**1. Introduction**

In this chapter, the research on the electrical conduction system of the human heart is concerned. The parts of this system are pacemaker centres made of special cells similar to embryonic cells which form the following concentrations: sino-atrial node (SA), atrioven‐ tricular node (AV) and His-Purkinje system [1]. The key elements of the conduction system considered include SA node and AV node. Modelling the formation and conduction of electrical impulses in the heart is one of the most developed areas of mathematical biolo‐ gy. For years, the most popular models of action potentials which occur in the heart have included the Hodgkin-Huxley model and the Purkinje cells model. These models, even though very effective, are not very suitable for studying the dynamics of the system, which is described in view of high dimensionality of the phase space and very broad spatial parameters. In this chapter, we propose ordinary differential system which is based on the van der Pol models and which makes it possible to reconstruct pathological behaviours in the system of the heart, such as the WPW syndrome and different types of AVNRT. The motivation for writing this chapter was that there is a problem with making the appropri‐ ate diagnosis and therefore with treating the disease effectively. This kind of problem is observed mainly in different types of AVNRT. The reasons of those problems are not fully known and the mechanisms of these pathologies are not fully understood. Also, the symptoms are often mistakenly taken for other heart diseases. It should be emphasized that the clinical pictures of these disease are also non-specific. In the past few years, we understood that the structure of the AV node has a multi-level architecture in which there may be many pathways (slow and fast) at different locations in the AV node [2, 3]. This helped to recognize many types of AVNRT, which previously were understood as one, although the mechanisms of action were different. In literature, there are no mathematical models that would specifically model the various types of AVNRT. There were attempts to model AVNRT only as a single pathology having the slow and fast pathways [4, 5]. Part of the population has abnormal accessory pathways: fast and slow, cf. [1, 6]. The pathways in the AV node are anatomical and functional contributions of the most popular supraventric‐ ular tachycardia, which is a re-entry tachycardia from the AV node. The atrioventricular nodal re-entrant tachycardia is caused by re-entries. A condition for AVNRT to occur is that two electric pathways occur in and around the AV node (for example, slow and fast pathways). This gives way to the occurrence of re-entry. We can distinguish five different forms of the AVNRT (typical: slow/fast, atypical: fast/slow and other forms: slow/slow, more than two re-entries waves, one fast pathway with depolarization of slow pathway) [7, 8, 9]. This depends on the multi-level architecture of the AV node. The AVNRT circuit involves larger areas including atrioventricular junction, adjacent atrial structures and in particular so-called atrial inputs including at least antero-superior and postero-inferior entries, and sometimes also the left atrial entry. Based on the van der Pol equation, we study the influence of feedback which occurs in the normal heart action mode as well as in patholog‐ ical modes. Particularly, it is important to introduce the time delay into this feedback. Delay values used in our research correspond to those which occur in the electrical conduction system, for example, in the case of an accessory conducting pathway, so-called WolffParkinson-White syndrome. In the conduction system of the heart, the only correct electri‐ cal conduction pathway between atria and ventricles is the bundle of His. Part of the population has abnormal accessory pathways through which electrical pulses are directly conducted from atria to ventricles. The extra electrical pathway is presented at birth and is fairly rare. The WPW syndrome is detected in about 4 out of every 100000 people. People of all ages, including infants, can experience the symptoms related to the WPW syn‐ drome. Additional path conducts electrical impulses to the ventricles in the case of WPW. This conduction takes place without proper synchronization. Usually, this extra conduc‐ tion pathway does not cause serious disturbances. Sometimes, however, there is a reflec‐ tion of the electrical impulse, and it returns to the atria after each heartbeat. This causes the coupling in which each atrial contraction is followed by chambers contraction and then atrial contraction again. The heart rate can reach about 200 beats per minute when the normal rhythm at rest is about 72 beats per minute.

As already mentioned, synchronization is very important for a correctly functioning heart in the sense of cardiovascular conduction. The impact of different types of couplings related to the pathologies which are presented in this research and how do they affect the synchroniza‐ tion of the two oscillators will be discussed in this chapter. The analysis of synchronization of various modifications of the van der Pol model has been considered in many works. For example, paper [10] presents synchronization areas near the main parametric resonance and transition conditions from regular to chaotic motion. In paper [11], authors analyse the phenomenon of complete synchronization in a network of four coupled oscillators. Reference [12] considered mechanisms of various bifurcation phenomena occurring in Bonhoffer van der Pol neurons coupled with a time delay through the characteristics of synaptic transmissions. The phenomenon of synchronization in van der Pol oscillators coupled by a time-varying resistor is researched in ref. [13]. However, in these papers there are no examples of application of this model for recreating a pathological behaviour of the electrical-conduction system of the human heart, and therefore the considered ranges of parameters are wider than those which can be used in medical applications. The van der Pol oscillator provides rich dynamical behaviour [14] and also synchronization phenomena. In order to better understand the mechanisms that cause these arrhythmias in paper are proposed models of AVNRT and WPW syndrome.
