Arnold tongues in human cardiorespiratory systems

Physiological Entrainment: A Key Mind-Body Mechanism for Cognitive, Motor and Affective Functioning, and Well-Being

BACKGROUND

The human sensorimotor system can naturally synchronize with environmental rhythms, such as light pulses or sound beats. Several studies showed that different styles and tempos of music, or other rhythmic stimuli, have an impact on physiological rhythms, including electrocortical brain activity, heart rate, and motor coordination. Such synchronization, also known as the "entrainment effect", has been identified as a crucial mechanism impacting cognitive, motor, and affective functioning.

OBJECTIVES

This review examines theoretical and empirical contributions to the literature on entrainment, with a particular focus on the physiological mechanisms underlying this phenomenon and its role in cognitive, motor, and affective functions. We also address the inconsistent terminology used in the literature and evaluate the range of measurement approaches used to assess entrainment phenomena. Finally, we propose a definition of "physiological entrainment" that emphasizes its role as a fundamental mechanism that encompasses rhythmic interactions between the body and its environment, to support information processing across bodily systems and to sustain adaptive motor responses.

METHODS

We reviewed the recent literature through the lens of the "embodied cognition" framework, offering a unified perspective on the phenomenon of physiological entrainment.

RESULTS

Evidence from the current literature suggests that physiological entrainment produces measurable effects, especially on neural oscillations, heart rate variability, and motor synchronization. Eventually, such physiological changes can impact cognitive processing, affective functioning, and motor coordination.

CONCLUSIONS

Physiological entrainment emerges as a fundamental mechanism underlying the mind-body connection. Entrainment-based interventions may be used to promote well-being by enhancing cognitive, motor, and affective functions, suggesting potential rehabilitative approaches to enhancing mental health.

Optimizing phase variability threshold for automated synchrogram analysis of cardiorespiratory interactions in amateur cyclists

We propose a procedure suitable for automated synchrogram analysis for setting the threshold below which phase variability between two marker event series is of such a negligible amount that the null hypothesis of phase desynchronization can be rejected. The procedure exploits the principle of maximizing the likelihood of detecting phase synchronization epochs and it is grounded on a surrogate data approach testing the null hypothesis of phase uncoupling. The approach was applied to assess cardiorespiratory phase interactions between heartbeat and inspiratory onset in amateur cyclists before and after 11-week inspiratory muscle training (IMT) at different intensities and compared to a more traditional approach to set phase variability threshold. The proposed procedure was able to detect the decrease in cardiorespiratory phase locking strength during vagal withdrawal induced by the modification of posture from supine to standing. IMT had very limited effects on cardiorespiratory phase synchronization strength and this result held regardless of the training intensity. In amateur athletes training, the inspiratory muscles did not limit the decrease in cardiorespiratory phase synchronization observed in the upright position as a likely consequence of the modest impact of this respiratory exercise, regardless of its intensity, on cardiac vagal control. This article is part of the theme issue 'Advanced computation in cardiovascular physiology: new challenges and opportunities'.

Dynamics and bifurcations in multistable 3-cell neural networks

We disclose the generality of the intrinsic mechanisms underlying multistability in reciprocally inhibitory 3-cell circuits composed of simplified, low-dimensional models of oscillatory neurons, as opposed to those of a detailed Hodgkin-Huxley type [Wojcik et al., PLoS One 9, e92918 (2014)]. The computational reduction to return maps for the phase-lags between neurons reveals a rich multiplicity of rhythmic patterns in such circuits. We perform a detailed bifurcation analysis to show how such rhythms can emerge, disappear, and gain or lose stability, as the parameters of the individual cells and the synapses are varied.

Frequency-dependent response in cortical network with periodic electrical stimulation

Electrical stimulation can shape oscillations in brain activity. However, the mechanism of how periodic electrical stimulation modulates brain oscillations by time-delayed neural networks is poorly understood at present. To address this question, we investigate the effects of periodic stimulations on the oscillations generated via a time-delayed neural network. We specifically study the effect of unipolar and asymmetric bidirectional pulse stimulations by altering amplitude and frequency in a systematic manner. Our findings suggest that electrical stimulations play a central role in altering oscillations in the time-delayed neural network and that these alterations are strongly dependent on the stimulus frequency. We observe that the time-delayed neural network responds differently as the stimulation frequency is altered, as manifested by changes in resonance, entrainment, non-linear oscillation, or oscillation suppression. The results also indicate that the network presents similar response activities with increasing stimulus frequency under different excitation-inhibition ratios. Collectively, our findings pave the way for exploring the potential mechanism underlying the frequency-dependent modulation of network activity via electrical stimulations and provide new insights into possible electrical stimulation therapies to the neurological and psychological disorders in clinical practice.

