Processes 2019, 7,20
pointes (TdP), which is a specific type of abnormal heart rhythm that can lead to sudden cardiac
death, see [5–10]. Furthermore, EADs are so-called mixed-mode oscillations (MMOs) [11], i.e., complex
oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs
correspond to the switching between small amplitude oscillations and relaxation oscillations.
In this paper, we will use the geometric singular perturbation theory [11,12] and bifurcation
analysis [13] to investigate reasons for the appearing of EADs. Here, we are focused on EADs related
to an enhancement in the calcium currents, see [14,15]. The main novelty is the combination of these
theories to study EADs and mainly the use of the needed time scale separation argument to derive
the parameter sensitivity of the considered system. Moreover, we will show that the mathematical
approach which is used for instance in [1] is limited to the study of EADs related to an inhibited
potassium current. We will see that the considered system exhibits up to four different time scales
depending on the different system parameters.
The paper is organised as follows. We start with a brief introduction into the topic of cardiac
APs and arrhythmia, i.e., afterdepolarisations, see Section 1.1. Then, in Section 1.2 we will go on
with the mathematical modelling of cardiac APs using a Hodgkin-Huxley type formalism. For our
mathematical and numerical analysis of the dynamics of our model, we will use the GSPT and
bifurcation analysis. Therefore, in Section 2.1 we will give a brief introduction into the topic of GSPT.
This theory we will utilise in Section 2.2 and it turns out that EADs related to an enhanced calcium
current are canard–induced MMOs. Nevertheless, in Section 2.3 we will show that the study of the
reduced system does not show all details of the occurrence of EADs. Therefore, we are also using
numerical bifurcation analysis. The desired bifurcation diagram we will derive utilising the MATLAB
toolboxes MATCONT and CL_MATCONT [16–18], which are numerical continuation packages for the
interactive bifurcation analysis of dynamical systems. Finally, in Section 3 we will discuss our results.
1.1. Biological and Mathematical Background
An AP is a temporary, characteristic variance in the membrane potential of an excitable biological
cell, e.g., neuron or cardiac muscle cell, from its resting potential. The molecular mechanism of an
AP is based on the interaction of voltage-sensitive ion channels. The reason for the formation and
the special properties of the AP is established in the properties of different groups of ion channels in
the plasma membrane. An initial stimulus activates the ion channels as soon as a certain threshold
potential is reached. Then, these ion channels break open and/or up such that this interaction allows
an ion current flow, which changes the membrane potential. A normal AP is always uniform and
the cardiac muscle cell AP is typically divided into four phases, i.e., the resting phase, the upstroke
phase, the (long) plateau phase and the repolarisation phase, see for more details [15]. The resting
phase is designated by high potassium (K + ) currents. After the initial stimulus the sodium (Na + )
conductance increases rapidly and the Na + current flux into the cardiac muscle cell until a spike
potential is achieved. Then, the Na + current inactivates rapidly followed by the activation of L-type
calcium (Ca 2+ ) current. The Ca 2+ current is more slowly than the Na + current and plays a key role in
maintaining the long plateau phase, which is characteristic for the cardiac muscle cell. While the Ca 2+
conductance increases the K + conductance decreases. The plateau phase is followed by a repolarisation
phase, where the intrinsic K + ion channels are activated and this is connected with the reduction of
the Ca 2+ conductance. Finally, the K + current increases until the resting phase is reached. If there are
depolarising variations of the membrane voltage, then we are speaking about afterdepolarisations.
These afterdepolarisations are divided into EADs and delayed afterdepolarisations (DADs). This
division depends on the timing obtaining of the AP. EADs occur either in the plateau or in the
repolarisation phase of the AP and are benefited by an elongation of the AP, while DADs occur
after the repolarisation phase is completed. EADs are resulting for example from a reduction of the
repolarising K + currents. Triggers for this are congenital disorders of the ion channels or the ingestion
of some medicament. The elongation of the AP can generate afterpolarisations by reactivation L-type
65
pointes (TdP), which is a specific type of abnormal heart rhythm that can lead to sudden cardiac
death, see [5–10]. Furthermore, EADs are so-called mixed-mode oscillations (MMOs) [11], i.e., complex
oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs
correspond to the switching between small amplitude oscillations and relaxation oscillations.
