Chaotic activity in a mathematical model of the vagally driven sinoatrial node

Bifurcations and Proarrhythmic Behaviors in Cardiac Electrical Excitations

The heart is a hierarchical dynamic system consisting of molecules, cells, and tissues, and acts as a pump for blood circulation. The pumping function depends critically on the preceding electrical activity, and disturbances in the pattern of excitation propagation lead to cardiac arrhythmia and pump failure. Excitation phenomena in cardiomyocytes have been modeled as a nonlinear dynamical system. Because of the nonlinearity of excitation phenomena, the system dynamics could be complex, and various analyses have been performed to understand the complex dynamics. Understanding the mechanisms underlying proarrhythmic responses in the heart is crucial for developing new ways to prevent and control cardiac arrhythmias and resulting contractile dysfunction. When the heart changes to a pathological state over time, the action potential (AP) in cardiomyocytes may also change to a different state in shape and duration, often undergoing a qualitative change in behavior. Such a dynamic change is called bifurcation. In this review, we first summarize the contribution of ion channels and transporters to AP formation and our knowledge of ion-transport molecules, then briefly describe bifurcation theory for nonlinear dynamical systems, and finally detail its recent progress, focusing on the research that attempts to understand the developing mechanisms of abnormal excitations in cardiomyocytes from the perspective of bifurcation phenomena.

Early Heart Rate Variability and Electroencephalographic Abnormalities in Acutely Brain-Injured Children Who Progress to Brain Death

OBJECTIVES

Heart rate variability is controlled by the autonomic nervous system. After brain death, this autonomic control stops, and heart rate variability is significantly decreased. However, it is unknown if early changes in heart rate variability are predictive of progression to brain death. We hypothesized that in brain-injured children, lower heart rate variability is an early indicator of autonomic system failure, and it predicts progression to brain death. We additionally explored the association between heart rate variability and markers of brain dysfunction such as electroencephalogram and neurologic examination between brain-injured children who progressed to brain death and those who survived.

DESIGN

Retrospective case-control study.

SETTING

PICU, single institution.

PATIENTS

Children up to 18 years with a Glasgow Coma Scale score of less than 8 admitted between August of 2016 and December of 2017, who had electrocardiographic data available for heart rate variability analysis, were included.

EXCLUSION CRITERIA

patients who died of causes other than brain death. Twenty-three patients met inclusion criteria: six progressed to brain death (cases), and 17 survived (controls). Five-minute electrocardiogram segments were used to estimate heart rate variability in the time domain (SD of normal-normal intervals, root mean square successive differences), frequency domain (low frequency, high frequency, low frequency/high frequency ratio), Poincaré plots, and approximate entropy.

INTERVENTIONS

None.

MEASUREMENTS AND MAIN RESULTS

Patients who progressed to brain death exhibited significantly lower heart rate variability in the time domain, frequency domain, and Poincaré plots (p < 0.01). The odds of death increased with decreasing low frequency (odds ratio, 4.0; 95% CI, 1.2-13.6) and high frequency (odds ratio, 2.5; 95% CI, 1.2-5.4) heart rate variability power (p < 0.03). Heart rate variability was significantly lower in those with discontinuous or attenuated/featureless electroencephalogram versus those with slow/disorganized background (p < 0.03).

CONCLUSIONS

These results support the concept of autonomic system failure as an early indicator of impending brain death in brain-injured children. Furthermore, decreased heart rate variability is associated with markers of CNS dysfunction such as electroencephalogram abnormalities.

Depression recognition according to heart rate variability using Bayesian Networks

BACKGROUND

Doctors mainly use scale tests and subjective judgment in the clinical diagnosis of depression. Researches have demonstrated that depression is associated with the dysfunction of the autonomic nervous system (ANS), where its modulation can be evaluated by heart rate variability (HRV). Depression patients have lower HRV than healthy subjects. Therefore, HRV may be used to distinguish depression patients from healthy people.

