Abstract
In this paper, a modified van der Pol equations are considered as a description of the heart action. Wide ranges of the model parameters yield interesting qualitative results, e.g. Hopf bifurcation, Bogdanov-Takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. The physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. When some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. My aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der Pol system. The coupling takes place in a system with healthy conductivity, between two nodes: SA and AV, but in some circumstances, a pathological coupling may occur in the heart. The van der Pol oscillator is a type of relaxation oscillator which can be synchronized. In this paper, synchronization properties of such a system are studied as well. For the purpose of a numerical analysis of the system in question, a numerical model was created.
Highlights
This paper is related to the research on the electrical conduction system of the human heart
We investigate a phenomenological model for the heartbeat consisting of two coupled van der Pol oscillators
We applied the Pecorra Caroll (PC) theory in our system, in spite of the fact that this theory is typical of chaotic behaviour
Summary
This paper is related to the research on the electrical conduction system of the human heart. In the SA block, the electrical impulse is delayed or blocked on the way to the atria, delaying atria depolarization Synchronization phenomena in van der Pol oscillators coupled by a time-varying resistor is researched in paper[4]. These articles do not offer any examples of application of this model for recreating pathological behaviour of the electrical-conduction system of the human heart, and the considered ranges of parameters are wider than those applicable for medical applications. The van der Pol oscillator provides rich dynamical behaviour, which we would like to exploit in the modelling of the heart action [7] synchronization phenomena
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