Abnormal synchronization patterns in the electrical stimulation-contractile response coupling decrease with noise

Synchronization theory predicts that if an oscillator interacts with a rhythmical external force, then it should react to a rhythmical force by adjusting its frequency. Furthermore, noise is present in nature, and it affects the nervous and cardiovascular systems. In this paper, we analyze the heart as an oscillator, where noisy periodic electrical stimulation can be regarded as an external forcing. This study aimed to investigate, from an experimental point of view, whether noise can induce synchronization of higher order in the mechanical heart response. A Langendorff heart preparation was used to obtain two variables of the mechanical response, intensity of contractile force and heart rate. The experiments show frequency locking in the electrical stimulation-contractile response coupling with and without noise induced. The role of noise in the response of effector organs invites further investigation.

Stabilization of dynamics of oscillatory systems by nonautonomous perturbation

Synchronization and stability under periodic oscillatory driving are well understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counterintuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronization where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilization phenomenon is numerically observed. Our findings help support the case that in general, deterministic nonautonomous perturbation is a very good candidate for stabilizing complex dynamics.

Cardiorespiratory interactions in humans and animals: rhythms for life

The cardiorespiratory system exhibits oscillations from a range of sources. One of the most studied oscillations is heart rate variability, which is thought to be beneficial and can serve as an index of a healthy cardiovascular system. Heart rate variability is dampened in many diseases including depression, autoimmune diseases, hypertension, and heart failure. Thus, understanding the interactions that lead to heart rate variability, and its physiological role, could help with prevention, diagnosis, and treatment of cardiovascular diseases. In this review, we consider three types of cardiorespiratory interactions: respiratory sinus arrhythmia (variability in heart rate at the frequency of breathing), cardioventilatory coupling (synchronization between the heart beat and the onset of inspiration), and respiratory stroke volume synchronization (the constant phase difference between the right and the left stroke volumes over one respiratory cycle). While the exact physiological role of these oscillations continues to be debated, the redundancies in the mechanisms responsible for its generation and its strong evolutionary conservation point to the importance of cardiorespiratory interactions. The putative mechanisms driving cardiorespiratory oscillations as well as the physiological significance of these oscillations will be reviewed. We suggest that cardiorespiratory interactions have the capacity to both dampen the variability in systemic blood flow as well as improve the efficiency of work done by the heart while maintaining physiological levels of arterial CO2. Given that reduction in variability is a prognostic indicator of disease, we argue that restoration of this variability via pharmaceutical or device-based approaches may be beneficial in prolonging life.

Cardiorespiratory coupling in young healthy subjects

OBJECTIVE

To quantify the presence of cardiorespiratory interaction in a group of 41 healthy subjects performing a subset of the Ewing test battery.

APPROACH

We measure the empirical distribution of the cardiorespiratory coupling time (RI), defined as the time from inspiration onset to R peaks in the ECG. The study protocol is a subset of the Ewing test battery. The respiratory function was measured with a thoracic belt and heart rate was obtained from a two channel ECG measurement. Both series of fiducial points were determined using custom software. Additionally, we determine the presence of cardiorespiratory coupling (CRC) and cardiorespiratory interaction (CRI) using Shannon entropy, synchrograms and coordigrams.

MAIN RESULTS

We observe that the RI distribution is asymmetric and nonuniform. These features are a manifestation of the causal relation between heart action and respiration. The preceding R peak strongly affects a position of inspiration onset. From the asymmetry of the RI distribution we conclude that this relation is stronger than the relation between inspiration onset and the following R peak. We use a suitable choice of surrogate data to prove that the result cannot be falsified. We observe a dual structure of the RI histograms, which may be related to the respiratory rhythmogenesis. We compare the sensitivity of RI histograms with other measures of CRI and CRC. In 46% of subjects, CRC appears in at least one stage of the examination, most often in resting states. In states of increased stress-orthostasis or physical (exercise)-the strength of coupling is visibly diminished. The nonuniform structure of the RI histogram is more sensitive to the presence of CRI than synchrograms or coordigrams are, as is well visible in the group averages. We also refer to the question of the most proper mathematical description of cardiorespiratory dynamics (phase domain or time domain). Finally, we formulate the hypothesis that the arterial blood pressure is a common driver of cardiac and respiratory rhythms.

