A Mean-Field Model for Extended Stochastic Systems with Distributed Time Delays

0.14 MB·English

by  Daniel Huber

A Mean-Field Model for Extended Stochastic Systems with
Distributed Time Delays
5
0
0 Daniel Huber and Lev Tsimring
2
Institute for Nonlinear Science

n University of California, San Diego
a
La Jolla, CA 92093-0402
J

6
2 ABSTRACT Jeong et al., 2002), and some significant changes of

Anetworkofnoisybistableelementswithglobaltime- the dynamical properties have been demonstrated
]
h delayed couplings is considered. A dichotomous mean (Nakamura et al., 1994; Bressloff and Coombes, 1998;
c fieldmodelhasrecentlybeendevelopeddescribingthe Choi and Huberman, 1985). The effects of uniform
e
collectivedynamicsinsuchsystemswithuniformtime time delays in globally coupled networks of phase
m
delays near the bifurcation points. Here the theory oscillators has been explored by Yeung and Strogatz
-
t is extended and applied to systems with nonuniform (1999). Tsimring and Pikovsky (2001) studied the
a
time delays. For strong enough couplings the systems dynamics of a single noise activated, bistable ele-
t
s exhibitdelay-independentstationarystatesanddelay- ment with time-delayed feedback. Combining the
.
t dependent oscillatory states. We find that the regions properties of these two systems Huber and Tsimring
a
m ofoscillatorystatesintheparameterspacearereduced (2003) studied the cooperative dynamics of anensem-
with increasing width of the time delay distribution ble of noisy bistable elements with delayed couplings
-
d function; that is, nonuniformity of the time delays in- and a mean field theory based on delay-differential
n creases the stability of the trivial equilibrium. How- master equations was developed. However, the the-
o
ever, for symmetric distribution functions the proper- ory assumed identical (i.e. uniform) time delays
c
[ ties of the oscillatory states depend only on the mean among all elements. Although for many systems
time delay. this approximation is justified (Salami et al., 2003;
1
Paulsson and Ehrenberg, 2001), most systems have
v KEY WORDS
distributed coupling delays. Thus, we study in this
0
Stochastic dynamics, mean-field dynamics, delay dif-
5 paper the generalized case of distributed time delays
ferential equation, self-organization
6 in a globally coupled network of bistable elements.
1
0
1 Introduction
5 2 The model
0
/ The understanding of the collective dynamics in ex-
t Ourgeneralizedmodelforthestudyofnoise-activated,
a tendedstochasticsystemswithlongrangeinteractions
collective dynamical phenomena in extended systems
m is relevant for many domains in physics, chemistry,
consists of N Langevin equations, each describing the
- biology and even social sciences. A popular and ef-
d overdamped noise driven motion of a particle in a
fective generic model for such systems is the globally
n bistablepotentialV = x2/2+x4/4,whosesymmetry
coupled bistable-element-network, whose dynamical −
o isdistortedbythe time-delayedcouplingstothe other
c properties have been studied in the absence (Zanette,
network elements,
: 1997) and presence (see Gammaitoni et al., 1998, and
v
references therein) of noise and whose relevance for
i N
X critical phenomena (Dawson, 1983), spin systems x˙ =x x3+ ε x (t τ )+√2Dξ(t), (1)
ar (Jung et al.,1992),neuralnetworks(Jung et al.,1992; i i− i N Xj=1 j − ij
Camperi and Wang, 1998; Koulakov et al., 2002), ge-
netic regulatory networks (Gardner et al., 2000) and where τij are the time delays depending on the two
decision making processes in social systems (Zanette, coupled elements i and j. The strength of the feed-
1997) has been pointed out. back is ε and D denotes the variance of the Gaussian
In recent years it has been realized that time fluctuations ξ(t), which are δ-correlatedand mutually
delays arising, for example, from the finite prop- independent ξ (t)ξ (t′) =δ(t t′)δ .
i j ij
h i −
agation speed of signals are ubiquitous in most WeuseanEulermethodtoexploremodel(1)nu-
physical and biological systems. The effects of merically and focus our interest on the collective dy-
the delays on the behavior of various dynam- namics of the bistable elements, i.e., on the dynamics
ical systems have been studied (Zanette, 2000; of the mean field, X =N−1 N .
i=1
P
0 laysbutalsoimplicatesthatthenumberofoperations,
trivial stationary state
whichhavetobecarriedouttostudysuchsystemsnu-
merically, can be reduced from (N2) to (N).
