Self-organized activation patterns in excitable media such as spiral waves and spatio-temporal chaos underlie dangerous cardiac arrhythmias. While the interaction of single spiral waves with different types of heterogeneity has been studied extensively, the effect of heterogeneity on fully developed spatio-temporal chaos remains poorly understood. We investigate how the complexity and stability properties of spatio-temporal chaos in the Bär–Eiswirth model of excitable media depend on the heterogeneity of the underlying medium. We employ different measures characterizing the chaoticity of the system and find that the spatial arrangement of multiple discrete lower excitability regions has a strong impact on the complexity of the dynamics. Varying the number, shape, and spatial arrangement of the heterogeneities, we observe strong emergent effects ranging from increases in chaoticity to the complete cessation of chaos, contrasting the expectation from the homogeneous behavior. The implications of our findings for the development and treatment of arrhythmias in the heterogeneous cardiac muscle are discussed.

1.
G.
Ertl
, “
Oscillatory kinetics and spatio-temporal self-organization in reactions at solid surfaces
,”
Science
254
,
1750
1755
(
1991
).
2.
W.
Jahnke
,
W.
Skaggs
, and
A.
Winfree
, “
Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the 2-variable Oregonator model
,”
J. Phys. Chem.
93
,
740
749
(
1989
).
3.
B.
Vasiev
,
P.
Hogeweg
, and
A.
Panfilov
, “
Simulation of Dictyostelium Discoideum aggregation via reaction-diffusion model
,”
Phys. Rev. Lett.
73
,
3173
3176
(
1994
).
4.
J.
Davidenko
,
A.
Pertsov
,
R.
Salomonsz
,
W.
Baxter
, and
J.
Jalife
, “
Stationary and drifting spiral waves of excitation in isolated cardiac muscle
,”
Nature
355
,
349
351
(
1992
).
5.
A.
Pertsov
,
J.
Davidenko
,
R.
Salomonsz
,
W.
Baxter
, and
J.
Jalife
, “
Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle
,”
Circ. Res.
72
,
631
650
(
1993
).
6.
R.
Gray
,
A.
Pertsov
, and
J.
Jalife
, “
Spatial and temporal organization during cardiac fibrillation
,”
Nature
392
,
75
78
(
1998
).
7.
F.
Witkowski
,
L.
Leon
,
P.
Penkoske
,
W.
Giles
,
M.
Spano
,
W.
Ditto
, and
A.
Winfree
, “
Spatiotemporal evolution of ventricular fibrillation
,”
Nature
392
,
78
82
(
1998
).
8.
R. H.
Clayton
,
O.
Bernus
,
E. M.
Cherry
,
H.
Dierckx
,
F. H.
Fenton
,
L.
Mirabella
,
A. V.
Panfilov
,
F. B.
Sachse
,
G.
Seemann
, and
H.
Zhang
, “
Models of cardiac tissue electrophysiology: Progress, challenges and open questions
,”
Prog. Biophys. Mol. Biol.
104
,
22
48
(
2011
).
9.
S.
Alonso
,
M.
Bär
, and
B.
Echebarria
, “
Nonlinear physics of electrical wave propagation in the heart: A review
,”
Rep. Prog. Phys.
79
,
096601
(
2016
).
10.
C.
Antzelevitch
and
J.
Fish
, “
Electrical heterogeneity within the ventricular wall
,”
Basic Res. Cardiol.
96
,
517
527
(
2001
).
11.
P.
Kohl
, “
Cardiac cellular heterogeneity and remodelling
,”
Cardiovasc. Res.
64
,
195
197
(
2004
).
12.
E. M.
Cherry
and
S. J.
Evans
, “
Properties of two human atrial cell models in tissue: Restitution, memory, propagation, and reentry
,”
J. Theor. Biol.
254
,
674
690
(
2008
).
13.
E.
Carmeliet
, “
Cardiac ionic currents and acute ischemia: From channels to arrhythmias
,”
Physiol. Rev.