SIGNIFICANCE

Analysis of the asymmetry of RI histograms is an interesting and sensitive method to study cardiorespiratory interaction and autonomic balance, in order to assess physical and mental health. The dual structure of the RI histograms which we have observed suggests the possible presence of a twofold mechanism for respiratory rhythmogenesis, as proposed by Galletly and Larsen.

On the difference of cardiorespiratory synchronisation and coordination

Cardiorespiratory phase synchronisation (CRS) is a type of cardiorespiratory coupling that manifests through a prediliction for heart beats to occur at specific points relative to the phase of the respiratory cycle. It has been under investigation for nearly 20 years, and while it seems to be mostly occurring in relaxed states such as deep sleep and anesthesia, no clear clinical implications have been established. Cardiorespiratory coordination (CRC) is a recent development in this field where the relationship between the respiratory onset and heart beat is analysed in the time domain and the possible relationship of each heart beat is considered for both the previous and the next respiratory onset. This ostensibly closely related effect must not only show relevant information content but also do so independent of CRS in order to be relevant for future studies. In this paper, we investigate CRC and its relation to CRS mainly using graphical and statistical methods on two exemplary datasets: measurements from a pregnant woman participating in a preeclampsia study and those from a man suffering from sleep apnea. We show fundamental differences between the results of both approaches and are able to show a formerly unknown dependency between the heart activity and respiratory rate, potentially indicating heartbeat-initiated inspiration. Despite their differences, methods developed for the quantification of CRS can be adapted to CRC. Completing the comparison is an investigation into the relationship between CRC and respiratory sinus arrhythmia. Similar to previous results for CRS, the two effects are found to be orthogonal, meaning that they can be observed independently or in conjunction.

Distinguishability notion based on Wootters statistical distance: Application to discrete maps

We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a statistical distance in chaotic unidimensional maps. Based on that definition, we provide a metric d¯ for an arbitrary discrete map. Moreover, from d¯, we associate a metric space with each invariant density of a given map, which results to be the set of all distinguished points when the number of iterations of the map tends to infinity. Also, we give a characterization of the wandering set of a map in terms of the metric d¯, which allows us to identify the dissipative regions in the phase space. We illustrate the results in the case of the logistic and the circle maps numerically and analytically, and we obtain d¯ and the wandering set for some characteristic values of their parameters. Finally, an extension of the metric space associated for arbitrary probability distributions (not necessarily invariant densities) is given along with some consequences. The statistical properties of distributions given by histograms are characterized in terms of the cardinal of the associated metric space. For two conjugate variables, the uncertainty principle is expressed in terms of the diameters of the associated metric space with those variables.

In vivo cardiac phase response curve elucidates human respiratory heart rate variability

Recovering interaction of endogenous rhythms from observations is challenging, especially if a mathematical model explaining the behaviour of the system is unknown. The decisive information for successful reconstruction of the dynamics is the sensitivity of an oscillator to external influences, which is quantified by its phase response curve. Here we present a technique that allows the extraction of the phase response curve from a non-invasive observation of a system consisting of two interacting oscillators--in this case heartbeat and respiration--in its natural environment and under free-running conditions. We use this method to obtain the phase-coupling functions describing cardiorespiratory interactions and the phase response curve of 17 healthy humans. We show for the first time the phase at which the cardiac beat is susceptible to respiratory drive and extract the respiratory-related component of heart rate variability. This non-invasive method for the determination of phase response curves of coupled oscillators may find application in many scientific disciplines.

Clocking convergence to a stable limit cycle of a periodically driven nonlinear pendulum

Convergence to a stable limit cycle of a periodically driven nonlinear pendulum is analyzed in this paper. The concept of the H-rank of a scalar sequence is used for the assessment of transient processes of the system. The circle map is used to illustrate the complex structure of the manifold of non-asymptotic convergence to a fixed point. It is demonstrated that the manifold of non-asymptotic convergence to a stable limit cycle also exists in the stroboscopic representation of the transient data of the periodically driven nonlinear pendulum. A simple method based on a short external impulse is proposed for the control of transient processes when the transition time to stable limit cycles must be minimized.

