An analysis of heart rhythm dynamics using a three (original) (raw)
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An analysis of heart rhythm dynamics using a three-coupled oscillator model
Chaos, Solitons & Fractals, 2009
Rhythmic phenomena represent one of the most striking manifestations of the dynamic behavior in biological systems. Understanding the mechanisms responsible for biological rhythms is crucial for the comprehension of the dynamics of life. Natural rhythms could be either regular or irregular over time and space. Each kind of dynamical behavior may be related to both normal and pathological physiological functioning. The cardiac conducting system can be treated as a network of self-excitatory elements and, since these elements exhibit oscillatory behavior, they can be modeled as nonlinear oscillators. This paper proposes a mathematical model to describe heart rhythms considering three modified Van der Pol oscillators connected with time delay couplings. Therefore, the heart dynamics is represented by a system of differential difference equations. Numerical simulations are carried out presenting qualitative agreement with the general heart rhythm behavior. Normal and pathological rhythms represented by the ECG signals are reproduced. Pathological rhythms are generated by either the coupling alterations that represents communications aspects in the heart electric system or forcing excitation representing external pacemaker excitation.
Nonlinear dynamics of the heartbeat
Physica D: Nonlinear Phenomena, 1985
Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V,) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3 : 2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1 : 1 (SA : AV) phase-locking of the oscillators occurs over a critical range of V,. Externally driving the SA oscillator at increasing rates results in 3 : 2 AV Wenckebach periodicity and a 2 : 1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1 : 1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.
Physica A: Statistical Mechanics and its Applications, 2004
We investigate a phenomenological model for the heartbeat consisting of two coupled Van der Pol oscillators. The coupling between these nodes can be both unidirectional or bidirectional, and an external driving produced by a pacemaker is also included in this model. In order to warrant a robust operation, it is desirable that both units oscillate in a synchronized way, even though in the presence of external in uences or parameter mismatches which are unavoidable in a physiological setting. We study the synchronization properties of such an association with respect to the nature and intensity of coupling. We analyze in particular the (generalized) synchronization of rhythms characterized by a chaotic modulation of the oscillator frequencies. We also investigate the shadowing breakdown of numerically generated chaotic trajectories of the coupled oscillator system via unstable dimension variability in its chaotic invariant set.
Modelling couplings among the oscillators of the cardiovascular system
Physiological Measurement, 2001
A mathematical model of the cardiovascular system is simulated numerically. The basic unit in the model is an oscillator that possesses a structural stability and robustness motivated by physiological understanding and by the analysis of measured time series. Oscillators with linear couplings are found to reproduce the main characteristic features of the experimentally obtained spectra. To explain the variability of cardiac and respiratory frequencies, however, it is essential to take into account the rest of the system, i.e. to consider the effect of noise. It is found that the addition of noise also results in epochs of synchronization, as observed experimentally. Preliminary analysis suggests that there is a mixture of linear and parametric couplings, but that the linear coupling seems to dominate.
ISRN Applied Mathematics, 2014
If theSAandAVoscillators are not synchronized, it may arise some kinds of blocking arrhythmias in the system of heart. In this paper, in order to examine the heart system more precisely, we apply the three-oscillator model of the heart system, and to prevent arrhythmias, perform the following steps. Firstly, we add a voltage with ranga1andωfrequency toSAnode. Then, we use delay time factor in oscillators and finally the appropriate control is designed. In this paper, we have explained how simulating and curing these arrhythmias are possible by designing a three-oscillator system for heart in the state of delay and without delay and by applying an appropriate control. In the end, we present the simulation results.
Fluctuations in a coupled-oscillator model of the cardiovascular system
2007
We present a model of the cardiovascular system (CVS) based on a system of coupled oscillators. Using this approach we can describe several complex physiological phenomena that can have a range of applications. For instance, heart rate variability (HRV), can have a new deterministic explanation. The intrinsic dynamics of the HRV is controlled by deterministic couplings between the physiological oscillators in our model and without the need to introduce external noise as is commonly done. This new result provides potential applications not only for physiological systems but also for the design of very precise electronic generators where the frequency stability is crucial. Another important phenomenon is that of oscillation death. We show that in our CVS model the mechanism leading to the quenching of the oscillations can be controlled, not only by the coupling parameter, but by a more general scheme. In fact, we propose that a change in the relative current state of the cardiovascular oscillators can lead to a cease of the oscillations without actually changing the strength of the coupling among them. We performed real experiments using electronic oscillators and show them to match the theoretical and numerical predictions. We discuss the relevance of the studied phenomena to real cardiovascular systems regimes, including the explanation of certain pathologies, and the possible applications in medical practice.
Three oscillator model of the heartbeat generator
Communications in Nonlinear Science and Numerical Simulation, 2009
The sinoatrial (SA) node is a group of self-oscillatory cells in the heart which beat rhythmically and initiate electric potentials, producing a wave of contraction that travels through the heart resulting in the circulation of blood. The SA node is an inhomogeneous collection of cells which have varying intrinsic frequencies. Experimental measurements of these frequencies have shown that the peripheral cells of the SA node have a higher natural frequency than do the interior cells. This is surprising to us since in 1:1 phase-locked motion of two oscillators of different frequency, the oscillator with the higher frequency leads the other oscillator by a phase angle. If the wave originates in the center of the SA node as one expects, then the interior cells would be leading in a 1:1 phase-locked motion and should therefore have a higher frequency than the peripheral cells. Our objective in this work is to explain this discrepancy between intuition and the measured results, and to determine possible advantages of having cells of lower frequency in the interior. Using a model of the SA node consisting of three coupled phase-only oscillators, we show that increased robustness of synchronized behavior (represented by a larger region of parameter space) comes as a result of the experimentally observed distribution of frequencies in the SA node. Associated with the loss of synchronized behavior is a complicated series of bifurcations called the ''devil's staircase". We use our system to derive a 1D discontinuous map which exhibits the devil's staircase, and we analyze its dynamics.
Effects of Time Delays in a Dynamical Heart Model
Journal of Chemistry and Chemical Engineering, 2014
In this work, a dynamical model of the heartbeat is studied using nonlinear dynamics and considering the time delays inherent in the system. Two fixed points are associated to sustained oscillations which might be interpreted as the diastole and systole. These parameters are associated with blood flow in human body called arterial pressure and are very important in the cardio-vascular diagnostic.
Nonlinear dynamics, chaos and complex cardiac arrhythmias
Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1987
Periodic stimulation of a nonlinear cardiac oscillatorin vitrogives rise to complex dynamics that is well described by one-dimensional finite difference equations. As stimulation parameters are varied, a large number of different phase locked and chaotic rhythms is observed. Similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites. Simplified models are analysed, which show some correspondence to clinical observations.