Table Of Content1
Dynamics of the Langevin model subjected to colored noise:
Functional-integral method
Hideo Hasegawa 1
8
0 Department of Physics, Tokyo Gakugei University
0
2 Koganei, Tokyo 184-8501, Japan
n
a
J (February 1, 2008)
8
Abstract
]
h
c We have discussed the dynamics of Langevin model subjected to colored noise,
e by using the functional-integral method (FIM) combined with equations of motion
m
for mean and variance of the state variable. Two sets of colored noise have been
-
t investigated: (a) one additive and one multiplicative colored noise, and (b) one
a
t additive and two multiplicative colored noise. The case (b) is examined with the
s
. relevance to a recent controversy on the stationary subthreshold voltage distribu-
t
a tion of an integrate-and-fire model including stochastic excitatory and inhibitory
m
synapses and anoisy input. We have studied the stationary probability distribution
- and dynamical responses to time-dependent (pulse and sinusoidal) inputs of the
d
n linear Langevin model. Model calculations have shown that results of the FIM are
o
in good agreement with those of direct simulations (DSs). A comparison is made
c
[ among various approximate analytic solutions such as the universal colored noise
approximation (UCNA). It has been pointed out that dynamical responses to pulse
3
v and sinusoidal inputs calculated by the UCNA are rather different from those of DS
3 and the FIM, although they yield the same stationary distribution.
6
5
2
.
8 PACS No. 05.10.Gg, 05.45.-a, 84.35.+i
0
7
0
Keywords: Langevin model, colored noise, functional-integral method
:
v
i
X
Address: 4-1-1, Nukui-kita machi, Koganei, Tokyo 184-8501, Japan
r
a
1
e-mail address: [email protected]
1 Introduction
Nonlinear stochastic dynamics of physical, chemical, biological and economical systems
hasbeenextensively studied(forarecent review, seeRef. [1]). Inmost theoreticalstudies,
Gaussian white noise is employed as random driving force because of its mathematical
simplicity. The white-noise approximation is appropriate to systems in which the time
scale characterizing the relaxation of the noise is much shorter than the characteristic
time scale of the system. There has been a growing interest in theoretical study of
nonlinear dynamical systems subjected to colored noise with the finite correlation time
(fora review oncolored noise, see Ref. [2]: relatedreferences therein). It has beenrealized
that colored noise gives rise to new intriguing effects such as the reentrant phenomenon
in a noise-induced transition [3] and a resonant activation in bistable systems [4].
The originalmodel for a system driven by colored noise is expressed by non-Markovian
stochastic differential equation. This problem may be transformed to a Markovian one, by
extending the number of relevant variables and including an additional differential equa-
tiondescribing the Orstein-Uhlenbeck (OU) process. It isdifficult to analytically solve the
Langevin model subjected to colored noise. For its analytical study, two approaches have
been adopted: (1) to construct the multi-dimensional Fokker-Planck equation (FPE) for
the multivariate probability distribution, and (2) to derive the effective one-dimensional
FPE equation. The presence of multi-variables in the approach (1) makes a calculation of
even the stationary distribution much difficult. In a recent study on the Langevin model
subjected to additive (non-Gaussian) colored noise [5], we employed the approach (1),
analyzing the multivariate FPE with the use of the second-order moment method. A typ-
ical example of the approach (2) is the universal colored noise approximation (UCNA) [6],
which interpolates between the limits of zero and infinite relaxation times, and which has
been widely adopted for a study of colored noise [2]. Another example of the approach (2)
is the path-integral and functional-integral methods [7]-[12] obtaining the effective FPE,
with which stationary properties such as the non-Gaussian stationary distribution have
been studied [2].
TheoreticalstudyontheLangevinmodeldrivenbycolorednoisehasbeenmostlymade
for its stationary properties such as the stationary probability distribution and the phase
diagram of noise-induced transition [2]. As far as we are aware of, little theoretical study
has been reportedondynamical properties such as the response to time-dependent inputs.
Refs. [13, 14] have discussed the filtering effect, in which the high-frequency response of
2
the system is shown to be improved by colored noise. The purpose of the present paper
is to extend the functional-integral method (FIM) such that we may discuss dynamical
properties of the Langevin model subjected to colored noise. We consider, in this paper,
two sets of colored noise: (a) one additive and one multiplicative colored noise, and (b)
one additive and two multiplicative colored noise. The case (b) is included to clarify,
to some extent, a recent controversy on the subthreshold voltage distribution of a leaky
integrate-and-fire modelincluding conductance-based stochastic excitatory and inhibitory
synapses as well as noisy inputs [15]-[18].
