In this work, we study the interplay between intrinsic biochemical noise and the diffusive coupling, in an array of three stochastic Brusselators that present a limit-cycle dynamics. The stochastic dynamics is simulated by means of the Gillespie algorithm. The intensity of the intrinsic biochemical noise is regulated by changing the value of the system volume (Ω), while keeping constant the chemical species' concentration. To characterize the system behavior, we measure the average spike amplitude (ASA), the order parameter R, the average interspike interval (ISI), and the coefficient of variation (CV) for the interspike interval. By analyzing how these measures depend on Ω and the coupling strength, we observe that when the coupling parameter is different from zero, increasing the level of intrinsic noise beyond a given threshold suddenly drives the spike amplitude, SA, to zero and makes ISI increase exponentially. These results provide numerical evidence that amplitude death (AD) takes place via a homoclinic bifurcation.

References

References
1.
Lara-Aparicio
,
M.
,
Barriga-Montoya
,
C.
,
Padilla-Longoria
,
P.
, and
Fuentes-Pardo
,
B.
,
2014
, “
Modeling Some Properties of Circadian Rhythms
,”
Math. Biosci. Eng.
,
11
(
2
), pp.
317
330
.
2.
Kopell
,
N.
,
Ermentrout
,
G. B.
,
Whittington
,
M. A.
, and
Traub
,
R. D.
,
2000
, “
Gamma Rhythms and Beta Rhythms Have Different Synchronization Properties
,”
PNAS
,
97
(
4
), pp.
1867
1872
.
3.
Whittington
,
M. A.
,
Traub
,
R. D.
, and
Jefferys
,
J. G. R.
,
1995
, “
Synchronized Oscillations in Interneuron Networks Driven by Metabotropic Glutamate Receptor Activation
,”
Nature
,
373
(
6515
), pp.
612
615
.
4.
Mirollo
,
R. E.
, and
Strogatz
,
S. H.
,
1990
, “
Synchronization of Pulse-Coupled Biological Oscillators
,”
SIAM J. Appl. Math.
,
50
(
6
), pp.
1645
1662
.
5.
Shiino
,
M.
, and
Frankowics
,
M.
,
1989
, “
Synchronization of Infinitely Many Coupled Limit-Cycle Type Oscillators
,”
Phys. Lett. A
,
136
(
3
), pp.
103
108
.
6.
Ermentrout
,
G. B.
,
1985
, “
Synchronization in a Pool of Mutually Coupled Oscillators With Random Frequencies
,”
J. Math. Biol.
,
22
, pp.
1
9
.
7.
Yamaguchi
,
Y.
, and
Shimizu
,
H.
,
1984
, “
Theory of Self-Synchronization in the Presence of Native Frequency Distribution and External Noises
,”
Phys. D
,
11
(
1–2
), pp.
212
226
.
8.
Tanu Singla
,
N. P.
, and
Parmananda
,
P.
,
2011
, “
Exploring the Dynamics of Conjugate Coupled Chua Circuits: Simulations and Experiments
,”
Phys. Rev. E
,
83
(
2 Pt. 2
), p.
026210
.
9.
Ramana Dodla
,
A. S.
, and
Johnston
,
G. L.
,
2006
, “
Phase-Locked Patterns and Amplitude Death in a Ring of Delay-Coupled Limit Cycle Oscillators
,”
Phys. Rev. E
,
69
(
5 Pt. 2
), p.
056217
.
10.
Strogatz
,
S. H.
,
Marcus
,
C. M.
,
Westervelt
,
R. M.
, and
Mirollo
,
R. E.
,
1989
, “
Collective Dynamics of Coupled Oscillators With Random Pinning
,”
Phys. D
,
36
(
1–2
), pp.
23
50
.
11.
Strogatz
,
S. H.
, and
Mirollo
,
R. E.
,
1988
, “
Phase-Locking and Critical Phenomena in Lattices of Coupled Nonlinear Oscillators With Random Intrinsic Frequencies
,”
Phys. D
,
31
(
2
), pp.
