Consider the periodic input with the frequency f such that
X
i
= 1
if iD mod f
− 1
B D
otherwise .
7 If the output series Y
i
is identical with X
i
, namely, if the relation X
i
= Y
i
is satisfied for all i, the correlation coefficient C takes the value 1. If the
output series Y
i
has no correlation with X
i
, the correlation coefficient C takes the value 0 in the
large n limit.
3. The case of d
p
= 0: stochastic resonance in the coupled system
In this section, we briefly describe the results for d
p
= 0 Kanamaru et al., 1999. As previously
mentioned, the frequency of the input pulse train is fixed at f = 0.1. Firstly, the system with N = 2 is
examined for simplicity. The dependence of the correlation coefficient C
on the noise intensity D is shown in Fig. 1, for the coupling strength w = 0, 0.1, and 0.5. For w = 0,
the system reduces to a single neuron, and the correlation coefficient C shows the characteristic
of Stochastic Resonance, namely, the existence of an optimal noise intensity D
which maximizes C. For w = 0.1 and 0.5, it is observed that D
in- creases with the increase of w.
Fig. 2. The dependence of the optimal noise intensity D on
the coupling strength w for the propagational time delay d
p
= 0.
In Fig. 2, the dependence of D on the coupling
strength w is investigated numerically. It is ob- served that the optimal noise intensity D
in- creases monotonically with the increase of w, and
it converges to about 0.0028. In the following, this limit value is denoted by D
. To consider the dependence of D
on the
number N of neurons, we introduce x
i
= u
i
, 6
i t
, j
i
= j
i
, 0
t
, and a two dimensional diag- onal matrix A with diagonal components A
11
= 1
and A
22
= 0, we rewrite the coupled FitzHugh –
Nagumo equation as
x
;
i
= Fx
i
+ wG
i
t + x
i
, 8
G
i
t = Á
à Í
à Ä
A 1
N − 1
j i
x
j
− x
i
if N ] 2 if N = 1
, 9
j
i
tj
j
t’ =Dd
ij
d t − t’,
i, j = 1,2,…,N, 10
where Fx
i
describes the dynamics of ith neu-
ron. Let us define the mean value X and the deviation dx
i
from X as X =
1 N
i
x
i
, 11
d x
i
= x
i
− X, 12
then, for large w, X and dx
i
obeys
Fig. 1. The dependence of the correlation C on the noise intensity D for the propagational time delay d
p
= 0.
X : =FX+
N i = 1
x
i
N +
O dx
i 2
, 13
d : x
1 i
= − w
1 − N
− 1
d x
1 i
+ j
i
−
N j = 1
j
j
N ,
14 d
: x
2 i
= d x
1 i
− bdx
2 i
, 15
where b is the parameter of the FN model Eq. 2. Thus the variances of dx
1 i
and dx
2 i
are esti- mated to be
dx
1 i
2
1 − N
− 1
2
2w D,
16 dx
2 i
2
dx
1 i
2
bb + 1 − N
− 1
− 1
w 1
w
2
. 17
Eq. 12, Eq. 13, Eq. 16, and Eq. 17 indi- cate that the dynamics of each neuron for large w
approaches the dynamics of one neuron with the scaled noise intensity DN. So, between the opti-
mal noise intensity D
N
for N neurons and D
1
for one neuron, the relation D
N
= ND
1
18 holds.
In Fig. 3, the numerically obtained optimal noise intensity D
is plotted against the num- ber N of neurons, where D
is estimated with the coupling strength w = 1.0, which is large
enough for the saturation of D . A good agree-
ment with the analytical result 18 is observed.
Fig. 4. The dependence of the correlation C on the noise intensity D for the propagational time delay d
p
= 10.
4. The case of d
