Stationary and dynamical properties of finite $N$-unit Langevin models subjected to multiplicative noises

Hideo Hasegawa (Tokyo Gakugei Univ.)

REVIEW

Stationary and dynamical properties of finite $N$-unit Langevin models subjected to multiplicative noises

Disordered Systems and Neural Networks · Submitted 6 Mar 2006 · 29 references

Pith's verdict: Solid incremental extension of the AMM with a useful diffusive/sigmoid comparison; the abstract overstates the agreement and should be qualified.

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arXiv IDcond-mat/0603115 (version 6)

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CitationHideo Hasegawa (Tokyo Gakugei Univ.). "Stationary and dynamical properties of finite $N$-unit Langevin models subjected to multiplicative noises." arXiv:cond-mat/0603115 (2006). https://pith.science/paper/LLCZIDBF

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Editor's note

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Quick take: this is a competent continuation of Hasegawa's AMM program, not a breakthrough. The genuinely new pieces are the sigmoid-coupling analysis, closed-form stationary expressions for μ, γ, ρ, and the asymptotic global distributions (stable laws for ν<1). The linear diffusive case is essentially exact for the moments, so the formulas there are solid. The soft spot is the abstract: it claims good agreement with direct simulations without noting that the AMM is a weak-noise approximation, and the paper's own Fig. 2(b) shows the AMM diverging at α≈0.71 for sigmoid coupling while DS persists to α≈0.77. That is a real limitation and should be front-and-center.

The moment equations (20)-(22) are derived cleanly from a second-moment truncation. For F=-λx, G=x with diffusive coupling, the closure is exact because all needed expectations are second order; the divergence at α=1 is the genuine divergence of the stationary second moment, not an artifact of truncation. The DS staying finite there is a finite-sample effect that the paper does not discuss. For sigmoid coupling, the truncation is uncontrolled—no small parameter—and the paper is honest about this in Sec. 2.5, but the abstract does not carry that honesty through.

The FPE section is a nice addition: it recovers q-Gaussian local distributions and shows the global variable has Gaussian (ν>1) or power-law (ν<1) asymptotics, consistent with earlier results. The diffusive vs sigmoid comparison (ρ depends on K but not J, γ depends on J but not K) is clearly laid out and is the most useful takeaway for modelers.

It is a self-contained continuation with heavy but appropriate self-citation. No code or data deposited, so reproducibility rests on the explicit equations and described simulations. That is acceptable for this kind of paper, though error tables would have made the reliability claims easier to check.

Who gets value: people using moment closures for stochastic networks or studying multiplicative-noise effects on synchronization. It deserves a serious referee, but the abstract needs qualification, and the divergence-vs-DS discrepancy should be addressed. My recommendation: send to peer review, ask for a revised abstract and a quantitative statement of the regime where AMM tracks DS.

Referee report

Significance

If the claimed agreement holds within a clearly delimited parameter region, the AMM is a useful reduction: it replaces N coupled stochastic differential equations with three ODEs, making dynamical and stationary studies of finite-N multiplicative-noise systems much cheaper. The paper provides explicit closed-form stationary formulas for the linear F, G case, a clear derivation of the moment hierarchy and its divergence condition in Eq. (45), and a range of systematic comparisons with direct simulations. The authors are also honest about the central failure mode, noting in Section 3.1.2 that the AMM diverges where simulations remain finite. The work is a continuation of the author's prior AMM papers, and its novelty lies in the extension to multiplicative noises and in the detailed diffusive-versus-sigmoid comparison. The main weakness is that the advertised domain of validity is stated more broadly in the abstract than the evidence supports, and no quantitative error measures are provided.

