Coupled neuronal networks have received considerable attention due to their important and extensive applications in science and engineering. This paper focuses on the nonlinear dynamics of delay-coupled bidirectional FitzHugh–Nagumo (FHN) neuronal networks through theoretical analysis, numerical computations, and circuit simulations. A variety of interesting dynamical behaviors of the network are explored, such as the coexistence of nontrivial equilibria and periodic solutions, different patterns of coexisting attractors, and even chaotic motions. An electronic circuit is designed and performed to validate the facticity of the complicated behaviors, such as multistability and chaotic attractors. It is shown that the circuit simulations reach an agreement with the obtained results.

References

1.
Fitzhugh
,
R.
,
1961
, “
Impulses and Physiological States in Theoretical Models of Nerve Membrane
,”
Biophys. J.
,
1
(
6
), pp.
445
466
.
2.
Nagumo
,
J.
,
Arimoto
,
S.
, and
Yoshizawa
,
S.
,
1962
, “
An Active Pulse Transmission Line Simulating Nerve Axon
,”
Proc. IRE
,
50
(
10
), pp.
2061
2070
.
3.
Hodgkin
,
A. L.
, and
Huxley
,
A. F.
,
1952
, “
A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve
,”
J. Physiol.
,
117
(
4
), pp.
500
544
.
4.
Bautin
,
A.
,
1975
, “
Qualitative Investigation of a Particular Nonlinear System
,”
J. Appl. Math. Mech.
,
39
(
4
), pp.
606
615
.
5.
Baltanas
,
J. P.
, and
Casado
,
J. M.
,
1998
, “
Bursting Behaviour of the FitzHugh-Nagumo Neuron Model Subject to Quasi-Monochromatic Noise
,”
Physica D
,
122
(
1–4
), pp.
231
240
.
6.
Ueta
,
T.
,
Miyazaki
,
H.
,
Kousaka
,
T.
, and
Kawakami
,
H.
,
2004
, “
Bifurcation and Chaos in Coupled BVP Oscillators
,”
Int. J. Bifurcation Chaos
,
14
(
4
), pp.
1305
1324
.
7.
Slavova
,
A.
, and
Zecca
,
P.
,
2007
, “
Complex Behavior of Polynomial FitzHugh-Nagumo Cellular Neural Network Model
,”
Nonlinear Anal.: Real World Appl.
,
8
(
4
), pp.
1331
1340
.
8.
Zhen
,
B.
, and
Xu
,
J.
,
2010
, “
Bautin Bifurcation Analysis for Synchronous Solution of a Coupled FHN Neural System With Delay
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
2
), pp.
442
458
.
9.
Abbasian
,
A. H.
,
Fallah
,
H.
, and
Razvan
,
M. R.
,
2013
, “
Symmetric Bursting Behaviors in the Generalized FitzHugh-Nagumo Model
,”
Biol. Cybern.
,
107
(
4
), pp.
465
476
.
10.
Mao
,
X. C.
,
2017
, “
Complicated Dynamics of a Ring of Nonidentical FitzHugh-Nagumo Neurons With Delayed Couplings
,”
Nonlinear Dyn.
,
87
(
4
), pp.
2395
2406
.
11.
Flunkert
,
V.
,
Fischer
,
I.
, and
Schöll
,
E.
,
2013
, “
Dynamics, Control and Information in Delay-Coupled Systems
,”
Philos. Trans. R. Soc. A
,
371
(
1999
), p.
20120465
.
12.
Marcus
,
C. M.
, and
Westervelt
,
R. M.
,
1989
, “
Stability of Analog Neural Network With Delay
,”
Phys. Rev. A
,
39
(
1
), pp.
347
359
.
13.
Ma
,
J.
, and
Tang
,
J.
,
2015
, “
A Review for Dynamics of Collective Behaviors of Network of Neurons
,”
Sci. China Tech. Sci.
,
58
(
12
), pp.
2038
2045
.
14.
Shepherd
,
G. M.
,
1983
,
Neurobiology
,
Oxford University Press
,
New York
.
15.
Sun
,
X. J.
, and
Li
,
G. F.
,
2017
, “
Synchronization Transitions Induced by Partial Time Delay in a Excitatory-Inhibitory Coupled Neuronal Network
,”
Nonlinear Dyn.
,
89
(
4
), pp.
2509
2520
.
16.
Campbell
,
S. A.
,
Edwards
,
R.
, and
Van Den Driessche
,
P.
,
2004
, “
Delayed Coupling Between Two Neural Network Loops
,”
SIAM J. Appl. Math
,
65
(
1
), pp.
316
335
.
17.
Wang
,
Q.
,
Zheng
,
Y.
, and
Ma
,
J.
,
2013
, “
Cooperative Dynamics in Neuronal Networks
,”
Chaos Solitons Fract.
,
56
(
SI
), pp.
19
27
.
18.
Song
,
Y. L.
, and
Xu
,
J.
,
2012
, “
In-phase and Antiphase Synchronization in a Delay-Coupled System With Applications to a Delay-Coupled FitzHugh-Nagumo System
,”
IEEE Trans. Neural Networks Learn. Syst.
,
23
(
10
), pp.
