PHYSICAL REVIEW E 78, 031110 共2008兲

Synchrony and variability induced by spatially correlated additive and multiplicative noise in the coupled Langevin model Hideo Hasegawa* Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan 共Received 18 February 2008; revised manuscript received 13 August 2008; published 8 September 2008兲 The synchrony and variability of the coupled Langevin model subjected to spatially correlated additive and multiplicative noise are discussed. We have employed numerical simulations and the analytical augmentedmoment method, which is the second-order moment method for local and global variables 关H. Hasegawa, Phys. Rev. E 67, 041903 共2003兲兴. It has been shown that the synchrony of an ensemble is increased 共decreased兲 by a positive 共negative兲 spatial correlation in both additive and multiplicative noise. Although the variability for local fluctuations is almost insensitive to spatial correlations, that for global fluctuations is increased 共decreased兲 by positive 共negative兲 correlations. When a pulse input is applied, the synchrony is increased for the correlated multiplicative noise, whereas it may be decreased for correlated additive noise coexisting with uncorrelated multiplicative noise. An application of our study to neuron ensembles has demonstrated the possibility that information is conveyed by the variance and synchrony in input signals, which accounts for some neuronal experiments. DOI: 10.1103/PhysRevE.78.031110

PACS number共s兲: 05.40.Ca, 05.10.Gg, 84.35.⫹i

I. INTRODUCTION

It has been realized that the coupled Langevin model is a valuable and useful model for a study of various phenomena observed in stochastic ensembles 共for a recent review, see Ref. 关1兴兲. Independent 共uncorrelated兲 additive and/or multiplicative noise has been widely adopted for theoretical analyses because of its mathematical simplicity. In natural phenomena, however, there exist some kinds of correlations in noise, such as spatial and temporal correlations, and a correlation between additive and multiplicative noise. In this paper we will pay attention to the spatial correlation in noise. The Langevin model has been usually discussed with the use of the Fokker-Planck equation 共FPE兲 for the probability distribution. In the case of correlated additive noise only, the probability distribution is expressed by the multivariate Gaussian probability with a covariance matrix. The effect of correlated additive noise has been extensively studied in neuroscience, where it is an important and essential problem to study the effect of correlations in noise and signals 共for a review, see Ref. 关2兴兲. It has been shown that the synchrony and variability in neuron ensembles are much influenced by the spatial correlations 关3–16兴. The spatial correlation in additive noise enhances the synchrony of firings in a neuron ensemble, while it works to diminish beneficial roles of independent noise, such as the stochastic and coherent resonances and the population 共pooling兲 effect 关2,10,11,14兴, related discussions being given in Sec. III. The problem becomes much difficult when multiplicative noise exists, for which the probability distribution generally becomes a non-Gaussian. Although an analytical expression of the stationary probability distribution for uncorrelated multiplicative noise is available, that for correlated multiplicative noise has not been obtained yet. Indeed, only a small amount of theoretical study of the effect of spatially corre-

*[email protected] 1539-3755/2008/78共3兲/031110共8兲

lated multiplicative noise has been reported for subjects such as the noise-induced phase separation 关17兴 and the Fisher information 关18–20兴, as far as the author is concerned. In a recent paper 关21兴, we have studied stationary and dynamical properties of the coupled Langevin model subjected to uncorrelated additive and multiplicative noise. We employed the augmented moment method 共AMM兲, which was developed for a study of stochastic systems with finite populations 关22,23兴. In the AMM, we consider global properties of ensembles, taking account of mean and fluctuations 共variances兲 of local and global variables. Although a calculation of the probability distribution for the spatially correlated multiplicative noise with the use of the FPE is very difficult as mentioned above, we may easily study its effects by using the AMM. It is the purpose of the present paper to apply the AMM to the coupled Langevin model including spatially correlated multiplicative noise and to study its effects on the synchrony and variability. The paper is organized as follows. In Sec. II, we discuss the AMM for the spatially correlated Langevin model. With the use of the analytical AMM and numerical methods, the synchrony and variability of the coupled Langevin model are investigated. In Sec. III, previous studies of the correlated multiplicative noise using the Gaussian approximation 关18–20兴 are critically discussed. An application of our study to neuron ensembles is also presented with model calculations. The final Sec. IV is devoted to our conclusion. II. FORMULATION A. Adopted model

We have assumed the N-unit coupled Langevin model subjected to spatially correlated additive and multiplicative noise. The dynamics of a variable xi 共i = 1 – N兲 is given by dxi = F共xi兲 + H共ui兲 + G共xi兲␩i共t兲 + ␰i共t兲, dt

共1兲

with 031110-1

©2008 The American Physical Society

PHYSICAL REVIEW E 78, 031110 共2008兲

HIDEO HASEGAWA

ui共t兲 =

冉 冊兺 w Z

x j共t兲 + Ii共t兲,

冉 冊

d␮ ␣2 = f 0 + h 0 + f 2␥ + 关g0g1 + 3共g1g2 + g0g3兲␥兴, 2 dt

共2兲

j共⫽i兲

共13兲 H共u兲 =

u

冑u2 + 1 ⌰共u兲.