−0.5 O O
3 Dichotomous Theory
−1 oscillatory states
ε
In this section we derive the dichotomous theory for
σ=0
−1.5 σ=20 the description of globally coupled bistable-element-
σ=30 networks with distributed time delays.
σ=40 The dichotomous theory, valid in the limit of
−2
smallcouplingstrengthandsmallnoise,neglectsintra-
0 0.1 0.2 0.3 0.4 0.5
well fluctuations of x . Thus, each bistable element
D i
can be replaced by a discrete two-state system which
canonlytakethevaluess = 1. Thenthecollective
1,2
Figure 1. The critical coupling of the Hopf bifur- ±
dynamicsoftheentiresystemisdescribedbythemas-
cation as a function of the noise strength D for dif-
ter equations for the occupation probabilities of these
ferent σ of the Gaussian time delay distribution with
states n . This approach has been successfully used
1,2
τ¯ = 100. The markers and the solid lines depict the
in studies of stochastic and coherence resonance (e.g.
critical couplings resulting from model (1) and model
McNamara and Wiesenfeld, 1989; Gammaitoni et al.,
(2), respectively.
1998;Jung et al.,1992;Tsimring and Pikovsky,2001),
and Huber and Tsimring (2003) used it to study the
special case of a network with uniform time delays.
For ε = 0, the elements are decoupled from each Theyfoundthatwhileawayfromthetransitionpoints
other. They randomly and independently jump from the system dynamics is well described by a Gaussian
one potentialwellto the other. Therefore,in this case approximation, near the bifurcation points a descrip-
the mean field X = 0. For small ε, the mean field tion in terms of a dichotomous theory is more ade-
| |
remains zero. However, for a strong enough feedback quate.
thesystemundergoesorderingtransitionsanddemon- In order to apply the dichotomous theory to a
strates multistability. That is, for a strong enough network with distributed time delays, we coarse grain
positivecouplingthesystemsundergoesapitchforkbi- system(2). Thecoarsegrainingisaccomplishedasfol-
furcation and adopts a non-zerostationary mean field lows: The range of possible time delays is divided up
X >0,andtransitionstoavarietyofstableoscillatory in M intervals Ik k = 1,2,...,M . The size of the
{ }
mean field states via Hopf bifurcations, are observed intervals ∆k is chosen, so that the number of bistable
for strong enough positive and negative feedbacks. oscillatorsassociatedwith a delay fitting in a particu-
In the general case, model (1), in which the time lar interval, is for each interval the same m = N/M.
delaysdependonboththe“transmitting”andthe“re- In this way oscillator groups are formed whose mean
ceiving”element,cannotdirectlybedescribedinterms field can be expressed as,
of a mean field theory. However, the system becomes
1
mathematically tractable if we assume that the time Ω (t) x (t), (3)
k j
≡ m
delaydoesonlydependonthe“transmitting”elements τXj∈Ik
j,
k−1
where I [τ ,τ [, τ = ∆ and j =1...N.
