Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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160 D. Fan and L. Hong
Taking square on the both sides of (13.6) and summing them up, we obtain
.!
4
B!
2
CD/
2
C!
2
.A!
2
CC/
2
D E
2
Œ!
2
.b
1
Cb
2
/
2
C.b
1
b
2
!
2
/
2
: (13.7)
Letting
p D A
2
2B;
q D B
2
C 2D 2AC E
2
;
u D C
2
2BD E
2
b
2
1
C b
2
2
; (13.8)
v D D
2
E
2
b
2
1
b
2
2
;
z D !
2
:
Then (13.7) becomes
z
4
C pz
3
C qz
2
C uz Cv D 0: (13.9)
Since the form of (13.9) is identical to that of (13.6) in [24], thus, we can get
Lemmas 1 and 2 analogously. The proofs are omitted.
Lemma 1. ([24]) If v <0,then(13.9) has at least one positive root.
Denote
h.z/ D z
4
C pz
3
C qz
2
C uz Cv: (13.10)
Then we have
h
0
.z/ D 4z
3
C 3p z
2
C 2qz Cu: (13.11)
Set
4z
3
C 3p z
2
C 2qz Cu D 0: (13.12)
Let y D z C
p
4
,then(13.12) becomes
y
3
C p
1
y C q
1
D 0; (13.13)
where p
1
D
q
2
3
16
p
2
;q
1
D
p
3
32
pq
8
C
u
4
:
Define
D
q
1
2
2
C
p
1
3
3
;
" D
1 C i
p
3
2
;
y
1
D
3
r
q
1
2
C
p
C
3
r
q
1
2
p
;
y
2
D
3
r
q
1
2
C
p
"C
3
r
q
1
2
p
"
2
;
13 Bifurcation Analysis in FHN Neurons with Delays 161
y
3
D
3
r
q
1
2
C
p
"
2
C
3
r
q
1
2
p
": (13.14)
Let
z
i
D y
i
p
4
;.iD 1; 2; 3/: (13.15)
Lemma 2. ([24]) Suppose that v 0, then we have the following:
(i) If 0,then(13.9) has positive roots if and only if z
1
>0and h.z
1
/<0
(ii) If <0,then(13.9) has positive roots if and only if there exists at least one
z
2fz
1
; z
2
; z
3
g such that z
>0and h.z
/ 0
Suppose that (13.9) has positive roots. Without loss of generality, we assume that it
has four positive roots, denoted by z
k
.k D 1; 2; 3; 4/.Then(13.7) has four positive
roots,
!
k
D
q
z
k
;.kD 1; 2; 3; 4/ (13.16)
By (13.16), we have
sin !
k
D
!
4
k
B!
2
k
C D
!
k
.b
1
C b
2
/
A!
3
k
C C!
k
b
1
b
2
!
2
k
E
h
.b
1
C b
2
/
2
!
2
k
C
b
1
b
2
!
2
k
2
i
;
cos !
k
D
!
4
k
B!
2
k
C D
b
1
b
2
!
2
k
C
A!
3
k
C C!
k
!
k
.b
1
C b
2
/
E
h
.b
1
C b
2
/
2
!
2
k
C
b
1
b
2
!
2
k
2
i
:
(13.17)
Thus, denoting
a
D
!
4
k
B!
2
k
C D
!
k
.b
1
C b
2
/
A!
3
k
C C!
k
b
1
b
2
!
2
k
E
h
.b
1
C b
2
/
2
!
2
k
C
b
1
b
2
!
2
k
2
i
;
b
D
!
4
k
B!
2
k
C D
b
1
b
2
!
2
k
C
A!
3
k
C C!
k
!
k
.b
1
C b
2
/
E
h
.b
1
C b
2
/
2
!
2
k
C
b
1
b
2
!
