A passive electrochemical coupling approach is proposed to induce spontaneous synchronization between chemical oscillators. The coupling exploits the potential difference between a catalyst redox couple in the Belousov–Zhabotinsky (BZ) reaction, without external feedback, to induce surface reactions that impact the kinetics of the bulk system. The effect of coupling in BZ oscillators under batch condition is characterized using phase synchronization measures. Although the frequency of the oscillators decreases nonlinearly over time, by a factor of 2 or more within 100 cycles, the coupling is strong enough to maintain synchronization. In such a highly drifting system, the Gibbs–Shannon entropy of the cyclic phase difference distribution can be used to quantify the coupling effect. We extend the Oregonator BZ model to account for the drifting natural frequencies in batch condition and for electrochemical coupling, and numerical simulations of the effect of acid concentration on synchronization patterns are in agreement with the experiments. Because of the passive nature of coupling, the proposed coupling scheme can open avenues for designing pattern recognition and neuromorphic computation systems using chemical reactions in a spontaneous process.

1.
I. R.
Epstein
, “
Coupled chemical oscillators and emergent system properties
,”
Chem. Commun.
50
,
10758
10767
(
2014
).
2.
A. F.
Taylor
,
M. R.
Tinsley
, and
K.
Showalter
, “
Insights into collective cell behaviour from populations of coupled chemical oscillators
,”
Phys. Chem. Chem. Phys.
17
,
20047
20055
(
2015
).
3.
I. R.
Epstein
and
K.
Showalter
, “
Nonlinear chemical dynamics: Oscillations, patterns, and chaos
,”
J. Phys. Chem.
100
,
13132
13147
(
1996
).
4.
Y.
Kuramoto
,
Chemical Oscillations, Waves & Turbulence
(
Springer
,
Berlin
,
1984
).
5.
M. A.
Budroni
,
G.
Pagano
,
D.
Conte
,
B.
Paternoster
,
R.
D’ambrosio
,
S.
Ristori
,
A.
Abou-Hassan
, and
F.
Rossi
, “
Synchronization scenarios induced by delayed communication in arrays of diffusively coupled autonomous chemical oscillators
,”
Phys. Chem. Chem. Phys.
23
,
17606
17615
(
2021
).
6.
G.
Philippou
and
D.
Luss
, “
Temperature patterns on a catalytic ribbon heated by a constant voltage
,”
Chem. Eng. Sci.
48
,
2313
2323
(
1993
).
7.
I. Z.
Kiss
,
Y.
Zhai
, and
J. L.
Hudson
, “
Emerging coherence in a population of chemical oscillators
,”
Science
296
,
1676
1678
(
2002
).
8.
R. J.
Field
,
E.
Koros
, and
R. M.
Noyes
, “
Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system
,”
J. Am. Chem. Soc.
94
,
8649
8664
(
1972
).
9.
R. J.
Field
and
R. M.
Noyes
, “
Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction
,”
J. Chem. Phys.
60
,
1877
1884
(
1974
).
10.
Oscillations and Traveling Waves in Chemical Systems, edited by R. J. Field and M. Burger (Wiley, New York, 1985).
11.
M.
Marek
and
I.
Stuchl
, “
Synchronization in two interacting oscillatory systems
,”
Biophys. Chem.
3
,
241
248
(
1975
).
12.
I.
Stuchl
and
M.
Marek
, “
Dissipative structures in coupled cells: Experiments
,”
J. Chem. Phys.
77
,
2956
2963
(
1982
).
13.
M. F.
Crowley
and
I. R.
Epstein
, “
Experimental and theoretical studies of a coupled chemical oscillator: Phase death, multistability and in-phase and out-of-phase entrainment
,”
J. Phys. Chem.
93
,
2496
2502
(
1989
).
14.
H.
Fujii
and
Y.
Sawada
, “
Phase-difference locking of coupled oscillating chemical systems
,”
J. Chem. Phys.
69
,
3830
3832
(
1978
).
15.