Major objections (3)

[Sec. 2.5, Eq. (45)] The defense of the second-moment closure is based on the observation that for F(x)=-lambda x and G(x)=x, the equation for <x^k> depends only on lower moments, so even a diverging <x^4> does not feed back into <x^2>. This argument does not apply to the sigmoid-coupling case or to general F and G, because the Taylor expansions leading to Eqs. (20)-(22) contain coefficients such as h2, f2, and g2 that multiply gamma, while the neglected terms involve third- and higher-order moments of delta x. The manuscript should either provide a bound on these neglected moments or explicitly restrict the central claim to parameter ranges where they are negligible; as written, the justification in Section 2.5 covers only the linear diffusive model.
[Sec. 3.1.2, Fig. 2(b)] The paper reports that for sigmoid coupling K=0.5 the AMM diverges at alpha=0.71 while direct simulations remain finite up to alpha=0.77, and for diffusive coupling the AMM diverges at alpha=1 where DS remain finite. These are discrepancies inside the parameter ranges presented in the paper, and they directly contradict the abstract's unqualified statement that AMM results are in good agreement with direct simulations. The manuscript should state the weak-noise domain of validity explicitly, and should quantify the agreement in the claimed regime, for example by reporting relative errors or a region plot in the (alpha, coupling) plane.
[Sec. 3.2, Figs. 4-6] The dynamical comparisons between AMM and direct simulations are presented only as overlapping curves; no quantitative error metric is given. Since the paper proposes the AMM as a low-cost replacement for direct simulations, the central claim would be substantially strengthened by reporting, for each displayed case, a time-averaged relative error or L2 difference between AMM and DS, together with the number of trials and the standard error of the DS averages. This is particularly important because the agreement visibly degrades for combined noises in Fig. 5(c)-(d).

Minor objections (6)

[Sec. 4, Eq. (52)] In the definition of Y(x), the integration variable is written as r, but the integrand depends on x; this should be x (or the dummy variable should be renamed consistently).
[Sec. 3.2 and Fig. 6 caption] The text introducing the sinusoidal input says 'when the external input given by Eq. (46) is applied', but Eq. (46) is the pulse input and the sinusoidal input is Eq. (47); the cross-reference should be corrected.
[Fig. 2(a) caption] The caption lists 'circles, triangles and square' for DS results but then mentions only J=0.0 and J=0.5; please specify which symbol corresponds to which coupling strength and which variable (gamma or rho).
[Sec. 4, Eq. (53)] The word 'Stratonivich' is a typo and should read 'Stratonovich'.
[Fig. 8 caption and Sec. 4] The figure caption refers to Eqs. (67), (68), and (70), while the text refers to Eqs. (61), (62), and (64) for the same analytic expression; the equation references should be aligned.
[Sec. 2.1, Eqs. (20)-(22)] The derivation of Eqs. (20)-(22) from Eqs. (12)-(15) is not shown in detail; including the key intermediate expansion or a precise pointer to the relevant equations in Refs. [4] and [10] would make the paper more self-contained.

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The paper

The paper claims

Three differential equations reproduce finite-N Langevin dynamics under multiplicative noise.

The reading

This paper claims that a coupled system of N Langevin units driven by multiplicative (state-dependent) noise can be summarized, in the weak-noise regime, by just three ordinary differential equations for the mean, the local variance, and the global variance. The reduction, called the augmented moment method, expands each variable around the population mean and keeps only second-order fluctuations. Against direct numerical simulations of the full N-unit system, the paper reports good agreement for stationary variances, responses to pulse and sinusoidal inputs, and the synchronization ratio, for both diffusive and sigmoid couplings. If this holds, expensive N-unit simulations can be replaced by a three-equation model, making the mathematics of multiplicative noise far more tractable for finite networks.

What carries the argument

The augmented moment method (AMM): expand $x_i = μ + δx_i$ in the exact moment equations and retain terms through $⟨δx_i δx_j⟩$, closing the moment hierarchy at second order. This produces three coupled ODEs, Eqs. (20)-(22), for the mean $μ$, the average local variance $γ = (1/N) Σ_i ⟨(x_i-μ)^2⟩$, and the global variance $ρ = ⟨(X-μ)^2⟩$. The method carries the argument because it converts N coupled stochastic equations into a low-dimensional deterministic system that can be integrated cheaply and analyzed through Jacobian eigenvalues; this particular closure is what the paper tests against direct simulation throughout.

