LF peak. This contradiction, however, is only apparent and can be explained by the difference between average activity and fluctuations. The peaks in power spectrum reflect the fluctuations in neural activities at that given frequency, not the mean value of the neural activity. The fluctuations are conceptually different from the mean activity, and the relationship between these two aspects is far from being fully understood (3, 31). In the model, fluctuations are especially determined by the mechanism gains but are also affected by the working point along the sigmoidal characteristic (see Fig. 2). All of the sensitivity analyses illustrated in Fig. 7 explore the effect of variations in the mechanism gains only, which cause large changes in fluctuations without significantly modifying the working point along the sigmoidal relationship. Indeed, throughout this study, the working point was always located in the central linear part of the sigmoidal curve. In contrast, maneuvers that modify the average neural activity (such as atropine infusion, tilting, or exercise) presumably cause a shift in the working point along the sigmoidal relationship. If the working point moves from the central part of the sigmoidal relationship toward saturation, fluctuations are reduced. By contrast, moving the working point from a saturation region toward the central region causes an increase in fluctuations. By way of example, during exercise or heart failure, average sympathetic activity increases but the LF power may be reduced (6, 46). Similarly, if vagal activity is partially abolished by atropine administration, the working point of the heart period shifts toward the lower saturation region (i.e., we have cardioacceleration), which dampens heart period variability. This effect should not be confused with a reduction in the vagal gain.

Furthermore, the sensitivity analysis in Fig. 8 reveals that the amplitude of the LF peak not only depends on efferent neural activity (sympathetic vs. vagal) but is also markedly affected by the individual sensitivity of different effectors. In particular, the present results suggest that the sensitivity of resistance vessels (arterioles) to sympathetic efferent activity plays a major role in the origin of the LF peak, whereas the other effectors (such as venous unstressed volume or contractility) are less important. The reason for this finding is that LF fluctuations originate from oscillations in blood pressure, which are detected by the baroreflex system and transmitted to heart period fluctuations. Pressure oscillations, in turn, are directly correlated with oscillations in resistance. Hence, if the ability of resistance vessels to respond to sympathetic influences is depressed, the power of the LF spectral component will similarly decline.

The previous considerations stress that the LF peak not only reflects efferent neural activity but is also sensibly affected by vessel reactivity. As remarked by Malpas (31), the latter aspect is often ignored in the clinical/physiological literature, where the ratio LF/HF is just considered a marker of efferent neural activity, thus neglecting all other components participating to the pressure control loop.

Of course, the present model implies some simplifications and omissions, which may be the target of future improvements and extensions. The main limitations are critically discussed below.

First, in the present study we focused attention on the autonomic pressure control as the main factor affecting the LF waves, whereas other possible mechanisms have been neglected. This choice was adopted because the aim of this work was just to investigate the role of autonomic control system in the genesis of HRV. However, as discussed by Koepchen in his review (24), several other factors, and their complex nonlinear interactions, may concur with the formation of LF waves besides autonomic regulation. These include central, humoral, and vasomotor factors. Moreover, not only the strength of these factors but also their spectral distribution may be important in the etiology of LF fluctuations. A possible effect of some of these mechanisms has been included in the present study only in an empirical way, in the form of VLF noise.

Second, the present description of the autonomic loops is necessarily simplified and omits some mechanisms that may have a role. In particular, experimental evidence suggests that cardiovascular afferent sympathetic fibers (located in the cardiac structures and in large thoracic vessels) are capable of mediating excitatory reflex actions, with positive feedback characteristics (29). This reflex seems to normally participate in the neural regulation of cardiovascular system, interacting with the negative feedback mechanisms (originating in the arterial baroreceptive and vagal afferents). Sympathosympathetic circuits can play a role in the genesis of HRV and in particular in the genesis of the LF oscillations, because these mechanisms are sympathetic in origin.

