Processes 2019, 7,20
8.
Vandersickel, N.; Panfilov, A.V. A study of early afterdepolarizations in human ventricular tissue.
In Proceedings of the 2015 Computing in Cardiology Conference (CinC), Nice, France, 6–9 September
2015; pp. 1213–1216.
9.
Sato, D.; Clancy, C.E.; Bers, D.M. Dynamics of sodium current mediated early afterdepolarizations.
J. Clin. Investig. 2017, 3, e00388. [CrossRef][PubMed]
10. Bergfeldt, L.; Lundahl, G.; Bergqvist, G.; Vahedi, F.; Gransberg, L. Ventricular repolarization duration and
dispersion adaptation after atropine induced rapid heart rate increase in healthy adults. J. Electrocardiol.
2017, 50, 424–432. [CrossRef]
11. Desroches, M.; Guckenheimer, J.; Krauskopf, B.; Kuehn, C.; Osinga, H.M.; Wechselberger, M. Mixed-Mode
Oscillations with Multiple Time Scales. SIAM Rev. 2012, 54, 211–288. [CrossRef]
12. C. Kuehn. Multiple Time Scale Dynamics; Applied Mathematical Sciences; Springer: Heidelberg, Germany;
New York, NY, USA, 2015; Volume 191.
13. Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer: New York, NY, USA, 1998.
14. Tsaneva-Atanasova, K.; Shuttleworth, T.J.; Yule, D.I.; Thompson, J.L.; Sneyd, J. Calcium Oscillations and
Membrane Transport: The Importance of Two Time Scales. Multiscale Model. Simul. 2005, 3, 245–264.
[CrossRef]
15. Erhardt, A.H. Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation
Parameters. Mathematics 2018, 6, 103. [CrossRef]
16. Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A. MATCONT: A MATLAB Package for Numerical Bifurcation
Analysis of ODEs. ACM Trans. Math. Softw. 2003, 29, 141–164. [CrossRef]
17. Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A.; Meijer, H.G.E.; Sautois, B. New features of the software MatCont
for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 2008, 14, 147–175. [CrossRef]
18. Govaerts, W.; Kuznetsov, Y.A.; Dhooge, A. Numerical Continuation of Bifurcations of Limit Cycles in
MATLAB. SIAM J. Sci. Comput. 2005, 27, 231–252. [CrossRef]
19. Hodgkin, A.L.; Huxley, A.F. A quantitative description of membrane current and its application to conduction
and excitation in nerve. J. Physiol. 1952, 117, 500–544. [CrossRef][PubMed]
20. Noble, D. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker
potentials. J. Physiol. 1962, 160, 317–352. [CrossRef][PubMed]
21. ten Tusscher, K.; Noble, D.; Noble, P.J.; Panfilov, A.V. A model for human ventricular tissue. Am. J. Physiol.
Heart. Circ. Physiol. 2004, 286, H1573–H1589. [CrossRef]
22. Szmolyan, P.; Wechselberger, M. Canards in R 3 . J. Differ. Equ. 2001, 177, 419–453. [CrossRef]
23. Vo, T.; Bertram, R.; Tabak, J.; Wechselberger, M. Mixed mode oscillations as a mechanism for pseudo-plateau
bursting. J. Comp. Neurosci. 2010, 28, 443–458. [CrossRef]
24. Wechselberger, M. Existence and Bifurcation of Canards in R 3 in the Case of a Folded Node. SIAM J. Appl.
Dyn. Syst. 2005, 4, 101–139. [CrossRef]
25. Rubin, J.; Wechselberger, M. Giant Squid-hidden Canard: The 3D Geometry of the Hodgkin-Huxley Model.
Biol. Cybern. 2007, 97, 5–32. [CrossRef][PubMed]
26. Vo, T.; Tabak, J.; Bertram, R.; Wechselberger, M. A geometric understanding of how fast activating potassium
channels promote bursting in pituitary cells. J. Comp. Neurosci. 2013, 36, 259–278. [CrossRef][PubMed]
27. Szmolyan, P.; Wechselberger, M. Relaxation oscillations in R 3 . J. Differ. Equ. 2004, 200, 69–104. [CrossRef]
28. Brøns, M.; Krupa, M.; Wechselberger, M. Mixed Mode Oscillations Due to the Generalized Canard
Phenomenon. Fields Inst. Commun. 2006, 49, 39–63.
29. Kügler, P.; Bulelzai, M.A.K.; Erhardt, A.H. Period doubling cascades of limit cycles in cardiac action potential
models as precursors to chaotic early Afterdepolarizations. BMC Syst. Biol. 2017, 11, 42. [CrossRef][PubMed]
30. Teka, W.; Tsaneva-Atanasova, K.; Bertram, R.; Tabak, J. From Plateau to Pseudo-Plateau Bursting: Making
the Transition. Bull. Math. Biol. 2011, 73, 1292–1311. [CrossRef][PubMed]
31. Tsaneva-Atanasova, K.; Osinga, H.M.; Rieb, T.; Sherman, A. Full system bifurcation analysis of endocrine
bursting models. J. Theor. Biol. 2010, 264, 1133–1146. [CrossRef][PubMed]
32. Benoit, E.; Callot, J.F.; Diener, F.; Diener, M. Chasse au canard. Collect. Math. 1981, 31–32, 37–119.
33. Luo, C.H.; Rudy, Y. A model of the ventricular cardiac action potential. Depolarization, repolarization,
and their interaction. Circ. Res. 1991, 68, 1501–1526. [CrossRef]
34. Sundnes, J.; Lines, G.T.; Nielsen, B.F.; Mardal, K.A.; Tveito, A. Computing the Electrical Activity in the Heart;
Springer: Heidelberg, Germany, 2006.