Extended reading notes

Core claim

The central claim is that the augmented moment method, a second-moment closure that turns the N-dimensional Langevin equations into three deterministic ODEs for the mean $μ$, the average local variance $γ$, and the global variance $ρ$, correctly describes stationary and dynamical properties of finite-N Langevin systems under multiplicative noise. For the linear model with $F(x) = -λx$ and $G(x) = x$, the closed equations are (20)-(22), specialized to diffusive couplings in (23)-(25) and sigmoid couplings in (32)-(34). With weak noise, these ODE solutions match direct simulations for stationary $γ$ and $ρ$ as functions of N, coupling strength J or K, and noise strength $α$, as well as for time-dependent responses to pulse and sinusoidal inputs. The paper also derives stationary distributions from the Fokker-Planck equation, finding local q-Gaussian distributions with power-law tails and a global distribution that crosses from Gaussian to power-law depending on $ν = λ/α^2$.

Load-bearing premise

The paper's central approximation is that all third- and higher-order correlations of the fluctuations around the mean can be dropped when closing the moment equations, so the method works only when those higher moments are negligible.

Editorial extensions

If this is right

For weak multiplicative noise, the stationary and time-dependent behavior of finite-N coupled Langevin systems can be computed by integrating three ODEs rather than simulating N stochastic units.
Local stationary distributions under multiplicative noise are generally non-Gaussian q-Gaussians with power-law tails of exponent $δ = 2λ/α^2 + 1$; the global distribution is Gaussian for $ν = λ/α^2 > 1$ and power-law for $0 < ν < 1$.
Diffusive coupling reduces the local variance $γ$ but leaves the global variance $ρ$ unchanged, whereas sigmoid coupling increases $ρ$ while leaving $γ$ nearly unchanged, so the synchronization ratio rises with either coupling type.
A pulse input changes not only the mean but also the variances, because multiplicative noise couples $μ$ into the variance equations through the $α^2 μ^2$ terms; this coupling is absent for additive noise alone.
The AMM's validity is confined to weak noise: it diverges at $α = 1$ where direct simulations stay finite, and higher even moments can diverge for $α^2 > 2λ/k$ even when the second moment is finite.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension is to apply the same three-moment closure to other smooth functions $G(x)$, such as $G(x) = √(1+x^2)$ or $G(x) = 1/(1+x^2)$, and compare the ODE predictions to direct simulations; the closure is not specific to linear drift.
Because the AMM fails near $α = 1$ while simulations remain finite, the discrepancy is likely due to neglected fourth-order cumulants; adding a fourth-moment correction term to Eqs. (20)-(22) is a concrete next step that could extend the method beyond the weak-noise regime.
The result that the global distribution $P(X)$ develops a power-law tail for $0 < ν < 1$ with an exponent independent of N suggests that collective fluctuations in large networks can remain non-Gaussian under multiplicative noise, an implication that could be probed in coupled neuron or opinion models.

Original abstract

We have studied the finite $N$-unit Langevin model subjected to multiplicative noises, by using the augmented moment method (AMM), as a continuation of our previous paper [H. Hasegawa, J. Phys. Soc. Jpn. {\bf 75} (2006) 033001]. Effects of couplings on stationary and dynamical properties of the model have been investigated. The difference and similarity between the results of diffusive and sigmoid couplings are studied in details. Time dependences of average and fluctuations in local and global variables calculated by the AMM are in good agreement with those of direct simulations (DSs). We also discuss stationary distributions of local and global variables with the use of the Fokker-Planck equation (FPE) method and DSs. It is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.91 · mentions code or data availability 0.02.

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arXiv listing abstract v6. Results and the main claim are underlined; show every role.