2
k
2
i
;
(13.18)
.j /
k
D
8
ˆ
<
ˆ
:
1
!
k
.arccosb
C 2j /; a
0;
1
!
k
.2 arccosb
C 2j /; a
<0;
(13.19)
where k D 1; 2; 3; 4 and j D 0;1;2;:::; then ˙i!
k
is a pair of purely imaginary
roots of (13.4) with D
.j /
k
. Clearly, the sequence
n
.j /
k
o
C1
j D0
is increasing, and
lim
j !C1
.j /
k
DC1.k D 1; 2; 3; 4/.
162 D. Fan and L. Hong
For convenience, we let
S
4
kD1
f
.j /
k
g
C1
j D0
Df
i
g
C1
iD0
, such that
0
<
1
<
2
< <
i
< ; (13.20)
where
0
D min
n
.0/
1
;
.0/
2
;
.0/
3
;
.0/
4
o
: (13.21)
Applying Lemmas 1, 2, and Corollary 2.4 of Ruan and Wei [16], we have the
following results.
Lemma 3. ([24]) Assume that (H) A>0; A.B E/ > C E.b
1
C b
2
/;
D>Eb
1
b
2
; ŒC E.b
1
Cb
2
/ŒA.B E/C CE.b
1
Cb
2
/ > A
2
.D Eb
1
b
2
/.
(i) If one of the followings holds (a) v <0;(b)v 0; D 0; z
1
>0and
h.z
1
/ 0;(c)v 0, D<0, and there exists a z
2fz
1
; z
2
; z
3
g such that
z
>0and h.z
/ 0, then all roots of (13.4) have negative real parts when
2 Œ0;
0
/.
(ii) If the conditions .a/ .c/ of (i) are not satisfied, then all roots of (13.4) have
negative real parts for all 0.
Proof. When D 0,(13.4) becomes
4
C A
3
C .B E/
2
C ŒC E.b
1
C b
2
/ C D Eb
1
b
2
D 0: (13.22)
By the Routh–Hurwite criterion, all roots of (13.22) have negative real
parts if and only if A>0, A.B E/ > C E.b
1
C b
2
/, D>Eb
1
b
2
,
ŒC E.b
1
C b
2
/ŒA.B E/ C C E.b
1
C b
2
/ > A
2
ŒD Eb
1
b
2
:
From Lemmas 1 and 2, we know that if .a/–.c/ of (i) are not satisfied, then (13.4)
has no roots with zero real part for all 0; If one of the .a/; .b/ and .c/ holds,
when ¤
.j /
k
(k D 1; 2; 3; 4; j D 0; 1; 2; : : :), (13.4) has no roots with zero real
part and
0
is the minimum value of so that (13.4) has purely imaginary roots.
This completes the proof. ut
Let
./ D ˛./ C i!./ (13.23)
be the root of (13.4) near D
.j /
k
satisfying
˛
.j /
k
D 0; !
.j /
k
D !
k
; (13.24)
then, from Lamma 2.5 of Hu and Huang [25] the following conclusion holds.
Lemma 4. ([25]) Suppose h
0
.z
k
/ ¤ 0,whereh.z/ is defined by (13.10).If D
.j /
k
,
then ˙i!
k
is a pair of simple purely imaginary roots of (13.4). Moreover,
d.Re.//
d
ˇ
ˇ
ˇ
ˇ
D
.j /
k
¤ 0; (13.25)
13 Bifurcation Analysis in FHN Neurons with Delays 163
and the sign of
d.Re.//
d
ˇ
ˇ
ˇ
ˇ
D
.j /
k
is consistent with that of h
0
.z
k
/.
Applying Lemmas 3 and 4, we obtain Theorem 1 immediately.
Theorem 1. Let
.j /
k
and
0
be defined by (13.19) and (13.21), respectively.
Suppose (H) hold,
(i) If the conditions (a) v <0;(b)v 0; D 0; z
1
>0and h.z
1
/ 0;
(c) v 0, D<0, and there exists a z
2fz
1
; z
2
; z
3
g such that z
>0
and h.z
/ 0 are not satisfied, then the zero solution of system (13.3) is
asymptotically stable for all 0.