N k ≡ k k+1 k l=1 l
x˙i =xi−x3i + Nε xj(t−τj)+√2Dξ(t). (2) meanAasnsudmthinegvtahraiatn∆ceko≪f tτ¯hP/eσt,imwehedreelaτ¯yadnidstσribauretitohne,
Xj=1
Eq. (2) can then be approximated by,
In order to check if such a simplification is jus-
M
tified, numerical simulations of model (1) and (2) are ε
x =x x3+ Ω (t τ )+√2Dξ(t). (4)
carried out and compared. In these simulations the i i− i M k − k
kX=1
distribution of the time delays is Gaussian, i.e., it is
fully determined by its mean τ¯ and variance σ. Fig. Thedynamicsofasingleelementx isdetermined
i
1, which compares the critical coupling strength of by the hopping rates p and p , i.e., by the proba-
12 21
the Hopf bifurcation for different σ, suggests that the bilities to hop over the potential barrier from s to s
1 2
above simplification is justified in order to study the and from s to s , respectively. In a globally coupled
2 1
stabilitypropertiesofabistable-element-networkwith system, in which the time delays depend only on the
time delays. transmitting elements,p andp areidenticalforall
12 21
Thissurprisingresultnotonlyrenderspossiblean elements and the master equations expressing the dy-
analyticaldescriptionofnetworkswithdistributedde- namics of Eq. (4) in terms of occupation probabilities
2
read, where
n˙1,k = p12n1,k+p21n2,k (5) 1 M 1 M
− I = sinωτ , I = cosωτ . (13)
n˙2,k = p12n1,k−p21n2,k. (6) s M kX=1 k c M Xk=1 k
Thehoppingprobabilitiesp aregivenbyKramers’
12,21 ForlargesystemsN ,thenumberofgroupsM
transition rate (Kramers, 1940) for the instantaneous →∞ →
and thus
potential well, which for our system in the limit of ∞
small noise D and coupling strength ε reads (cf. ∞ ∞
Tsimring and Pikovsky, 2001), I = P(τ)sinωτdτ, I = P(τ)cosωτdτ, (14)
s c
Z Z
√2 3α 1 4α 0 0
p12,21 = ∓ exp ∓ , (7)
2π (cid:18)− 4D (cid:19) where P(τ) is the time delay distribution function.
We can express the time delay distribution func-
where α=(ε/M) M Ω (t τ ).
k=1 k − k tion interms ofcumulantmoments κn (Van Kampen,
For large oscPillator groups (m → ∞), Ωk = 2003) and solve the integrals in (14):
n s + n s = n n and n + n = 1
1,k 1 2,k 2 2,k 1,k 1,k 2,k
−
holds. With these terms, we can find the following
I =sin(g )exp(g ), I =cos(g )exp(g ), (15)
s 1 2 c 1 2
set of equations:
where
Ω˙ (t)=p p (p +p )Ω (t). (8)
k 12 21 21 12 k
− −
∞ (iω)2m+1
The Jacobianmatrix of this system is given through, g = κ , (16)
1 2m+1
i(2m+1)!
mX=0
a1+b a2 ... aM ∞ (iω)2m
J =c a1 a2+b ... aM , (9) g2 = κ2m. (17)
............................ mX=1 (2m)!
 a1 a2 ... aM +b
  Consequently,
where c = −√2exp(−1/4D)/(4MπD), ak = ε(3D− Is =tan(g1). (18)
4)exp( λτ ) and b = 4MD. With this Jacobian the I
k c
−
characteristic equation, determining the stability of
Since for symmetric distribution functions all odd cu-
the trivial equilibrium X =0, becomes
mulant moments except the first one κ =τ¯ are zero,
1
I /I = tanωτ¯ holds. That is, in the case of a sym-
M s c
(bc λ)M−1cb+ aj λ=0. (10) metric distribution of the time delays, the frequencies
− Xj=1 − of the unstable modes in Eq. (12) depend only on the
    mean time delay.
The trivial equilibrium loses its stability and under- Let us now determine the critical coupling of the
goes a pitchfork bifurcation, describing the transition Hopf bifurcation. For large time delays τ¯ τ (τ
k k
≫
toasteadynonzeromeanfield,whentherealsolutions is the inverse Kramers escape rate from one well into
of the characteristic equation (i.e. the eigenvalues of the other) the low-order solutions of the transcendent
the Jacobian) become positive. Thus setting λ = 0 equation (12) yield frequencies ω 1. Thus the real
≪
and solving Eq. (10) for ε yields the critical coupling partofequation(10)canbelinearizednearω =0and
for the pitchfork instability, the criticalcoupling ofthe Hopf bifurcationbecomes,
4D
4Dπω
ε = . (11)
p ε = .