We have studied the finite $N$-unit Langevin model subjected to multiplicative noises, by using the augmented moment method (AMM), as a continuation of our previous paper [H. Hasegawa, J. Phys. Soc. Jpn. {\bf 75} (2006) 033001]. Effects of couplings on stationary and dynamical properties of the model have been investigated. The difference and similarity between the results of diffusive and sigmoid couplings are studied in details. Time dependences of average and fluctuations in local and global variables calculated by the AMM are in good agreement with those of direct simulations (DSs). We also discuss stationary distributions of local and global variables with the use of the Fokker-Planck equation (FPE) method and DSs. It is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We have studied the finite $N$-unit Langevin model subjected to multiplicative noises, by using the augmented moment method (AMM), as a continuation of our previous paper [H. Hasegawa, J. Phys. Soc.	contribution 0.88	0.89	0.04	asserted 0.96
s1	Jpn. {\bf 75} (2006) 033001].	background 0.98	0.05	0.02	unclear 0.97
s2	Effects of couplings on stationary and dynamical properties of the model have been investigated.	contribution 0.98	0.91	0.13	asserted 0.89
s3	The difference and similarity between the results of diffusive and sigmoid couplings are studied in details.	contribution 0.95	0.92	0.10	asserted 0.95
s4	Time dependences of average and fluctuations in local and global variables calculated by the AMM are in good agreement with those of direct simulations (DSs).	result 1.00	0.95	0.20	asserted 1.00
s5	We also discuss stationary distributions of local and global variables with the use of the Fokker-Planck equation (FPE) method and DSs.	method 0.82	0.91	0.09	asserted 0.85
s6	It is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account.	result 0.97	0.92	0.51	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from simulation or computation 0.86.

What was examined, and what was not

sources read

arXiv listing abstract v6

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** (Color online) (a) The N dependences of the stationary γ and ρ for various diffusive couplings (DC): circles, triangles and square denote results of DS for J = 0.0, J = 0.2 and J = 0.5, respectively, and solid curves express those of the AMM. (b) The N dependence of the stationary γ and ρ for various sigmoid couplings (SC) with H(x) = x: circles, triangles and square denote results of DS for K = 0.0, K = 0.2 and K = 0.5, respectively, solid curves express those of the AMM (λ = 1.0, α = 0.5 and β = 1.0). Dashed curves are drawn only for a guide of the eye

**Figure 2.** (Color online) (a) The α dependences of the stationary γ (solid curves) and ρ (chain curves) for the diffusive couplings (DC): circles, triangles and square denote results of DS for J = 0.0 and J = 0.5, respectively, and solid and chain curves express those of the AMM. (b) The α dependence of the stationary γ (solid curves) and ρ (chain curves) for sigmoid couplings (SC) of K = 0.5 with H(x) = x: squares denote results of DS, and solid and chain curves express those of the AMM. (N = 10, λ = 1.0 and β = 1.0). Dashed curves are drawn only for a guide of the eye

**Figure 3.** (Color online) (a) The J dependences of the stationary γ (solid curves) and ρ (chain curves) for the diffusive couplings (DC): circles denote results of DS, and solid and chain curves express those of the AMM. (b) The K dependence of the stationary γ (solid curves) and ρ (chain curves) for sigmoid couplings (SC) with H(x) = x: squares denote results of DS, and solid and chain curves express those of the AMM. (N = 10, λ = 1.0, α = 0.5 and β = 1.0). Dashed curves are drawn only for a guide of the eye

**Figure 4.** (Color online) Responses of (a) µ(t), (b) γ(t) and (c) ρ(t) to the pulse input with J = 0.0, N = 10, λ = 1.0, and β = 1.0: solid and dashed curve denote results of AMM and DS, respectively, for α = 0.5, and chain curves express that of AMM for α = 0.0

**Figure 8.** (Color online) Distributions P(X) of global variable X calculated by direct simulation (DS) for N = 1 (dashed curves), N = 10 (solid curves) and N = 100 (chain curves) with (a) I = 0, (b) I = 1.0 and (c) I = 2.0 (λ = 1.0, α = 1.0 and β = 0.5). Results calculated with the use of Eqs. (67), (68) and (70) are plotted in (a) with a shift of X = −2 for a clarity of the figure

**Figure 9.** (Color online) Distributions of p(x) and P(X) of local and global variables, respectively, with (a) I = 0.0 and (b) I = 1.0 for diffusive coupling (DC) with J = 0.0 (dashed curves) and J = 1.0 (solid curves), and those with (c) I = 0.0 and I = 1.0 for sigmoid coupling (SC) with K = 0.0 (dashed curves) and K = 0.5 (solid curves). (N = 10, λ = 1.0, α = 0.5 and β = 1.0). 19

Showing 6 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603115 by the authors.