143
168
.
12.
Ryu
,
J.-W.
,
Kim
,
J.-H.
,
Son
,
W.-S.
, and
Hwang
,
D.-U.
,
2017
, “
Amplitude Death in a Ring of Nonidentical Nonlinear Oscillators With Unidirectional Coupling
,”
Chaos
,
27
, p.
083119
.
13.
Banerjee
,
T.
, and
Ghosh
,
D.
,
2014
, “
Transition From Amplitude to Oscillation Death Under Mean-Field Diffusive Coupling
,”
Phys. Rev. E
,
89
, p.
052912
.
14.
Sharma
,
A.
, and
Shrimali
,
M. D.
,
2012
, “
Amplitude Death With Mean-Field Diffusion
,”
Phys. Rev. E
,
85
(
5 Pt. 2
), p.
057204
.
15.
Garima Saxena
,
A. P.
, and
Ramaswany
,
R.
,
2012
, “
Amplitude Death: The Emergence of Stationary in Coupled Nonlinear Systems
,”
Phys. Rep.
,
521
(
5
), pp.
205
228
.
16.
Du
,
W. G. L.
, and
Mei
,
D.
,
2012
, “
Coherence and Spike Death Induced by Bounded Noise and Delayed Feedback in an Excitable System
,”
Eur. Phys. J. B
,
85
(
182
), pp.
1
7
.
17.
Sheng-Jun
,
W.
,
Xin-Jian
,
X.
,
Zhi-Xi
,
W.
,
Zi-Gang
,
H.
, and
Ying-Hai
,
W.
,
2008
, “
Influence of Synaptic Interaction on Firing Synchronization and Spike Death in Excitatory Neuronal Networks
,”
Phys. Rev. E
,
78
(
6 Pt. 1
), p.
061906
.
18.
Konishi
,
K.
,
2004
, “
Amplitude Death in Oscillators Coupled by a One-Way Ring Time-Delay Connection
,”
Phys. Rev. E
,
70
(
6 Pt. 2
), p.
066201
.
19.
Pisarchik
,
A. N.
,
2003
, “
Oscillation Death in Coupled Nonautonomous Systems With Parametrical Modulation
,”
Phys. Lett. A
,
318
(
1–2
), pp.
65
70
.
20.
Aronson
,
D. G.
,
Ermentrout
,
G. B.
, and
Kopell
,
N.
,
1990
, “
Amplitude Response of Coupled Oscillators
,”
Phys. D
,
41
(
3
), pp.
403
449
.
21.
Mirollo
,
R. E.
, and
Strogatz
,
S. H.
,
1990
, “
Amplitude Death in an Array of Limit-Cycle Oscillators
,”
J. Stat. Phys.
,
60
(
1–2
), pp.
245
262
.
22.
Matthews
,
P. C.
, and
Strogatz
,
S. H.
,
1990
, “
Phase Diagram for the Collective Behavior of Limit-Cycle Oscillators
,”
Phys. Rev. Lett.
,
65
(
14
), pp.
1701
1704
.
23.
Ermentrout
,
G. B.
,
1990
, “
Oscillator Death in Populations of ‘All to All’ Coupled Nonlinear Oscillator
,”
Phys. D
,
41
(
2
), pp.
219
231
.
24.
Koseska
,
A.
,
Volkov
,
E.
, and
Kurths
,
J. K.
,
2013
, “
Oscillation Quenching Mechanisms: Amplitude vs. Oscillation Death
,”
Phys. Rep.
,
531
(
4
), pp.
173
199
.
25.
Matjaz Perc
,
M. G.
, and
Marhl
,
M.
,
2007
, “
Periodic Calcium Waves in Coupled Cells Induced by Internal Noise
,”
Chem. Phys. Lett.
,
437
(
1–3
), pp.
143
147
.
26.
To
,
T.-L.
