Processes 2019, 7,20
and the canard theory [11,32] provides a strategy for the investigation of the complex dynamics of
dynamical systems. This strategy yields simultaneously the parameter dependence for the occurrence
of complex oscillatory behaviour of the studied system as well as the nature of these oscillations.
Further, our approach shows also that the reduction of the system complexity is associated with the
loose of information. In addition, if the singular limit is not satisfied, then the GSPT breaks down.
However, the GSPT provides a powerful approach to study simpler subsystems and to combine the
results of the studies, which yields a better understanding of the original system. This one can use for
a specifically targeted examination of the processes, e.g., with the bifurcation theory. The bifurcation
theory shows the behaviour of the system nicely with respect to one system parameter. This is
also possible for several bifurcation parameters, cf. [15], but it becomes more complicated and time
consuming. Moreover, the visualisation becomes more difficult if the phase space and/or the parameter
space of the system increase. Therefore, one has to be more careful regarding the interpretation of
the result.
Finally, we want to remark that for every new gating variable we have one more system parameter
with influence on the appearing of EADs. Even more for each new ion current depending on a specific
conductance, there is a further system parameter playing a huge role. Moreover, the investigation of
such system using GSPT, yielding on the one hand the important system parameters, as we saw here,
on the other hand we have to study ’only’ subsystem of a reduced dimension, which is easier to handle.
This approach we can use of course to investigate higher dimensional model C m ˙
V = −I ion + I stim ,
as in [33]orin[2]. High dimensional system are not only more challenging to study, in fact one has
more possibilities to control oscillatory dynamics in such systems. Therefore, it is highly interesting
and important to study high dimensional systems in theory as well as in applications. To this aim the
GSPT and the bifurcation theory are important components. The numerical efforts will be higher but
this will be a acceptable price which is to pay. Finally, we want to emphasise that the future project is
the extension from the cellular level to the tissue level, cf. e.g., [21,34–37].
Acknowledgments: The author wishes to thank the anonymous referees for their careful reading of the original
manuscript and their comments that eventually led to an improved presentation.
Conflicts of Interest: The author declares no conflict of interest.
References
1.
Xie, Y.; Izu, L.T.; Bers, D.M.; Sato, D. Arrhythmogenic Transient Dynamics in Cardiac Myocytes. Biophys. J.
2014, 106, 1391–1397. [CrossRef][PubMed]
2.
Clayton, R.H.; Bernus, O.; Cherry, E.M.; Dierckx, H.; Fenton, F.H.; Mirabella, L.; Panfilov, A.V.; Sachse, F.B.;
Seemann, G.; Zhang, H. Models of cardiac tissue electrophysiology: Progress, challenges and open questions.
Prog. Biophys. Mol. Biol. 2011, 104, 22–48. [CrossRef][PubMed]
3.
Fink, M.; Niederer, S.A.; Cherry, E.M.; Fenton, F.H.; Koivumäki, J.T.; Seemann, G.; Thul, R.; Zhang, H.;
Sachse, F.B.; Beard, D.; et al. Cardiac cell modelling: Observations from the heart of the cardiac physiome
project. Prog. Biophys. Mol. Biol. 2011, 104, 2–21. [CrossRef][PubMed]
4.
Landstrom, A.P.; Dobrev, D.; Wehrens, X.H.T. Calcium Signaling and Cardiac Arrhythmias. Circ. Res. 2017,
120, 1969–1993. [CrossRef][PubMed]
5.
Roden, D.M.; Viswanathan, P.C. Genetics of acquired long QT syndromeg. J. Clin. Investig. 2005, 115, 2025–2032.
[CrossRef][PubMed]
6.
Nieuwenhuyse, E.V.; Seemann, G.; Panfilov, A.V.; Vandersickel, N. Effects of early afterdepolarizations on
excitation patterns in an accurate model of the human ventricles. PLoS ONE 2017, 12, e0188867. [CrossRef]
[PubMed]
7.
Zimik, S.; Vandersickel, N.; Nayak, A.R.; Panfilov, A.V.; Pandit, R. A Comparative Study of Early
Afterdepolarization-Mediated Fibrillation in Two Mathematical Models for Human Ventricular Cells.