The description of factors involved in HF waves is also simplified and neglects some mechanisms described in the literature. In particular, the baroreflex system includes not only arterial (or high pressure) baroreceptors but also cardiopulmonary (low pressure) baroreceptors (located in the atria, ventricles, and pulmonary veins). Activity in these receptors may be significantly affected by changes in venous pressure induced by respiration; hence, these receptors may parallel the effect of lung stretch receptors. Moreover, several authors have hypothesized that an important constituent of HF waves may be irradiation of impulses from the respiratory centers to cardiac vagal motor neurons (central mechanism).

Furthermore, in the present simulations only metronomic breathing was simulated, with a respiratory period as great as 5 s. As a consequence, the HF peak and the changes in vagal activity are more evident than during normal respiration, because of a synchronism between all respiratory components (28). This consideration justifies why, with basal parameter values, the model predicts a HF spectral component much greater than the LF component.

Finally, the mechanical effect of respiration on cardiac output is imputable not only to venous compression (i.e., the respiratory pump included in the present model) but also to ventricular interdependence. Ventricles share a common septal wall and are situated in a space limited by pericardial constraints. Filling and emptying of one ventricle are thus directly influenced by changes in the volume or pressure in the other ventricle. As a consequence of this phenomenon, called ventricular interdependence, the inspiratory increase in right ventricle filling (due to the fall in intrathoracic pressure) results in a larger decrease in left ventricle filling from the pulmonary circulation (8, 37). Thereby, ventricular interdependence mechanically contributes to the respiratory fluctuations in systemic arterial pressure. The mechanical coupling between ventricles is increased in pericardial diseases (such as cardiac tamponade and constrictive pericarditis), producing an exacerbation of respiratory-induced hemodynamic events (pulsus paradoxus).

In the present study, the model has been validated by demonstrating that the autonomic pressure reflex may induce oscillations in heart period with physiological characteristics and by studying the size of the LF and HF components at different values of the parameters for the autonomic control. Of course, a broader validation is necessary in future works. This should include the following major items: 1) analysis of arterial pressure fluctuations and comparison of the corresponding spectra at different locations along the vascular system, 2) cross-spectral analysis among different quantities in the model, with special emphasis on phase differences among fluctuations, 3) characterization of model behavior with classic methods for nonlinear analysis (such as a study of bifurcations, computation of embedded dimensions and entrainments among oscillators). All these topics may be the target of future works.

An important aspect that deserves discussion is the choice of species used to assign the numerical parameter values. Of course, the results obtained in this work depend on this choice, and scaling to other species requires caution.

In the present study, parameters of the cardiovascular system (resistances, compliances, volumes, cardiac output, etc.) have been assigned to simulate hemodynamics of a normal man (see Ref. 48 for a thorough description). By contrast, as discussed in QUALITATIVE MODEL DESCRIPTION, the parameters of the autonomic regulation have been assigned mainly on the basis of experiments performed in dogs. The latter choice has two main implications. First, the use of a parameter setting based on dog experiments may lead to spectra more similar to those observed in dogs than in humans. Second, as discussed by Malpas (31), the dynamic characteristics of the autonomic loops (time delays, time constants) are significantly different between animal species. As is well known from the automatic control theory, time delays are particularly important in the genesis of instability phenomena. Small animals (such as rabbits and rats) exhibit time delays shorter than those measured in the dog's baroreflex arc: these differences determine a higher resonance frequency for the control loop. For instance, the LF band is at ~0.1 Hz in the human, 0.14 Hz in the dog, 0.3 Hz in the rabbit, and 0.4 Hz in the rat (see Ref. 31 for extensive references). Hence, we emphasize that new "ad hoc" values should be assigned for the baroreflex loop parameters (especially time delays) if one intends to use the present model to simulate experiments in smaller animals.

In conclusion, the present study demonstrates that LF and HF peaks in the heart period spectrum spontaneously emerge from a model of short-term cardiovascular regulation based on actual physiological knowledge. The model suggests that the LF peak reflects a resonance of the pressure control loop, especially connected with sympathetic regulation and with the reactivity of resistance vessels, but it is also intensified by other LF sources of noise and is attenuated by a strong vagal control. The HF peak is dominated by the vagal control of heart period. The model can be used to achieve deeper understanding of the mechanisms causing HRV, and of the significance of spectral changes, on the basis of rigorous quantitative hypotheses and physiological considerations.