A mathematical model of spontaneous calcium(II) oscillations in astrocytes

Astrocytes exhibit oscillations and waves of Ca²⁺ ions within their cytosol and it appears that this behavior helps facilitate the astrocyte's interaction with its environment, including its neighboring neurons. Often changes in the oscillatory behavior are initiated by an external stimulus such as glutamate, recently however, it has been observed that oscillations are also initiated spontaneously. We propose here a mathematical model of how spontaneous Ca²⁺ oscillations arise in astrocytes. This model uses the calcium-induced calcium release and inositol cross-coupling mechanisms coupled with a receptor-independent method for producing inositol (1,4,5)-trisphosphate as the heart of the model. By computationally mimicking experimental constraints we have found that this model provides results that are qualitatively similar to experiment.     

Heart rate variability biofeedback as a method for assessing baroreflex function: a preliminary study of resonance in the cardiovascular system

This study describes the use of a biofeedback method for the noninvasive study of baroreflex mechanisms. Five previously untrained healthy male participants learned to control oscillations in heart rate using biofeedback training to modify their heart rate variability at specific frequencies. They were instructed to match computer-generated sinusoidal oscillations with oscillations in heart rate at seven frequencies within the range of 0.01–0.14 Hz. All participants successfully produced high-amplitude target-frequency oscillations in both heart rate and blood pressure. Stable and predictable transfer functions between heart rate and blood pressure were obtained in all participants. The highest oscillation amplitudes were produced in the range of 0.055–0.11 Hz for heart rate and 0.02–0.055 Hz for blood pressure. Transfer functions were calculated among sinusoidal oscillations in the target stimuli, heart rate, blood pressure, and respiration for frequencies at which subjects received training. High and low target-frequency oscillation amplitudes at specific frequencies could be explained by resonance among various oscillatory processes in the cardiovascular system. The exact resonant frequencies differed among individuals. Changes in heart rate oscillations could not be completely explained by changes in breathing. The biofeedback method also allowed us to quantity characteristics of inertia, delay, and speed sensitivity in baroreflex system. We discuss the implications of these findings for using heart rate variability biofeedback as an aid in diagnosing various autonomic and cardiovascular system disorders and as a method for treating these disorders.     

Analysis of Heart Rate Variability Based on the Graph Method

A method of analysis of heart rate variability based on the graph theory principle was suggested. The main parameters of the heart rate graph structure were determined and analyzed using models of harmonic oscillations, white noise, and various functional tests (including controllable respiration and mental load). The efficiency of the use of parameters of the heart rate graph for diagnosing some functional states was considered. A correlation of the parameters of the heart rate graph structure with the frequency characteristics of heart rate variability was studied. A general model of changes in the heart rate graph structure parameters at different levels of mental activity was constructed in terms of entropy changes.     

Seidel–Herzel model of human baroreflex in cardiorespiratory system with stochastic delays

The stochastic versus deterministic solution of the Seidel–Herzel model describing the baroreceptor control loop (which regulates the short-time heart rate) are compared with the aim of exploring the heart rate variability. The deterministic model solutions are known to bifurcate from the stable to sustained oscillatory solutions if time delays in transfer of signals by sympathetic nervous system to the heart and vasculature are changed. Oscillations in the heart rate and blood pressure are physiologically crucial since they are recognized as Mayer waves. We test the role of delays of the sympathetic stimulation in reconstruction of the known features of the heart rate. It appears that realistic histograms and return plots are attainable if sympathetic time delays are stochastically perturbed, namely, we consider a perturbation by a white noise. Moreover, in the case of stochastic model the bifurcation points vanish and Mayer oscillations in heart period and blood pressure are observed for whole considered space of sympathetic time delays.      

Role of additive stochastic modulation of the heart activity in the formation of 0.1-Hz blood flow oscillations in the human cardiovascular system

In the framework of our previous hypothesis about the participation of structural and hydrodynamic properties of the vascular bed in the formation of the 0.1-Hz component of blood flow oscillations in the human cardiovascular system and on the basis of the reduced hydrodynamic model, the role of additive stochastic perturbations of the operation of the single-chamber pump that simulates the heart was investigated. It was shown that aperiodic noise modulation of the rigidity of the walls of the pump or its valves generates low-frequency oscillations of pressure and blood flow velocity of arterial vascular bed with the maximum amplitude at a frequency close to 0.1 Hz.     