METHODS

HRV signals were collected from 76 female subjects composed of 38 depression patients and 38 healthy people. Time domain, frequency domain, and non-linear features were extracted from the HRV signals of these subjects, who were subjected to the Ewing test as an ANS stimulus. Then, these multiple features were input into Bayesian networks, served as a classifier, to distinguish depression patients from healthy people. Hence, accuracy, sensitivity, and specificity were calculated to evaluate the performance of the classifier.

RESULTS

Recognition results indicate 86.4% accuracy, 89.5% sensitivity, and 84.2% specificity. The individuals subjected to the Ewing test showed better recognition results than those at individual test states (resting state, deep breathing state, Valsalva state, and standing state) of the Ewing test. The root mean square of successive differences (RMSSD) of the HRV exhibits a significant relevance with recognition.

CONCLUSION

Bayesian networks can be applied to the recognition of depression patients from healthy people and the recognition results demonstrate the significant association between depression and HRV. The Ewing test is a good ANS stimulus for acquiring the difference of HRV between depression patients and healthy people to recognize depression. The RMSSD of the HRV is important in recognition and may be a significant index in distinguishing depression patients from healthy people.

The functions of atrial strands interdigitating with and penetrating into sinoatrial node: a theoretical study of the problem

The sinoatrial node (SAN)-atrium system is closely involved with the activity of heart beating. The impulse propagation and phase-locking behaviors of this system are of theoretical interest. Some experiments have revealed that atrial strands (ASs) interdigitate with and penetrate into the SAN, whereby the SAN-atrium system works as a complex network. In this study, the functions of ASs are numerically investigated using realistic cardiac models. The results indicate that the ASs penetrating into the central region of the SAN play a major role in propagating excitation into the atrium. This is because the threshold SAN-AS coupling for an AS to function as an alternative path for propagation is lower at the center than at the periphery. However, ASs penetrating into the peripheral region have a great effect in terms of enlarging the 1:1 entrainment range of the SAN because the automaticity of the SAN is evidently reduced by ASs. Moreover, an analytical formula for approximating the enlargement of the 1:1 range is derived.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Cycle length restitution in sinoatrial node cells: a theory for understanding spontaneous action potential dynamics

Normal heart rhythm (sinus rhythm) is governed by the sinoatrial node, a specialized and highly heterogeneous collection of spontaneously active myocytes in the right atrium. Sinoatrial node dysfunction, characterized by slow and/or asynchronous pacemaker activity and even failure, is associated with cardiovascular disease (e.g. heart failure, atrial fibrillation). While tremendous progress has been made in understanding the molecular and ionic basis of automaticity in sinoatrial node cells, the dynamics governing sinoatrial nodel cell synchrony and overall pacemaker function remain unclear. Here, a well-validated computational model of the mouse sinoatrial node cell is used to test the hypothesis that sinoatrial node cell dynamics reflect an inherent restitution property (cycle length restitution) that may give rise to a wide range of behavior from regular periodicity to highly complex, irregular activation. Computer simulations are performed to determine the cycle length restitution curve in the computational model using a newly defined voltage pulse protocol. The ability of the restitution curve to predict sinoatrial node cell dynamics (e.g., the emergence of irregular spontaneous activity) and susceptibility to termination is evaluated. Finally, ionic and tissue level factors (e.g. ion channel conductances, ion concentrations, cell-to-cell coupling) that influence restitution and sinoatrial node cell dynamics are explored. Together, these findings suggest that cycle length restitution may be a useful tool for analyzing cell dynamics and dysfunction in the sinoatrial node.

Poincaré plot indexes of heart rate variability detect dynamic autonomic modulation during general anesthesia induction

PURPOSE

Beat-to-beat heart rate variability (HRV) is caused by the fluctuating balance of sympathetic and parasympathetic tone. The Poincaré plot has been used to evaluate HRV. In this study, we validate that this new method may qualitatively and quantitatively assess the sympathovagal fluctuation in patients during induction of anesthesia with sevoflurane.

METHODS

Twenty-eight young patients were allocated for the study. The patients received a tilt test and on the next day they sustained anesthesia induced with inhaled anesthetics. Electrocardiography signals from the patients were relayed to an analogue-digital converter. The Poincaré plot is quantified by measuring SD1, SD2, and SD1/SD2. Power spectral analyses were performed and LF, HF and HF/LF were calculated.