Evaluating the physiological significance of respiratory sinus arrhythmia: looking beyond ventilation-perfusion efficiency

We conducted a theoretical study of the physiological significance of respiratory sinus arrhythmia (RSA), a phenomenon used as an index of cardiac vagal tone and wellbeing, whereby the heart rate (HR) increases during inspiration and decreases during expiration. We first tested the hypothesis that RSA improves gas exchange efficiency but found that although gas exchange efficiency improved with slow and deep breathing and with increased mean heart rate, this was unrelated to RSA. We then formulated and tested a new hypothesis: that RSA minimizes the work done by the heart while maintaining physiological levels of arterial carbon dioxide. We tested the new hypothesis using two methods. First, the HR for which the work is minimized was calculated using techniques from optimal control theory. This calculation was done on simplified models that we derived from a previously published model of gas exchange in mammals. We found that the calculated HR was remarkably similar to RSA and that this became more profound under slow and deep breathing. Second, the HR was prescribed and the work done by the heart was calculated by conducting a series of numerical experiments on the previously published gas exchange model. We found that cardiac work was minimized for RSA-like HR functions, most profoundly under slow and deep breathing. These findings provide novel insights into potential reasons for and benefits of RSA under physiological conditions.

Computational models for the study of heart-lung interactions in mammals

The operation and regulation of the lungs and the heart are closely related. This is evident when examining the anatomy within the thorax cavity, in the brainstem and in the aortic and carotid arteries where chemoreceptors and baroreceptors, which provide feedback affecting the regulation of both organs, are concentrated. This is also evident in phenomena such as respiratory sinus arrhythmia where the heart rate increases during inspiration and decreases during expiration, in other types of synchronization between the heart and the lungs known as cardioventilatory coupling and in the association between heart failure and sleep apnea where breathing is interrupted periodically by periods of no-breathing. The full implication and physiological significance of the cardiorespiratory coupling under normal, pathological, or extreme physiological conditions are still unknown and are subject to ongoing investigation both experimentally and theoretically using mathematical models. This article reviews mathematical models that take heart-lung interactions into account. The main ideas behind low dimensional, phenomenological models for the study of the heart-lung synchronization and sleep apnea are described first. Higher dimensions, physiology-based models are described next. These models can vary widely in detail and scope and are characterized by the way the heart-lung interaction is taken into account: via gas exchange, via the central nervous system, via the mechanical interactions, and via time delays. The article emphasizes the need for the integration of the different sources of heart-lung coupling as well as the different mathematical approaches.

Multi-frequency activation of neuronal networks by coordinated reset stimulation

We computationally study whether it is possible to stimulate a neuronal population in such a way that its mean firing rate increases without an increase of the population's net synchronization. For this, we use coordinated reset (CR) stimulation, which has previously been developed to desynchronize populations of oscillatory neurons. Intriguingly, delivered to a population of predominantly silent FitzHugh-Nagumo or Hindmarsh-Rose neurons at sufficient stimulation amplitudes, CR robustly causes a multi-frequency activation: different Arnold tongues such as 1 : 1 or n : m entrained neuronal clusters emerge, which consist of phase-shifted sub clusters. Owing to the clustering pattern the neurons' timing is well balanced, so that in total there is no synchronization. Our findings may contribute to the development of novel and safe stimulation treatments that specifically counteract cerebral hypo-activity without promoting pathological synchronization or inducing epileptic seizures.

Chaotic dynamics of cardioventilatory coupling in humans: effects of ventilatory modes

Cardioventilatory coupling (CVC), a transient temporal alignment between the heartbeat and inspiratory activity, has been studied in animals and humans mainly during anesthesia. The origin of the coupling remains uncertain, whether or not ventilation is a main determinant in the CVC process and whether the coupling exhibits chaotic behavior. In this frame, we studied sedative-free, mechanically ventilated patients experiencing rapid sequential changes in breathing control during ventilator weaning during a switch from a machine-controlled assistance mode [assist-controlled ventilation (ACV)] to a patient-driven mode [inspiratory pressure support (IPS) and unsupported spontaneous breathing (USB)]. Time series were computed as R to start inspiration (RI) and R to the start of expiration (RE). Chaos was characterized with the noise titration method (noise limit), largest Lyapunov exponent (LLE) and correlation dimension (CD). All the RI and RE time series exhibit chaotic behavior. Specific coupling patterns were displayed in each ventilatory mode, and these patterns exhibited different linear and chaotic dynamics. When switching from ACV to IPS, partial inspiratory loading decreases the noise limit value, the LLE, and the correlation dimension of the RI and RE time series in parallel, whereas decreasing intrathoracic pressure from IPS to USB has the opposite effect. Coupling with expiration exhibits higher complexity than coupling with inspiration during mechanical ventilation either during ACV or IPS, probably due to active expiration. Only 33% of the cardiac time series (RR interval) exhibit complexity either during ACV, IPS, or USB making the contribution of the cardiac signal to the chaotic feature of the coupling minimal. We conclude that 1) CVC in unsedated humans exhibits a complex dynamic that can be chaotic, and 2) ventilatory mode has major effects on the linear and chaotic features of the coupling. Taken together these findings reinforce the role of ventilation in the CVC process.