(ii) If one of the conditions (a), (b) and (c) of (i) is satisfied, then the zero solution
of system (13.3) is asymptotically stable when 2 Œ0;
0
/.
(iii) If one of the conditions (a), (b) and (c) of (i) is satisfied, and h
0
.z
k
/ ¤ 0,
then the system (13.3) undergoes a Hopf bifurcation at (0,0,0,0) when D
i
(i D 0; 1; 2; : : :).
13.3 The Direction and Stability of Hopf Bifurcation
In the previous section, we have obtained some conditions to ensure that the system
(13.3) undergoes a single Hopf bifurcation at the origin when passes through cer-
tain critical values. In this section, we study the direction, stability, and the period
of the bifurcating periodic solutions. The method we used is based on the normal
form method and the center manifold theory introduced by Hassard et al. [12].
Without loss of generality, we denote the critical value
j
.j D 0; 1; 2; : : :/ by
Q DQ
1
CQ
2
at which system (13.3) undergoes a Hopf bifurcation,where Q
1
< Q
2
and
DQ C D .Q
1
C/CQ
2
,then D 0 is Hopf bifurcation value of system (13.3).
We choose the phase space as C D C.ŒQ
2
;0;C
4
/, where for convenience in
computation we use C
4
instead of R
4
. Letting X.t/ D .x
1
.t/; x
2
.t/; x
3
.t/; x
4
.t//
T
and X
t
./ D X.t C / 2 C.Q
2
0/, we can transform system (13.3)into
an operator of the form:
P
X.t/ D L
.X
t
/ C G.; X
t
/; (13.26)
with
L
./ D B.0/ C B
1
.Q
1
/ C B
2
.Q
2
/; (13.27)
and
G.; / D
0
B
B
B
B
B
@
3
1
.0/
C
1
3
3
3
.Q
1
/ C
0
3
3
.0/
C
2
3
3
1
.Q
2
/ C
0
1
C
C
C
C
C
A
; (13.28)
164 D. Fan and L. Hong
where ./ D .
1
./;
2
./;
3
./;
4
.//
T
2 C ,and
B D
2
6
6
4
a 10 0
1 b
1
00
00 a 1
00 1 b
2
3
7
7
5
;B
1
D
2
6
6
4
00C
1
0
000 0
000 0
000 0
3
7
7
5
;B
2
D
2
6
6
4
0000
0000
C
2
000
0000
3
7
7
5
:
By the Riesz representation theorem, there exists a 4 4 matrix function .; /,
whose elements are of bounded variation, such that
L
./ D
Z
0
Q
2
Œd.; / ./; for 2 C: (13.29)
For 2 C , define the operator A. / as
A././ D
8
<
:
d./
d
;2 ŒQ
2
;0/;
R
0
Q
2
Œd.; /./; D 0:
(13.30)
We further define the operator R./ as
R././ D
0; 2 ŒQ
2
;0/;
G.; /; D 0:
(13.31)
Then, the system (13.26) is equivalent to the following operator equation
P
X
t
D A./X
t
C R./X
t
: (13.32)
Letting C
D C.Œ0; Q
2
; C
4
/,for 2 C
, A
is defined by
A
.s/ D
8
<
:
d'.s/
ds
;s2 .0; Q
2
;
R
0
Q
2
'./d.; 0/; s D 0;
(13.33)
and a bilinear form
h .s/ ; ./iD N'.0/.0/
Z
0
Q
2
Z
D0
N'. /d././d; (13.34)
where ./ D .; 0/.ThenA.0/ and A
are adjoint operators.
From the discussion in Sect. 13.2, we know that ˙i!
k
are eigenvalues of A.0/
and therefore they are also eigenvalues of A
, without loss of generality, we write
˙i!
k
as ˙i!
