The Lavrova-Vanag (LV) model of the periodical Belousov-Zhabotinsky (BZ) reaction has been investigated at pulsed self-perturbations, when a sharp spike of the BZ reaction induces a short inhibitory pulse that perturbs the BZ reaction after some time τ since each spike. The dynamics of this BZ system is strongly dependent on the amplitude Cinh of the perturbing pulses. At Cinh > Ccr, a new pseudo-steady state (SS) emerges far away from the limit cycle of the unperturbed BZ oscillator. The perturbed BZ system spends rather long time in the vicinity of this pseudo-SS, which serves as a trap for phase trajectories. As a result, the dynamics of the BZ system changes qualitatively. We observe new modes with packed spikes separated by either long “silent” dynamics or small-amplitude oscillations around pseudo-SS, depending on Cinh. Networks of two or three LV-BZ oscillators with strong pulsatile coupling and self-inhibition are able to generate so-called “cognitive” modes, which are very sensitive to small changes in Cinh. We demonstrate how the coupling between the BZ oscillators in these networks should be organized to find “cognitive” modes.

1.
U.
Feudel
,
A. N.
Pisarchik
, and
K.
Showalter
, “
Multistability and tipping: From mathematics and physics to climate and brain-minireview and preface to the focus issue
,”
Chaos
28
,
033501
(
2018
).
2.
V.
Horvath
,
P. L.
Gentili
,
V. K.
Vanag
, and
I. R.
Epstein
, “
Pulse-coupled chemical oscillators with time delay
,”
Angew. Chem. Int. Ed.
51
,
6878
6881
(
2012
).
3.
A.
Saha
and
U.
Feudel
, “
Riddled basins of attraction in systems exhibiting extreme events
,”
Chaos
28
,
033610
(
2018
).
4.
A. F.
Taylor
 et al., “
Dynamical quorum sensing and synchronization in large populations of chemical oscillators
,”
Science
323
,
614
617
(
2009
).
5.
N.
Tompkins
 et al., “
Testing Turing’s theory of morphogenesis in chemical cells
,”
Proc. Natl. Acad. Sci. USA
111
,
4397
4402
(
2014
).
6.
V.
Horvath
and
I. R.
Epstein
, “
Pulse-coupled Belousov-Zhabotinsky oscillators with frequency modulation
,”
Chaos
28
,
045108
(
2018
).
7.
P.
Kumar
and
P.
Parmananda
, “
Control, synchronization, and enhanced reliability of aperiodic oscillations in the mercury beating heart system
,”
Chaos
28
,
045105
(
2018
).
8.
D.
Yengi
,
M. R.
Tinsley
, and
K.
Showalter
, “
Autonomous cycling between excitatory and inhibitory coupling in photosensitive chemical oscillators
,”
Chaos
28
,
045114
(
2018
).
9.
N. V.
Barabash
and
V. N.
Belykh
, “
Synchronization thresholds in an ensemble of Kuramoto phase oscillators with randomly blinking couplings
,”
Radiophys. Quant. Electron.
60
,
761
768
(
2018
).
10.
M. I.
Bolotov
,
L. A.
Smirnov
,
G. V.
Osipov
, and
A. S.
Pikovsky
, “
Breathing chimera in a system of phase oscillators
,”
JETP Lett.
106
,
393
399
(
2017
).
11.
O.
Burylko
,
Y.
Kazanovich
, and
R.
Borisyuk
, “
Winner-take-all in a phase oscillator system with adaptation
,”
Sci. Rep.
8
,
416
(
2018
).
12.
R.
Cestnik
and
M.
Rosenblum
, “
Inferring the phase response curve from observation of a continuously perturbed oscillator
,”
Sci. Rep.
8
,
13606
(
2018
).
13.
H.
Daido
, “
Superslow relaxation in identical phase oscillators with random and frustrated interactions
,”
Chaos
28
,
045102
(
2018
).
14.
S.
Eydam
and
M.
Wolfrum
, “
Mode locking in systems of globally coupled phase oscillators
,”
Phys. Rev. E
96
,
052205
(
2017
).
15.
V.
Klinshov
,
S.
Yanchuk
,
A.
Stephan
, and
V.
Nekorkin
, “
Phase response function for oscillators with strong forcing or coupling
