This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

1.
L. M.
Pecora
and
T. L.
Carroll
, “
Master stability functions for synchronized coupled systems
,”
Phys. Rev. Lett.
80
,
2109
2112
(
1998
).
2.
I.
Belykh
,
E.
de Lange
, and
M.
Hasler
, “
Synchronization of bursting neurons: What matters in the network topology
,”
Phys. Rev. Lett.
94
(
18
),
188101
(
2005
).
3.
L.
Tang
,
X.
Wu
,
J.
,
J. A.
Lu
, and
R. M.
D’Souza
, “
Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators
,”
Phys. Rev. E
99
,
012304
(
2019
).
4.
S.
Boccaletti
,
V.
Latora
,
Y.
Moreno
,
M.
Chavez
, and
D. U.
Hwang
, “
Complex networks: Structure and dynamics
,”
Phys. Rep.
424
,
175
308
(
2006
).
5.
M.
De Domenico
,
A.
Solé-Ribalta
,
E.
Cozzo
,
M.
Kivelä
,
Y.
Moreno
,
M. A.
Porter
,
S.
Gómez
, and
A.
Arenas
, “
Mathematical formulation of multilayer networks
,”
Phys. Rev. X
3
,
041022
(
2013
).
6.
M.
Kivelä
,
A.
Arenas
,
M.
Barthelemy
,
J. P.
Gleeson
,
Y.
Moreno
, and
M. A.
Porter
, “
Multilayer networks
,”
J. Complex Netw.
2
(
3
),
203
271
(
2014
).
7.
S.
Boccaletti
,
G.
Bianconi
,
R.
Criado
,
C. I.
Del Genio
,
J.
Gómez-Gardenes
,
M.
Romance
,
I.
Sendina-Nadal
,
Z.
Wang
, and
M.
Zanin
, “
The structure and dynamics of multilayer networks
,”
Phys. Rep.
544
(
1
),
1
122
(
2014
).
8.
L. R.
Varshney
,
B. L.
Chen
,
E.
Paniagua
,
D. H.
Hall
, and
D. B.
Chklovskii
, “
Structural properties of the Caenorhabditis elegans neuronal network
,”
PLoS Comput. Biol.
7
(
2
),
e1001066
(
2011
).
9.
C. G.
Antonopoulos
, “
Dynamic range in the C. elegans brain network
,”
Chaos
26
,
013102
(
2016
).
10.
T.
Maertens
,
E.
Schöll
,
J.
Ruiz
, and
P.
Hövel
, “
Multilayer network analysis of C. elegans: Looking into the locomotory circuitry
,”
Neurocomputing
427
,
238
261
(
2021
).
11.
P. J.
Mucha
and
T.
Richardson
, “
Community structure in time-dependent, multiscale, and multiplex networks
,”
Science
328
,
876
(
2010
).
12.
L.
Barrett
,
S. P.
Henzi
, and
D.
Lusseau
, “
Taking sociality seriously: The structure of multi-dimensional social networks as a source of information for individuals
,”
R. Soc.
367
(
1599
),
2108
2118
(
2012
).
13.
J.
Menche
,
A.
Sharma
,
M.
Kitsak
,
S. D.
Ghiassian
,
M.
Vidal
,
J.
Loscalzo
, and
A. L.
Barabási
, “
Uncovering disease-disease relationships through the incomplete interactome
,”
Science
347
,
1257601
(
2015
).
14.
M.
Zanin
and
F.
Lillo
, “
Modelling the air transport with complex networks: A short review
,”
Eur. Phys. J. Spec. Top.
215
,
5
21
(
2013
).
15.
A.
Cardillo
,
M.
Zanin
,
J.
Gómez-Gardenes
,
M.
Romance
,
A. J.
García del Amo
, and
S.
Boccaletti
, “
Modeling the multi-layer nature of the European air transport network: Resilience and passengers re-scheduling under random failures
,”
Eur. Phys. J. Spec. Top.
215
,
23
33
(
2013
).
16.
T.
Tanaka
and
T.
Aoyagi
, “
Multistable attractors in a network of phase oscillators with three-body interactions
,”
Phys. Rev. Lett.
106
,
224101
(
2011
).
17.
P. S.
Skardal
and
A.
Arenas
, “
Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes
,”
Phys. Rev. Lett.
122
,
248301
(
2019
).
18.
C.