共3兲

Here F共x兲 and G共x兲 are arbitrary functions of x, Z 共=N − 1兲 denotes the coordination number, Ii共t兲 an input signal from external sources, w the coupling strength, and ⌰共u兲 the Heaviside function: ⌰共u兲 = 1 for u ⬎ 0 and ⌰共u兲 = 0 otherwise. We have included additive and multiplicative noise by ␰i共t兲 and ␩i共t兲, respectively, expressing zero-mean Gaussian white noise with correlations given by 具␩i共t兲␩ j共t⬘兲典 = ␣ 关␦ij + c M 共1 − ␦ij兲兴␦共t − t⬘兲,

共4兲

具␰i共t兲␰ j共t⬘兲典 = ␤2关␦ij + cA共1 − ␦ij兲兴␦共t − t⬘兲,

共5兲

具␩i共t兲␰ j共t⬘兲典 = 0,

共6兲

2

where the brackets 具·典 denote the average, ␣ 共␤兲 expresses the magnitude of multiplicative 共additive兲 noise, and c M 共cA兲 stands for the degree of the spatial correlation in multiplicative 共additive兲 noise. Although our results to be present in the following are valid for any choice of H共x兲, we have adopted a simple analytic expression given by Eq. 共3兲 in this study. We assume that external inputs have a variability defined by Ii共t兲 = I共t兲 + ␦Ii共t兲,

共7兲

2h1w d␥ = 2f 1␥ + 共N␳ − ␥兲 + 2共g21 + 2g0g2兲␣2␥ + ␥I + ␤2 dt Z + ␣2g20 ,

1 d␳ = 2f 1␳ + 2h1w␳ + 2共g21 + 2g0g2兲␣2␳ + 共␥I + ␤2 + ␣2g20兲 dt N +

Z 共SI␥I + cA␤2 + c M ␣2g20兲, N

共8兲

具␦Ii共t兲␦I j共t⬘兲典 = ␥I关␦ij + SI共1 − ␦ij兲兴␦共t − t⬘兲,

共9兲

where ␥I and SI denote the variance and degree of the spatial correlation, respectively, in external signals. We will investigate the response of the coupled Langevin model to correlated external inputs given by Eqs. 共7兲–共9兲.

F共x兲 = − ␭x,

共16兲

G共x兲 = x,

共17兲

␣ 2␮ d␮ = − ␭␮ + h0 + , dt 2

共18兲

Eqs. 共13兲–共15兲 become

冉 冊

2h1wN ␥ d␥ = − 2␭␥ + ␳− + 2␣2␥ + P, dt Z N

共19兲

共P + ZR兲 d␳ = − 2␭␳ + 2h1w␳ + 2␣2␳ + , dt N

共20兲

P = ␥ I + ␤ 2 + ␣ 2␮ 2 ,

共21兲

R = S I␥ I + c A␤ 2 + c M ␣ 2␮ 2 ,

共22兲

with

B. Augmented moment method

In the AMM 关22,23兴, we define the three quantities of ␮共t兲, ␥共t兲, and ␳共t兲 expressed by

共15兲

where f ᐉ = 共1 / ᐉ!兲关⳵ᐉF共␮兲 / ⳵xᐉ兴, gᐉ = 共1 / ᐉ!兲关⳵ᐉG共␮兲 / ⳵xᐉ兴, hᐉ = 共1 / ᐉ!兲关⳵ᐉH共u兲 / ⳵uᐉ兴, and u = w␮ + I. Original N-dimensional stochastic differential equations 共DEs兲 given by Eqs. 共1兲–共3兲 are transformed to the three-dimensional deterministic DEs given by Eqs. 共13兲–共15兲. For ␥I = SI = cA = c M = 0, equations of motion given by Eqs. 共13兲–共15兲 reduce to those obtained in our previous study 关21兴. When we adopt F共x兲 and G共x兲 given by

with 具␦Ii共t兲典 = 0,

共14兲

where h0 = H共w␮ + I兲, and P and R express uncorrelated and correlated contributions, respectively. We employ Eqs. 共18兲–共22兲 in the remainder of this paper.