,”
Europhys. Lett.
118
,
50006
(
2017
).
16.
I. R.
Epstein
and
K.
Showalter
, “
Nonlinear chemical dynamics: Oscillations, patterns, and chaos
,”
J. Phys. Chem.
100
,
13132
13147
(
1996
).
17.
I. R.
Epstein
, “
Coupled chemical oscillators and emergent system properties
,”
Chem. Commun.
50
,
10758
10767
(
2014
).
18.
K.
Showalter
and
I. R.
Epstein
, “
From chemical systems to systems chemistry: Patterns in space and time
,”
Chaos
25,
097613
(
2015
).
19.
E. M.
Izhikevich
and
G. M.
Edelman
, “
Large-scale model of mammalian thalamocortical systems
,”
Proc. Natl. Acad. Sci. U.S.A.
105
,
3593
3598
(
2008
).
20.
B.
Szatmary
and
E. M.
Izhikevich
, “
Spike-timing theory of working memory
,”
PLoS Comput. Biol.
6
,
e1000879
(
2010
).
21.
H.
Haken
,
Brain Dynamics: Sychronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise
(
Springer
,
Berlin
,
2007
).
22.
V.
Horvath
,
D. J.
Kutner
,
J. T.
Chavis
, and
I. R.
Epstein
, “
Pulse-coupled BZ oscillators with unequal coupling strengths
,”
Phys. Chem. Chem. Phys.
17
,
4664
4676
(
2015
).
23.
A. I.
Lavrova
and
V. K.
Vanag
, “
Two pulse-coupled non-identical, frequency-different BZ oscillators with time delay
,”
Phys. Chem. Chem. Phys.
16
,
6764
6772
(
2014
).
24.
I. S.
Proskurkin
,
A. I.
Lavrova
, and
V. K.
Vanag
, “
Inhibitory and excitatory pulse coupling of two frequency-different chemical oscillators with time delay
,”
Chaos
25
,
064601
(
2015
).
25.
V. K.
Vanag
and
V. O.
Yasuk
, “
Dynamic modes in a network of five oscillators with inhibitory all-to-all pulse coupling
,”
Chaos
28
,
033105
(
2018
).
26.
V. K.
Vanag
,
P. S.
Smelov
, and
V. V.
Klinshov
, “
Dynamical regimes of four almost identical chemical oscillators coupled via pulse inhibitory coupling with time delay
,”
Phys. Chem. Chem. Phys.
18
,
5509
5520
(
2016
).
27.
M. I.
Rabinovich
,
R.
Huerta
,
P.
Varona
, and
V. S.
Afraimovich
, “
Transient cognitive dynamics, metastability, and decision making
,”
PLoS Comput. Biol.
4
,
e1000072
(
2008
).
28.
M. I.
Rabinovich
,
P.
Varona
,
I.
Tristan
, and
V. S.
Afraimovich
, “
Chunking dynamics: Heteroclinics in mind
,”
Front. Comput. Neurosci.
8
,
22
(
2014
).
29.
M.
Rabinovich
,
R.
Huerta
, and
G.
Laurent
, “
Neuroscience - Transient dynamics for neural processing
,”
Science
321
,
48
50
(
2008
).
30.
I. S.
Proskurkin
and
V. K.
Vanag
, “
New type of excitatory pulse coupling of chemical oscillators via inhibitor
,”
Phys. Chem. Chem. Phys.
17
,
17906
17913
(
2015
).
31.
N.
Kopell
, in
Neural Control of Rhythms
, edited by
A. H.
Cohen
,
S.
Rossignol
, and
S.
Grillner
(
Wiley
,
New York
,
1988
).
32.
See http://www.pdesolutions.com for more information about FlexPDE (
2018
).
33.
P. S.
Smelov
and
V. K.
Vanag
, “
A ‘reader’ unit of the chemical computer
,”
R. Soc. Open Sci.
5
,
171495
(
2018
).
34.
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
).
35.
V. K.
Vanag
and
I. R.
Epstein
, “
A model for jumping and bubble waves in the Belousov-Zhabotinsky-aerosol OT system
,”
J. Chem. Phys.
131
,
104512
(
2009
).
36.
E. M.
Izhikevich
,
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
(
MIT Press
,
Cambridge
,
2007
).
37.
R.
Fitzhugh
, “
Impulses and physiological states in theoretical models of nerve membrane
,”
Biophys. J.
1
,
445
466
(
1961
).
38.
J.
Nagumo
,
S.
Arimoto
, and
S.
Yoshizawa
, “
An active pulse transmission line simulating nerve axon
,”
Proc. IRE
50
,
2061
2070
(
1962
).

Supplementary Material

You do not currently have access to this content.