RESULTS

The LF power and the SD2 of the Poincaré plot increased while subjects were tilt-up from the supine position. Additionally, a significant correlation were found between LF and SD2, HF and SD1 (p < 0.05), and LF/HF and SD2/SD1 (p < 0.01). Sevoflurane inhalation for 10 minutes had no effect on heart rate, but diminished LF, total power and SD1, SD2 of the Poincaré plot respectively. However, the LF, SD2 and LF/HF increased; the HF, SD1 and SD1/SD2 ratio decreased after intubation stimulation.

CONCLUSION

Poincaré plot and power spectral analysis of HRV during tilt test and sevoflurane induction significantly correlate. Poincaré plot analysis is easier and more sensitive at evaluating the sympathovagal balance and observing the beat-to-beat HRV.

Phase-locking behaviors in an ionic model of sinoatrial node cell and tissue

Phase-locking behaviors in sinoatrial node (SAN) are closely related to cardiac arrhythmias. An ionic model considering structural heterogeneity of SAN is numerically investigated. The bifurcations between phase-locking zones are interpreted by the map derived from the phase resetting curve. Furthermore, the validity of the circle map in describing phase locking of the actual SAN system is evaluated and explained. We reveal also how the phase-locking behaviors in heterogeneous tissue depend on the location of stimulating site and the coupling strength of the tissue. All these results may be of suggestive uses for understanding and controlling practical SAN dynamics.

Propagation of bursting oscillations

We investigate a system of partial differential equations of reaction-diffusion type which displays propagation of bursting oscillations. This system represents the time evolution of an assembly of cells constituted by a small nucleus of bursting cells near the origin immersed in the middle of excitable cells. We show that this system displays a global attractor in an appropriated functional space. Numerical simulations show the existence in this attractor of recurrent solutions which are waves propagating from the central source. The propagation seems possible if the excitability of the neighbouring cells is above some threshold.

Phase Resetting in One-Dimensional Model of the Sinoatrial Node

In this paper, we use a one-dimensional model of the rabbit sinoatrial node (SAN), and we investigate the response of the model to hyperpolarizing and depolarizing stimulus. Depending on the stimulus timing, either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior of the model is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. The main focus of this paper is to compare the dynamic properties of the spatially extended system and the single cell model. The detailed analysis of the results provides new insights in the understanding of the transition from the theoretical single cell models to the spatially extended systems.

Phase response characteristics of sinoatrial node cells

In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the sinoatrial node of the heart.

An index related to the autocorrelation function of RR intervals for the analysis of heart rate variability

Heart rate variability (HRV) is concerned with analysis of the variations in the intervals between heartbeats, known as RR intervals. Commonly used HRV indices may be insensitive in detecting some dynamic changes related to complex autocorrelation functions of the RR intervals. For example, indices SD1 and SD2 of the Poincaré plot can be expressed by the variance and first auto-covariance of the signal. The acceleration change index is related to the autocorrelation functions of the series only at the first three lags. We extend the idea of characterizing the sign of differences of a time series to propose a new index called VRL, which is the variance of the run length of the sign of the lagged differentiated time series. The theoretical study shows that VRL is directly related to the autocorrelation functions of the RR series at larger lags. Simulated data are used to validate the theoretical results and assess the power of testing group differences measured with VRL and other HRV indices. The performance of VRL is also evaluated for classifying subjects with normal sinus rhythm and congestive heart failure using the RR intervals taken from the PhysioNet database. We apply the index to RR intervals from an animal study of long-term exposure to particulate matter. The VRL values for the young mice susceptible to atherosclerosis in the control and exposure groups decreased gradually with different slopes after several weeks of exposure. The exposure effect changes in this HRV index estimated by fitting a generalized additive model are significant after 7 weeks of exposure.

Do existing measures of Poincaré plot geometry reflect nonlinear features of heart rate variability?