References

Works cited (29)

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B. Lindner, J. Garc ´ia-Ojalvo, A. Neiman, and L. Schimansky-Gel ´ier, Physics Report 392 (2004) 321

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H. Hasegawa, J. Phys. Soc. Jpn. 75 (2006) 033001; the third term of Eq. (17) in this reference should be ( w/Z) ∑ k[⟨xixk⟩ + ⟨xjxk⟩ − 2⟨xixj⟩]

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M. A. Mu˜ noz: in ’Advances in Condensed Matter and Statistical Mechanics ’, edit by E. Korutcheva and R. Cuerno (Nova Science Publishers, New York, 2004), p. 34

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R. Kawai, X. Sailer, L. Schimansky-Geier, and C. Van den Broeck : Phys. Rev E 69 (2004) 051104

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M. A. Mu˜ noz, F. Colaiori, and C. Castellano: Phys. Rev. E 72 (2005) 056102

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L. H. A. Monteiro, M. A. Bussab, J. G. C. Berlinck, J. Theor. Bio l. 219 (2002) 83

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[24]

H. Haken: Advanced Synergetics (Springer-Verlag, Berlin, 1983); the multiplicative- noise term in FPE given by Eq. (7) is derived from eqs. (4.2.24) and (4.3 .18) in this reference with gkm = α G(xk) δkm

View on Pith 1983
[25]

First we consider the quantity given by P (t) = N −2∑ ij < [xi(t)−xj(t)]2 >= 2[γ(t)− ρ(t)], which is zero for the completely synchronous state and which is 2( 1 − 1/N)γ (≡ P0) for the completely asynchronous states. The synchronization r atio given by Eq. (41) is deﬁned by S(t) (= 1 − P (t)/P0(t)), which is zero (one) for completely asynchronous (synchron...

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C. Tsallis: J. Stat. Phys. 52 (1988) 479

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S. Abe and A. K. Rajagopal: J. Phys. A: Math. Gen. 33 (2000) 8723

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H. Hasegawa: cond-mat/0601358. 17 Figure 1: (Color online) (a) The N dependences of the stationary γ and ρ for various diﬀusive couplings (DC): circles, triangles and square denote result s of DS for J = 0 .0, J = 0.2 and J = 0.5, respectively, and solid curves express those of the AMM. (b) Th e N dependence of the stationary γ and ρ for various sigmoid c...

View on Pith Pith review arXiv

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

B. Lindner, J. Garc ´ia-Ojalvo, A. Neiman, and L. Schimansky-Gel ´ier, Physics Report 392 (2004) 321

View on Pith 2004

[2] [2]

H. Risken: The Fokker-Planck Equation: Methods of Solution and Applic ations, Springer Series in Synergetics, Vol. 18 (Springer Verlag, Berlin, 199 2)

View on Pith

[3] [3]

R. Rodriguez and H. C. Tuckwell, Phys. Rev. E 54 (1996) 5585

View on Pith 1996

[4] [4]

H. Hasegawa: Phys. Rev E 67 (2003) 041903

View on Pith 2003

[5] [5]

H. Hasegawa, Phys. Rev E 68 (2003) 041909

View on Pith 2003

[6] [6]

H. Hasegawa, Phys. Rev. E 70 (2004) 021911

View on Pith 2004

[7] [7]

H. Hasegawa, Phys. Rev. E 70 (2004) 021912

View on Pith 2004

[8] [8]