1659
1670
.
19.
Stepan
,
G.
,
1989
,
Retarded Dynamical Systems: Stability and Characteristic Function
,
Longman
,
UK
.
20.
Hu
,
H. Y.
, and
Wang
,
Z. H.
,
2002
,
Dynamics of Controlled Mechanical Systems With Delayed Feedback
,
Springer-Verlag
,
Berlin
.
21.
Hu
,
H. Y.
, and
Wang
,
Z. H.
,
2009
, “
Singular Perturbation Methods for Nonlinear Dynamic Systems With Time Delays
,”
Chaos Solitons Fract.
,
40
(
1
), pp.
13
27
.
22.
Samaey
,
G.
,
Engelborghs
,
K.
, and
Roose
,
D.
,
2002
, “
Numerical Computation of Connecting Orbits in Delay Differential Equations
,”
Numer. Algorithms
,
30
(
3/4
), pp.
335
352
.
23.
Buric
,
N.
, and
Todorovic
,
D.
,
2003
, “
Dynamics of FitzHugh-Nagumo Excitable Systems With Delayed Coupling
,”
Phys. Rev. E
,
67
(
6
), p.
066222
.
24.
Wang
,
Q.
,
Lu
,
Q.
,
Chen
,
G.
,
Feng
,
Z.
, and
Duan
,
L.
,
2009
, “
Bifurcation and Synchronization of Synaptically Coupled FHN Models With Time Delay
,”
Chaos Solitons Fract.
,
39
(
2
), pp.
918
925
.
25.
Fan
,
D.
, and
Hong
,
L.
,
2010
, “
Hopf Bifurcation Analysis in a Synaptically Coupled FHN Neuron Model With Delays
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
7
), pp.
1873
1886
.
26.
Zhen
,
B.
, and
Xu
,
J.
,
2010
, “
Fold-Hopf Bifurcation Analysis for a Coupled Fitzhugh-Nagumo Neural System With Time Delay
,”
Int. J. Bifurcation Chaos
,
20
(
12
), pp.
3919
3934
.
27.
Tehrani
,
N. F.
, and
Razvan
,
M.
,
2015
, “
Bifurcation Structure of Two Coupled FHN Neurons With Delay
,”
Math. Biosci.
,
270
(
Pt A
), pp.
41
56
.
28.
Krupa
,
M.
, and
Touboul
,
J. D.
,
2016
, “
Complex Oscillations in the Delayed FitzHugh-Nagumo Equation
,”
J. Nonlinear Sci.
,
26
(
1
), pp.
43
81
.
29.
Rehan
,
M.
,
Hong
,
K. S.
, and
Aqil
,
M.
,
2011
, “
Synchronization of Multiple Chaotic FitzHugh-Nagumo Neurons With Gap Junctions Under External Electrical Stimulation
,”
Neurocomputing
,
74
(
17
), pp.
3296
3304
.
30.
Kandel
,
E. R.
,
Schwartz
,
J. H.
, and
Jessell
,
T. M.
,
2000
,
Principles of Neural Science
,
McGraw-Hill
,
New York
.
31.
Mao
,
X. C.
, and
Wang
,
Z. H.
,
2016
, “
Stability, Bifurcation, and Synchronization of Delay-Coupled Ring Neural Networks
,”
Nonlinear Dyn.
,
84
(
2
), pp.
1063
1078
.
32.
Gopalsamy
,
K.
, and
He
,
X.
,
1994
, “
Delay-Independent Stability in Bidirectional Associative Memory Networks
,”
IEEE Trans. Neural Networks
,
5
(
6
), pp.
998
1002
.
33.
Ge
,
J. H.
, and
Xu
,
J.
,
2010
, “
Computation of Synchronized Periodic Solution in a BAM Network With Two Delays
,”
IEEE Trans. Neural Networks
,
21
(
3
), pp.
439
450
.
34.
Hassard
,
B. D.
,
Kazarinoff
,
N. D.
, and
Wan
,
Y. H.
,
1981
,
Theory and Application of Hopf Bifurcation
,
Cambridge University Press
,
Cambridge, UK
.
35.
Ermentrout
,
B.
,
1997
,
XPPAUT 3.0—The Differential Equations Tool
,
University of Pittsburgh
,
Pittsburgh, PA
.
36.
Ablay
,
G.
,
2015
, “
Novel Chaotic Delay Systems and Electronic Circuit Solutions
,”
Nonlinear Dyn.
,
81
(
4
), pp.
1795
1804
.
37.
Duan
,
S.
, and
Liao
,
X.
,
2007
, “
An Electronic Implementation for Liao's Chaotic Delayed Neuron Model With Non-Monotonous Activation Function
,”
Phys. Lett. A
,
369
(
1–2
), pp.
37
43
.
38.
Bao
,
B.
,
Qian
,
H.
,
Xu
,
Q.
,
Chen
,
M.
,
Wang
,
J.
, and
Yu
,
Y.
,
2017
, “
Coexisting Behaviors of Asymmetric Attractors in Hyperbolic-Type Memristor Based Hopfield Neural Network
,”
Front. Comput. Neurosci.
,
11
, p.
81
.
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