Xu
,
X.
Wang
, and
P. S.
Skardal
, “
Bifurcation analysis and structural stability of simplicial oscillator populations
,”
Phys. Rev. Res.
2
,
023281
(
2020
).
19.
M. S.
Anwar
and
D.
Ghosh
, “
Stability of synchronization in simplicial complexes with multiple interaction layers
,”
Phys. Rev. E
106
,
034314
(
2022
).
20.
I.
Leyva
,
R.
Sevilla-Escoboza
,
I.
Sendiña-Nadal
,
R.
Gutiérrez
,
J. M.
Buldú
, and
S.
Boccaletti
, “
Inter-layer synchronization in nonidentical multi-layer network
,”
Sci. Rep.
7
,
45475
(
2017
).
21.
M. C.
Eser
,
E. S.
Medeiros
,
M.
Riza
, and
A.
Zakharova
, “
Edges of inter-layer synchronization in multilayer networks with time-switching links
,”
Chaos
31
,
103119
(
2021
).
22.
M. S.
Anwar
,
S.
Rakshit
,
D.
Ghosh
, and
E. M.
Bollt
, “
Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functions
,”
Phys. Rev. E
105
,
024303
(
2022
).
23.
M. T.
Schaub
,
N.
O’Clery
,
Y. N.
Billeh
,
J. C.
Delvenne
,
R.
Lambiotte
, and
M.
Barahona
, “
Graph partitions and cluster synchronization in networks of oscillators
,”
Chaos
26
,
094821
(
2016
).
24.
R.
Berner
,
J.
Sawicki
, and
E.
Schöll
, “
Birth and stabilization of phase clusters by multiplexing of adaptive networks
,”
Phys. Rev. Lett.
124
,
088301
(
2020
).
25.
F. D.
Rossa
,
L.
Pecora
,
K.
Blaha
,
A.
Shirin
,
I.
Klickstein
, and
F.
Sorrentino
, “
Symmetries and cluster synchronization in multilayer networks
,”
Nat. Commun.
11
,
3179
(
2020
).
26.
Y.
Zhang
,
V.
Latora
, and
A. E.
Motter
, “
Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions
,”
Commun. Phys.
4
,
195
(
2021
).
27.
S.
Panahi
,
I.
Klickstein
, and
F.
Sorrentino
, “
Cluster synchronization of networks via a canonical transformation for simultaneous block diagonalization of matrices
,”
Chaos
31
,
111102
(
2021
).
28.
S.
Kundu
and
S.
Majhi
, “
Chimera states in two-dimensional networks of locally coupled oscillators
,”
Phys. Rev. E
97
,
022201
(
2018
).
29.
S.
Majhi
,
B. K.
Bera
,
D.
Ghosh
, and
M.
Perc
, “
Chimera states in neuronal networks: A review
,”
Phys. Life Rev.
28
,
100
121
(
2019
).
30.
S.
Kundu
and
D.
Ghosh
, “
Higher-order interactions promote chimera states
,”
Phys. Rev. E
105
,
L042202
(
2022
).
31.
R. G.
Andrzejak
and
A.
Espinoso
, “
Chimera states in multiplex networks: Chameleon-like across-layer synchronization
,”
Chaos
33
,
053112
(
2023
).
32.
A. D.
Kachhvah
and
S.
Jalan
, “
Explosive synchronization and chimera in interpinned multilayer networks
,”
Phys. Rev. E
104
,
L042301
(
2021
).
33.
S. N.
Chowdhury
,
S.
Rakshit
,
J. M.
Buldu
,
D.
Ghosh
, and
C.
Hens
, “
Antiphase synchronization in multiplex networks with attractive and repulsive interactions
,”
Phys. Rev. E
103
,
032310
(
2021
).
34.
S. N.
Chowdhury
,
S.
Rakshit
,
C.
Hens
, and
D.
Ghosh
, “
Interlayer antisynchronization in degree-biased duplex networks
,”
Phys. Rev. E
107
,
034313
(
2023
).
35.
I.
Leyva
and
I.
Sendiña-Nadal
, “
Relay synchronization in multiplex networks
,”
Sci. Rep.
8
,
8629
(
2018
).
36.
F.
Drauschke
,
J.
Sawick
,
R.
Berner
,
I.
Omelchenko
, and
E.