A. F.
Taylor
,
M. R.
Tinsley
,
F.
Wang
,
Z.
Huang
, and
K.
Showalter
, “
Dynamical quorum sensing and synchronization in large populations of chemical oscillators
,”
Science
323
,
614
617
(
2009
).
16.
M. R.
Tinsley
,
S.
Nkomo
, and
K.
Showalter
, “
Chimera and phase-cluster states in populations of coupled chemical oscillators
,”
Nat. Phys.
8
,
662
665
(
2012
).
17.
A. F.
Taylor
,
M. R.
Tinsley
,
F.
Wang
, and
K.
Showalter
, “
Phase clusters in large populations of chemical oscillators
,”
Angew. Chem. Int. Ed.
50
,
10161
10164
(
2011
).
18.
J. F.
Totz
,
J.
Rode
,
M. R.
Tinsley
,
K.
Showalter
, and
H.
Engel
, “
Spiral wave chimera states in large populations of coupled chemical oscillators
,”
Nat. Phys.
14
,
282
285
(
2018
).
19.
N.
Tompkins
,
N.
Li
,
C.
Girabawe
,
M.
Heymann
,
G.
Ermentrout
,
I. R.
Epstein
, and
S.
Fraden
, “
Testing Turing’s theory of morphogenesis in chemical cells
,”
Proc. Natl. Acad. Sci. U.S.A.
111
,
4397
4402
(
2014
).
20.
M. M.
Norton
,
N.
Tompkins
,
B.
Blanc
,
M. C.
Cambria
,
J.
Held
, and
S.
Fraden
, “
Dynamics of reaction-diffusion oscillators in star and other networks with cyclic symmetries exhibiting multiple clusters
,”
Phys. Rev. Lett.
123
,
148301
(
2019
).
21.
M.
Wickramasinghe
and
I. Z.
Kiss
, “Synchronization of electrochemical oscillators,” in Engineering of Chemical Complexity (World Scientific, New Jersey, 2013), pp. 215–236.
22.
D. K.
Verma
,
H.
Singh
,
A. Q.
Contractor
, and
P.
Parmananda
, “
Synchronization in autonomous mercury beating heart systems
,”
J. Phys. Chem. A
118
,
4647
4651
(
2014
).
23.
T.
Singla
,
F.
Montoya
,
M.
Rivera
,
S.
Tajima
,
S.
Nakabayashi
, and
P.
Parmananda
, “
Synchronization using environmental coupling in mercury beating heart oscillators
,”
Chaos
26
,
063103
(
2016
).
24.
P.
Kumar
,
D. K.
Verma
,
P.
Parmananda
, and
S.
Boccaletti
, “
Experimental evidence of explosive synchronization in mercury beating-heart oscillators
,”
Phys. Rev. E
91
,
062909
(
2015
).
25.
A.
Biswas
,
D.
Das
, and
P.
Parmananda
, “
Scaling dependence and synchronization of forced mercury beating heart systems
,”
Phys. Rev. E
95
,
042202
(
2017
).
26.
A.
Biswas
,
P.
Kumar
,
D.
Das
, and
P.
Parmananda
, “
Oscillatory activity regulation in an ensemble of autonomous mercury beating heart oscillators
,”
Phys. Rev. E
99
,
032223
(
2019
).
27.
P.
Kumar
and
P.
Parmananda
, “
Control, synchronization, and enhanced reliability of aperiodic oscillations in the mercury beating heart system
,”
Chaos
28
,
045105
(
2018
).
28.
S.
Tajima
,
H.
Singh
,
S.
Nakabayashi
,
T.
Singla
, and
P.
Parmananda
, “
The emergence of synchrony behavior in weakly coupled electrochemical oscillators via a ‘metallic plate
,”’
J. Electroanal. Chem.
769
,
16
20
(
2016
).
29.
B. K.
Bera
,
D.
Ghosh
,
P.
