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CHAPTER 4

Electrical activity of the heart

4.1 Introduction
Doubtless, the cardiovascular system is one of the most complex structures in the
human body responsible for vital functions. Current research interest in elucidating its
mechanisms, not yet fully understood, is increasing due to the associated conditions,
which are often deadly such that the cardiovascular diseases are considered the main
cause of death worldwide (Thomas et al., 2004; Yancy, 2000; World Health
Organization, 2017). Numerous models for studying this topic can be found in litera-
ture, treating various branches of the main cardiovascular challenges. An extensive
review to the direct (forward, from cell to body) and inverse electrocardiography pro-
blems is presented by Pullan et al. (2005), who provides a glimpse in the mathematics
and associated computational methods used to numerically simulate and interpret the
heart’s physiological function. Same topic is also detailed in the studies by Boulakia
et al. (2008) and Szilagyi et al. (2002), which give numerical results on the parameters
estimation, and later on, Section 4.2 provides a brief discussion on the direct and
inverse electrocardiogram (ECG) problems.
Starting from the cell, the nonlinear dynamics of the ionic channels and the ionic
currents driven by concentration gradients between the inner and outer cell media is
described by mathematical models of the heart’s pacemaker activity (Di Francesco and
Noble, 1985), which may be solved numerically for the nonlinear electrical activity of
the heart (Murillo and Cai, 2004).
In the beginning, the physics of the heart’s sinoatrial (SA) node, the natural pace-
maker, was associated with a van der Pol oscillator (Van der Pol and Van der Mark,
1928). Later on, this assumption still remains valid and is considered close enough to
the real behavior of the SA node to keep developing new models (Sato et al., 1994;
Zduniak et al., 2014). The SA node activity is related also with more elaborated oscil-
lator models such as FitzHugh Nagumo, for the normal activity of the heart,
Nagumo (1962), and Landau Ginzburg, for different pathologies and abnormal
regimes (e.g., arrhythmias and fibrillations; Gong and Christini, 2004; Takembo et al.,
2019). These are used to reproduce the reentrant or spiral waves that mimic the action
potential (AP) propagation throughout the myocardium. The physical modeling and
numerical simulation of the heart still represents a great challenge due to the complex
phenomena underlying the myocardial tissue electrophysiology. The heart muscle
contractions are generated through the SA pulses. The nodal cells that make the

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DOI: https://doi.org/10.1016/B978-0-12-817897-3.00004-X All rights reserved. 93