Schöll
, “
Effect of topology upon relay synchronization in triplex neuronal networks
,”
Chaos
30
,
051104
(
2020
).
37.
K. A.
Blaha
,
K.
Huang
,
F.
Della Rossa
,
L.
Pecora
,
M.
Hossein-Zadeh
, and
F.
Sorrentino
, “
Cluster synchronization in multilayer networks: A fully analog experiment with LC oscillators with physically dissimilar coupling
,”
Phys. Rev. Lett.
122
,
014101
(
2019
).
38.
T.
Njougouo
,
V.
Camargo
,
P.
Louodop
,
F.
Fagundes Ferreira
,
P. K.
Talla
, and
H. A.
Cerdeira
, “
Dynamics of multilayer networks with amplification
,”
Chaos
30
,
123136
(
2020
).
39.
E.
Rybalova
,
G.
Strelkova
,
E.
Schöll
, and
V.
Anishchenko
, “
Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps
,”
Chaos
30
,
061104
(
2020
).
40.
H. J.
Park
and
K.
Friston
, “
Structural and functional brain networks: From connections to cognition
,”
Science
342
(
6158
),
1238411
(
2013
).
41.
N.
Frolov
,
V.
Maksimenko
, and
A.
Hramov
, “
Revealing a multiplex brain network through the analysis of recurrences
,”
Chaos
30
,
121108
(
2020
).
42.
A. E.
Hramov
,
V. A.
Maksimenko
, and
A. N.
Pisarchikad
, “
Physical principles of brain-computer interfaces and their applications for rehabilitation, robotics and control of human brain states
,”
Phys. Rep.
918
(
25
),
1
133
(
2021
).
43.
N.
Kopell
and
B.
Ermentrout
, “
Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks
,”
Proc. Natl. Acad. Sci. U.S.A.
101
(
43
),
15482
15487
(
2004
).
44.
J. A.
Prasad
and
Y.
Chudasama
, “
Viral tracing identifies parallel disynaptic pathways to the hippocampus
,”
J. Neurosci.
33
(
19
),
8494
8503
(
2013
).
45.
F.
Kajtor
,
J.
Hullay
,
L.
Farago
, and
K.
Haberland
, “
Electrical activity of the hippocampus of patients with temporal lobe epilepsy
,”
AMA Arch NeurPsych.
80
(
1
),
25
38
(
1958
).
46.
M. M.
Halassa
and
S.
Kastner
, “
Thalamic functions in distributed cognitive control
,”
Nat. Neurosci.
20
,
1669
1679
(
2017
).
47.
J.
Sawicki
,
I.
Omelchenko
,
A.
Zakharova
, and
E.
Schöll
, “
Delay controls chimera relay synchronization in multiplex networks
,”
Phys. Rev. E
98
,
062224
(
2018
).
48.
J.
Sawicki
,
J. M.
Koulen
, and
E.
Schöll
, “
Synchronization scenarios in three-layer networks with a hub
,”
Chaos
31
,
073131
(
2021
).
49.
M.
Shafiei
,
S.
Jafari
,
F.
Parastesh
,
M.
Ozer
,
T.
Kapitaniak
, and
M.
Perc
, “
Time delayed chemical synapses and synchronization in multilayer neuronal networks with ephaptic inter-layer coupling
,”
Commun. Nonlinear Sci. Numer. Simul.
84
,
105175
(
2020
).
50.
S.
Srirama
,
K.
Rajagopal
,
A.
Karthikeyan
, and
A.
Akgul
, “
Memristive field effect in a single and multilayer neural network with different connection topologies
,”
Appl. Math. Comput.
457
,
128171
(
2023
).
51.
G.
Vivekanandhan
,
S.
Mirzaei
,
M.
Mehrabbeik
,
K.
Rajagopal
, and
S.
Jafari
, “
The simplest multilayer network of Rulkov neuron maps: A dynamical analysis under different neuronal interactions
,”
Europhys. Lett.
140
,
61002
(
2022
).
52.
L. M.
Pecora
,
T. L.
Carroll
,
G. A.
Johnson
,
D. J.
Mar
, and
J. F.
Heagy
, “
Fundamentals of synchronization in chaotic systems, concepts, and applications
,”
Chaos
7
,
520
(
1997
).
53.
D.
Irving
and
F.