Intracellular Ca2+ oscillations, a potential pacemaking mechanism in early embryonic heart cells

Early (E9.5-E11.5) embryonic heart cells beat spontaneously, even though the adult pacemaking mechanisms are not yet fully established. Here we show that in isolated murine early embryonic cardiomyocytes periodic oscillations of cytosolic Ca(2+) occur and that these induce contractions. The Ca(2+) oscillations originate from the sarcoplasmic reticulum and are dependent on the IP(3) and the ryanodine receptor. The Ca(2+) oscillations activate the Na(+)-Ca(2+) exchanger, giving rise to subthreshold depolarizations of the membrane potential and/or action potentials. Although early embryonic heart cells are voltage-independent Ca(2+) oscillators, the generation of action potentials provides synchronization of the electrical and mechanical signals. Thus, Ca(2+) oscillations pace early embryonic heart cells and the ensuing activation of the Na(+)-Ca(2+) exchanger evokes small membrane depolarizations or action potentials.     

Induction of oscillatory ventilation pattern using dynamic modulation of heart rate through a pacemaker

For disease states characterized by oscillatory ventilation, an ideal dynamic therapy would apply a counteracting oscillation in ventilation. Modulating respiratory gas transport through the circulation might allow this. We explore the ability of repetitive alternations in heart rate, using a cardiac pacemaker, to elicit oscillations in respiratory variables and discuss the potential for therapeutic exploitation. By incorporating acute cardiac output manipulations into an integrated mathematical model, we observed that a rise in cardiac output should yield a gradual rise in end-tidal CO2 and, subsequently, ventilation. An alternating pattern of cardiac output might, therefore, create oscillations in CO2 and ventilation. We studied the effect of repeated alternations in heart rate of 30 beats/min with periodicity of 60 s, on cardiac output, respiratory gases, and ventilation in 22 subjects with implanted cardiac pacemakers and stable breathing patterns. End-tidal CO2 and ventilation developed consistent oscillations with a period of 60 s during the heart rate alternations, with mean peak-to-trough relative excursions of 8.4 +/- 5.0% (P < 0.0001) and 24.4 +/- 18.8% (P < 0.0001), respectively. Furthermore, we verified the mathematical prediction that the amplitude of these oscillations would depend on those in cardiac output (r = 0.59, P = 0.001). Repetitive alternations in heart rate can elicit reproducible oscillations in end-tidal CO2 and ventilation. The size of this effect depends on the magnitude of the cardiac output response. Harnessed and timed appropriately, this cardiorespiratory mechanism might be exploited to create an active dynamic responsive pacing algorithm to counteract spontaneous respiratory oscillations, such as those causing apneic breathing disorders.     

Models of Cheyne-Stokes respiration with cardiovascular pathologies

Cheyne-Stokes respiration (CSR) is a periodic breathing pattern, characterized by short intervals of very little or no breathing (apnea), each followed by an interval of very heavy breathing (hyperpnea). This work presents a new compartmental model of the human cardio-respiratory system, simulating the factors that determine the concentrations of carbon dioxide in the compartments of the cardiovascular system and the lungs. The parameter set on which a Hopf bifurcation gives birth to stable CSR oscillations has been determined. The model predicts that the onset of CSR oscillations may result from an increase in any of: ventilation-perfusion ratio, feedback control gain, transport delay, left heart volume, lung congestion, or cardiovascular efficiency. The model is employed to investigate the relationship between CSR and serious cardiovascular pathologies, such as congestive heart failure and encephalitis, as well as the effects of acclimatization to higher altitudes. In all cases, the model is consistent with medical observations.     

Neural control of heart rate: the role of neuronal networking

Neural control of heart rate, particularly its sympathetic component, is generally thought to reside primarily in the central nervous system, though accumulating evidence suggests that intrathoracic extracardiac and intrinsic cardiac ganglia are also involved. We propose an integrated model in which the control of heart rate is achieved via three neuronal “levels” representing three control centers instead of the conventional one. Most importantly, in this model control is effected through networking between neuronal populations within and among these layers. The results obtained indicate that networking serves to process demands for systemic blood flow before transducing them to cardiac motor neurons. This provides the heart with a measure of protection against the possibility of “overdrive” implied by the currently held centrally driven system. The results also show that localized networking instabilities can lead to sporadic low frequency oscillations that have the characteristics of the well-known Mayer waves. The sporadic nature of Mayer waves has been unexplained so far and is of particular interest in clinical diagnosis.     

Do the oscillations of cardiovascular parameters persist during voluntary apnea in humans?