0
,thatis,
A.0/q./ D i!
0
q./; A
q
.s/ Di!
0
q
.s/:
13 Bifurcation Analysis in FHN Neurons with Delays 165
It is not difficult to verify that the vectors q./ D .q
1
;q
2
;q
3
;1/
T
e
i!
0
. 2 ŒQ
2
;0/ and q
.s/ D
1
N
D
.q
1
;q
2
;q
3
;1/e
i!
0
s
.s 2 Œ0; Q
2
/ are the eigen-
vectors of A.0/ and A
corresponding to the eigenvalue i!
0
and i!
0
, respectively.
By direct computation, we obtain
q
1
D
1
C
2
e
i!
0
Q
2
Œ1 C .i!
0
a/.i!
0
C b
2
/;
q
2
D
1
C
2
.i!
0
C b
1
/
e
i!
0
Q
2
Œ1 C .i!
0
a/.i!
0
C b
2
/; (13.35)
q
3
D i!
0
C b
2
:
and
q
1
D
1
C
1
e
i!
0
Q
1
Œ1 C .i!
0
C a/.i!
0
b
2
/;
q
2
D
1
C
1
.i!
0
b
1
/
e
i!
0
Q
1
Œ1 C .i!
0
C a/.i!
0
b
2
/; (13.36)
q
3
D i!
0
b
2
:
Then, we can choose
D D .
N
q
1
q
1
C
N
q
2
q
2
C
N
q
3
q
3
C 1/
Z
0
Q
2
Z
D0
N
q
. /d./q./d: (13.37)
Then hq
.s/; q./iD1:
Following the algorithms given in Hassard et al. [12] and using a computation
process similar to that of Wei and Ruan [15], we can obtain the coefficients which
will be used in determining the important qualities:
g
20
D g
11
D g
02
D 0;
g
21
D
2
D
h
N
q
1
3q
2
1
Nq
1
C C
1
q
2
3
Nq
3
e
i!
0
Q
1
C
N
q
3
3q
2
3
Nq
3
C C
2
q
2
1
Nq
1
e
i!
0
Q
2
i
:
(13.38)
Because each g
ij
in (13.38) is expressed by the parameters and delay in system
(13.3), we can compute the following quantities:
c
1
.0/ D
i
2!
0
Q
g
11
g
20
2jg
11
j
2
jg
02
j
2
3
C
g
21
2
D
g
21
2
;
2
D
Re.c
1
.0//
Re.
0
.Q//
;
ˇ
2
D 2Re.c
1
.0//;
T
2
D
Im.c
1
.0// C
2
Im.
0
.Q//
!
0
: (13.39)
166 D. Fan and L. Hong
From the discussion in Sects. 13.2 and 13.3, we have the following result
immediately.
Theorem 2. The direction of the Hopf bifurcation of the system (13.3) at the origin
when D
i
.i D 0; 1; 2; : : :/ is supercritical (subcritical) if
2
>0.
2
<0/,
that is, there exist the bifurcating periodic solutions for >
i
. <
i
/;
The bifurcating periodic solutions on the center manifold are stable (unstable)
if ˇ
2
<0.ˇ
2
>0/; The period of the bifurcating periodic solutions increases
(decreases) if T
2
>0.T
2
<0/.
Acknowledgment This research is supported by the National Science Foundation of China
under Grant No. 10772140 and Harbin Institute Technology (Weihai) Science Foundation with
No. HIT(WH)ZB200812.
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Chapter 14
On the Feedback Controlling of the Neuronal
System with Time Delay
Hao Liu, Wuyin Jin, Chi Zhang, Ruicheng Feng, and Aihua Zhang
Abstract For an individual Hindmarsh–Rose model neuron with time delay and
dynamic threshold, the dynamic considerations of firing depending on time delay
and synaptic intensity are studied in this chapter. It is found out that with the scaling
of the delay time and synaptic intensity ", neural firing patterns transform among
tonic spiking, busting, and resting firing states each other; as an example, with two
groups of the time delays and synaptic intensity " together, the neuronal chaotic
firing behavior could be controlled to period-1 or period-3 activities, respectively.