Parmananda
,
G. V.
Osipov
, and
S. K.
Dana
, “
Coexisting synchronous and asynchronous states in locally coupled array of oscillators by partial self-feedback control
,”
Chaos
27
,
073108
(
2017
).
30.
F.
Montoya
,
M.
Rivera
,
J.
Escalona
, and
P.
Parmananda
, “
Construction of Arnold tongue structures for coupled periodic oscillators
,”
Phys. Lett. A
377
,
3124
3127
(
2013
).
31.
R.
Nagao
,
W.
Zou
,
J.
Kurths
, and
I. Z.
Kiss
, “
Restoring oscillatory behavior from amplitude death with anti-phase synchronization patterns in networks of electrochemical oscillations
,”
Chaos
26
,
094808
(
2016
).
32.
M.
Wickramasinghe
and
I. Z.
Kiss
, “
Spatially organized dynamical states in chemical oscillator networks: Synchronization, dynamical differentiation, and chimera patterns
,”
PLoS One
8
,
e80586
(
2013
).
33.
Y.
Liu
,
M.
Sebek
,
F.
Mori
, and
I. Z.
Kiss
, “
Synchronization of three electrochemical oscillators: From local to global coupling
,”
Chaos
28
,
045104
(
2018
).
34.
M.
Wickramasinghe
,
E. M.
Mrugacz
, and
I. Z.
Kiss
, “
Dynamics of electrochemical oscillators with electrode size disparity: Asymmetrical coupling and anomalous phase synchronization
,”
Phys. Chem. Chem. Phys.
13
,
15483
15491
(
2011
).
35.
I. Z.
Kiss
and
J. L.
Hudson
, “
Phase synchronization of nonidentical chaotic electrochemical oscillators
,”
Phys. Chem. Chem. Phys.
4
,
2638
2647
(
2002
).
36.
I. Z.
Kiss
,
W.
Wang
, and
J. L.
Hudson
, “
Complexity of globally coupled chaotic electrochemical oscillators
,”
Phys. Chem. Chem. Phys.
2
,
3847
3854
(
2000
).
37.
I. Z.
Kiss
,
Y.
Zhai
, and
J. L.
Hudson
, “
Predicting mutual entrainment of oscillators with experiment-based phase models
,”
Phys. Rev. Lett.
94
,
248301
(
2005
).
38.
M. J.
Hankins
,
V.
Gáspár
, and
I. Z.
Kiss
, “
Abrupt and gradual onset of synchronized oscillations due to dynamical quorum sensing in the single-cathode multi-anode nickel electrodissolution system
,”
Chaos
29
,
033114
(
2019
).
39.
A. S.
Mikhailov
,
D. H.
Zanette
,
Y. M.
Zhai
,
I. Z.
Kiss
, and
J. L.
Hudson
, “
Cooperative action of coherent groups in broadly heterogeneous populations of interacting chemical oscillators
,”
Proc. Natl. Acad. Sci. U.S.A.
101
,
10890
10894
(
2004
).
40.
J. L.
Ocampo-Espindola
,
C.
Bick
, and
I. Z.
Kiss
, “
Weak chimeras in modular electrochemical oscillator networks
,”
Front. Appl. Math. Stat.
5
,
38
(
2019
).
41.
L.
Schmidt
,
K.
Schönleber
,
K.
Krischer
, and
V.
García-Morales
, “
Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling
,”
Chaos
24
,
013102
(
2014
).
42.
Y.
Jia
and
I. Z.
Kiss
, “
Spontaneously synchronized electrochemical micro-oscillators with nickel electrodissolution
,”
J. Phys. Chem. C
116
,
19290
19299
(
2012
).
43.
Y.
Liu
and
I. Z.
Kiss
, “
Localization of current oscillations and synchronization patterns in microchip-based dual electrode flow cell without resistance balancing
,”
Eur. Phys. J. Spec. Top.
227
,
2659
2673
(
2019
).
44.
M. F.
Crowley
and
R. J.
Field
, “Electrically coupled Belousov-Zhabotinsky oscillators: A potential chaos generator,” in Nonlinear Phenomena in Chemical Dynamics, edited by C. Vidal and A. Pacault (Springer, Berlin, 1981), pp. 147–153.