79
,
917
1017
(
1999
); available at https://www.ncbi.nlm.nih.gov/pubmed/10390520.
14.
S.
Nattel
, “
Experimental evidence for proarrhythmic mechanisms of antiarrhythmic drugs
,”
Cardiovasc. Res.
37
,
567
577
(
1998
).
15.
P.
Kohl
,
P.
Camelliti
,
F. L.
Burton
, and
G. L.
Smith
, “
Electrical coupling of fibroblasts and myocytes: Relevance for cardiac propagation
,”
J. Electrocardiol.
38
,
45
50
(
2005
).
16.
M.
Miragoli
,
G.
Gaudesius
, and
S.
Rohr
, “
Electrotonic modulation of cardiac impulse conduction by myofibroblasts
,”
Circ. Res.
98
,
801
(
2006
).
17.
M. S.
Spach
, “
Mounting evidence that fibrosis generates a major mechanism for atrial fibrillation
,”
Circ. Res.
101
,
743
745
(
2007
).
18.
S.
Rohr
, “
Myofibroblasts in diseased hearts: New players in cardiac arrhythmias?
,”
Heart Rhythm
6
,
848
856
(
2009
).
19.
I. V.
Kazbanov
,
K. H. W. J.
ten Tusscher
, and
A. V.
Panfilov
, “
Effects of heterogeneous diffuse fibrosis on arrhythmia dynamics and mechanism
,”
Sci. Rep.
6
,
20835
(
2016
).
20.
T. K.
Shajahan
,
A. R.
Nayak
, and
R.
Pandit
, “
Spiral-wave turbulence and its control in the presence of inhomogeneities in four mathematical models of cardiac tissue
,”
PLoS One
4
,
e4738
(
2009
).
21.
V.
Zykov
,
A.
Krekhov
, and
E.
Bodenschatz
, “
Fast propagation regions cause self-sustained reentry in excitable media
,”
Proc. Natl. Acad. Sci. U.S.A.
114
,
1281
1286
(
2017
).
22.
E.
Boccia
,
U.
Parlitz
, and
S.
Luther
, “
Spontaneous termination of reentrant activity under myocardial acute ischemia: Role of cellular conductivity and its relation to ischemic heterogeneities
,”
Commun. Nonlinear Sci. Numer. Simul.
48
,
115
122
(
2017
).
23.
E.
Boccia
,
S.
Luther
, and
U.
Parlitz
, “
Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue
,”
Philos. Trans. R. Soc. A
375
,
20160289
(
2017
).
24.
D.
Barkley
, “
Linear stability analysis of rotating spiral waves in excitable media
,”
Phys. Rev. Lett.
68
,
2090
2093
(
1992
).
25.
D.
Barkley
, “
Euclidean symmetry and the dynamics of rotating spiral waves
,”
Phys. Rev. Lett.
72
,
164
167
(
1994
).
26.
D.
Allexandre
and
N. F.
Otani
, “
Preventing alternans-induced spiral wave breakup in cardiac tissue: An ion-channel-based approach
,”
Phys. Rev. E
70
,
061903
(
2004
).
27.
I. V.
Biktasheva
,
D.
Barkley
,
V. N.
Biktashev
, and
A. J.
Foulkes
, “
Computation of the drift velocity of spiral waves using response functions
,”
Phys. Rev. E
81
,
066202
(
2010
).
28.
I. V.
Biktasheva
,
A. V.
Holden
, and
V. N.
Biktashev
, “
Localization of response functions of spiral waves in the FitzHugh-Nagumo system
,”
Int. J. Bifurcation Chaos
16
,
1547
1555
(
2006
).
29.
V. G.
Fast
and
A. M.
Pertsov
, “
Drift of a vortex in the myocardium
,”
Biofizika
35
,
478
482
(
1990
).
30.
C. W.
Zemlin
and
A. M.
Pertsov
, “
Anchoring of drifting spiral and scroll waves to impermeable inclusions in excitable media
,”
Phys. Rev. Lett.
109
,
038303
(
2012
).