1 ␮共t兲 = 具X共t兲典 = 兺 具xi共t兲典, N i

共10兲

1 兺 具关xi共t兲 − ␮共t兲兴2典, N i

共11兲

1. Synchrony

共12兲

In order to quantitatively discuss the synchronization, we first consider the quantity S⬘共t兲 given by

␥共t兲 =

␳共t兲 = 具关X共t兲 − ␮共t兲兴2典,

C. Synchrony and variability

where X共t兲 = 共1 / N兲兺ixi共t兲, ␮共t兲 expresses the mean, and ␥共t兲 and ␳共t兲 denote fluctuations in local 共xi兲 and global variables 共X兲, respectively. By using the FPE, we obtain equations of motion for ␮共t兲, ␥共t兲 and ␳共t兲 which are given by 共argument t is suppressed, details being given in the Appendix A兲

S⬘共t兲 =

1 兺 具关xi共t兲 − x j共t兲兴2典 = 2关␥共t兲 − ␳共t兲兴. N2 ij

共23兲

When all neurons are in the same state, xi共t兲 = X共t兲 for all i 共the completely synchronous state兲, we obtain S⬘共t兲 = 0 in Eq.

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PHYSICAL REVIEW E 78, 031110 共2008兲

共23兲. On the contrary, in the asynchronous state where ␳ = ␥ / N, it is given by S⬘共t兲 = 2共1 − 1 / N兲␥共t兲 ⬅ S0⬘共t兲 关21,22兴. We may define the normalized ratio for the synchrony given by 关22,23兴

冉 冊冉

N S⬘共t兲 = Z S0⬘共t兲

S共t兲 ⬅ 1 −

冊

␳共t兲 1 − , ␥共t兲 N

共24兲

increased with increasing spatial correlations and/or coupling. This is more clearly seen in the limit of no spatial correlations 关Eq. 共31兲兴 or no couplings 关Eq. 共32兲兴. The local and global variabilities CV and DV defined by Eqs. 共25兲 and 共26兲, respectively, are generally expressed in terms of P and R, and they are given for w = 0.0 by

which is 0 and 1 for completely asynchronous 共S⬘ = S0⬘兲 and synchronous states 共S⬘ = 0兲, respectively.

CV =

2. Variability

DV = CV

The local variability is conventionally given by CV共t兲 =

冑具关␦xi共t兲兴2典 冑␥共t兲 =

␮共t兲

␮共t兲

冑具关␦X共t兲兴2典 冑␳共t兲 ␮共t兲

=

␮共t兲

= CV共t兲

共25兲

,

冑

␳共t兲 , ␥共t兲

共26兲

where ␦X共t兲 = X共t兲 − ␮共t兲.

␥=

␭1 = − ␭ +

共27兲

共h1w/Z兲共P + ZR兲 P + , 2共␭ − ␣2 + h1w/Z兲 2共␭ − ␣2 + h1w/Z兲共␭ − ␣2 − h1w兲 共28兲

␳=

P + ZR , 2N共␭ − ␣2 − h1w兲

共29兲

where ␮ in P and R of Eqs. 共21兲 and 共22兲 is given by Eq. 共27兲. We note in Eq. 共27兲 that ␮ is increased as I is increased with an enhancement factor of 1 / 共␭ − ␣2 / 2兲. A local fluctuation ␥ is increased with increasing input fluctuations 共␥I兲 and/or noise 共␣ , ␤兲 as Eq. 共28兲 shows. In the limit of SI = cA = c M = R = w = 0.0, Eqs. 共28兲 and 共29兲 lead to ␳ / ␥ = 1 / N, which expresses the central-limit theorem. From Eqs. 共24兲, 共28兲, and 共29兲, we obtain h1wP + Z共␭ − ␣ 兲R P关Z共␭ − ␣2兲 − h1w共Z − 1兲兴 + h1wZR

=

h 1w Z共␭ − ␣ 兲 − h1w共Z − 1兲

=

S I␥ I + c A␤ 2 + c M ␣ 2␮ 2 ␥ I + ␤ 2 + ␣ 2␮ 2

2

共for w = 0兲,

共32兲

where P and R in Eq. 共30兲 are given by Eqs. 共21兲 and 共22兲, respectively. Equation 共30兲 shows that the synchrony S is

冊

1/2

共34兲

共35兲

2h1w , Z

共36兲

␭3 = − 2␭ + 2␣2 + 2h1w.

共37兲

The first eigenvalue of ␭1 arises from an equation of motion for ␮, which is decoupled from the rest of variables. The stability condition for ␮ is given by h1w ⬍ 共␭ − ␣2/2兲.

共38兲

The stability condition for ␥ and ␳ is given by − Z共␭ − ␣2兲 ⬍ h1w ⬍ 共␭ − ␣2兲.

