Blowout bifurcation and onoﬀ intermittency
in a pulse neural network with two modules
Takashi Kanamaru
∗
and Masatoshi Sekine
Tokyo University of Agriculture and Technology,Tokyo 1848588,Japan
International Symposium on Oscillation,Chaos and Network Dynamics in Nonlinear Science 2004,p.45,Kyoto,JAPAN
Let us consider a pulse neural network composed of
excitatory neurons and inhibitory neurons written as
˙
θ
(i)
E
k
= (1 −cos θ
(i)
E
k
) +(1 +cos θ
(i)
E
k
)
×(r
E
k
+ξ
(i)
E
k
(t) +I
E
k
E
k
(t) −I
E
k
I
k
(t)
+
l
=k
(I
E
k
E
l
(t) −I
E
k
I
l
(t))),(1)
˙
θ
(i)
I
k
= (1 −cos θ
(i)
I
k
) +(1 +cos θ
(i)
I
k
)
×(r
I
k
+ξ
(i)
I
k
(t) +I
I
k
E
k
(t) −I
I
k
I
k
(t)
+
l
=k
(I
I
k
E
l
(t) −I
I
k
I
l
(t))),(2)
I
XY
(t) =
g
XY
2N
Y
N
Y
j=1
m
1
κ
Y
exp
−
t −t
(j)
m
κ
Y
,(3)
ξ
(i)
X
(t)ξ
(j)
Y
(t
) = Dδ
XY
δ
ij
δ(t −t
),(4)
where X or Y = E
k
or I
k
.This network is mod
eled by the slowly connected class 1 neural network[1].
Note that this network is composed of multiple mod
ules,and the kth module is composed of excitatory
ensemble E
k
and inhibitory ensemble I
k
.The each
neuron is connected to the other neurons with the
synaptic coupling written by Eq.(3) where the ﬁring
time t
(j)
m
is deﬁned as the time when the phase θ
(j)
Y
of
the jth neuron in the ensemble Y exceeds the value
π.The connection strengths in the identical module
are set as g
E
k
E
k
= g
EE
,g
I
k
I
k
= g
II
,and g
E
k
I
k
=
g
I
k
E
k
= g
ext
,and the connection strengths between
diﬀerent modules are set as g
X
k
Y
l
≡
XY
(k
= l).
To examine the dynamics of the network with one
module,the analysis with the FokkerPlanck equation
is applicable,and various synchronized ﬁrings includ
ing chaotic ones are observed[2].
For the network with two modules,the blowout bi
furcation and the onoﬀ intermittency[3] are observed
as shown in Fig.1.The onoﬀ intermittency induces
0.15
0.1
0.05
0
0.05
0.1
0.15
0
10000
20000
30000
40000
50000
JE0
JE1
−
t
FIG.1:The time series of the diﬀerences of the ﬁring rate
J
E
k
in the limit of N
E
k
,N
I
k
→∞.
chaotic itinerancy for networks with multiple modules.
The raster plot of the ﬁring times of excitatory neu
rons in the network with three modules is shown in
Fig.2.It is observed that the synchronized clusters
are organized and rearranged chaotically.
0
100
200
300
400
500
0
100
200
300
400
500
t
t
0
5000
0
5000
0
5000
J
E0
JE1
−
J
E1
J
E2
−
J
E0
JE2
−
0
0.1
0.1
0
0.1
0.1
0
0.1
0.1
module0module1module2
01 sync.
12 sync.
(a)
(b)
FIG.2:Chaotic itinerancy observed in the network with
three modules where N
E
k
= N
I
k
= 5000.(a) The raster
plot of the ﬁring times of excitatory neurons in each mod
ule.(b) The time series of the diﬀerences of the ﬁring rate
J
E
k
.
This research was partially supported by a Grant
inAid for Encouragement of Young Scientists (B)
(No.14780260) from the Ministry of Education,Cul
ture,Sports,Science,and Technology,Japan.
∗
Corresponding author:kanamaru@sekinelab.ei.
tuat.ac.jp
[1] E.M.Izhikevich,Int.J.of Bif.and Chaos,vol.10,
pp.1171–1266,2000.
[2] T.Kanamaru and M.Sekine,Neural Comput.,in press,
2004.
[3] E.Ott and J.C.Sommerer,Phys.Lett.A,vol.188,
pp.39–47,1994.
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