The aim of this study was to ascertain the persistence of heart rate and blood pressure oscillations at the onset of voluntary apnea in humans and to assess the dependence of the fluctuations parameters on the chemoreceptor activity. In 24 young subjects (10 males, 14 females, mean age 20.4 years) heart rate (represented by its reciprocal value--RR-intervals), systolic blood pressure (SBP) and diastolic blood pressure (DBP) during controlled breathing (CB) of atmospheric air and oxygen followed by apnea were recorded continuously. The cosine functions were then fitted by nonlinear regression analysis to the heart rate, SBP and DBP oscillations during CB and at the onset of apnea. The parameters of oscillations were different during atmospheric air breathing compared to oxygen breathing. During oxygen breathing there was an increase of the RR-interval oscillations--relative bradycardia and enhanced magnitude of respiratory sinus arythmia. During apnea, the base level of the blood pressure oscillations was higher after breathing of atmospheric air compared to oxygen breathing. At least one cosine-like wave oscillation was present at the onset of apnea in the heart rate, SBP and DBP and the second wave was present in all assessed parameters in at least 70% of recordings. The oscillations in RR-intervals are, to some extent, independent of blood pressure oscillations. No significant gender differences were found either in the duration of breath holding or in the RR and SBP oscillations parameters.     

Contribution of cardiogenic oscillations to gas exchange in constant-flow ventilation

The contribution of cardiogenic oscillations to gas exchange during constant-flow ventilation was examined in 11 dogs. With the use of two variations of cardiopulmonary bypass to maintain the systemic and pulmonary circulation, the influence of cardiogenic oscillations was removed by arresting the heart. Cardiac arrest by ventricular fibrillation was associated with a mean decrease in alveolar ventilation of 43% in five dogs on right and left heart bypass. However, successful defibrillation and return of the prearrest level of alveolar ventilation could not be achieved; thus we studied six dogs on left heart bypass. Alveolar ventilation decreased an average of 37% with cardiac arrest, and defibrillation resulted in a return of alveolar ventilation to 81% of the prearrest value. These results are consistent with previous predictions that cardiogenic oscillations are an important mechanism of gas transport during constant-flow ventilation.     

Quantitative estimation of connection of the heart rate rhythm with motor activity in rat fetuses

In rat fetuses over E17-20 with preserved placental circulation with use of mathematical analysis there were revealed value and character of connections of slow wave oscillations of the heart rhythm with motor activity for 30 min of observation. In the software PowerGraph 3.3.8, normalization and filtration of the studied signals were performed in three frequency diapasons: D1-0.02–0.2 Hz (5–50 s), D2-0.0083-0.02 Hz (50 s-2 min), and D3-0.0017–0.0083 Hz (2–10 min). The EMG curves filtrated by diapasons or piezograms were compared with periodograms in the corresponding diapasons of the heart rhythm variations. In the software “Origin 8.0”, quantitative estimation of the degree of intersystemic interrelations for each frequency diapason was performed by Pearson correlation of coefficient, by the correlation connection value, and by the time shift of maximum of cross-correlation function. It has been established that in the frequency D1, regardless of age, the connection of heart rhythm oscillations with motor activity is expressed weakly. In the frequency diapason D2, the connection in most cases is located in the zone of weak and moderate correlations. In the multiminute diapason (D3), the connection is more pronounced. The number of animals that have a significant value of the correlation connection rises. The fetal motor activity fires in the decasecond diapason in all age groups are accompanied by short-time decelerations of the heart rhythms. In the minute diapason, there is observed a transition from positive connections in E17 and E18 to the negative ones in E19-20. Results of the study are considered in association with age-related changes of ratios of positive and negative oscillations of the heart rhythm change depending on the character of motor activity.     

Respiratory modulation of the autonomic nervous system during Cheyne-Stokes respiration

Cheyne-Stokes respiration (CSR) is associated with increased mortality among patients with heart failure. However, the specific link between CSR and mortality remains unclear. One possibility is that CSR results in excitation of the sympathetic nervous system. This review relates evidence that CSR exerts acute effects on the autonomic nervous system during sleep, and thereby influences a number of cardiovascular phenomena, including heart rate, blood pressure, atrioventricular conduction, and ventricular ectopy. In patients in sinus rhythm, heart rate and blood pressure oscillate during CSR in association with respiratory oscillations, such that both peak heart rate and blood pressure occur during the hyperpneic phase. Inhalation of CO2 abolishes both CSR and the associated oscillations in heart rate and blood pressure. In contrast, O2 inhalation sufficient to eliminate hypoxic dips has no significant effect on CSR, heart rate, or blood pressure. In patients with atrial fibrillation, ventricular rate oscillates in association with CSR despite the absence of within-breath respiratory arrhythmia. The comparison of RR intervals between the apneic and hyperpneic phases of CSR indicates that this breathing disorder exerts its effect on ventricular rate by inducing cyclical changes in atrioventricular node conduction properties. In patients with frequent ventricular premature beats (VPBs), VPBs occur more frequently during the hyperpneic phase than the apneic phase of CSR. VPB frequency is also higher during periods of CSR than during periods of regular breathing, with or without correction of hypoxia. In summary, CSR exerts multiple effects on the cardiovascular system that are likely manifestations of respiratory modulation of autonomic activity. It is speculated that the rhythmic oscillations in autonomic tone brought about by CSR may ultimately contribute to the sympatho-excitation and increased mortality long observed in patients with heart failure and CSR.     

