Following an electrical stimulus, the transmembrane voltage of cardiac tissue rises rapidly and remains at a constant value before returning to the resting value, a phenomenon known as an action potential. When the pacing rate of a periodic train of stimuli is increased above a critical value, the action potential undergoes a period-doubling bifurcation, where the resulting alternation of the action potential duration is known as alternans in medical literature. Existing cardiac models treat alternans either as a smooth or as a border-collision bifurcation. However, recent experiments in paced cardiac tissue reveal that the bifurcation to alternans exhibits hybrid smooth∕nonsmooth behaviors, which can be qualitatively described by a model of so-called unfolded border-collision bifurcation. In this paper, we obtain analytical solutions of the unfolded border-collision model and use it to explore the crossover between smooth and nonsmooth behaviors. Our analysis shows that the hybrid smooth∕nonsmooth behavior is due to large variations in the system’s properties over a small interval of the bifurcation parameter, providing guidance for the development of future models.

2.
Rosenbaum
,
D. S.
,
Jackson
,
L. E.
,
Smith
,
J. M.
,
Garan
,
H.
,
Ruskin
,
J. N.
, and
Cohen
,
R. J.
, 1994, “
Electrical Alternans and Vulnerability to Ventricular Arrhythmias
,”
N. Engl. J. Med.
0028-4793,
330
, pp.
235
241
.
3.
Pastore
,
J. M.
,
Girouard
,
S. D.
,
Laurita
,
K. R.
,
Akar
,
F. G.
, and
Rosenbaum
,
D. S.
, 1999, “
Mechanism Linking T-Wave Alternans to the Genesis of Cardiac Fibrillation
,”
Circulation
0009-7322,
99
, pp.
1385
1394
.
4.
Plonsey
,
R.
, and
Barr
,
R. C.
, 2000,
Bioelectricity: A Quantitative Approach
,
Kluwer
,
New York
, 2000.
5.
Gilmour
,
R. F.
, Jr.
and
Chialvo
,
D. R.
, 1999, “
Electrical Restitution, Critical Mass, and the Riddle of Fibrillation
,”
J. Cardiovasc. Electrophysiol.
1045-3873,
10
, pp.
1087
1089
.
6.
Garfinkel
,
A.
,
Kim
,
Y. H.
,
Voroshilovsky
,
O.
,
Qu
,
Z.
,
Kil
,
J. R.
,
Lee
,
M. H.
,
Karagueuzian
,
H. S.
,
Weiss
,
J. N.
, and
Chen
,
P. S.
, 2000, “
Preventing Ventricular Fibrillation by Flattening Cardiac Restitution
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
97
, pp.
6061
6066
.
7.
Panfilov
,
A.
, 1998, “
Spiral Breakup as a Model of Ventricular Fibrillation
,”
Chaos
1054-1500,
8
, pp.
57
64
.
8.
Nolasco
,
J. B.
, and
Dahlen
,
R. W.
, 1968, “
A Graphic Method for the Study of Alternation in Cardiac Action Potentials
,”
J. Appl. Physiol.
0021-8987,
25
, pp.
191
196
.
9.
Chialvo
,
D. R.
,
Michaels
,
D. C.
, and
Jalife
,
J.
, 1990, “
Supernormal Excitability as a Mechanism of Chaotic Dynamics of Activation in Cardiac Purkinje-Fibers
,”
Circ. Res.
0009-7330,
66
, pp.
525
545
.
10.
Fox
,
J. J.
,
Bodenschatz
,
E.
, and
Gilmour
,
R. F.
, 2002, “
Period-Doubling Instability and Memory in Cardiac Tissue
,”
Phys. Rev. Lett.
0031-9007,
89
, pp.
138101
.
11.
Hall
,
G. M.
, and
Gauthier
,
D. J.
, 2002, “
Experimental Control of Cardiac Muscle Alternans
,”
Phys. Rev. Lett.
0031-9007,
88
, pp.
198102
.
12.
Tolkacheva
,
E. G.
,
Romeo
,
M. M.
,
Guerraty
,
M.
, and
Gauthier
,
D. J.
, 2004, “
Condition for Alternans and Its Control in a Two-Dimensional Mapping Model of Paced Cardiac Dynamics
,”
Phys. Rev. E
1063-651X,
69
, pp.
031904
.
13.
Hall
,
K.
,
Christini
,
D. J.
,
Tremblay
,
M.
,
Collins
,
J. J.
,
Glass
,
L.
, and
Billette
,
J.
, 1997, “
Dynamic Control of Cardiac Alternans
,”
Phys. Rev. Lett.
0031-9007,
78
, pp.
4518
4521
.
14.
Strogatz
,
S. H.
, 1994,
Nonlinear Dynamics and Chaos
,
Westview
,
Cambridge
.
15.
Sun
,
J.
,
Amellal
,
F.
,
Glass
,
L.
, and
Billette
,
J.
, 1995, “
Alternans and Period-Doubling Bifurcations in Atrioventricular Nodal Conduction
,”
J. Theor. Biol.
0022-5193,
173
, pp.
79
91
.
16.
Hassouneh
,
M. A.
, and
Abed
,
E. H.
, 2004, “
Border Collision Bifurcation Control of Cardiac Alternans
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274,
14
, pp.
3303
3315
.
17.