78
8.
Vandersickel, N.; Panfilov, A.V. A study of early afterdepolarizations in human ventricular tissue.
In Proceedings of the 2015 Computing in Cardiology Conference (CinC), Nice, France, 6–9 September
2015; pp. 1213–1216.
9.
Sato, D.; Clancy, C.E.; Bers, D.M. Dynamics of sodium current mediated early afterdepolarizations.
J. Clin. Investig. 2017, 3, e00388. [CrossRef][PubMed]
10. Bergfeldt, L.; Lundahl, G.; Bergqvist, G.; Vahedi, F.; Gransberg, L. Ventricular repolarization duration and
dispersion adaptation after atropine induced rapid heart rate increase in healthy adults. J. Electrocardiol.
2017, 50, 424–432. [CrossRef]
11. Desroches, M.; Guckenheimer, J.; Krauskopf, B.; Kuehn, C.; Osinga, H.M.; Wechselberger, M. Mixed-Mode
Oscillations with Multiple Time Scales. SIAM Rev. 2012, 54, 211–288. [CrossRef]
12. C. Kuehn. Multiple Time Scale Dynamics; Applied Mathematical Sciences; Springer: Heidelberg, Germany;
New York, NY, USA, 2015; Volume 191.
13. Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer: New York, NY, USA, 1998.
14. Tsaneva-Atanasova, K.; Shuttleworth, T.J.; Yule, D.I.; Thompson, J.L.; Sneyd, J. Calcium Oscillations and
Membrane Transport: The Importance of Two Time Scales. Multiscale Model. Simul. 2005, 3, 245–264.
[CrossRef]
15. Erhardt, A.H. Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation
Parameters. Mathematics 2018, 6, 103. [CrossRef]
16. Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A. MATCONT: A MATLAB Package for Numerical Bifurcation
Analysis of ODEs. ACM Trans. Math. Softw. 2003, 29, 141–164. [CrossRef]
17. Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A.; Meijer, H.G.E.; Sautois, B. New features of the software MatCont
for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 2008, 14, 147–175. [CrossRef]
18. Govaerts, W.; Kuznetsov, Y.A.; Dhooge, A. Numerical Continuation of Bifurcations of Limit Cycles in
MATLAB. SIAM J. Sci. Comput. 2005, 27, 231–252. [CrossRef]
19. Hodgkin, A.L.; Huxley, A.F. A quantitative description of membrane current and its application to conduction
and excitation in nerve. J. Physiol. 1952, 117, 500–544. [CrossRef][PubMed]
20. Noble, D. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker
potentials. J. Physiol. 1962, 160, 317–352. [CrossRef][PubMed]
21. ten Tusscher, K.; Noble, D.; Noble, P.J.; Panfilov, A.V. A model for human ventricular tissue. Am. J. Physiol.
Heart. Circ. Physiol. 2004, 286, H1573–H1589. [CrossRef]
22. Szmolyan, P.; Wechselberger, M. Canards in R 3 . J. Differ. Equ. 2001, 177, 419–453. [CrossRef]
23. Vo, T.; Bertram, R.; Tabak, J.; Wechselberger, M. Mixed mode oscillations as a mechanism for pseudo-plateau
bursting. J. Comp. Neurosci. 2010, 28, 443–458. [CrossRef]
24. Wechselberger, M. Existence and Bifurcation of Canards in R 3 in the Case of a Folded Node. SIAM J. Appl.
Dyn. Syst. 2005, 4, 101–139. [CrossRef]
25. Rubin, J.; Wechselberger, M. Giant Squid-hidden Canard: The 3D Geometry of the Hodgkin-Huxley Model.
Biol. Cybern. 2007, 97, 5–32. [CrossRef][PubMed]
26. Vo, T.; Tabak, J.; Bertram, R.; Wechselberger, M. A geometric understanding of how fast activating potassium
channels promote bursting in pituitary cells. J. Comp. Neurosci. 2013, 36, 259–278. [CrossRef][PubMed]
27. Szmolyan, P.; Wechselberger, M. Relaxation oscillations in R 3 . J. Differ. Equ. 2004, 200, 69–104. [CrossRef]
28. Brøns, M.; Krupa, M.; Wechselberger, M. Mixed Mode Oscillations Due to the Generalized Canard
Phenomenon. Fields Inst. Commun. 2006, 49, 39–63.
29. Kügler, P.; Bulelzai, M.A.K.; Erhardt, A.H. Period doubling cascades of limit cycles in cardiac action potential
models as precursors to chaotic early Afterdepolarizations. BMC Syst. Biol. 2017, 11, 42. [CrossRef][PubMed]
30. Teka, W.; Tsaneva-Atanasova, K.; Bertram, R.; Tabak, J. From Plateau to Pseudo-Plateau Bursting: Making
the Transition. Bull. Math. Biol. 2011, 73, 1292–1311. [CrossRef][PubMed]
31. Tsaneva-Atanasova, K.; Osinga, H.M.; Rieb, T.; Sherman, A. Full system bifurcation analysis of endocrine
bursting models. J. Theor. Biol. 2010, 264, 1133–1146. [CrossRef][PubMed]
32. Benoit, E.; Callot, J.F.; Diener, F.; Diener, M. Chasse au canard. Collect. Math. 1981, 31–32, 37–119.
33. Luo, C.H.; Rudy, Y. A model of the ventricular cardiac action potential. Depolarization, repolarization,
and their interaction. Circ. Res. 1991, 68, 1501–1526. [CrossRef]
34. Sundnes, J.; Lines, G.T.; Nielsen, B.F.; Mardal, K.A.; Tveito, A. Computing the Electrical Activity in the Heart;
Springer: Heidelberg, Germany, 2006.
78