共39兲

Then for ␭ − ␣ ⬍ h1w ⬍ ␭ − ␣ / 2, ␥ and ␳ are unstable, but ␮ remains stable. It is note that there is a limitation in a parameter value of c, as given by 2

−

共30兲

共for SI = cA = c M = 0兲 共31兲

␣2 + h1w, 2

␭2 = − 2␭ + 2␣2 −

2

S=

共33兲

for w = 0,

The local variability CV only weakly depends on the spatial correlation through the coupling, and it is independent of the correlation for w = 0. In contrast, the global variability DV is increased 共decreased兲 for positive 共negative兲 correlations. In the limit of SI = cA = c M = w = 0.0, Eq. 共34兲 yields DV = CV / 冑N expressing a smaller global variability in a larger-N ensemble 共the population or pooling effect兲 关2,10,11,14兴. The stability condition around the stationary state given by Eqs. 共27兲–共29兲 may be examined from eigenvalues of the Jacobian matrix of Eqs. 共18兲–共20兲, which are given by

The stationary solution of Eqs. 共18兲–共20兲 is given by h0 , 共␭ − ␣2/2兲

1/2

共1 + ZSI兲␥I + 共1 + ZcA兲␤2 + 共1 + Zc M 兲␣2␮2 N共␥I + ␤2 + ␣2␮2兲

D. Stationary properties

␮=

冊

for w = 0.

where ␦xi共t兲 = xi共t兲 − ␮共t兲. Similarly, the global variability is defined by DV共t兲 =

冉

冉

1 ␥ I + ␤ 2 + ␣ 2␮ 2 ␮ 2共␭ − ␣2兲

2

冉

冊

1 S I␥ I + c A␤ 2 + c M ␣ 2␮ 2 艋 艋 1, Z ␥ I + ␤ 2 + ␣ 2␮ 2

共40兲

which arises from the condition given by 0 艋 ␳ 艋 ␥ 关see Eqs. 共12兲 and 共23兲兴. When ␤ = ␥I = 0, for example, a physically conceivable value of c M is given by −1 / Z 艋 c M 艋 1. The c M dependence of the synchrony S is shown in Fig. 1 where ␣ = 0.1, ␤ = 0.1, cA = 0.1, ␥I = SI = 0, and N = 100. Equation 共40兲 yields the condition that −0.44⬍ c M ⬍ 1.0 with ␮ = 0.5 for a given set of parameters. We note that the synchrony is increased with increasing s and that the effect of the correlated variability is more considerable for larger ␮ and w, as Eq. 共30兲 shows.

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HIDEO HASEGAWA 0.5

0.8

µ=0.5

0.2

0.1 0 -0.1-1

0 cM

α=0.1 β=0.1 cA=0.1 N=100

1

FIG. 1. 共Color online兲 c M dependence of the stationary synchrony S for w = 0.5 共solid curves兲 and w = 0.0 共dashed curves兲 with ␣ = 0.1, ␤ = 0.1, cA = 0.1, ␥I = SI = 0, and N = 100, ␮ being treated as a parameter. E. Dynamical properties

In order to study the dynamical properties of our model given by Eqs. 共1兲–共3兲, we have performed direct simulations 共DSs兲 by using the Heun method 关24,25兴 with a time step of 0.0001: DS results are averages of 100 trials. AMM calculations have been performed for Eqs. 共18兲–共20兲 by using the fourth-order Runge-Kutta method with a time step of 0.01. We consider a set of typical parameters of ␭ = 1.0, ␣ = 0.1, ␤ = 0.1, w = 0.5, ␥I = SI = 0, and N = 100. We apply a pulse input given by I共t兲 = A⌰共t − 40兲⌰共60 − t兲 + Ab ,

共41兲

with A = 0.4 and Ab = 0.1, where ⌰共x兲 denotes the Heaviside function: ⌰共x兲 = 1 for x 艌 0 and 0 otherwise. Figures 2共a兲–2共d兲 show time courses of ␮共t兲, ␥共t兲, S共t兲, and CV共t兲 for the correlated multiplicative noise 共cA = 0.0 and c M = 0.5兲. ␮共t兲 and S共t兲 are increased by an applied input at 40艋 t ⬍ 60 shown by the chain curve in Fig. 2共a兲, by which ␥共t兲 is slightly increased. The variability CV共t兲 is decreased because of an increased ␮共t兲. The results of the AMM shown by the solid curves are in fairly good agreement with those of DSs shown by the dashed curves. In contrast, Figs. 3共a兲–3共d兲 show time courses of ␮共t兲, ␥共t兲, S共t兲, and CV共t兲 for the correlated additive noise 共cA = 0.1 and c M = 0.0兲. With an applied pulse input, ␮共t兲 is in0.8

0.4 0.3 (c) 0.2 0.4 0.1 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0.02 1 (b) 0.8 (d) 0.6 0.01 0.4 0.2 00 20 40 60 80 100 00 20 40 60 80 100 t t

I, µ

S

CV

(a)

γ

0.6

(a)

S

CV

0.2

0.4 0.3 (c) 0.2 0.4 0.1 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0.02 1 (b) 0.8 (d) 0.6 0.01 0.4 0.2 00 20 40 60 80 100 00 20 40 60 80 100 t t 0.6

γ

S

0.3 µ=0.1

w=0.5 w=0.0

µ

0.4

FIG. 2. 共Color online兲 Time courses of 共a兲 ␮共t兲, 共b兲 ␥共t兲, 共c兲 S共t兲, and 共d兲 CV共t兲 with the correlated multiplicative noise 共cA = 0.0, c M = 0.5, ␣ = 0.1, ␤ = 0.1兲 for a pulse input given by Eq. 共41兲 with A = 0.4 and Ab = 0.1: the solid and dotted curves express results of the AMM and DS, respectively: the chain curve in 共a兲 expresses an input of I共t兲 共␭ = 1.0, SI = ␥I = 0.0, and N = 100兲.

