A novel input–output linearization minimum sliding mode error feedback control (I/OMSMEFC) is proposed for the synchronization between two uncoupled FitzHugh–Nagumo (FHN) neurons with different ionic currents and external electrical stimulations. To estimate and offset the system uncertainties and external disturbances, the concept of equivalent control error is introduced, which is the key to utilization of I/OMSMEFC. A cost function is formulated on the basis of the principle of minimum sliding mode covariance constraint; then the equivalent control error is estimated and fed back. It is shown that the proposed I/OMSMEFC can compensate various kinds of system uncertainties and external disturbances. Meanwhile, it can reduce the steady-state error more than the conventional sliding mode control (SMC). In addition, the sliding mode after the I/OMSMEFC will tend to be the ideal SMC, resulting in improved control performance and quantity. Sufficient conditions are given based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented I/OMSMEFC for the chaotic synchronization accurately.

Reference

Reference
1.
Liu
,
Y. J.
, and
Yang
,
Q. G.
,
2010
, “
Dynamics of a New Lorenz-Like Chaotic System
,”
Nonlinear Anal.: Real World Appl.
,
11
(
4
), pp.
2563
2572
.
2.
Harb
,
A. M.
, and
Abdel-Jabbar
,
N.
,
2003
, “
Controlling Hopf Bifurcation and Chaos in a Small Power System
,”
Chaos, Solitons Fractals
,
18
(
5
), pp.
1055
1063
.
3.
Ma
,
J.
,
Wang
,
C. N.
,
Tang
,
J.
, and
Xia
,
Y. F.
,
2009
, “
Suppression of the Spiral Wave and Turbulence in the Excitability-Modulated Media
,”
Int. J. Theor. Phys.
,
48
(
1
), pp.
150
157
.
4.
Liu
,
Y. J.
,
2012
, “
Circuit Implementation and Finite-Time Synchronization of the 4D Rabinovich Hyperchaotic System
,”
Nonlinear Dyn.
,
67
(
1
), pp.
89
96
.
5.
Ellacott
,
S. W.
,
Mason
,
J. C.
, and
Anderson
,
I. J.
,
1997
,
Mathematics of Neural Networks: Models, Algorithms and Applications
,
Kluwer
,
Norwell, MA
, pp.
47
97
.
6.
Purves
,
D.
,
Augustine
,
G. J.
,
Fitzpatrick
,
D.
,
Hall
,
W. C.
,
Lamatia
,
A.-S.
,
McNamarax
,
J. O.
, and
Williams
,
S. M.
,
2004
,
Neuroscience
,
3rd ed.
,
Sinauer Associates
,
Sunderland, MA
.
7.
Hodgkin
,
A. L.
, and
Huxley
,
A. F.
,
1952
, “
A Quantitative Description of Membrane and Its Application to Conduction and Excitation in Nerve
,”
J. Physiol.
,
117
(
4
), pp.
500
544
.
8.
FitzHugh
,
R.
,
1961
, “
Impulses and Physiological States in Theoretical Models of Nerve Membrane
,”
J. Biophys.
,
1
(
6
), pp.
445
466
.
9.
Nagumo
,
J.
,
Arimoto
,
S.
, and
Yoshizawa
,
S.
,
1962
, “
An Active Pulse Transmission Line Simulating Nerve Axon
,”
Proc. IRE
,
50
(
10
), pp.
2061
2070
.
10.
Bautin
,
A. N.
,
1975
, “
Qualitative Investigation of a Particular Nonlinear System
,”
J. Appl. Math. Mech.
,
39
(
4
), pp.
606
615
.
11.
Chen
,
C. H.
,
Sheu
,
L. J.
,
Chen
,
H.-K.
,
Chen
,
J. H.
,
Wang
,
H. C.
,
Chao
,
Y.-C.
, and
Lin
,
Y.-K.
,
2009
, “
A New Hyper-Chaotic System and Its Synchronization
,”
Nonlinear Anal.: Real World Appl.
,
10
(
4
), pp.
2088
2096
.
12.
Wei
,
D. Q.
,
Luo
,
X. S.
,
Zhang
,
B.
, and
Qin
,
Y. H.
,
2010
, “
Controlling Chaos in Space-Clamped FitzHugh–Nagumo Neuron by Adaptive Passive Method
,”
Nonlinear Anal.: Real World Appl.
,
11
(
3
), pp.
1752
1759
.
13.
Wang
,
J.