Thepaperisorganizedasfollows. TheFIMisappliedtotheabove-mentionedcases(a)
and (b) in Secs. 2 and 3, respectively, where the stationary distribution and the response
to time-dependent inputs are studied. In Sec. 4, we will discuss the recent controversy
on the subthreshold voltage distribution of a leaky integrate-and-fire model [15]-[18]. A
comparison is made among results of some approximate analytical theories such as the
UCNA [2, 6]. The final Sec. 5 is devoted to conclusion.
2 Langevin model subjected to one additive and one
multiplicative colored noise
2.1 Effective Langevin equation
We have considered the Langevin model subjected to additive and multiplicative colored
noise given by
dx
= F(x)+η (t)+G(x)η (t), (1)
0 1
dt
with
dη (t) η √2D
m m m
= + ξ (t), (m = 0 and 1) (2)
m
dt −τ τ
m m
where F(x) and G(x) denote arbitrary functions of x: η (t) and η (t) stand for additive
0 1
and multiplicative noise, respectively: τ and D express the relaxation times and the
m m
strengths of colored noise for additive (m = 0) and multiplicative noise (m = 1): η (t)
m
express independent zero-mean Gaussian white noise with correlations given by
′ ′
ξ (t)ξ (t) = δ δ(t t). (3)
m n mn
h i −
The distribution and correlation of η are given by
m
τ
p(η ) exp m η2 , (4)
m ∝ −2D m
(cid:18) m (cid:19)
3
′
D t t
′ ′ m
c (t,t) = η (t)η (t) = δ exp | − | . (5)
mn m n mn
h i (cid:18) τm (cid:19) − τm !
By applying the FIM to the Langevin model given by Eqs. (1) and (2), we obtain the
effective FPE given by [7, 11] (details being given in the Appendix)
∂ ∂ ∂2 ∂ ∂
p(x,t) = F˜(x)p(x,t)+D˜ p(x,t)+D˜ G(x) G(x)p(x,t), (6)
∂t −∂x 0∂x2 1∂x ∂x
from which the effective Langevin model is derived as
dx
˜ ˜ ˜
= F(x)+ 2D ξ (t)+ 2D G(x)ξ (t), (7)
0 0 1 1
dt
q q
with
F˜ = F, (8)
D
D˜ = 0 , (9)
0 (1 τ F′ )
0
− h i
D
D˜ = 1 . (10)
1 [1 τ ( F′ FG′/G )]
1
− h i−h i
′ ′
Here F = dF/dx and G = dG/dx, and the bracket expresses the average over p(x,t)
h·i
to be discussed shorty [Eq. (11)]. It is noted that we will temporally evaluated F′ etc.
h i
in order to discuss dynamics of the system, while they are conventionally evaluated for
the stationary value as F′(x ) etc. with x = x(t = ) [7, 11].
s s
h ∞ i
2.2 Equations of motion for mean and variance
With the use of the effective FPE given by Eq. (7), an equation of motion for the average
of Q(x):
Q = Q(x)p(x,t)dx, (11)
h i
Z
is given by [19]
d Q
′ ˜ ˜ ′′ ˜ ′ ′
h i = QF +D Q +D (QG)G , (12)
0 1
dt h i h i h i
which yields (for Q = x,x2)
d x
˜ ˜ ′
h i = F +D GG , (13)
1
dt h i h i
d x2
h i = 2 xF˜ +2D˜ +2D˜ G2 +xG′G . (14)
0 1
dt h i h i
4
Mean (µ) and variance (γ) are defined by
µ = x , (15)
h i
γ = x2 x 2. (16)
h i−h i
Expanding Eqs. (13) and (14) around the mean value of µ, and retaining up to the second
order of (δx )2 , we get equations of motion for µ and γ expressed by [19]
i
h i
dµ
= f˜ +f˜γ +D˜ [g g +3(g g +g g )γ], (17)
0 2 1 0 1 1 2 0 3
dt
dγ
= 2f˜γ +4D˜ γ(g2 +2g g )+2D˜ g2 +2D˜ , (18)
dt 1 1 1 0 2 1 0 0
where f˜ = (1/ℓ!)∂ℓF˜(µ)/∂xℓ and g = (1/ℓ!)∂ℓG(µ)/∂xℓ. It is noted that D˜ and D˜ in
ℓ ℓ 0 1
Eqs. (17) and (18) are given by Eqs. (9) and (10), respectively.