FIG. 3. 共Color online兲 Time courses of 共a兲 ␮共t兲, 共b兲 ␥共t兲, 共c兲 S共t兲, and 共d兲 CV共t兲 with the correlated additive noise 共cA = 0.1, c M = 0.0, ␣ = 0.1, ␤ = 0.1兲 for a pulse input given by Eq. 共41兲 with A = 0.4 and Ab = 0.1: the solid and dashed curves denote results of the AMM and DSs, respectively: the chain curve in 共a兲 expresses an input of I共t兲 共␭ = 1.0, SI = ␥I = 0.0, and N = 100兲.

creased and ␥共t兲 is a little increased, as in the case of Figs. 2共a兲 and 2共b兲. However, the synchrony S共t兲 is decreased in Fig. 3共c兲, while it is increased in Fig. 2共c兲. This difference arises from the fact that a decrease in S共t兲 in the former case is mainly due to an increase in P of the denominator of Eq. 共30兲, while in the latter case, its increase arises from an increase in R of the numerator of Eq. 共30兲. This point is more easily realized for w = 0, for which Eq. 共32兲 yields S=

c M ␣ 2␮ 2 ␤ 2 + ␣ 2␮ 2

共for cA = 0兲

共42兲

=

c A␤ 2 ␤ + ␣ 2␮ 2

共for c M = 0兲.

共43兲

2

The situation is almost the same even for finite w, as Figs. 2共c兲 and 3共c兲 show. In both Figs. 2共d兲 and 3共d兲, CV共t兲 is decreased by an applied input because of an increased ␮共t兲. III. DISCUSSION A. Comparison with related studies

We have investigated the stationary and dynamical properties of the spatially correlated Langevin model given by Eqs. 共1兲–共3兲. In Ref. 关26兴, we discussed the Fisher information in the Langevin model subjected to uncorrelated additive and multiplicative noise, which is a typical microscopic model showing the nonextensive behavior 关27兴. It is interesting to calculate the Fisher information of the Langevin model with correlated multiplicative noise. Such a calculation needs to solve the FPE of the Langevin model given by Eq. 共A1兲 because the Fisher information is expressed in terms of derivatives of the probability distribution. For additive noise only 共␣ = c M = 0兲, the stationary probability distribution p共兵xk其兲 is expressed by the multivariate Gaussian distribution given by

冋

p共兵xk其兲 ⬀ exp −

with the covariance matrix Q expressed by

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册

1 兺 共xi − ␮兲共Q−1兲ij共x j − ␮兲 , 2 ij

共44兲

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PHYSICAL REVIEW E 78, 031110 共2008兲

Qij = ␴2关␦ij + cA共1 − ␦ij兲兴,

共45兲

where ␮ = H / ␭ and ␴2 = ␤2 / 2␭. From Eqs. 共44兲 and 共45兲, we obtain the Fisher information given by 关18兴 g=

N . ␴2关1 + 共N − 1兲cA兴

共46兲

way as additive noise, we obtain the AMM equations given by Eqs. 共18兲–共20兲, but without the third term of ␣2␮ / 2 in Eq. 共18兲. By using the Gaussian approximation given by Eq. 共51兲, Abbott and Dayan 共AD兲 关18兴 obtained the Fisher information expressed by 关Eq. 4.7 of Ref. 关18兴兴

When multiplicative noise exists, a calculation of even stationary distribution becomes difficult, and it is generally not given by the Gaussian. The stationary distribution for uncorrelated additive and multiplicative noise 关G共x兲 = x, cA = c M = ␥I = 0.0兴 is given by 关21,26,28,29兴 p共兵xk其兲 ⬀ 兿 共␤2 + ␣2x2i 兲−共␣

2/␭+1/2兲 共2H/␣␤兲tan−1共␣x /␤兲 i

e

gAD =

共47兲 In the limit of ␣ = 0.0 and ␤ ⫽ 0.0 共i.e., uncorrelated additive noise only兲, Eq. 共47兲 becomes the Gaussian distribution given by p共兵xk其兲 ⬀ 兿 e−共␭/␤

2兲共x − i

␮兲2

.