In this paper, we will use the geometric singular perturbation theory [11,12] and bifurcation
analysis [13] to investigate reasons for the appearing of EADs. Here, we are focused on EADs related
to an enhancement in the calcium currents, see [14,15]. The main novelty is the combination of these
theories to study EADs and mainly the use of the needed time scale separation argument to derive
the parameter sensitivity of the considered system. Moreover, we will show that the mathematical
approach which is used for instance in [1] is limited to the study of EADs related to an inhibited
potassium current. We will see that the considered system exhibits up to four different time scales
depending on the different system parameters.
The paper is organised as follows. We start with a brief introduction into the topic of cardiac
APs and arrhythmia, i.e., afterdepolarisations, see Section 1.1. Then, in Section 1.2 we will go on
with the mathematical modelling of cardiac APs using a Hodgkin-Huxley type formalism. For our
mathematical and numerical analysis of the dynamics of our model, we will use the GSPT and
bifurcation analysis. Therefore, in Section 2.1 we will give a brief introduction into the topic of GSPT.
This theory we will utilise in Section 2.2 and it turns out that EADs related to an enhanced calcium
current are canard–induced MMOs. Nevertheless, in Section 2.3 we will show that the study of the
reduced system does not show all details of the occurrence of EADs. Therefore, we are also using
numerical bifurcation analysis. The desired bifurcation diagram we will derive utilising the MATLAB
toolboxes MATCONT and CL_MATCONT [16–18], which are numerical continuation packages for the
interactive bifurcation analysis of dynamical systems. Finally, in Section 3 we will discuss our results.
1.1. Biological and Mathematical Background
An AP is a temporary, characteristic variance in the membrane potential of an excitable biological
cell, e.g., neuron or cardiac muscle cell, from its resting potential. The molecular mechanism of an
AP is based on the interaction of voltage-sensitive ion channels. The reason for the formation and
the special properties of the AP is established in the properties of different groups of ion channels in
the plasma membrane. An initial stimulus activates the ion channels as soon as a certain threshold
potential is reached. Then, these ion channels break open and/or up such that this interaction allows
an ion current flow, which changes the membrane potential. A normal AP is always uniform and
the cardiac muscle cell AP is typically divided into four phases, i.e., the resting phase, the upstroke
phase, the (long) plateau phase and the repolarisation phase, see for more details [15]. The resting
phase is designated by high potassium (K + ) currents. After the initial stimulus the sodium (Na + )
conductance increases rapidly and the Na + current flux into the cardiac muscle cell until a spike
potential is achieved. Then, the Na + current inactivates rapidly followed by the activation of L-type
calcium (Ca 2+ ) current. The Ca 2+ current is more slowly than the Na + current and plays a key role in
maintaining the long plateau phase, which is characteristic for the cardiac muscle cell. While the Ca 2+
conductance increases the K + conductance decreases. The plateau phase is followed by a repolarisation
phase, where the intrinsic K + ion channels are activated and this is connected with the reduction of
the Ca 2+ conductance. Finally, the K + current increases until the resting phase is reached. If there are
depolarising variations of the membrane voltage, then we are speaking about afterdepolarisations.
These afterdepolarisations are divided into EADs and delayed afterdepolarisations (DADs). This
division depends on the timing obtaining of the AP. EADs occur either in the plateau or in the
repolarisation phase of the AP and are benefited by an elongation of the AP, while DADs occur
after the repolarisation phase is completed. EADs are resulting for example from a reduction of the
repolarising K + currents. Triggers for this are congenital disorders of the ion channels or the ingestion
of some medicament. The elongation of the AP can generate afterpolarisations by reactivation L-type
65