H. Hasegawa, Phys. Rev. E 70 (2004) 066107

View on Pith 2004

[9] [9]

H. Hasegawa, Phys. Rev. E 72 (2005) 056139

View on Pith 2005

[10] [10]

H. Hasegawa, J. Phys. Soc. Jpn. 75 (2006) 033001; the third term of Eq. (17) in this reference should be ( w/Z) ∑ k[⟨xixk⟩ + ⟨xjxk⟩ − 2⟨xixj⟩]

View on Pith 2006

[11] [11]

M. A. Mu˜ noz: in ’Advances in Condensed Matter and Statistical Mechanics ’, edit by E. Korutcheva and R. Cuerno (Nova Science Publishers, New York, 2004), p. 34

View on Pith 2004

[12] [12]

A. Schenzle and H. Brandt, Phys. Rev. A 20 (1979) 1628

View on Pith 1979

[13] [13]

H. Sakaguchi: J. Phys. Soc. Jpn. 70 (2001) 3247

View on Pith 2001

[14] [14]

C. Anteneodo and C. Tsallis: J. Math. Phys. 44 (2003) 5194

View on Pith 2003

[15] [15]

C. Anteneodo and R. Riera, Phys. Rev. E 72 (2005) 026106

View on Pith 2005

[16] [16]

C. Van den Broeck, J. M. R. Parrondo, and R. Toral, Phys. Rev . Lett. 73 (1994) 3395

View on Pith 1994

[17] [17]

C. Van den Broeck, J. M. R. Parrondo, R. Toral, and R. Kawai: P hys. Rev. E 55 (1997) 4084. 16

View on Pith 1997

[18] [18]

T. Birner, K. Lippert, R. M¨ uller, A. K¨ uhnel, and U. Behn, Phys . Rev E 65 (2002) 046110

View on Pith 2002

[19] [19]

R. Kawai, X. Sailer, L. Schimansky-Geier, and C. Van den Broeck : Phys. Rev E 69 (2004) 051104

View on Pith 2004

[20] [20]

M. A. Mu˜ noz, F. Colaiori, and C. Castellano: Phys. Rev. E 72 (2005) 056102

View on Pith 2005

[21] [21]

G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett. 84 (2000) 2770

View on Pith 2000

[22] [22]

H. Hasegawa: Physica A 365 (2006) 383

View on Pith 2006

[23] [23]

L. H. A. Monteiro, M. A. Bussab, J. G. C. Berlinck, J. Theor. Bio l. 219 (2002) 83

View on Pith 2002

[24] [24]

H. Haken: Advanced Synergetics (Springer-Verlag, Berlin, 1983); the multiplicative- noise term in FPE given by Eq. (7) is derived from eqs. (4.2.24) and (4.3 .18) in this reference with gkm = α G(xk) δkm

View on Pith 1983

[25] [25]

First we consider the quantity given by P (t) = N −2∑ ij < [xi(t)−xj(t)]2 >= 2[γ(t)− ρ(t)], which is zero for the completely synchronous state and which is 2( 1 − 1/N)γ (≡ P0) for the completely asynchronous states. The synchronization r atio given by Eq. (41) is deﬁned by S(t) (= 1 − P (t)/P0(t)), which is zero (one) for completely asynchronous (synchron...

View on Pith

[26] [26]

C. Tsallis: J. Stat. Phys. 52 (1988) 479

View on Pith 1988

[27] [27]

C. Tsallis, R. S. Mendes, and A. R. Plastino: Physica A 261 (1998) 534

View on Pith 1998

[28] [28]

S. Abe and A. K. Rajagopal: J. Phys. A: Math. Gen. 33 (2000) 8723

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H. Hasegawa: cond-mat/0601358. 17 Figure 1: (Color online) (a) The N dependences of the stationary γ and ρ for various diﬀusive couplings (DC): circles, triangles and square denote result s of DS for J = 0 .0, J = 0.2 and J = 0.5, respectively, and solid curves express those of the AMM. (b) Th e N dependence of the stationary γ and ρ for various sigmoid c...

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