EPJ manusc ip No.
(will be inse ed by he edi o )
S ochas ic esonance in ini e a ays o bis able
elemen s wi h local coupling
Manuel Mo illo1, Jos´e G´omez O d´o˜nez1, Jos´e M. Casado1, Jes´us Casado-Pascual1and Da id
Cube o2 a
1F´ısica Te´o ica. Facul ad de F´ısica. Uni e sidad de Se illa. Apa ado de Co eos 1065. Se illa 41080.
Spain
2Depa amen o de F´ısica Aplicada I, EUP, Uni e sidad de Se illa, Calle Vi gen de ´
A ica 7, 41011
Se illa, Spain
Abs ac . In his a icle, we in es iga e he s ochas ic esonance (SR) e ec on
a ini e a ay o noisy bis able sys ems wi h nea es -neighbo coupling d i en
by a weak ime-pe iodic d i ing o ce. The a ay is cha ac e ized by a collec i e
a iable. By means o nume ical simula ions, he signal- o-noise a io (SNR) and
he gain a e es ima ed as unc ions o he noise and he in e ac ion coupling
s eng h. A s ong enhancemen o he SR phenomenon o his collec i e a iable
in compa ison wi h SR in single uni bis able sys ems is obse ed. Gains la ge
han uni y a e ob ained o some pa ame e alues, indica ing ha he sys em is
ope a ing in a non-linea egime, albei he smallness o he d i ing ampli ude.
The la ge alues o he SNR obse ed a e basically due o he ac ha he ou pu
luc ua ions a e small and sho li ed, in compa ison wi h hei ypical alues in a
linea egime. A non-mono onic beha io o he SNR wi h he coupling s eng h
is also ob ained.
1 In oduc ion
The s udy o he esponse o complex sys ems o med by coupled nonlinea noisy elemen s
subjec o he ac ion o a ime-pe iodic o ce b ings in o he pic u e a new ing edien wi h
espec o he esponse o a single uni : he e ec o he in e ac ions. The in e play o noise,
in insic nonlinea dynamics, d i ing o ces and in e ac ions ende he esponse a e y ich
s ochas ic p ocess. Ce ain aspec s o he esponse a e associa ed wi h wha is called s ochas ic
esonance (SR) [1,2], a phenomenon o in e es o scien i ic ields o he han Physics (as
discussed, o ins ance, in [3]). SR is cha ac e ized by he non-mono onic beha io o some
quan i ie s wi h he noise s eng h. An enhancemen o SR e ec s wi h espec o hose obse ed
in single uni s ha e been epo ed in se e al pape s [4–8]. Mo e ecen ly [9–11], we ha e analyzed
he collec i e esponse o a ini e se o globally coupled bis able sys ems. We demons a ed
ha he esponse o he collec i e a iable shows SR e ec s which a e indeed much enhanced
wi h espec o hose in single bis able uni s. Ac ually, in hose a ays, SR gains la ge han
uni y we e obse ed, we belie e o he i s ime, o sub h eshold sinusoidal d i ing o ces.
Those indings indica ed ha he a ays we e indeed ope a ing in nonlinea egimes. Simila
esul s ha e been epo ed o single bis able uni s d i en by a sup a h eshold sinusoidal o ce
[12], by sub h eshold mul i equency o ces [13–18], o by a sub h eshold sinusoidal o ce in he
p esence o a s ong, high- equency monoch oma ic signal [19]. Ne e heless, o he bes o ou
aWe acknowledge he suppo o he Di ecci´on Gene al de Ense˜nanza Supe io o Spain (BFM2005-
02884) and he Jun a de Andaluc´ıa.
2 Will be inse ed by he edi o
knowledge, gains la ge han uni y ha e ne e been obse ed when he single bis able uni is
d i en by jus a sub h eshold sinusoidal o ce.
In a la e wo k [20], he esponse o he collec i e a iable o a ini e a ay o globally coupled
uni s o a a he weak d i ing o ce was analyzed. Two aspec s o he esponse we e s udied:
SR and noise induced phase equency synch oniza ion. The ac ha SR e ec s we e g ea ly
ampli ied and ha e y good phase equency synch oniza ion was ound in he esponse o a
weak d i e led us o indica e ha he a ays we e indeed ope a ing in nonlinea egimes.
In his wo k, we ex end he esul s o he p e ious wo k o o he opologies o he a ay.
