processes
Article
Early Afterdepolarisations Induced by an
Enhancement in the Calcium Current
André H. Erhardt
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway;
andreerh@math.uio.no
Received: 20 November 2018; Accepted: 29 December 2018; Published: 4 January 2019
Abstract: Excitable biological cells, such as cardiac muscle cells, can exhibit complex patterns of
oscillations such as spiking and bursting. Moreover, it is well known that an enhancement in calcium
currents may yield certain kind of cardiac arrhythmia, so-called early afterdepolarisations (EADs).
The presence of EADs strongly correlates with the onset of dangerous cardiac arrhythmia. In this
paper we study mathematically and numerically the dynamics of a cardiac muscle cell with respect
to the calcium current by investigating a simplistic system of differential equations. For the study
of this phenomena, we use bifurcation theory, numerical bifurcation analysis, geometric singular
perturbation theory and computational methods to investigate a nonlinear multiple time scales
system. It will turn out that EADs related to an enhanced calcium current are canard–induced and
that we have to combine these theories to derive a better understanding of the dynamics behind
EADs. Moreover, a suitable time scale separation argument determines the important and sensitive
system parameters which are related to the occurrence of EADs.
Keywords: nonlinear dynamics; multiple time scales; geometric singular perturbation theory;
bifurcation analysis; canard-induced EADs; calcium current
MSC: 37G15; 37N25; 65P30; 92B05
1. Introduction
The aim of this manuscript is the mathematical and numerical investigation of a four dimensional
version of the model introduced in [1] with respect to an enhancement in the calcium current,
which is already used to study early afterdepolarisations (EADs)—a special type of cardiac
arrhythmia—induced by a reduced potassium current. We will show reasons for the occurrence of
EADs via an enhancement in the calcium current, using numerical bifurcation analysis and geometric
singular perturbation theory (GSPT). One main advantage of the GSPT, which is an analytic technique
for multi-scale problems that combines asymptotic theory with dynamical techniques, is the study
of a reduced model, i.e., a subsystem. This approach is very useful and shows some mechanisms
yielding EADs. Moreover, this ansatz is very valuable to identify the sensitive parameters of the
system. Nevertheless, it turns out that not all details can be explained using GSPT. Thus, a combination
of both theories—bifurcation theory and geometric singular perturbation theory—is needed. We will
explain our approach for this simplified model, but of course we can use this ansatz also for more
complex models, cf. [2,3].
In general, EADs are additional small amplitude spikes during the plateau or the repolarisation
phase of the action potential (AP), i.e., pathological voltage oscillations during one of these phases.
They are caused by ion channel diseases, oxidative stress or drugs and are often associated with
deficiencies in potassium currents or enhancements in calcium currents [4]. Furthermore, the presence
of EADs strongly correlates with the onset of dangerous cardiac arrhythmias, including torsades de
Processes 2019, 7, 20; doi:10.3390/pr7010020
www.mdpi.com/journal/processes
64
Article
Early Afterdepolarisations Induced by an
Enhancement in the Calcium Current
André H. Erhardt
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway;
andreerh@math.uio.no
Received: 20 November 2018; Accepted: 29 December 2018; Published: 4 January 2019
Abstract: Excitable biological cells, such as cardiac muscle cells, can exhibit complex patterns of
oscillations such as spiking and bursting. Moreover, it is well known that an enhancement in calcium
currents may yield certain kind of cardiac arrhythmia, so-called early afterdepolarisations (EADs).
The presence of EADs strongly correlates with the onset of dangerous cardiac arrhythmia. In this
paper we study mathematically and numerically the dynamics of a cardiac muscle cell with respect
to the calcium current by investigating a simplistic system of differential equations. For the study
of this phenomena, we use bifurcation theory, numerical bifurcation analysis, geometric singular
perturbation theory and computational methods to investigate a nonlinear multiple time scales
system. It will turn out that EADs related to an enhanced calcium current are canard–induced and
that we have to combine these theories to derive a better understanding of the dynamics behind
EADs. Moreover, a suitable time scale separation argument determines the important and sensitive
system parameters which are related to the occurrence of EADs.
Keywords: nonlinear dynamics; multiple time scales; geometric singular perturbation theory;
bifurcation analysis; canard-induced EADs; calcium current
MSC: 37G15; 37N25; 65P30; 92B05
1. Introduction
The aim of this manuscript is the mathematical and numerical investigation of a four dimensional
version of the model introduced in [1] with respect to an enhancement in the calcium current,
which is already used to study early afterdepolarisations (EADs)—a special type of cardiac
arrhythmia—induced by a reduced potassium current. We will show reasons for the occurrence of
EADs via an enhancement in the calcium current, using numerical bifurcation analysis and geometric
singular perturbation theory (GSPT). One main advantage of the GSPT, which is an analytic technique
for multi-scale problems that combines asymptotic theory with dynamical techniques, is the study
of a reduced model, i.e., a subsystem. This approach is very useful and shows some mechanisms
yielding EADs. Moreover, this ansatz is very valuable to identify the sensitive parameters of the
system. Nevertheless, it turns out that not all details can be explained using GSPT. Thus, a combination
of both theories—bifurcation theory and geometric singular perturbation theory—is needed. We will
explain our approach for this simplified model, but of course we can use this ansatz also for more
complex models, cf. [2,3].
In general, EADs are additional small amplitude spikes during the plateau or the repolarisation
phase of the action potential (AP), i.e., pathological voltage oscillations during one of these phases.
They are caused by ion channel diseases, oxidative stress or drugs and are often associated with
deficiencies in potassium currents or enhancements in calcium currents [4]. Furthermore, the presence
of EADs strongly correlates with the onset of dangerous cardiac arrhythmias, including torsades de
Processes 2019, 7, 20; doi:10.3390/pr7010020
www.mdpi.com/journal/processes
64