In the case of F(x) = λx + I and G(x) = x where λ and I denote the relaxation
−
rate and an input, respectively, the FIM yields equations of motion for µ and γ given by
dµ
= λµ+D˜ µ+I, (19)
1
dt −
dγ
= 2λγ +4D˜ γ +2D˜ µ2 +2D˜ , (20)
1 1 0
dt −
with
D
˜ 0
D = , (21)
0
(1+λτ )
0
D
˜ 1
D = . (22)
1
[1+(τ I/µ)]
1
We have to solve Eqs. (19)-(22) for µ, γ, D˜ and D˜ in a self-consistent way.
0 1
Stationary values of µ and γ are implicitly given by
I
µ = , (23)
s ˜
(λ D )
1
−
(D˜ +D˜ µ2)
γ = 0 1 s , (24)
s (λ 2D˜ )
1
−
˜ ˜
with D and D given by Eqs. (21) and (22), respectively, with µ = µ . Equations
0 1 s
(23) and (24) show that µ and γ diverge for D˜ > λ and D˜ > λ/2, respectively. The
s s 1 1
divergence of moments is common in systems subjected to multiplicative noise, because
its stationary distribution has a long-tail power-law structure [19]-[21]. From Eqs. (22)
5
and (23), we get
D
D˜ = 1 , (25)
1 ˜
[1+τ (λ D )]
1 1
−
1
= (1+λτ ) (1+λτ )2 4τ D , (26)
1 1 1 1
2τ − −
1 (cid:20) q (cid:21)
which yields
D τ D τ2D2
D˜ = 1 1+ 1 1 2 1 1 + ,
1 (cid:18)1+λτ1(cid:19)" (1+λτ1)2 − (1+λτ1)4 ··#
D
1 D˜APP. for τ D /(1+λτ )2 1 (27)
≃ (1+λτ ) ≡ 1 1 1 1 ≪
1
Equation(27)impliesthattheapproximationofD˜APP isvalidbothfor(i)τ (1/λ,1/D )
1 1 ≪ 1
and (ii) τ (1/λ, D /λ2).
1 1
≫
2.3 Stationary distribution
From the effective FPE given by Eq. (7), we get the stationary distribution p(x) given by
1
lnp(x) = ln[D˜ +D˜ G(x)2]+Z(x), (28)
0 1
− 2
(cid:18) (cid:19)
with
F(x)
Z(x) = dx. (29)
[D˜ +D˜ G(x)2]
Z 0 1
Because of the presence of multiplicative noise, the stationary distribution generally has
non-Gaussian power-law structure [19]-[21].
Inthewhite-noise limit(τ = 0), thestationarydistributionfortheLangevinequation
m
given by Eq. (1) with η = √2D is expressed by Eqs. (28) and (29) with D˜ = D
m m m m
in the Stratonovich representation. Then the stationary distribution for colored noise is
expressed by
p(x;D ,D ,τ ,τ ) = p(x;D˜ ,D˜ ,0,0) p (x;D˜ ,D˜ ), (30)
0 1 0 1 0 1 wn 0 1
≡
where p (x;D˜ ,D˜ ) expresses the stationary distribution for white noise.
wn 0 1
In the case of F(x) = λx+I and G(x) = x, we get
−
p(x) (D˜ +D˜ x2)−(λ/2D˜1+1/2)exp[Y(x)], (31)
0 1
∝
with
I D˜
Y(x) = .tan−1 1x , (32)
D˜ D˜ vuD˜0
0 1 u
t
q
6
D
D˜ = 0 , (33)
0
(1+λτ )
0
D
D˜ = 1 , (34)
1
[1+(τ I/µ )]
1 s
where µ expresses the stationary value of µ [Eq. (23) ].
s
2.4 Model calculations
2.4.1 Stationary properties
In order to demonstrate the feasibility of our analytical theory, we have performed model
calculations. Direct simulations (DSs) for Eqs. (1) and (2) have been performed by using
thefourth-orderRunge-Kuttamethodforperiodof1000withatimemeshof0.01. Results
of DSs are the average over hundred thousands trials otherwise noticed. All quantities
are dimensionless.