45.
M. F.
Crowley
and
R. J.
Field
, “Electrically coupled Belousov-Zhabotinskii oscillators: Experimental observation of chaos in a chemical system and identification of its source in the Field-Noyes equations,” in Nonlinear Oscillations in Biology and Chemistry, edited by H. G. Othmer (Springer, Berlin, 1986), pp. 68–97.
46.
M. F.
Crowley
and
R. J.
Field
, “
Electrically coupled Belousov–Zhabotinskii oscillators. 1. Experiments and simulations
,”
J. Phys. Chem.
90
,
1907
1915
(
1986
).
47.
W.
Hohmann
,
N.
Schinor
,
M.
Kraus
, and
F. W.
Schneider
, “
Electrically coupled chemical oscillators and their action potentials
,”
J. Phys. Chem. A
103
,
5742
5748
(
1999
).
48.
A.
Guderian
,
A. F.
Münster
,
M.
Kraus
, and
F. W.
Schneider
, “
Electrochemical chaos control in a chemical reaction: Experiment and simulation
,”
J. Phys. Chem. A
102
,
5059
5064
(
1998
).
49.
G.
Dechert
,
K.-P.
Zeyer
,
D.
Lebender
, and
F. W.
Schneider
, “
Recognition of phase patterns in a chemical reactor network
,”
J. Phys. Chem.
100
,
19043
19048
(
1996
).
50.
W.
Hohmann
,
M.
Kraus
, and
F. W.
Schneider
, “
Recognition in excitable chemical reactor networks. Experiments and model-simulations
,”
J. Phys. Chem. A
101
,
7364
7370
(
1997
).
51.
W.
Hohmann
,
M.
Kraus
, and
F. W.
Schneider
, “
Learning and recognition in excitable chemical reactor networks
,”
J. Phys. Chem. A
102
,
3103
3111
(
1998
).
52.
W.
Hohmann
,
M.
Kraus
, and
F. W.
Schneider
, “
Pattern recognition by electrical coupling of eight chemical reactors
,”
J. Phys. Chem. A
103
,
7606
7611
(
1999
).
53.
A.
Adamatzky
,
B.
De Lacy Costello
, and
T.
Asai
,
Reaction-Diffusion Computers
(
Elsevier Science
,
2005
).
54.
J. J.
Tyson
, “
Scaling and reducing the Field-Koros-Noyes mechanism of the Belousov-Zhabotinskii reaction
,”
J. Phys. Chem.
86
,
3006
3012
(
1982
).
55.
K.
Sriram
, “
Effects of positive electrical feedback in the oscillating Belousov–Zhabotinsky reaction: Experiments and simulations
,”
Chaos Solitons Fractals
28
,
1055
1066
(
2006
).
56.
C. S.
Pittendrigh
and
S.
Daan
, “
Circadian oscillations in rodents: A systematic increase of their frequency with age
,”
Science
186
,
548
550
(
1974
).
57.
I. Z.
Kiss
, “
Synchronization engineering
,”
Curr. Opin. Chem. Eng.
21
,
1
9
(
2018
).
58.
M.
Sebek
and
I. Z.
Kiss
, “
Spatiotemporal patterns on a ring network of oscillatory electrochemical reaction with negative global feedback
,”
Isr. J. Chem.
58
,
753
761
(
2018
).
59.
M.
Dueñas-Díez
and
J.
Pérez-Mercader
, “
How chemistry computes: Language recognition by non-biochemical chemical automata. From finite automata to turing machines
,”
iScience
19
,
514
526
(
2019
).
60.
M.
Dueñas-Díez
and
J.
Pérez-Mercader
, “
Native chemical computation. A generic application of oscillating chemistry illustrated with the Belousov-Zhabotinsky reaction. A review
,”
Front. Chem.
9
,
204
(
2021
).
You do not currently have access to this content.