,
Henson
,
M. A.
,
Herzog
,
E. D
, and
Doyle
,
F. J.
, III
,
2007
, “
A Molecular Model for Intercellular Synchronization in the Mammalian Circadian Clock
,”
Biophys. J.
,
92
(
11
), pp.
3792
3803
.
27.
Jordi García-Ojalvo
,
M. B. E.
, and
Strogatz
,
S. H.
,
2004
, “
Modeling a Synthetic Multicellular Clock: Repressilators Coupled by Quorum Sensing
,”
PNAS
,
101
(
30
), pp.
10955
10960
.
28.
Gracheva
,
M. E.
, and
Gunton
,
J. D.
,
2003
, “
Intercellular Communication Via Intracellular Calcium Oscillations
,”
J. Theor. Biol.
,
221
(
4
), pp.
513
518
.
29.
Gracheva
,
M. E.
,
Toral
,
R.
, and
Gunton
,
J. D.
,
2001
, “
Stochastic Effects in Intercellular Calcium Spiking in Hepatocytes
,”
J. Theor. Biol.
,
212
(
1
), pp.
111
125
.
30.
Glass
,
L.
,
2001
, “
Synchronization and Rhythmic Processes in Physiology
,”
Nature
,
410
(
6825
), pp.
277
284
.
31.
Feudel
,
U.
,
Neiman
,
A.
,
Braun
,
H.
,
Huber
,
M.
, and
Moss
,
F.
,
2000
, “
Homoclinic Bifurcation in a Hodgkin-Huxley Model of Thermally Sensitive Neurons
,”
Chaos
,
10
(
1
), pp.
231
239
.
32.
Elowitz
,
M. B.
, and
Leibler
,
S.
,
2000
, “
A Synthetic Oscillatory Network of Transcriptional Regulators
,”
Nature
,
403
(
6767
), pp.
335
338
.
33.
Wei Wang
,
G. P.
, and
Cerdeira
,
H. A.
,
1993
, “
Dynamical Behavior of the Firings in a Coupled Neuronal System
,”
Phys. Rev. E
,
47
(
4
), pp.
2893
2898
.
34.
Dongmei Zhang
,
L. G.
, and
Peltier
,
W. R.
,
1993
, “
Deterministic Chaos in the Belousov-Zhabotinsky Reaction: Experiments and Simulations
,”
Chaos
,
3
(
4
), pp.
723
745
.
35.
Crowley
,
M. F.
, and
Epstein
,
I. R.
,
1989
, “
Experimental and Theoretical Studies of a Coupled Chemical Oscillator: Phase Multistability, and In-Phase and Out-of-Phase Entrainment
,”
J. Phys. Chem.
,
93
(
6
), pp.
2496
2502
.
36.
Sherman
,
J. R. A.
, and
Keizer
,
J.
,
1988
, “
Emergence of Organized Bursting in Clusters of Pancreatic A-Cells by Channel Sharing
,”
Biophys. J.
,
54
(
3
), pp.
411
425
.
37.
Winfree
,
A. T.
,
1967
, “
Biological Rhythms and the Behavior of Populations of Coupled Oscillators
,”
J. Theoret. Biol.
,
16
(
1
), pp.
15
42
.
38.
Hou
,
Z.
, and
Xin
,
H.
,
2003
, “
Oscillator Death on Small-World Networks
,”
Phys. Rev. E
,
68
(
5 Pt. 2
), p.
055103
.
39.
Kuramoto
,
Y.
,
1991
, “
Collective Synchronization of Pulse-Coupled Oscillators and Excitable Units
,”
Phys. D
,
50
(
1
), pp.
15
30
.
40.
Bar-Eli
,
K.
,
1985
, “
On the Stability of Coupled Chemical Oscillators
,”
Phys. D
,
14
(
2
), pp.
242
252
.
41.
Murray
,
J. D.
,
1974
, “
On a Model for the Temporal Oscillations in the Belousov-Zhabotinsky Reaction
,”
J. Chem. Phys.
,
61
(
9
), pp.
3610
3613
.
42.
Dey Supravat
,
D. D.
, and
Parmananda
,
P.