4−3D H (3D−4) N1√2exp(−1/4D)Is− 1− N1 πωIc
A Hopf bifurcation indicating the transition to an os- (cid:0) (cid:2) (cid:3) (19(cid:1))
cillatory mean field state occurs when the real part of Then, for large systems N the critical coupling
→ ∞
the complex eigenvalues becomes positive. Therefore, is,
the properties of the corresponding instabilities (i.e. 4D
ε = , (20)
H
frequencies and coupling strengths at the bifurcation (4 3D)I
c
−
points) can be found by substituting λ = µ+iω into
with I = 3sin(ωτ¯)sin(5ωσ/3)/(5ωσ) and I =
Eq. (10),separatingrealandimaginarypartsandset- c c
cos(ωτ¯)exp( ω2σ2/2) for a uniform and a Gaussian
ting µ=0. For the frequencies of the unstable modes −
distribution, respectively.
we find,
Eq. (12) and(20)havea multiplicity ofsolutions
√2 I indicating thatmultistability occursinthe systembe-
s
ωτ¯= exp( 1/4D)τ¯ , (12)
yond a certain coupling strength.
− π − I
c
3
1 0.55
oscillatory states σ=0
non−trivial stationary states σ=20 0.5
0.5 σ=30
σ=40 0.45
ε 0 trivial stationary state 〈τ〉f [1/ ] 0.03.54
theory
−0.5 0.3 Langevin, σ=20
oscillatory states Langevin, σ=30
0.25
Langevin, σ=40
−1
0.1 0.2 0.3 0.4 0.5 0.2
D 0 0.1 0.2 0.3 0.4 0.5
D
Figure 2. Phase diagram of the globally coupled
bistable-element-network with uniformly distributed Figure 3. The frequencies of the unstable modes
time delays derived from the theoretical model (solid at the bifurcation points resulting from the Langevin
lines) and numerical simulations of the Langevin model(markers)andthethedichotomousmodel(solid
model(markers). The phase diagramis shownfor dif- line). For symmetric distributions the frequencies de-
ferent standard deviations σ of the delay distribution pend only on the mean time delay (see Eq. 12 and
function. The mean time delay is τ¯=100. 18).
4 Results are ordered as follows, 0 < ε < ε1 < ε2 ...
st H+ H+
If the feedback is negative, the system has oscilla-
Eq. (11), (12) and (20) are used to determine the tory solutions with periods fl (2l + 1)/(2τ¯) for
≈
phase diagram and the frequencies of the unstable ε < εlH− {l = 0,1,...}, where 0 > ε0H− > ε1H−...
oscillatory modes f = ω/(2π) of a bistable-element- In the aboveannotationa (+/-)index means that the
networkwith uniformly distributed time delays1. The corresponding value is associated with a negative and
theoreticalpredictionsareverifiedwithnumericalsim- positive feedback, respectively.
ulations of the Langevin model (1). The results are
shown in Fig. 2 and Fig. 3.
5 Summary and conclusions
Thefiguresshowthatnearthe bifurcationpoints
the predictionsby the dichotomoustheory arereason-
We generalized a dichotomous theory based on delay-
ably good for small noise (D .0.3) and consequently
differential master-equations to account for the dy-
in this regime the Langevin models (1) and (2) are
namics of globally coupled networks of bistable ele-
dynamically equivalent.
ments with nonuniform time-delays. As in the case of
Fig. 2 also shows that the regions in the phase
uniform time delays these systems possess a nonzero
space where mean field oscillations occur are reduced
stationarymeanfieldforastrongenoughpositivefeed-
with increasing width σ of the time delay distribution
backwhosepropertiesaretimedelayindependentand
function, meaning that nonuniformity of the time de-
a multiplicity of time delay dependent stable oscilla-
lays inhibits the occurence of Hopf bifurctions, i.e.,
tory states for both positive and negative feedback.
increases the stability of the trivial equilibrium.