共48兲

i

In the opposite limit of ␣ ⫽ 0.0, ␤ = 0.0, and H ⬎ 0 共i.e., uncorrelated multiplicative noise only兲, Eq. 共47兲 reduces to ␣ p共兵xk其兲 ⬀ 兿 x−共2 i

2/␭+1兲

2

e−2H/␣ xi⌰共xi兲,

共49兲

i

1

␴2M ␮2cM

=

␴2M ␮2

N

Qij = ␴2M ␮i␮ j关␦ij + c M 共1 − ␦ij兲兴,

共51兲

where ␮i 共=具xi典兲 denotes the average of xi and ␴2M a variance due to multiplicative noise. This is equivalent to assume that the multiplicative-noise term in the FPE given by Eq. 共A1兲 is approximated as

⳵ ␣2 ⳵ 兺 兺 关␦ij + cM共1 − ␦ij兲兴 ⳵xi xi ⳵x j x jp共兵xk其兲 2 i j ⯝

␣2 ⳵ ⳵ 关␦ij + c M 共1 − ␦ij兲兴 具xi典 具x j典p共兵xk其兲 兺 兺 2 i j ⳵xi ⳵x j

=

⳵ ⳵ ␣2 兺 兺 ␮i␮ j关␦ij + cM共1 − ␦ij兲兴 ⳵xi ⳵x j p共兵xk其兲. 共52兲 2 j i

If we adopt the Gaussian approximation given by Eq. 共52兲, with which multiplicative noise may be treated in the same

2N ␮2

共for N → ⬁兲

共for c M = 0兲,

共54兲

共55兲

共56兲

=

1 c␴2

共for N → ⬁兲

共57兲

=

N ␴2

共for c = 0.0兲.

共58兲

共50兲

where q = 共2␭ + 3␣2兲 / 共2␭ + ␣2兲 and ␴2q = ␣2␮2 / 2␭ 关26兴. Unfortunately, we have not succeeded in obtaining the analytic expression for the stationary distribution of the Langevin model including correlated multiplicative noise. In some previous studies 关18–20兴, the stationary distribution for correlated multiplicative noise only 共G共x兲 = x, cA = ␤ = 0.0, and H = ␮兲 is assumed to be expressed by the Gaussian distribution given by Eq. 共44兲 with the covariance matrix given by

2N ␮2

N ␴ 关1 + 共N − 1兲c兴

g=

4

2N␭q Nq g= 2 = 2 2 , ␣␮ ␴q

+

+

共53兲

with K = N−1兺i关d ln H共␮i兲 / d␮i兴2 = 1 / ␮2. Equation 共55兲 is not in agreement with the exact expression given by Eq. 共50兲 for uncorrelated multiplicative noise only. Instead of using the Langevin model, we may alternatively calculate the Fisher information of a spatially correlated nonextensive system by using the maximum-entropy method. In our recent paper 关30兴, we have obtained the analytic, stationary probability distribution which maximizes the Tsallis entropy 关27兴 under the constraints for a given set of the variance 共␴2兲 and covariance 共c␴2兲. The Fisher information is expressed by 关30兴

yielding the Fisher information given by 4

NK + 2NK + 共N − 1兲c M 兴

=

.

i

␴2M 关1

2

The Fisher information given by Eq. 共56兲 is increased 共decreased兲 by a negative 共positive兲 correlation. This implies from the Cramér-Rao theorem that an unbiased estimate of fluctuations is improved by a negative spatial correlation, by which the synchrony is decreased as shown by Eqs. 共30兲 and 共32兲. The N and c dependences of the Fisher information given by Eq. 共56兲 are different from those of gAD given by Eq. 共53兲, although they are the same as those for additive noise only 关Eq. 共46兲兴. It is noted that the Gaussian approximation given by Eq. 共51兲 or 共52兲 assumes the Gaussian distribution, although multiplicative noise generally yields the non-Gaussian distribution as shown by Eqs. 共47兲 and 共49兲. The spurious second term 共2NK兲 in Eq. 共53兲, which is independent of c M and ␴2M , arises from an inappropriate Gaussian approximation. In discussing the Fisher information of spatially correlated nonextensive systems, we must take into account the detailed structure of the non-Gaussian distribution. B. Application to neuronal ensembles

When ␥I and SI in Eq. 共9兲 are allowed to be time dependent, they may carry input information. This is easily real-

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γ

CV

1 0.8 (c) Synchrony driven 0.6 0.4 0.2 00 20 40 60 80 100 4 3 (d) 2 1 00 20 40 60 80 100 t

SI, S

0.8 (a) 0.6 0.4 0.2 00 20 40 60 80 100 0.4 0.3 (b) 0.2 0.1 00 20 40 60 80 100 t

CV

I, µ

1 0.8 (c) 0.6 0.4 0.2 00 20 40 60 80 100 4 3 (d) 2 1 00 20 40 60 80 100 t

S

0.8 0.6 (a) 0.4 0.2 00 20 40 60 80 100 0.4 (b) Fluctuation driven 0.3 0.2 0.1 00 20 40 60 80 100 t

γI, γ

I, µ

HIDEO HASEGAWA

FIG. 4. 共Color online兲 Time courses of 共a兲 ␮共t兲, 共b兲 ␥共t兲, 共c兲 S共t兲, and 共d兲 CV共t兲 for a fluctuation-driven input of ␥I共t兲 given by Eq. 共62兲 with SI = 0.1, B = 0.4, and Bb = 0.1: the solid and dotted curves express results of the AMM and DS, respectively: the chain curves in 共a兲 and 共b兲 express inputs of I共t兲 and ␥I共t兲, respectively 共␭ = 1.0, cA = 0.0, c M = 0.5, ␣ = 0.1, ␤ = 0.1, N = 100兲.