31.
A. V.
Panfilov
and
B.
Vasiev
, “
Vortex initiation in a heterogeneous excitable medium
,”
Physica D
49
,
107
113
(
1991
).
32.
P.
Sadeghi
and
H. H.
Rotermund
, “
Gradient induced spiral drift in heterogeneous excitable media
,”
Chaos
21
,
013125
(
2011
).
33.
K. H. W. J.
ten Tusscher
and
A. V.
Panfilov
, “
Reentry in heterogeneous cardiac tissue described by the Luo-Rudy ventricular action potential model
,”
Am. J. Physiol.: Heart Circ. Physiol.
284
,
H542
H548
(
2003
).
34.
P.
Comtois
and
S.
Nattel
, “
Impact of tissue geometry on simulated cholinergic atrial fibrillation: A modeling study
,”
Chaos
21
,
013108
(
2011
).
35.
S.
Sridhar
,
S.
Sinha
, and
A. V.
Panfilov
, “
Anomalous drift of spiral waves in heterogeneous excitable media
,”
Phys. Rev. E
82
,
051908
(
2010
).
36.
M.
Bär
and
M.
Eiswirth
, “
Turbulence due to spiral breakup in a continuous excitable medium
,”
Phys. Rev. E
48
,
R1635
R1637
(
1993
).
37.
M. C.
Strain
and
H. S.
Greenside
, “
Size-dependent transition to high-dimensional chaotic dynamics in a two-dimensional excitable medium
,”
Phys. Rev. Lett.
80
,
2306
(
1998
).
38.
D.
Barkley
, “
A model for fast computer simulation of waves in excitable media
,”
Physica D
49
,
61
70
(
1991
).
39.
M.
Bär
and
M.
Or-Guil
, “
Alternative scenarios of spiral breakup in a reaction-diffusion model with excitable and oscillatory dynamics
,”
Phys. Rev. Lett.
82
,
1160
(
1999
).
40.
P.
Bittihn
, “
Complex structure and dynamics of the heart
,” Springer Theses (
Springer International Publishing
,
2015
).
41.
G.
Benettin
,
L.
Galgani
,
A.
Giorgilli
, and
J.-M.
Strelcyn
, “
Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 1: Theory
,”
Meccanica
15
,
9
20
(
1980
).
42.
G.
Benettin
,
L.
Galgani
,
A.
Giorgilli
, and
J.-M.
Strelcyn
, “
Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 2: Numerical application
,”
Meccanica
15
,
21
30
(
1980
).
43.
A.
Iyer
and
R.
Gray
, “
An experimentalist's approach to accurate localization of phase singularities during reentry
,”
Ann. Biomed. Eng.
29
,
47
59
(
2001
).
44.
F.
Spreckelsen
,
D.
Hornung
,
O.
Steinbock
,
U.
Parlitz
, and
S.
Luther
, “
Stabilization of three-dimensional scroll waves and suppression of spatiotemporal chaos by heterogeneities
,”
Phys. Rev. E
92
,
042920
(
2015
).
45.
F.
Xie
,
Z.
Qu
, and
A.
Garfinkel
, “
Dynamics of reentry around a circular obstacle in cardiac tissue
,”
Phys. Rev. E
58
,
6355
(
1998
).
46.
Z.
Qu
, “
Critical mass hypothesis revisited: Role of dynamical wave stability in spontaneous termination of cardiac fibrillation
,”
Am. J. Physiol.: Heart Circ. Physiol.
290
,
H255
H263
(
2005
).
47.
M. J.
Bishop
,
E. J.
Vigmond
, and
G.
Plank
, “
The functional role of electrophysiological heterogeneity in the rabbit ventricle during rapid pacing and arrhythmias
,”
Am. J. Physiol.: Heart Circ. Physiol.
304
,
H1240
H1252
(
2013
).
48.
I.
Gussak
,
C.
Antzelevitch
,
A. A. M.
Wilde
,
P. A.
Friedman
,
M. J.
Ackerman
, and
W.-K.