Berger
,
C. M.
,
Dobrovolny
,
H.
,
Zhao
,
X.
,
Schaeffer
,
D. G.
,
Krassowska
,
W.
, and
Gauthier
,
D. J.
, 2005, “
Evidence for a Border-Collision Bifurcation in Paced Cardiac Tissue
,” Southeastern Section of the APS, Gainesville, FL, Nov. 10–12.
18.
Fenton
,
F.
, 2006, “
Beyond Slope One: Alternans Suppression and Other Understudied Properties of APD Restitution
,”
KITP Miniprogram on Cardiac Dynamics
,
Kavli Institute for Theoretical Physics
,
Santa Barbara, CA
, Jul. 28.
19.
Cherry
,
E. M.
, and
Fenton
,
F. H.
, 2007, “
A Tale of Two Dogs: Analyzing Two Models of Canine Ventricular Electrophysiology
,”
Am. J. Physiol. Heart Circ. Physiol.
0363-6135,
292
, pp.
H43
H55
.
20.
di Bernardo
,
M.
,
Budd
,
C.
,
Champneys
,
A.
, and
Kowalczyk
,
P.
, 2007,
Bifurcation and Chaos in Piecewise-Smooth Dynamical Systems: Theory and Applications
,
Springer-Verlag
,
New York
.
21.
Zhusubaliyev
,
Z. T.
, and
Mosekilde
,
E.
, 2003,
Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems
,
World Scientific
,
Singapore
.
22.
Zhao
,
X.
,
Schaeffer
,
D. G.
,
Berger
,
C. M.
, and
Gauthier
,
D. J.
, 2007, “
Small Signal Amplification of Period-Doubling Bifurcations in Smooth Iterated Maps
,”
Nonlinear Dyn.
0924-090X,
48
, pp.
381
389
.
23.
Zhao
,
X.
, and
Schaeffer
,
D. G.
, 2007, “
Alternate Pacing Border-Collision Period-Doubling Bifurcations
,”
Nonlinear Dyn.
0924-090X,
50
, pp.
733
742
.
24.
Zhao
,
X.
,
Schaeffer
,
D. G.
,
Berger
,
C. M.
,
Gauthier
,
D. J.
, and
Krassowska
,
W.
, 2006, “
Evidence of an Unfolded Border-Collision Bifurcation in Paced Cardiac Tissue
,”
KITP Miniprogram on Cardiac Dynamics
,
Kavli Institute for Theoretical Physics
,
Santa Barbara, CA
, Jul. 13.
25.
Berger
,
C. M.
,
Zhao
,
X.
,
Schaeffer
,
D. G.
,
Dobrovolny
,
H.
,
Krassowska
,
W.
, and
Gauthier
,
D. J.
, 2007, “
Period-Doubling Bifurcation to Alternans in Paced Cardiac Tissue: Crossover From Smooth to Border-Collision Characteristics
,”
Phys. Rev. Lett.
0031-9007,
99
, pp.
058101
.
26.
Wiesenfeld
,
K.
, 1985, “
Virtual Hopf Phenomenon: A New Precursor of Period-Doubling Bifurcations
,”
Phys. Rev. A
1050-2947,
32
, pp.
1744
1751
.
27.
Wiesenfeld
,
K.
and
McNamara
,
B.
, 1986, “
Small-Signal Amplification in Bifurcating Dynamical Systems
,”
Phys. Rev. A
1050-2947,
33
, pp.
629
642
;
Wiesenfeld
,
K.
and
McNamara
,
B.
, 1986, “
Erratum: Small-Signal Amplification in Bifurcating Dynamical Systems [Phys. Rev. A 33, 629 (1986)]
,”
Phys. Rev. A
1050-2947,
33
, p.
3578
(E).
28.
Kravtsov
,
Yu. A.
, and
Surovyatkina
,
E. D.
, 2003, “
Nonlinear Saturation of Prebifurcation Noise Amplification
,”
Phys. Lett. A
0375-9601,
319
, pp.
348
351
.
29.
Surovyatkina
,
E. D.
, 2004, “
Rise and Saturation of the Correlation Time Near Bifurcation Threshold
,”
Phys. Lett. A
0375-9601,
329
, pp.
169
172
.
30.
Heldstab
,
J.
,
Thomas
,
H.
,
Geisel
,
T.
, and
Randons
,
G.
, 1983, “
Linear and Nonlinear Response of Discrete Dynamical Systems I. Periodic Attractors
,”
Z. Phys. B: Condens. Matter
0722-3277,
50
, pp.
141
150
.
31.
Guevara
,
M. R.
,
Ward
,
G.
,
Shrier
,
A.
, and
Glass
,
L.
, 1984, “
Electrical Alternans and Period-Doubling Bifurcations
,”
Proceedings of IEEE Computers in Cardiology
,
IEEE Computers Society
,
Silver Spring, MD
, pp.
167
170
.
32.
Shukla
,
A.
, and
Zhao
,
X.
, 2007, “
Special Issue on Discontinuous Dynamical Systems
,”
Nonlinear Dyn.
0924-090X,
50
, pp.
373
742
.
33.
Dankowicz
,
H.
, 2007, “
On the Purposeful Coarsening of Smooth Vector Fields
,”
Nonlinear Dyn.
0924-090X,
50
, pp.
511
522
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