The τ dependence of the ratio of D˜ /D is depicted in Fig.1, where results calculated
1 1 1
bytheFIMandtheapproximation(APP)givenbyEq. (27)areshownbysolidanddashed
curves, respectively, for λ = 1.0, D = 0.01, D = 0.2 and τ = 0.01. D˜ calculated by
0 1 0 1
the APP is in good agreement with that by the FIM, and the effective noise strength is
decreased with increasing τ . The difference between D˜ /D of the FIM and APP, plotted
1 1 1
by the chain curve, is zero at τ = 0 with a maximum at τ 0.5, and decreased at larger
1 1
∼
τ (> 1). The APP is fairly good for small τ and large τ , as discussed after Eq. (27).
1 1 1
Figure 2 (a)-(f) show the stationary distribution p(x) calculated by the FIM (solid
curves), DS (dashed curves), with the APP (chain curves) and in the white-noise limit
(WN: double-chain curves) when τ is changed for fixed values of λ = 1.0, D = 0.01,
1 0
D = 0.2 and τ = 0.01: (a), (c) and (e) in normal scale, and (b), (d) and (f) in log
1 0
scale. Calculations show that with increasing τ , p(x) becomes narrower, deviating from
1
results of WN. Results of the FIM and APP are in fairly good agreement with those of
DS: results of the APP is indistinguishable from those of the FIM. Figures 2(b), (d) and
(f) plotting p(x) in log scale show that results of the FIM and APP are in fairly good
agreement with that of DS up to the order of 10−2 for τ = 1.0 and of 10−4 for τ = 5.0.
1 1
2.4.2 Dynamical properties
We have investigated the response to an applied pulse input given by
I(t) = AΘ(t t )Θ(t t)+B, (35)
b e
− −
7
where A = 0.5, B = 0.1, t = 100 and t = 200, and Θ(t) is the Heaviside function. Time
b e
courses of µ(t) and γ(t) are shown in Fig. 3(a)-(f), where τ is changed for fixed values
1
of λ = 1.0, D = 0.01, D = 0.2 and τ = 0.01. With increasing τ , µ(t) and γ(t) induced
0 1 0 1
by an applied pulse at t < t < t are decreased. This is because they are given by
b e
(A+B)
µ(t) = , (36)
(λ D˜ )
1
−
D˜ (A+B)2D˜
0 1
γ(t) = + , for t < t < t (37)
(λ 2D˜ ) (λ 2D˜ )(λ D˜ )2 b e
1 1 1
− − −
where D˜ is decreased with increasing τ as Fig. 1 shows. The results of the FIM and
1 1
APP are again in good agreement with that of the DS: the FIM yields slightly better
results than the APP as shown in Figs. 3(b) and 3(f).
Next we study the response to a sinusoidal input given by
I(t) = Csinωt, (38)
where C = 0.5, ω = 2π/T and T = 100. Figures 4(a), (c) and (f) show time courses
p p
of µ(t), and Figs. 4(b), (d) and (f) those of γ(t) when τ is changed for fixed values of
1
λ = 1.0, D = 0.01, D = 0.2 and τ = 0.01. With increasing τ , the magnitude of µ(t) is
0 1 0 1
decreased. This is understood from an analysis with the use of Eq. (19), which yields
C
µ(t) = sin(ωt φ), (39)
(λ D˜ )2 +ω2 −
1
−
q
with
ω
φ = tan−1 . (40)
λ D˜ !
1
−
Equation (39) shows that with increasing ω (i.e. decreasing T ), the magnitude of µ(t) is
p
decreased, representing a character of the low-pass filter.