FIG. 5. 共Color online兲 Time courses of 共a兲 ␮共t兲, 共b兲 ␥共t兲, 共c兲 S共t兲, and 共d兲 CV共t兲 for a synchrony-driven input of SI共t兲 given by Eq. 共63兲 with ␥I = 0.1, C = 0.4, and Cb = 0.1: the solid and dotted curves express results of the AMM and DS, respectively: the chain curves in 共a兲 and 共c兲 express inputs of I共t兲 and SI共t兲, respectively 共␭ = 1.0, cA = 0.0, c M = 0.5, ␣ = 0.1, ␤ = 0.1, N = 100兲.

ized if the AMM equations given by Eqs. 共18兲–共20兲 are explicitly expressed in terms of ␮, ␥, and S as

When we regard a variable xi in the Langevin model given by Eqs. 共1兲–共3兲 as the firing rate ri 共⬎0兲 of a neuron i in a neuron ensemble, our model expresses the neuronal model proposed in Refs. 关31,32兴. It belongs to the firing-rate 共rate-code兲 models such as the Wilson-Cowan 关33兴 and Hopfield models 关34兴, in which a neuron is regarded as a transducer from input rate signals to output rate ones. Alternative neuronal models are spiking-neuron 共temporal-code兲 models such as the Hodgkin-Huxley 关35兴, FitzHugh-Nagumo 关36,37兴, and integrate-and-fire 共IF兲 models 关38兴. Various attempts have been proposed to obtain the firing-rate model, starting from spiking-neuron models 关39–43兴. It is difficult to analytically calculate the firing rate based on spiking-neuron models, except for the IF-type model 关38兴. It has been shown with the use of the IF model that information transmission is possible by noise-coded signals 关44,45兴 and that the modulation of the synchrony is possible without a change in firing rate 关46兴. Model calculations shown in Figs. 4 and 5 have demonstrated the possibility that information may be conveyed by ␥I共t兲 and SI共t兲, which is partly supported by results of the IF model 关44–46兴. Some relevant results have been reported in neuronal experiments 关47–52兴. In motor tasks of monkey, firing rate and synchrony are considered to encode behavioral events and cognitive events, respectively 关47兴. During visual tasks, rate and synchrony are suggested to encode task-related signals and expectation, respectively 关48兴. A change in synchrony may amplify behaviorally relevant signals in V4 of monkeys 关49兴. The synchrony is modified without a change in firing rate in some experiments 关47,49,51兴. The synchrony-dependent firing-rate signal is shown to propagate in iteratively constructed networks in vitro 关50兴.

d␮ ␣ 2␮ = − ␭␮ + h0 + , dt 2

共59兲

d␥ = − 2␭␥ + 2h1w␥S + 2␣2␥ + ␥I共t兲 + ␣2␮2 + ␤2 , dt 共60兲 1 dS S = − 关␥I共t兲 + ␣2␮2 + ␤2兴 + 关␥I共t兲SI共t兲 + cM ␣2␮2 + cA␤2兴 dt ␥ ␥ +

冉 冊

2h1w 共1 + ZS兲共1 − S兲, Z

共61兲

which are derived with the use of Eq. 共24兲. In order to numerically examine the possibility that input information is conveyed by ␥I共t兲 and SI共t兲, we first apply a fluctuation-driven input given by

␥I共t兲 = B⌰共t − 40兲⌰共60 − t兲 + Bb ,

共62兲

with B = 0.4, Bb = 0.1, I共t兲 = 0.1, and SI共t兲 = 0.1 for ␭ = 1.0, cA = 0.0, c M = 0.5, ␣ = 0.1, ␤ = 0.1, and N = 100. Time courses of ␮共t兲, ␥共t兲, S共t兲, and CV共t兲 are shown in Figs. 4共a兲–4共d兲: the chain curves in Figs. 4共a兲 and 4共b兲 express I共t兲 and ␥I共t兲, respectively. When the magnitude of ␥I共t兲 is increased at 40艋 t ⬍ 60, ␥共t兲 and CV共t兲 are much increased, while there is no changes in ␮共t兲 because it is decoupled from the rest of variables in Eq. 共18兲. S共t兲 is slightly modified only at t ⬃ 40 and t ⬃ 60 where the ␥I共t兲 is on and off. Next we apply a synchrony-driven input SI共t兲 given by SI共t兲 = C⌰共t − 40兲⌰共60 − t兲 + Cb ,

共63兲

with C = 0.4, Cb = 0.1, I共I兲 = 0.1, and ␥I共t兲 = 0.1. Figures 5共a兲–5共d兲 show time courses of ␮共t兲, ␥共t兲, S共t兲, and CV共t兲: the chain curves in Figs. 5共a兲 and 5共c兲 express I共t兲 and SI共t兲, respectively. An increase in synchrony-driven input at 40 艋 t ⬍ 60 induces a significant increase in S共t兲 and slight increases in ␥共t兲 and CV共t兲, but no changes in ␮共t兲.