Secondary rhythms of automatically functioning systems

In wakeful rats at the age from newborn to 3 weeks there were studied interrelations of rhythmical oscillations of heart rate, respiration rate, and periodic spontaneous motor activity. In all studied systems, these oscillations are coordinated and occur in several frequency diapasons. Frequencies in the near-minute diapason (the period from 30 to 90 s) are the most constant and do not change with age. Regular oscillations are the most stable in newborns. Bursts of motor activity usually are irregular, but episodically the distinct near-minute rhythm is traced. Studies of the character of interaction of the heart activity, respiration, and periodic somatomotor excitation allow stating that the main coordination pathway of spontaneous autorhythmic functions in early postnatal ontogenesis is formation of secondary rhythms. Possible ways of formation of the secondary rhythms are considered.     

A two variable delay model for the circadian rhythm of Neurospora crassa

A two variable model with delay in both the variables, is proposed for the circadian oscillations of protein concentrations in the fungal species Neurospora crassa. The dynamical variables chosen are the concentrations of FRQ and WC-1 proteins. Our model is a two variable simplification of the detailed model of Smolen et al. (J. Neurosci. 21 (2001) 6644) modeling circadian oscillations with interlocking positive and negative feedback loops, containing 23 variables. In our model, as in the case of Smolen's model, a sustained limit cycle oscillation takes place in both FRQ and WC-1 protein in continuous darkness, and WC-1 is anti-phase to FRQ protein, as observed in experiments. The model accounts for various characteristic features of circadian rhythms such as entrainment to light dark cycles, phase response curves and robustness to parameter variation and molecular fluctuations. Simulations are carried out to study the effect of periodic forcing of circadian oscillations by light-dark cycles. The periodic forcing resulted in a rich bifurcation diagram that includes quasiperiodicity and chaotic oscillations, depending on the magnitude of the periodic changes in the light controlled parameter. When positive feedback is eliminated, our model reduces to the generic one dimensional delay model of Lema et al. (J. Theor. Biol. 204 (2000) 565), delay model of the circadian pace maker with FRQ protein as the dynamical variable which represses its own production. This one-dimensional model also exhibits all characteristic features of circadian oscillations and gives rise to circadian oscillations which are reasonably robust to parameter variations and molecular noise.     

Regulation of cAMP dynamics by Ca2+ and G protein-coupled receptors in the pancreatic beta-cell: a computational approach

In this report we describe a mathematical model for the regulation of cAMP dynamics in pancreatic beta-cells. Incretin hormones such as glucagon-like peptide 1 (GLP-1) increase cAMP and augment insulin secretion in pancreatic beta-cells. Imaging experiments performed in MIN6 insulinoma cells expressing a genetically encoded cAMP biosensor and loaded with fura-2, a calcium indicator, showed that cAMP oscillations are differentially regulated by periodic changes in membrane potential and GLP-1. We modeled the interplay of intracellular calcium (Ca(2+)) and its interaction with calmodulin, G protein-coupled receptor activation, adenylyl cyclases (AC), and phosphodiesterases (PDE). Simulations with the model demonstrate that cAMP oscillations are coupled to cytoplasmic Ca(2+) oscillations in the beta-cell. Slow Ca(2+) oscillations (<1 min(-1)) produce low-frequency cAMP oscillations, and faster Ca(2+) oscillations (>3-4 min(-1)) entrain high-frequency, low-amplitude cAMP oscillations. The model predicts that GLP-1 receptor agonists induce cAMP oscillations in phase with cytoplasmic Ca(2+) oscillations. In contrast, observed antiphasic Ca(2+) and cAMP oscillations can be simulated following combined glucose and tetraethylammonium-induced changes in membrane potential. The model provides additional evidence for a pivotal role for Ca(2+)-dependent AC and PDE activation in coupling of Ca(2+) and cAMP signals. Our results reveal important differences in the effects of glucose/TEA and GLP-1 on cAMP dynamics in MIN6 beta-cells.     