Sorrentino
, “
Synchronization of dynamical hypernetworks: Dimensionality reduction through simultaneous block-diagonalization of matrices
,”
Phys. Rev. E
86
,
056102
(
2012
).
54.
L.
Tang
,
K.
Smith
,
K.
Daley
, and
I.
Belykh
, “
When multilayer links exchange their roles in synchronization
,”
Phys. Rev. E
106
,
024214
(
2022
).
55.
Y.
Zhang
and
A. E.
Motter
, “
Symmetry-independent stability analysis of synchronization patterns
,”
SIAM Rev.
62
(
4
),
817
836
(
2020
).
56.
S.
Panahi
,
N.
Amaya
,
I.
Klickstein
,
G.
Novello
, and
F.
Sorrentino
, “
Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks
,”
Phys. Rev. E
105
,
014313
(
2022
).
57.
T.
Jing
,
D.
Zhang
, and
X.
Zhang
, “
New criteria for synchronization of multilayer neural networks via aperiodically intermittent control
,”
Comput. Intell. Neurosci.
2022
,
8157794
(
2022
).
58.
N. F.
Rulkov
, “
Regularization of synchronized chaotic bursts
,”
Phys. Rev. Lett.
86
,
183
(
2001
).
59.
N. F.
Rulkov
, “
Modeling of spiking-bursting neural behavior using two-dimensional map
,”
Phys. Rev. E
65
,
041922
(
2002
).
60.
B.
Ibarz
and
J. M.
Casado
, “
Map-based models in neuronal dynamics
,”
Phys. Rep.
501
,
1
74
(
2011
).
61.
C.
Wang
and
H.
Cao
, “
Parameter space of the Rulkov chaotic neuron model
,”
Commun. Nonlinear Sci. Numer. Simul.
19
,
2060
2070
(
2014
).
62.
A. E.
Pereda
and
E.
Macagno
, “
Electrical transmission: Two structures, same functions?
,”
Dev. Neurobiol.
77
(
5
),
517
521
(
2017
).
63.
B.
Ibarz
,
H.
Cao
, and
M. A. F.
Sanjuán
, “
Bursting regimes in map-based neuron models coupled through fast threshold modulation
,”
Phys. Rev. E
77
,
051918
(
2008
).
64.
P.
Ge
and
H.
Cao
, “
Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses
,”
Chaos
29
,
023129
(
2019
).
65.
H.
Sun
and
H.
Cao
, “
Synchronization of two identical and non-identical Rulkov models
,”
Commun. Nonlinear Sci. Numer. Simul.
40
,
15
27
(
2016
).
66.
P.
Ge
and
H.
Cao
, “
Intermittent evolution routes to the periodic or the chaotic orbits in Rulkov map
,”
Chaos
31
,
093119
(
2021
).
67.
K. T.
Alligood
,
T.
Sauer
,
J. A.
Yorke
, and
D.
Chillingworth
,
Chaos: An Introduction to Dynamical Systems
(
Springer
,
2008
).
68.
M.
Winkler
,
J.
Sawicki
,
I.
Omelchenko
,
A.
Zakharova
,
V.
Anishchenko
, and
E.
Schöll
, “
Relay synchronization in multiplex networks of discrete maps
,”
Europhys. Lett.
126
,
50004
(
2019
).
69.
B.
Zhang
,
R.
Du
,
S.
Wang
, and
S.
Qu
, “
Transition of synchronization of coupled maps in modular networks
,”
Int. J. Modern Phys. C
31
(
1
),
2050011
(
2020
).
70.
E.
Rybalova
,
G.
Strelkova
, and
V.
Anishchenko
, “
Impact of sparse inter-layer coupling on the dynamics of a heterogeneous multilayer network of chaotic maps
,”
Chaos, Solitons Fractals
142
,
110477
(
2021
).
71.
L. D.
Iasemidis
and
J.
Sackellares
, “
Review: Chaos theory and epilepsy
,”
The Neuroscientist
2
(
2
),
118
126
(
1996
).
72.
R. S.
Fisher
,
B. W. V.
Emde
,
W.
Blume
,
C.
Elger
,
P.
Genton
,
P.
Lee
, and
J.
Engel Jr
, “
Epileptic seizures and epilepsy: Definitions proposed by the international league against epilepsy (ILAE) and the international bureau for epilepsy (IBE)
,”
Epilepsia
46
(
4
),
470
472
(
2005
).
You do not currently have access to this content.