For symmetric time delay distributions the fre-
In Fig. 4 the bifurcation diagrams including quencies of the oscillations depend only on the mean
higher order solutions of Eq. (12) and (20) are pre- time delay. However, the critical couplings of the cor-
sented. This Figure shows that for a strong enough respondingHopfbifurcationsdependalsoonthewidth
positive (ε = εp) coupling the trivial equilibrium of the time delay distribution; that is, the critical
loses its stability via a pitchfork bifurcation while coupling strengths and consequently the stability of
for a strong enough negative coupling (ε = ε1H−) a the trivial equilibrium are increased for broadly dis-
Hopf bifurcation determined by the primary solution tributed time delays. This may be important for time
of Eq. (12) and (20) occurs. The higher order so- delay systems such as neural networks and genetic
lutions of these equations provide the multistability regulatory networks, since the degree of time delay
of the system. For a positive feedback several oscil- nonuniformity, which is often related to the diversity
latory states with frequencies fk k/τ¯ are observed in the connectivity of the underlying network, affects
≈
for ε > εkH+ {k = 1,2,...}. The transition points the accessibility of the nontrivial dynamical states.
1Thisshouldnotbeconfusedwithuniformtimedelays,which The bifurcations of the trivial equilibrium are
meansthatthedelayforeachcouplingisthesame. well described by the dichotomous theory in the limit
4
σ=0 σ=10
2 2
pitchfork
1 1
ε 0 ε 0
−1 −1
−2 −2
0.2 0.4 0.6 0.2 0.4 0.6
D
σ=40
2 8
1 6
〉]
τ
ε 0 〈1/ 4
f [
−1 2
−2 0
0.2 0.4 0.6 0.2 0.4 0.6
D D
Figure 4. Upper panels and lower left panel: Phase diagrams of systems with uniformly distributed time delays,
where τ¯= 100 and σ = 0, 10, 40. The dotted line depicts the critical coupling of the pitchfork bifurcation and the
other lines depict those of the primary Hopf bifurcation as well as some higher order solutions (i.e. solution 1-16)of
Eq. (12)and (20). Lowerrightpanel: The frequencies ofthe correspondingunstable modes whichdo notdepend on
σ, but slightly vary with the noise strength D.
5
of small noise and coupling strength. Far away L. S. Tsimring and A. Pikovsky,Phys. Rev. Lett. 87,
from the transition points a theoretical description 250602 (2001).
of the mean field dynamics can be found using a
D. Huber and L. S. Tsimring, Phys. Rev. Lett. 91,
Gaussian approximation (Desai and Zwanzig, 1978;
260601 (2003).
Huber and Tsimring, 2003). However, a theoretical
approach for the description of the dynamics in the
M. Salami, C. Itami, T. Tsumoto, and F. Kimura,
regime of strong noise near the bifurcation points is PNAS 100, 6174 (2003).
still lacking.
This paper discusses the dynamics of globally J. Paulsson and M. Ehrenberg, Q. Rev. Biophys. 34,
coupled systems with time delays. It is assumed that 1 (2001).
all elements are coupled with uniform (i.e. identical)
B. McNamara and K. Wiesenfeld, Phys. Rev. A 39,
strength ε. However,many networks have sparse cou-
4854 (1989).
plings and nonuniform coupling strength, which may
endow the system with a complexer dynamics. This H. Kramers, Physica (Utrecht) 7, 284 (1940).
issue will be discussed in an upcoming paper.
N. Van Kampen, Stochastic Processes in Physics and
Chemistry (Elesiver Scinece B.V., Amsterdam, The
Acknowledgments
Netherlands, 2003).
This work was supported by the Swiss National Sci- R. C. Desai and R. Zwanzig, J. Stat. Phys. 19, 1
ence Foundation (D.H.) and by the U.S. Department
(1978).
ofEnergy,OfficeofBasicEnergySciencesundergrant
DE-FG-03-96ER14592(L.T.).
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