Heart rate variability (HRV) is concerned with the analysis of the intervals between heartbeats. An emerging analysis technique is the Poincaré plot, which takes a sequence of intervals and plots each interval against the following interval. The geometry of this plot has been shown to distinguish between healthy and unhealthy subjects in clinical settings. The Poincaré plot is a valuable HRV analysis technique due to its ability to display nonlinear aspects of the interval sequence. The problem is, how do we quantitatively characterize the plot to capture useful summary descriptors that are independent of existing HRV measures? Researchers have investigated a number of techniques: converting the two-dimensional plot into various one-dimensional views; the fitting of an ellipse to the plot shape; and measuring the correlation coefficient of the plot. We investigate each of these methods in detail and show that they are all measuring linear aspects of the intervals which existing HRV indexes already specify. The fact that these methods appear insensitive to the nonlinear characteristics of the intervals is an important finding because the Poincaré plot is primarily a nonlinear technique. Therefore, further work is needed to determine if better methods of characterizing Poincaré plot geometry can be found.

Alternans and higher-order rhythms in an ionic model of a sheet of ischemic ventricular muscle

Life-threatening arrhythmias such as ventricular tachycardia and fibrillation often occur during acute myocardial ischemia. During the first few minutes following coronary occlusion, there is a gradual rise in the extracellular concentration of potassium ions ([K(+)](0)) within ischemic tissue. This elevation of [K(+)](0) is one of the main causes of the electrophysiological changes produced by ischemia, and has been implicated in inducing arrhythmias. We investigate an ionic model of a 3 cmx3 cm sheet of normal ventricular myocardium containing an ischemic zone, simulated by elevating [K(+)](0) within a centrally-placed 1 cmx1 cm area of the sheet. As [K(+)](0) is gradually raised within the ischemic zone from the normal value of 5.4 mM, conduction first slows within the ischemic zone and then, at higher [K(+)](0), an arc of block develops within that area. The area distal to the arc of block is activated in a delayed fashion by a retrogradely moving wavefront originating from the distal edge of the ischemic zone. With a further increase in [K(+)](0), the point eventually comes where a very small increase in [K(+)](0) (0.01 mM) results in the abrupt transition from a global period-1 rhythm to a global period-2 rhythm in the sheet. In the peripheral part of the ischemic zone and in the normal area surrounding it, there is an alternation of action potential duration, producing a 2:2 response. Within the core of the ischemic zone, there is an alternation between an action potential and a maintained small-amplitude response ( approximately 30 mV in height). With a further increase of [K(+)](0), the maintained small-amplitude response turns into a decrementing subthreshold response, so that there is 2:1 block in the central part of the ischemic zone. A still further increase of [K(+)](0) leads to a transition in the sheet from a global period-2 to a period-4 rhythm, and then to period-6 and period-8 rhythms, and finally to a complete block of propagation within the ischemic core. When the size of the sheet is increased to 4 cmx4 cm (with a 2 cmx2 cm ischemic area), one observes essentially the same sequence of rhythms, except that the period-6 rhythm is not seen. Very similar sequences of rhythms are seen as [K(+)](0) is increased in the central region (1 or 2 cm long) of a thin strand of tissue (3 or 4 cm long) in which propagation is essentially one-dimensional and in which retrograde propagation does not occur. While reentrant rhythms resembling tachycardia and fibrillation were not encountered in the above simulations, well-known precursors to such rhythms (e.g., delayed activation, arcs of block, two-component upstrokes, retrograde activation, nascent spiral tips, alternans) were seen. We outline how additional modifications to the ischemic model might result in the emergence of reentrant rhythms following alternans. (c) 2000 American Institute of Physics.

Fascinating rhythm: a primer on chaos theory and its application to cardiology

Nonlinear dynamics is an exciting new way of looking at peculiarities that in the past have been ignored or explained away. We have attempted to give a general introduction to the basics of the mathematics, applications to cardiology, and a brief review of the new tools needed to use the concepts of nonlinear mathematics. The careful mathematical approach to problems in cardiac electrical dynamics and blood flow is opening a window on behaviors and mechanisms previously inaccessible.