In [8], a ays o nonlinea bis able uni s wi h nea es neighbo coupling a e conside ed. In ha
wo k he esponse o a single oscilla o is ound o be enhanced due o he coupling o he
o he uni s in he chain. In [21], he au ho s conside an a ay o d i en spins wi h Glaube
dynamics and nea es neighbo in e ac ion. They s udy no only he esponse o a single spin,
bu also he beha io o he global magne iza ion o he a ay. Thei analysis o he sys em
esponse is based on he linea esponse app oxima ion. In bo h pape s [8,21], he coupling
s eng h be ween he elemen s o he chain a e sugges ed as sui able pa ame e s besides he
noise s eng h o e icien ly pe o m desi ed ope a ions. In his wo k, a he han dealing wi h a
globally coupled ne wo k, we will conside as in [8] ha he bis able uni s ha e nea es neighbo
in e ac ions. We will s ill be in e es ed in ini e a ays. While in [8] he au ho s ocus on he
esponse o a single oscilla o , we will concen a e he e as in [9,21] on he beha io o a collec i e
o global a iable cha ac e izing he en i e a ay a he han on a single indi idual. By con as
wi h he ange o pa ame e s analyzed in [21], we will be dealing he e wi h si ua ions whe e a
linea esponse app oxima ion is no alid.
The es o he pape is as ollows. In Sec ion 2, we in oduce he model sys em, ix no a ion
and indica e he ele an a iables ha we will use o quan i y he SR e ec . In Sec ion 3, we
p esen he main esul s o ou nume ical simula ions. In Sec ion 4 we commen on he main
conclusions o ou pape .
2 The model sys em
We conside a se o Niden ical bis able elemen s cha ac e ized by he a iables xi( ) (i=
1,...,N) wi h nea es neighbo in e ac ions. The dynamics is gi en by s ochas ic e olu ion
equa ions (in dimensionless o m) o he ype
˙xi( ) = xi( )−x3
i( ) + θ
2[xi−1( ) + xi+1( )−2xi( )] + √2Dξi( ) + F( ),(1)
subjec o he condi ions xN+1( ) = x1( ), x0( ) = xN( ). The ex e nal d i ing o ce is pe iodic
in ime wi h pe iod T, i. e., F( ) = F( +T). The e m ξi( ) ep esen s a whi e noise wi h ze o
a e age and hξi( )ξj(s)i=δij δ( −s). In he N→ ∞ limi , his model becomes he classical φ4
model analyzed in [22]. In he absence o d i ing, i has also been used o model he dynamics
o ol age pulses along myelina ed ne es [23]. In he con ex o he a che e ec , a simila
model has been s udied in [24].
We de ine a collec i e a iable S( ) as
S( ) = 1
N
N
X
j=1
xj( ).(2)
We will concen a e on he SR e ec s associa ed wi h he collec i e a iable, when he sys em
size, N, is kep ini e and he ampli ude o he d i ing e m is small. By small we mean ha
when he d i ing o ce ac s on a single isola ed uni , SR is well desc ibed by linea esponse
heo y.
We will use he signal- o-noise a io (SNR) o he collec i e beha io as he quan i ie o
he SR e ec s. I s de ini ion equi es he e alua ion o he one- ime co ela ion unc ion de ined
as
L(τ) = 1
TZT
0
d hS( )S( +τ)i∞.(3)
Will be inse ed by he edi o 3
The no a ion h...iindica es an a e age o e he noise ealiza ions and he subindex ∞indica es
he long ime limi o he noise a e age, i. e., i s alue a e wai ing o long enough o he
ansien s o die ou . As indica ed in ou p e ious wo k [9], we can w i e
L(τ) = Lcoh(τ) + Lincoh(τ),(4)
whe e he cohe en pa , Lcoh(τ),
Lcoh(τ) = 1
TZT
0
d hS( )i∞hS( +τ)i∞,(5)
is pe iodic in τwi h he pe iod o he d i ing o ce, while he incohe en pa , Lincoh(τ) a ising
om he luc ua ions o he ou pu S( ) a ound i s a e age alue, decays o ze o as τinc eases.
The ou pu SNR, Rou , is
Rou = lim
ǫ→0+RΩ+ǫ
Ω−ǫdω ˜
L(ω)
˜
Lincoh(Ω)=˜
Lcoh(Ω)
˜
Lincoh(Ω),(6)
whe e Ω= 2π/T is he undamen al equency o he d i ing o ce F( ), ˜
Lcoh(Ω) is he co -
esponding Fou ie coe icien in he Fou ie se ies expansion o Lcoh(τ), and ˜
Lincoh(Ω) is he
Fou ie ans o m a equency Ωo Lincoh(τ).