A model for mechano-electrical feedback effects on atrial flutter interval variability

Atrial flutter is a supraventricular arrhythmia, based on a reentrant mechanism mainly confined to the right atrium. Although atrial flutter is considered a regular rhythm, the atrial flutter interval (i.e., the time interval between consecutive atrial activation times) presents a spontaneous beat-to-beat variability, which has been suggested to be related to ventricular contraction and respiration by mechano-electrical feedback. This paper introduces a model to predict atrial activity during atrial flutter, based on the assumption that atrial flutter variability is related to the phase of the reentrant activity in the ventricular and respiratory cycles. Thus, atrial intervals are given as a superimposition of phase-dependent ventricular and respiratory modulations. The model includes a simplified atrioventricular (AV) branch with constant refractoriness and conduction times, which allows the prediction of ventricular activations in a closed-loop with atrial activations. Model predictions are quantitatively compared with real activation series recorded in 12 patients with atrial flutter. The model predicts the time course of both atrial and ventricular time series with a high beat-to-beat agreement, reproducing 96+/-8% and 86+/-21% of atrial and ventricular variability, respectively. The model also predicts the existence of phase-locking of atrial flutter intervals during periodic ventricular pacing and such results are observed in patients. These results constitute evidence in favor of mechano-electrical feedback as a major source of cycle length variability during atrial flutter.

Cardiac vagal control and respiratory sinus arrhythmia during hypercapnia in humans

Normoxic hypercapnia may increase high-frequency (HF) power in heart rate variability (HRV) and also increase respiratory sinus arrhythmia (RSA). Low-frequency (LF) power may remain unchanged. In this study, 5-min ECG recordings (N = 10) were analyzed in time and frequency domains while human subjects breathed normoxic 5% CO2 (5%CO2) or room air (RA). Tidal volume (VT), inhalatory (TI), and exhalatory (TE) times of breaths in the final minute were measured. ECG time domain measures were unaffected by CO2 inhalation (P > 0.05). Following natural logarithmic transformation (LN), LFLN was unaltered (RA: 7.14 +/- 0.95 vs. 5%CO2: 7.35 +/- 1.12, P > 0.05), and HFLN increased (RA: 7.65 +/- 1.37 vs. 5%CO2: 8.58 +/- 1.11, P < 0.05) with CO2 inhalation. When changes in total power (NU) were corrected, LF(NU) decreased (RA: 34.4 +/- 22.9 vs. 5%CO2: 23.8 +/- 23.1, P < 0.01), and HFNU increased (RA: 56.5 +/- 22.3 vs. 5%CO2: 66.8 +/- 22.9, P < 0.01) with CO2 inhalation. TI (RA: 2.0 +/- 1.0 vs. 5%CO2: 1.9 +/- 0.8 s) and TE (RA: 2.5 +/- 1.1 vs. 5%CO2: 2.4 +/- 0.9 s) remained unchanged, but VT increased with CO2 inhalation (RA: 1.1 +/- 0.3 vs. 5%CO2: 2.0 +/- 0.8 L, P < 0.001). Heart rates during inhalation (RA: 35.2 +/- 4.4, 5%CO2: 34.5 +/- 4.8 beats min(-1)) were different from heart rates during exhalation (RA: 28.8 +/- 4.4, 5%CO2: 29.1 +/- 3.1 beats min(-1)). Hypercapnia did not increase the clustering of heart beats during inhalation, and we suggest that the HF component may not adequately reflect RSA.