IV. CONCLUSION

With the use of DSs and the AMM 关22,23兴, the effects of spatially correlated additive and multiplicative noise have been discussed on the synchrony and variability in the coupled Langevin model. Our calculations have shown the following: 共i兲 the synchrony is increased 共decreased兲 by the positive 共negative兲 correlation in additive and multiplicative

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PHYSICAL REVIEW E 78, 031110 共2008兲

noise 关Eq. 共30兲兴, 共ii兲 although an applied pulse input works to increase the synchrony for correlated multiplicative noise, it is possible to decrease the synchrony when correlated additive noise coexists with uncorrelated multiplicative one, 共iii兲 the local variability CV is almost independent of spatial correlations, while global variability DV is increased 共decreased兲 with increasing the positive 共negative兲 correlation, and 共iv兲 information may be carried by variance and synchrony in input signals. Item 共iv兲 is consistent with the results of Refs. 关44–46兴 and elucidates some phenomena observed in neuronal experiments 关47–52兴. Although we have applied the AMM to the Langevin model in this paper, it is possible to apply it to other types of stochastic neuronal models such as the FitzHugh and Hodgkin-Huxley models subjected to correlated additive and multiplicative noise, which is left as our future study. ACKNOWLEDGMENT

d具xi典 ␣2 = 具F共xi兲 + H共ui兲典 + 具G⬘共xi兲G共xi兲典, 2 dt d具xix j典 = 具xi关F共x j兲 + H共u j兲兴典 + 具x j关F共xi兲 + H共ui兲兴典 dt +

+ c M ␣2具G共xi兲G共x j兲典兴.

共A3兲

In the AMM 关22,23兴, the three quantities of ␮, ␥, and ␳ are defined by Eqs. 共10兲–共12兲. We use the expansion given by xi = ␮ + ␦xi

共A4兲

and the relations given by

APPENDIX: DERIVATION OF THE AMM EQUATIONS

The FPE for the Langevin equation given by Eqs. 共1兲–共3兲 in the Stratonovich representation is expressed by 关17,53兴

⳵ ⳵ p共兵xk其,t兲 = − 兺 兵关F共xi兲 + H共ui兲兴p共兵xk其,t兲其 ⳵t i ⳵xi ⳵2 ␥I 兵关␦ij + SI共1 − ␦ij兲兴p共兵xk其,t兲其 兺兺 2 i j ⳵ x i⳵ x j

+

␣2 兺 兺 关␦ij + cM共1 − ␦ij兲兴 2 i j

⫻

⳵ ⳵ G共xi兲 关G共x j兲p共兵xk其,t兲兴 , ⳵xi ⳵x j

冎

d具ri典 d␮ 1 , = 兺 dt N i dt

共A5兲

d具共␦ri兲2典 d␥ 1 , = 兺 dt dt N i

共A6兲

d具␦ri␦r j典 d␳ 1 . = 兺兺 dt dt N2 i j

共A7兲

For example, Eq. 共A5兲 for d␮ / dt is calculated as follows:

⳵2 ␤2 + 兺兺 兵关␦ij + cA共1 − ␦ij兲兴p共兵xk其,t兲其 2 i j ⳵ x i⳵ x j

再

␣2 关具xiG⬘共x j兲G共x j兲典 + 具x jG⬘共xi兲G共xi兲典兴 2

+ ␦ij关␥I + ␤2 + ␣2具G共xi兲2典兴 + 共1 − ␦ij兲关SI␥I + cA␤2

This work is partly supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

+

共A2兲

共A1兲

1 兺 具F共ri兲典 = f 0 + f 2␥ , N i

共A8兲

1 兺 具H共ui兲典 = h0 , N i

共A9兲

1 兺 具G⬘共ri兲G共ri兲典 = g0g1 + 3共g0g3 + g1g2兲␥ . 共A10兲 N i

where ui = 共w / Z兲兺 j共⫽i兲x j + I and H⬘共ui兲 is absorbed in a new definition of ␥I in its second term. Equations of motion for moments 具xi典 and 具xix j典 are derived with the use of the FPE 关21兴:

Equations 共A6兲 and 共A7兲 are calculated in a similar way. Then, we have obtained equations of motion for ␮共t兲, ␥共t兲, and ␳共t兲 given by Eqs. 共13兲–共15兲.

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