,
Zhang
,
T.
, and
Deng
,
B.
,
2007
, “
Synchronization of FitzHugh–Nagumo Neurons in External Electrical Stimulation Via Nonlinear Control
,”
Chaos, Solitons Fractals
,
31
(
1
), pp.
30
38
.
14.
Wei
,
X.
,
Wang
,
J.
, and
Deng
,
B.
,
2009
, “
Introducing Internal Model to Robust Output Synchronization of FitzHugh-Nagumo Neurons in External Electrical Stimulation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
7
), pp.
3108
3119
.
15.
Ambrosio
,
B.
, and
Aziz-Alaoui
,
M. A.
,
2012
, “
Synchronization and Control of Coupled Reaction-Diffusion Systems of the FitzHugh-Nagumo Type
,”
Comput. Math. Appl.
,
64
(
5
), pp.
934
943
.
16.
Yu
,
T. H.
,
Wang
,
J.
,
Deng
,
B.
,
Wei
,
X. L.
,
Che
,
Y. Q.
,
Wong
,
Y. K.
,
Chan
,
W. L.
, and
Tsang
,
K. M.
,
2012
, “
Adaptive Backstepping Sliding Mode Control for Chaos Synchronization of Two Coupled Neurons in the External Electrical Stimulation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
3
), pp.
1344
1354
.
17.
Yang
,
C. C.
, and
Lin
,
C. L.
,
2012
, “
Robust Adaptive Sliding Mode Control for Synchronization of Space-Clamped FitzHugh-Nagumo Neurons
,”
Nonlinear Dyn.
,
69
(
4
), pp.
2089
2096
.
18.
Li
,
Y.
, and
Jiang
,
W.
,
2011
, “
Hopf and Bogdanov-Takens Bifurcations in a Coupled FitzHugh-Nagumo Neural System With Delay
,”
Nonlinear Dyn.
,
65
(
1
), pp.
161
173
.
19.
Utkin
,
V. L.
,
1977
, “
Variable Structure System With Sliding Modes
,”
IEEE Trans. Autom. Control
,
22
(
2
), pp.
212
221
.
20.
Zribi
,
M.
,
Smaoui
,
N.
, and
Salim
,
H.
,
2010
, “
Synchronization of the Unified Chaotic Systems Using Sliding Mode Controller
,”
Chaos, Solitons Fractals
,
42
(5), pp.
3197
3209
.
21.
Li
,
W.
,
Liu
,
Z.
, and
Miao
,
J.
,
2010
, “
Adaptive Synchronization for a Unified Chaotic System With Uncertainty
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
10
), pp.
3015
3021
.
22.
Pourmahmood
,
M.
,
Khanmohammadi
,
S.
, and
Alizadeh
,
G.
,
2011
, “
Synchronization of Two Different Uncertain Chaotic Systems With Unknown Parameters Using a Robust Adaptive Sliding Mode Controller
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
7
), pp.
2853
2868
.
23.
Aghababa
,
P. M.
,
Khanmohammadi
,
S.
, and
Alizadeh
,
G.
,
2011
, “
Finite-Time Synchronization of Two Different Chaotic Systems With Unknown Parameters Via Sliding Mode Technique
,”
Appl. Math. Modell.
,
35
(
6
), pp.
3080
3091
.
24.
Gao
,
Y.
,
2004
, “
Chaos and Bifurcation in the Space-Clamped FitzHugh-Nagumo System
,”
Chaos, Solitons Fractals
,
21
(
4
), pp.
943
956
.
25.
Chou
,
M. H.
, and
Lin
,
Y. T.
,
1996
, “
Exotic Dynamic Behavior of the Forced FitzHugh-Nagumo Equations
,”
Comput. Math. Appl.
,
32
(
10
), pp.
109
119
.
26.
Elmali
,
H.
, and
Olgac
,
N.
,
1992
, “
Robust Output Tracking of MIMO Nonlinear Systems Via Sliding Mode Technique
,”
Automatica
,
28
(
1
), pp.
145
151
.
27.
Guan
,
P.
,
Liu
,
X. J.
, and
Liu
,
J. Z.
,
2005
, “
Flexible Satellite Attitude Control Via Sliding Mode Technique
,” The 44th
IEEE
Conference on Decision and Control, and the European Control Conference
, Seville, Spain, Dec. 12–15, pp.
1258
1263
.
This content is only available via PDF.
You do not currently have access to this content.