,
2011
, “
Intrinsic Noise Induced Resonance in Presence of Sub-Threshold Signal in Brusselator
,”
Chaos
,
21
(
3
), p.
033124
.
43.
Ly
,
C.
, and
Ermentrout
,
G. B.
,
2010
, “
Coupling Regularizes Individual Units in Noisy Populations
,”
Phys. Rev. E
,
81
(
1 Pt 1
), p.
011911
.
44.
Nandi
,
A.
, and
Lindner
,
B.
,
2010
, “
Intrinsic Common Noise in a System of Two Coupled Brusselators
,”
Chem. Phys.
,
375
(
2–3
), pp.
348
358
.
45.
Makarov
,
V. A.
,
Nekorkin
,
V. I.
, and
Velarde
,
M. G.
,
2001
, “
Spiking Behavior in a Noise-Driven System Combining Oscillatory and Excitatory Properties
,”
Phys. Rev. Lett.
,
86
(
15
), pp.
3431
3434
.
46.
Pikovsky
,
A. S.
, and
Kurths
,
J.
,
1997
, “
Coherence Resonance in a Noise-Driven Excitable System
,”
Phys. Rev. Lett.
,
78
(
5
), pp.
775
778
.
47.
Hairer
,
E.
,
Norsett
,
S. P.
, and
Wanner
,
G.
,
1993
,
Solving Ordinary Differential Equations I
,
Springer
,
Berlin
.
48.
René Lefever
,
G. N.
, and
Borckmans
,
P.
,
1988
, “
The Brusselator: It Does Oscillate All the Same
,”
J. Chem. Soc., Faraday Trans. 1
,
84
(
4
), pp.
1013
1023
.
49.
Gillespie
,
D. T.
,
1976
, “
A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions
,”
J. Comput. Phys.
,
22
(
4
), pp.
403
434
.
50.
Gillespie
,
D. T.
,
1977
, “
Exact Stochastic Simulation of Coupled Chemical Reactions
,”
J. Phys. Chem.
,
81
(
25
), pp.
2340
2361
.
51.
Strogatz
,
S. H.
,
2014
,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry and Engineering
,
2nd ed.
,
Westview Press
, Philadelphia, PA.
52.
Orlik
,
M.
,
2012
,
Self-Organization in Electrochemical Systems I: General Principles of Self-Organization. Temporal Instabilities
,
Springer Science & Business Media
, Berlin.
53.
Gray
,
P.
,
Nicolis
,
G.
,
Baras
,
F.
, and
Scott
,
S. K.
,
1990
,
Spatial Inhomogeneities and Transient Behaviour in Chemical Kinetics
,
Manchester University Press
,
Manchester, UK
.
54.
Gonze
,
D.
,
Bernard
,
S.
,
Waltermann
,
C.
,
Kramer
,
A.
, and
Herzel
,
H.
,
2005
, “
Spontaneous Synchronization of Coupled Circadian Oscillators
,”
Biophys. J.
,
89
(
1
), pp.
120
129
.
55.
Volkov
,
E. I.
, and
Stolyarov
,
M.
,
1994
, “
Temporal Variability in a System of Coupled Mitotic Timers
,”
Biol. Cybern.
,
71
(
5
), pp.
451
459
.
56.
Nini
,
A.
,
Feingold
,
A.
,
Slovin
,
H.
, and
Bergman
,
H.
,
1995
, “
Neurons in the Globus Pallidus Do Not Show Correlated Activity in the Normal Monkey, But Phase-Locked Oscillations Appear in the MPTP Model of Parkinsonism
,”
J. Neurophysiol.
,
74
(
4
), pp.
1800
1805
.
57.
Hammond
,
C.
,
Bergman
,
H.
, and
Brown
,
P.
,
2007
, “
Pathological Synchronization in Parkinson's Disease: Networks, Models and Treatments
,”
Trends Neurosci.
,
30
(
7
), pp.
357
364
.
You do not currently have access to this content.