We will also discuss he SR gain, G, de ined as [9]
G=Rou
Rinp
,(7)
whe e Rinp is he SNR o he andom inpu p ocess o med by he a i hme ic mean o he
indi idual noise e ms ξi( ) plus he de e minis ic d i ing o ce F( ), namely, F( ) + ξ( ) wi h
ξ( ) = N−1PN
i=1 ξi( ). The gain can be seen as a dimensionless pa ame e ha compa es he
ou pu SNR o ha o he inpu and, in his sense, i measu es he quali y o he ou pu ela i e
o he inpu .
3 Resul s
Following he nume ical p ocedu e de ailed in ou p e ious wo ks [15,20], we ha e es ima ed
he cohe en and incohe en pa o he collec i e co ela ion unc ion L(τ), by in eg a ing
he Lange in equa ions, Eq. (1) and a e aging o e se e al housand noise ealiza ions. Wi h
his in o ma ion, we e alua e nume ically he in eg als de ining he Fou ie coe icien s a he
d i ing equency and, using Eqs. (6) and (7), he collec i e SNR and gain o a wide ange
o pa ame e alues. In all he cases discussed below, we ha e used a weak d i ing ec angula
o ce gi en by
F( ) = (−1)n( )A, (8)
whe e n( ) = ⌊2 /T⌋,⌊z⌋is he loo unc ion o z, i.e., he g ea es in ege less han o equal
o z. The o ce ampli ude will be aken o be A= 0.1, much smalle han he ba ie heigh o
an isola ed bis able uni while he undamen al equency will be Ω= 0.01.
As no iced in [9], in he case o nonin e ac ing uni s (θ= 0), he SNR o he collec i e
ou pu is N imes la ge han ha o a isola ed uni d i en by he same o ce. None heless, as
discussed in [9], he gain associa ed wi h he collec i e ou pu is jus he same as he one o a
single, isola ed, uni . Thus, o he weak o ces ha we a e conside ing he e, he collec i e gain
does no exceed uni y, in ag eemen wi h he p edic ions o he linea esponse heo y [15,25,
26]. The in oduc ion o in e ac ions be ween he bis able uni s d as ically changes his pic u e
and he enhancemen o he SNR leads o he possibili y o obse ing gains la ge han uni y
o he weak d i ing o ces conside ed. These ac s can be obse ed in Fig. 1.
4 Will be inse ed by he edi o
0
0.5
1
1.5
2
2.5
3
Rou
θ=0
θ=0.2
θ=0.5
θ=1.0
θ=1.5
0.1 0.2 0.3 0.4
D
0
0.5
1
1.5
2
2.5
Gou
Fig. 1. The collec i e SNR, Rou , (uppe panel) and he collec i e gain, G, (lowe panel) s. he noise
s eng h D o an a ay o N= 10 bis able uni s d i en by a ec angula d i ing o ce wi h ampli ude
A= 0.1, undamen al equency Ω= 0.01 and se e al alues o he coupling pa ame e : θ= 0 (c osses),
0.2 (ci cles), 0.5 (diamonds), 1. (squa es), and 1.5 ( iangles). Lines a e a guide o he eye.
0 500 1000
-0.5
0
0.5
θ=0 (D=0.1)
θ=0.5 (D=0.16)
θ=1 (D=0.24)
θ=1.5 (D=0.28)
0 50 100
0
0.02
0.04
0.06
0.08
θ=0 (D=0.1)
θ=0.5 (D=0.16)
θ=1 (D=0.24)
θ=1.5 (D=0.28)
Lcoh Lincoh
Fig. 2. The cohe en pa , Lcoh ( ), (le panel) and he incohe en pa , Lincoh( ), ( igh panel) o
he co ela ion unc ion o he collec i e a iable o se e al alues o he coupling pa ame e : θ= 0
(solid line), 0.5 (do ed), 1 (dashed), and 1.5 (do -dashed), co esponding o he noise s eng h alues
D= 0.1, 0.16, 0.24, and 0.28, espec i ely. These noise alues co espond o he peaks obse ed in Rou
in he uppe panel o Fig. 1. O he pa ame e alues: N= 10, ampli ude A= 0.1 and undamen al
equency Ω= 0.01.