3 Langevin model subjected to one additive and two
multiplicative colored noise
3.1 Effective Langevin equation
We have assumed the Langevin model subjected to one additive (η ) and two multiplica-
0
tive colored noise (η , η ), as given by
1 2
dx
= F(x)+η (t)+G (x)η (t)+G (x)η (t), (41)
0 1 1 2 2
dt
8
with
dη (t) 1 √2D
m m
= η + ξ (t), (m = 0,1,2) (42)
m m
dt −τ τ
m m
where F(x), G (x) and G (x) express arbitrary functions of x, and ξ are independent
1 2 m
zero-mean white noise with correaltion:
′ ′
ξ (t)ξ (t) = δ δ(t t). (43)
m n mn
h i −
Applying the FIM [11] to the model under consideration, we get the effective FPE
given by (details being given in the Appendix):
∂ ∂ ∂2
p(x,t) = F˜(x)p(x,t)+D˜ p(x,t)
0
∂t −∂x ∂x2
∂ ∂ ∂ ∂
˜ ˜
+ D G (x) G (x)p(x,t)+D G (x) G (x)p(x,t), (44)
1 1 1 2 2 2
∂x ∂x ∂x ∂x
from which we get the effective Langevin equation:
dx
= F(x)+ 2D˜ ξ (t)+ 2D˜ G (x)ξ (t)+ 2D˜ G (x)ξ (t), (45)
0 0 1 1 1 2 2 2
dt
q q q
with
D
D˜ = 0 , (46)
0 (1 τ F′ )
0
− h i
D
D˜ = m , (m = 1,2) (47)
m [1 τ ( F′ FG′ /G )]
− m h i−h m mi
′ ′
where F = dF/dx and G = dG /dx, and the bracket stands for the average over
m m h·i
p(x,t):
Q(x) = Q(x)p(x,t)dx. (48)
h i
Z
3.2 Equations of motion for mean and variance
By using the effective FPE given by Eq. (45), we can obtain equations of motion for mean
(µ) and variance (γ) defined by
µ = x , (49)
h i
γ = x2 x 2. (50)
h i−h i
When F(x) and G (x) are given by
m
F(x) = λx+I, (51)
−
G = a (x e ), (m = 1,2) (52)
m m m
−
9
where λ is the relaxation rate, I an input, and a and e constants, we get equations of
m m
motion for µ and γ given by [19]
dµ
= λµ+I +D˜ (µ e )+D˜ (µ e ), (53)
1 1 2 2
dt − − −
dγ
= 2λγ +4(D˜ +D˜ )γ +2D˜ (µ e )2 +2D˜ (µ e )2 +2D˜ , (54)
1 2 2 1 2 2 0
dt − − −
with
D
D˜ = 0 , (55)
0
(1+λτ )
0
D
D˜ = m . (for m = 1,2) (56)
m
[1+τ ( λe +I)/(µ e )]
m m m
− −
It is necessary to self-consistently solve Eqs. (53)-(56) for µ, γ, D˜ , D˜ and D˜ .
0 1 2
Stationary values of µ and γ are implicitly given by
(I D˜ e D˜ e )
1 1 2 2
µ = − − , (57)
s (λ D˜ D˜ )
1 2
− −
[D˜ +D˜ (µ e )2 +D˜ (µ e )2]
0 1 s 1 2 s 2
γ = − − , (58)
s [λ 2(D˜ +D˜ )]
1 2
−
with D˜ (m = 0,1,2) given by Eqs. (55) and (56) with µ = µ . Equations (57) and
m s
(58) show that µ and γ diverge for (D˜ +D˜ ) > λ and (D˜ +D˜ ) > λ/2, respectively.
s s 1 2 1 2
Equation (27) suggests that the approximation given by
D
D˜ m D˜APP, for τ D /(1+λτ )2 1 (m = 1,2) (59)
m ≃ (1+λτ ) ≡ m m m m ≪
m
may be valid both for small τ and large τ , as will be numerically shown in Fig. 5 [22].
m m
3.3 Stationary distribution
From the effective FPE of Eq. (45), we get the stationary distribution p(x) given by [19]
1
lnp(x) = ln[D˜ +D˜ G2(x)+D˜ G2(x)]+Z(x), (60)
− 2 0 1 1 2 2
(cid:18) (cid:19)
with
F(x)
Z(x) = dx. (61)
[D˜ +D˜ G2(x)+D˜ G2(x)]
Z 0 1 1 2 2
In the white-noise limit (τ = 0), the stationary distribution of the Langevin model
m
given by Eq. (41) with η = √2D (m = 0,1,2), is expressed by [19]-[21]
m m
1
lnp(x) = Z(x) ln[D +D G2(x)+D G2(x)], (62)
− 2 0 1 1 1 1
(cid:18) (cid:19)
10