14.1 Introduction
As we all know, there is a growing evidence that the hysteresis phenomenon exists
in the dynamical system inevitably, i.e., the systemical development direction de-
pends not only on the current state but also on their passed state; specifically, there
is always a time-lag response to the input of system. Many scientists, including neu-
roscientists and biologists, have done a lot of research on dynamical system with
time delay, acquiring a great deal of significant achievements, moreover, controlling
the dynamical system by time delay skillfully [1–3]. For example, it is important to
control chaos with feedback of time delay.
Due to the time delay, the system appears to have abundant dynamical behaviors
and its characteristics have been altered, such as a delay of cells, the propagation
delay, and synaptic delay in the biological systems [4]. Then, it is necessary and
important to study the time delays in transmition of neuro-information among the
neurons. Initially the neuron system with time delay is studied, which consists of
W. Jin (
)
School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050,
People’s Republic of China
and
Key Laboratory of Digital Manufacturing Technology and Application, The Ministry of Education,
Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China
e-mail: jinwuyin@263.net
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
14,
c
Springer Science+Business Media, LLC 2010
169
170 H. Liu et al.
an individual neuron or a pair of neurons with delayed coupling, obtaining a sense
of results especially in research of stability of system; Yu and Peng Jian-hua have
controlled the chaotic activities to one 5 spikes/burst orbit embedded in the chaotic
attractor by one nonlinear time-continuous feedback perturbation stimulus of mem-
brane potential [5]. Adding the time delay to the feedback signal, the dynamical
system could be controlled to some target orbits. Besides the method of OGY, the
adaptive control, OPF control, transfer and displace control, periodic disturbances
with parameters, and periodic vibration are widely used in dynamical system [6–8].
Pyragas proposes two kinds of control strategy to the continuous system based
on the idea of OGY method for discrete data, i.e., by using the feedback controlling,
add one bit disturbance with continuous time on the system. The disturbance cannot
change the unstable period orbits (UPOs), but these UPOs could stabilize under
some conditions [9, 10]. The feedback controlling with time delay has been widely
adapted to control dynamical system and has been realized in experimental study.
For neural system, it is useful and also important to use explicitly the time de-
lays in the description of the transfer of information between the neurons, owing
to a single neuron that might influence a recurrent loop through an autosynapse
and/or through synaptic connections involving other neurons, and synaptic com-
munication between neurons depends on propagation of action potentials along the
axons. Diez-Martinez and Segundo studied experimentally the pacemaker neuron in
the crayfish stretch receptor organ and showed that as the transmission delay time
was increased the discharge patterns went from periodic spikes to trains of spikes
separated by silent intervals [4,11].
14.2 HR Model Neuron with Time Delay
Many biological systems operate under the influence of time-delayed feedback
mechanisms, and excitable cells can exhibit dissimilar firing patterns to various time
delay . The influence of time delay on chaotic system has attracted particular atten-
tion extensively, and an individual neuron can exhibit different intrinsic oscillatory
activities introduced by external currents [12, 13]. However, it is also interesting to
analyze discharge activities induced by the time-delayed synaptic interaction be-
tween neurons without external current. In this case, the properties of discharge
activities of each neuron depend on the synaptic intensity " and the time delay .
Here, the one time-delay HR model neuron with external stimulus I is intro-
duced, and the change of discharge patterns of the neuron are studied on synaptic
intensity " and the time delay , as well as their dynamical behavior, and the model
is described as follows:
8
ˆ
ˆ
<
ˆ
ˆ
:
Px D y ax
3
C bx
2
z C I C ".x.t //
Py D c dx
2
y
Pz D rŒS.x / z
; (14.1)