The non-mono onic beha io o he SNR o he collec i e a iable wi h he noise s eng h
obse ed in he uppe panel o Fig. 1 is indica i e o he SR phenomenon. In he uppe panel
o Fig. 1, we depic he beha io o he global Rou o an a ay o N= 10 iden ical pa icles
wi h nea es neighbo coupling. The alues o he in e ac ion pa ame e ange om small alues
(θ= 0.2) o a he la ge ones (θ= 1.5). The Rou peak alue depends on he coupling s eng h
in such a way ha as θis inc eased, he noise alue a which Rou eaches i s maximum is
shi ed sligh ly o highe alues.
I is in e es ing o compa e he ime beha io o he cohe en , Lcoh( ), and incohe en ,
Lincoh( ), pa s o he collec i e co ela ion unc ion. In he le panel o Fig. 2, we depic he
ime beha io o he cohe en pa o se e al alues o he in e ac ion s eng h θ. Fo each alue
o θ, he noise s eng h alue is ha a which he SNR is maximal. The pe iodici y o Lcoh( )
is clea ly demons a ed. I s ampli ude is only sligh ly dependen on he in e ac ion s eng h.
I s Fou ie componen a he undamen al d i ing equency is p ecisely he nume a o o Rou .
Thei peak alues a e no much di e en om hose ob ained in a single bis able uni d i en
by he same o ce a he same noise s eng h.
In he igh panel o Fig. 2, he beha io s o he co esponding incohe en pa s a e de-
pic ed. I is ema kable he as decay o he luc ua ions as well as hei small alues. As he
denomina o o he SNR is he Fou ie componen o hose decaying unc ions a he d i ing
equency, i is clea ha hose con ibu ions a e small. Consequen ly, he alues o he SNR a e
expec ed o be much enhanced wi h espec o hose alues ypical o SR in he linea egime.
The enhancemen o SR e ec s in a ays o in e ac ing bis able uni s wi h espec o hose in
Will be inse ed by he edi o 5
0100 200 300 400 500
0
0.1
0.2
0.3
0.4 θ=0.5; equilib ium
θ=0.5; A=0.1
Lincoh
Fig. 3. Solid line: The incohe en pa o he co ela ion unc ion o he global a iable o an a ay o
N= 10 bis able uni s d i en by a ec angula d i ing o ce wi h ampli ude A= 0.1 and undamen al
equency Ω= 0.01. Dashed line: he equilib ium ime co ela ion unc ion o he global a iable o
he same a ay in he absence o d i ing. In bo h cases he coupling pa ame e is θ= 0.5 and he noise
s eng h alue D= 0.16.
indi idual uni s is hen basically a consequence o he s ong educ ion o he luc ua ion le el
wi h espec o he one ound in an single d i en uni .
I should also be no ed in he lowe panel o Fig. 1 ha he gain can be la ge han
uni y o some anges o noise s eng h alues. This ea u e is a clea indica ion ha he SR
phenomenon obse ed in he a ay o he pa ame e alues conside ed can no be desc ibed
wi hin he limi s o a linea esponse heo y. To u he unde s and why linea esponse heo y
ails in he cases conside ed he e, i seems use ul o compa e he beha io o he incohe en
pa o he one- ime co ela ion unc ion o he global a iable and ha o he equilib ium
co ela ion unc ion o he same global a iable in an un-d i en sys em. An example o such
compa ison is depic ed in Fig. 3 o a coupling s eng h θ= 0.5. The g aph clea ly indica es
ha he equilib ium luc ua ions a e much la ge and longe las ing han he luc ua ions abou
he a e age beha io in he d i en sys em. In he linea esponse heo y desc ip ion o SR, (see,
o ins ance, [1] and e e ences he ein), i is assumed ha he co ela ion unc ion o he
luc ua ions a ound he a e age beha io in a d i en sys em can be sa ely app oxima ed by
hei co esponding equilib ium alues in a un-d i en one. This is clea ly no he case o he
he sys em a hand. A de ailed s udy o he alidi y condi ions o linea esponse heo y can
be ound in [27,28].
The peak alues o Rou also show a non-mono onic beha io wi h θas depic ed in Fig. 4.