Effects of hypercapnia and hypoxemia on respiratory sinus arrhythmia in conscious humans during spontaneous respiration

Normally, at rest, the amplitude of respiratory sinus arrhythmia (RSA) appears to correlate with cardiac vagal tone. However, recent studies showed that, under stress, RSA dissociates from vagal tone, indicating that separate mechanisms might regulate phasic and tonic vagal activity. This dissociation has been linked to the hypothesis that RSA improves pulmonary gas exchange through preferential distribution of heartbeats in inspiration. We examined the effects of hypercapnia and mild hypoxemia on RSA-vagal dissociation in relation to heartbeat distribution throughout the respiratory cycle in 12 volunteers. We found that hypercapnia, but not hypoxemia, was associated with significant increases in heart rate (HR), tidal volume, and RSA amplitude. The RSA amplitude increase remained statistically significant after adjustment for respiratory rate, tidal volume, and HR. Moreover, the RSA amplitude increase was associated with a paradoxical rise in HR and decrease in low-frequency-to-high-frequency mean amplitude ratio derived from spectral analysis, which is consistent with RSA-vagal dissociation. Although hypercapnia was associated with a significant increase in the percentage of heartbeats during inspiration, this association was largely secondary to increases in the inspiratory period-to-respiratory period ratio, rather than RSA amplitude. Additional model analyses of RSA were consistent with the experimental data. Heartbeat distribution did not change during hypoxemia. These results support the concept of RSA-vagal dissociation during hypercapnia; however, the putative role of RSA in optimizing pulmonary perfusion matching requires further experimental validation.

Mechanism of cardioventilatory coupling: insights from cardiac pacing, vagotomy, and sinoaortic denervation in the anesthetized rat

Cardioventilatory coupling (CVC), a temporal alignment between the heartbeat and inspiratory activity, is a major determinant of breath-to-breath variation in observed respiratory rate (f(o)). The cardiac-trigger hypothesis attributes this to adjustments of respiratory timing by baroreceptor afferent impulses to the central respiratory pattern generator. A mathematical model of this hypothesis indicates that apparent CVC in graphical plots of ECG R wave vs. inspiratory time is dependent on the heart rate (HR), the rate of the intrinsic respiratory oscillator (f(i)), and the strength of the hypothetical cardiovascular afferent impulse. Failure to account for HR and f(i) may explain the inconsistent results from previous attempts to identify the neural pathways involved in CVC. Cognizant of these interactions, we factored in the HR-to-f(i) ratio in our examination of the role of the vagus nerve and arterial baroreceptors in CVC by cardiac pacing 29 anesthetized Sprague-Dawley rats and incrementally changing the HR. With the assumption of a relatively constant f(i), CVC could be examined across a range of HR-to-f(o) ratios before and after vagotomy, sinoaortic denervation, and vagotomy + sinoaortic denervation. We confirmed the relation between CVC, HR-to-f(o) ratio, and breath-to-breath respiratory period variability and demonstrated the loss of these relations after baroreceptor elimination. Sham experiments (n = 8) showed that these changes were not due to surgical stress. Our data support the notion that inspiratory timing can be influenced by cardiac afferent activity. We conclude that the putative cardiovascular input arises from the arterial baroreceptors and that the vagus nerve is not critical for CVC.

A minimal model for the respiratory sinus arrhythmia

The cardiac and respiratory rhythms in humans are known to be coupled by several mechanisms. In particular, the first rhythm is deeply modulated by the second. In this report we propose a simple operational model for heart rate variability which, taking such modulation into account, reproduces the main features of some experimental sequences of RR intervals recorded from healthy subjects in the resting condition. Also, peer analysis of the model performance allows us to answer the question whether the observed behaviour should be ascribed to phase synchronisation of the heart beating to the respiratory rhythm. Lastly, the changes of the model activity brought about by changing its relevant parameters are analysed and discussed.

Simplified models for gas exchange in the human lungs

This paper presents a hierarchy of models with increasing complexity for gas exchange in the human lungs. The models span from a single compartment, inflexible lung to a single compartment, flexible lung with pulmonary gas exchange. It is shown how the models are related to well-known models in the literature. A long-term purpose of this work is to study nonlinear phenomena seen in the cardio-respiratory system (for example, synchronization between ventilation rate and heart rate, and Cheyne-Stokes respiration). The models developed in this paper can be regarded as the controlled system (plant) and provide a mathematical framework to link between "molecular-level", and "systems-level" models. It is shown how changes in molecular level affect the alveolar partial pressure. Two assumptions that have previously been made are re-examined: (1) the hidden assumption that the air flow through the mouth is equal to the rate of volume change in the lungs, and, (2) the assumption that the process of oxygen binding to hemoglobin is near equilibrium. Conditions under which these assumptions are valid are studied. All the parameters in the models, except two, are physiologically realistic. Numerical results are consistent with published experimental observations.