Shen
, eds.,
Electrical Diseases of the Heart
(
Springer London
,
London
,
2008
).
49.
S.
Luther
,
F. H.
Fenton
,
B. G.
Kornreich
,
A.
Squires
,
P.
Bittihn
,
D.
Hornung
,
M.
Zabel
,
J.
Flanders
,
A.
Gladuli
,
L.
Campoy
,
E. M.
Cherry
,
G.
Luther
,
G.
Hasenfuss
,
V. I.
Krinsky
,
A.
Pumir
,
R. F.
Gilmour
, and
E.
Bodenschatz
, “
Low-energy control of electrical turbulence in the heart
,”
Nature
475
,
235
239
(
2011
).
50.
W.
Li
,
A. H.
Janardhan
,
V. V.
Fedorov
,
Q.
Sha
,
R. B.
Schuessler
, and
I. R.
Efimov
, “
Low-energy multistage atrial defibrillation therapy terminates atrial fibrillation with less energy than a single shock
,”
Circ.: Arrhythmia Electrophysiol.
4
,
917
925
(
2011
).
51.
C. M.
Ambrosi
,
C. M.
Ripplinger
,
I. R.
Efimov
, and
V. V.
Fedorov
, “
Termination of sustained atrial flutter and fibrillation using low-voltage multiple-shock therapy
,”
Heart Rhythm
8
,
101
108
(
2011
).
52.
W.-J.
Rappel
,
J. A. B.
Zaman
, and
S. M.
Narayan
, “
Mechanisms for the termination of atrial fibrillation by localized ablation
,”
Circ.: Arrhythmia Electrophysiol.
8
,
1325
1333
(
2015
).
53.
S.
Takagi
,
A.
Pumir
,
D.
Pazo
,
I.
Efimov
,
V. P.
Nikolski
, and
V. I.
Krinsky
, “
Unpinning and removal of a rotating wave in cardiac muscle
,”
Phys. Rev. Lett.
93
,
058101
(
2004
).
54.
C. M.
Ripplinger
,
V. I.
Krinsky
,
V. P.
Nikolski
, and
I. R.
Efimov
, “
Mechanisms of unpinning and termination of ventricular tachycardia
,”
Am. J. Physiol.: Heart Circ. Physiol.
291
,
H184
H192
(
2006
).
55.
P.
Bittihn
,
G.
Luther
,
E.
Bodenschatz
,
V. I.
Krinsky
,
U.
Parlitz
, and
S.
Luther
, “
Far field pacing supersedes anti-tachycardia pacing in a generic model of excitable media
,”
New J. Phys.
10
,
103012
(
2008
).
56.
A.
Pumir
,
S.
Sinha
,
S.
Sridhar
,
M.
Argentina
,
M.
Hörning
,
S.
Filippi
,
C.
Cherubini
,
S.
Luther
, and
V. I.
Krinsky
, “
Wave-train-induced termination of weakly anchored vortices in excitable media
,”
Phys. Rev. E
81
,
010901
(
2010
).
57.
A.
Isomura
,
M.
Hörning
,
K.
Agladze
, and
K.
Yoshikawa
, “
Eliminating spiral waves pinned to an anatomical obstacle in cardiac myocytes by high-frequency stimuli
,”
Phys. Rev. E
78
,
066216
(
2008
).
58.
S.
Morgan
,
I.
Biktasheva
, and
V.
Biktashev
, “
Control of scroll-wave turbulence using resonant perturbations
,”
Phys. Rev. E
78
,
046207
(
2008
).
59.
S.
Alonso
,
F.
Sagues
, and
A.
Mikhailov
, “
Taming winfree turbulence of scroll waves in excitable media
,”
Science
299
,
1722
1725
(
2003
).
60.
T.
Lilienkamp
,
J.
Christoph
, and
U.
Parlitz
, “
Features of chaotic transients in excitable media governed by spiral and scroll waves
,”
Phys. Rev. Lett.
119
,
054101
(
2017
).
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