PLoS ONE 2015, 10, e0130632. [CrossRef][PubMed]
77
and the canard theory [11,32] provides a strategy for the investigation of the complex dynamics of
dynamical systems. This strategy yields simultaneously the parameter dependence for the occurrence
of complex oscillatory behaviour of the studied system as well as the nature of these oscillations.
Further, our approach shows also that the reduction of the system complexity is associated with the
loose of information. In addition, if the singular limit is not satisfied, then the GSPT breaks down.
However, the GSPT provides a powerful approach to study simpler subsystems and to combine the
results of the studies, which yields a better understanding of the original system. This one can use for
a specifically targeted examination of the processes, e.g., with the bifurcation theory. The bifurcation
theory shows the behaviour of the system nicely with respect to one system parameter. This is
also possible for several bifurcation parameters, cf. [15], but it becomes more complicated and time
consuming. Moreover, the visualisation becomes more difficult if the phase space and/or the parameter
space of the system increase. Therefore, one has to be more careful regarding the interpretation of
the result.
Finally, we want to remark that for every new gating variable we have one more system parameter
with influence on the appearing of EADs. Even more for each new ion current depending on a specific
conductance, there is a further system parameter playing a huge role. Moreover, the investigation of
such system using GSPT, yielding on the one hand the important system parameters, as we saw here,
on the other hand we have to study ’only’ subsystem of a reduced dimension, which is easier to handle.
This approach we can use of course to investigate higher dimensional model C m ˙
V = −I ion + I stim ,
as in [33]orin[2]. High dimensional system are not only more challenging to study, in fact one has
more possibilities to control oscillatory dynamics in such systems. Therefore, it is highly interesting
and important to study high dimensional systems in theory as well as in applications. To this aim the
GSPT and the bifurcation theory are important components. The numerical efforts will be higher but
this will be a acceptable price which is to pay. Finally, we want to emphasise that the future project is
the extension from the cellular level to the tissue level, cf. e.g., [21,34–37].
Acknowledgments: The author wishes to thank the anonymous referees for their careful reading of the original
manuscript and their comments that eventually led to an improved presentation.
Conflicts of Interest: The author declares no conflict of interest.
References
1.
Xie, Y.; Izu, L.T.; Bers, D.M.; Sato, D. Arrhythmogenic Transient Dynamics in Cardiac Myocytes. Biophys. J.
2014, 106, 1391–1397. [CrossRef][PubMed]
2.
Clayton, R.H.; Bernus, O.; Cherry, E.M.; Dierckx, H.; Fenton, F.H.; Mirabella, L.; Panfilov, A.V.; Sachse, F.B.;
Seemann, G.; Zhang, H. Models of cardiac tissue electrophysiology: Progress, challenges and open questions.
Prog. Biophys. Mol. Biol. 2011, 104, 22–48. [CrossRef][PubMed]
3.
Fink, M.; Niederer, S.A.; Cherry, E.M.; Fenton, F.H.; Koivumäki, J.T.; Seemann, G.; Thul, R.; Zhang, H.;
Sachse, F.B.; Beard, D.; et al. Cardiac cell modelling: Observations from the heart of the cardiac physiome
project. Prog. Biophys. Mol. Biol. 2011, 104, 2–21. [CrossRef][PubMed]
4.
Landstrom, A.P.; Dobrev, D.; Wehrens, X.H.T. Calcium Signaling and Cardiac Arrhythmias. Circ. Res. 2017,
120, 1969–1993. [CrossRef][PubMed]
5.
Roden, D.M.; Viswanathan, P.C. Genetics of acquired long QT syndromeg. J. Clin. Investig. 2005, 115, 2025–2032.
[CrossRef][PubMed]
6.
Nieuwenhuyse, E.V.; Seemann, G.; Panfilov, A.V.; Vandersickel, N. Effects of early afterdepolarizations on
excitation patterns in an accurate model of the human ventricles. PLoS ONE 2017, 12, e0188867. [CrossRef]
[PubMed]
7.
Zimik, S.; Vandersickel, N.; Nayak, A.R.; Panfilov, A.V.; Pandit, R. A Comparative Study of Early
Afterdepolarization-Mediated Fibrillation in Two Mathematical Models for Human Ventricular Cells.
PLoS ONE 2015, 10, e0130632. [CrossRef][PubMed]
77