Modeling the impact of store-operated Ca2+ entry on intracellular Ca2+ oscillations

Calcium (Ca2+) oscillations play fundamental roles in various cell signaling processes and have been the subject of numerous modeling studies. Here we have implemented a general mathematical model to simulate the impact of store-operated Ca2+ entry on intracellular Ca2+ oscillations. In addition, we have compared two different models of the inositol 1,4,5-trisphosphate (IP3) receptor (IP3R) and their influences on intracellular Ca2+ oscillations. Store-operated Ca2+ entry following Ca2+ depletion of endoplasmic reticulum (ER) is an important component of Ca2+ signaling. We have developed a phenomenological model of store-operated Ca2+ entry via store-operated Ca2+ (SOC) channels, which are activated upon ER Ca2+ depletion. The depletion evokes a bi-phasic Ca2+ signal, which is also produced in our mathematical model. The IP3R is an important regulator of intracellular Ca2+ signals. This IP3 sensitive Ca2+ channel is also regulated by Ca2+. We apply two IP3R models, the Mak-McBride-Foskett model and the De Young and Keizer model, with significantly different channel characteristics. Our results show that the two separate IP3R models evoke intracellular Ca2+ oscillations with different frequencies and amplitudes. Store-operated Ca2+ entry affects the oscillatory behavior of these intracellular Ca2+ oscillations. The IP3 threshold is altered when store-operated Ca2+ entry is excluded from the model. Frequencies and amplitudes of intracellular Ca2+ oscillations are also altered without store-operated Ca2+ entry. Under certain conditions, when intracellular Ca2+ oscillations are absent, excluding store-operated Ca2+ entry induces an oscillatory response. These findings increase knowledge concerning store-operated Ca2+ entry and its impact on intracellular Ca2+ oscillations.     

Spectral analysis of heart rate variability in the beluga (<Emphasis Type="Italic">Delphinapterus leucas</Emphasis>) during exposure to acoustic noise

The spectral parameters of heart rate variability are a measure of activation of the sympathetic and parasympathetic branches of the mammalian autonomic nervous system. In this study, spectral analysis was used for the first time to evaluate the impact of acoustic noise (one of the major anthropogenic factors) on a cetacean. We analyzed cardiac intervals in a captive beluga (a member of the Odontoceti whales) in response to a 10-min band-pass acoustic noise at an intensity of 150–165 dB and frequency of 19–38 kHz. The beluga’s response to acoustic noise, when examined shortly after the animal’s capture, was characterized by a sharp tachycardia (the first phase) followed by a decrease in the heart rate (the second phase). Based on spectral analysis, the frequency range of heart rate oscillations in the beluga decreased during the period of tachycardia while shifting to a lower frequency range (below 0.01 Hz) as compared with the control conditions. Accordingly, the spectral power of low-frequency components was reduced. During the second phase, the range of heart rate variability oscillations expanded and fully recovered only after the noise had been turned off. After one year in captivity, no significant changes in the heart rate parameters (both in time and frequency domain) were recorded in response to a similar noise exposure. Therefore, the changes in the heart rate spectral components in the studied beluga exposed to acoustic noise were comparable to those recorded in terrestrial mammals and in humans in stressful and emotionally negative situations. The spectral characteristics of heart rate oscillations can be used as a quantitative measure of beluga whales’ response to acoustic noise as a stress factor.     

A model for the dynamics of bipolar disorders

Bipolar disorders are characterized by recurrent, alternating episodes of mania and depression. To examine the dynamical bases of this cyclical illness we consider a minimal model for bipolar disorders based on the observation that the two poles of the disease are mutually exclusive. We assume that the propensities to mania and depression, which are correlated with the activity of two putative neural circuits that promote, respectively, the manic or the depressive state, inhibit each other. When mutual inhibition is sufficiently strong, the model predicts bistability: the bipolar system is then either in a depressive or in a manic state and can display abrupt switches between these stable states. We consider two simple mechanisms which, when added to mutual inhibition, allow the model to pass from bistability to oscillations. Self-sustained oscillations provide a mechanism for the spontaneous, recurrent switching between mania and depression. The model can generate oscillations with a variety of waveforms, including simple periodic oscillations with comparable or unequal durations of the manic and depressive episodes, or small-amplitude oscillations around one of the two states preceding large-amplitude periodic changes in the propensities to mania or depression. The model provides a theoretical framework that covers the bipolar spectrum, i.e., cycling between the two poles of the disease, or evolution to either mania or depression or to an intermediate state without alternating between the two poles of the disease. The model accounts for the clinical observation that antidepressants can trigger the transition to mania or increase the frequency of bipolar cycling.