As θis aised om ze o up o 0.5, he e is an inc ease in he peak o he SNR alue. This is due
o he combina ion o wo e ec s: i) he as e decay o he co ela ion unc ion as θis inc eased
(see igh panel in Fig. 2) wi h he subsequen dec ease o he denomina o in he SNR, and ii)
he la ge ampli ude o he cohe en pa (see le panel in Fig. 2). On he o he hand, as he
θ alues is u he inc eased, he con ibu ion o he cohe en pa dec eases, while ha o he
incohe en pa inc eases and, consequen ly, he SNR dec eases as he coupling e m inc eases.
The gain also shows a non-mono onic beha io wi h θas depic ed in Fig. 4.
The complexi y o he N-dimensional po en ial su ace whe e he in e ac ing pa icles mo e
ende s he ask o gi e a simple explana ion o he obse ed non-mono onic beha io wi h θ
a di icul one. Fo an N-dimensional su ace in he absence o in e ac ions and d i ing o ces,
he whole su ace is symme ical abou he o igin wi h ba ie s o equal heigh s along each axis.
The obse ed non-mono onic beha io can be a ionalized in e ms o he pe iodic ocking o
he bis able po en ial independen ly along each axis. Each pa icle jumps o e i s co esponding
ba ie independen ly o he o he pa icles, unde he in luence o he d i ing o ce and he
noise. The cen al limi heo em can be sa ely used o independen subuni s, so ha he inc ease
in he SNR alues is jus a size e ec .
On he o he hand, o coupled sys ems, he indi idual s ochas ic p ocesses xi( ) a e no
longe independen and he cen al limi heo em alone is no enough o unde s and he epo ed
esul s. The de o ma ion o he po en ial su ace due o he ex e nal d i ing and he in e ac ion

6 Will be inse ed by he edi o
0 0.5 1 1.5
θ
0
1
2
3
Rou
Gou
Fig. 4. The peak alues o Rou and G s. he coupling s eng h θ o N= 10, A= 0.1 and undamen al
equency Ω= 0.01.
e m migh e y well educe he heigh o he ba ie s, elimina e some o hem and al e he
loca ion o he minima. Then, one can no ule ou he possibili y o he exis ence o new
pa hs acili a ing he ansi ions be ween he a ac o s. This being he case, a educ ion o
he luc ua ion le els besides he one coming om he sys em size is o be expec ed. Fo a
ixed small d i ing ampli ude, he amoun o dis o ion o he po en ial elie mus depend
on he s eng h o he coupling e m θ. Fo e y small alues o θ, pa icles loca ed beyond
nea es neighbo posi ions a e expec ed o be weakly co ela ed, wi h he co ela ion leng h
inc easing as θinc eases. One migh expec an inc ease on he SNR alues as θinc eases
om ze o, as he incohe en pa o he co ela ion unc ion basically dec eases. This is due o
he inc easing easiness o ansi ions be ween a ac o s along he new pa hs. As θis u he
inc eased, he dis o ion o he po en ial elie will inc ease, bu a he same ime, mos o he
pa icles along he chain will s a o be s ongly co ela ed. In o he wo ds, he mo ion o he
collec i e a iable will esemble mo e and mo e he mo ion o a pa icle on a single bis able
po en ial. The chain will beha e mo e and mo e like a igid objec . Jumps o e he ba ie s
become inc easingly mo e di icul and wi h a dec ease o he SNR alues. The e mus be an
in e media e in e ac ion s eng h alue so ha a maximum SNR o he ou pu is achie ed.
I is in e es ing o s udy wha happens in he case o a sinusoidal d i ing o ce wi h he
same ampli ude A= 0.1 and equency Ω= 0.01. In Fig. 5 we depic Rou (lowe panel) and
G(uppe panel) s. D o se e al alues o he coupling s eng h. As i can be seen, he SNR
alues a e subs an ially smalle han he ones obse ed o a ec angula d i ing o ce (compa e
wi h Fig. 1). The gain is always below uni y by con as wi h he ec angula d i ing o ce,
whe e he gain can each alues la ge han 1.
In Fig. 6 we show he ime dependence o he cohe en and incohe en pa s o he co ela ion
unc ion o an a ay o N= 10 pa icles d i en by he sinusoidal o ce. By compa ison wi h
Fig. 2 we see ha he eason why he SNR alues a e much smalle o a sinusoidal d i ing
han o a ec angula one is mainly ha Lincoh( ) is la ge in he single equency case han
in he mul i- equency one. Concluding ha because he gain in he case o sinusoidal d i ing
is less han 1, a linea esponse heo y desc ip ion is adequa e is no igh . As shown in Fig. 7,
Lincoh( ) o a sinusoidal d i ing is oo di e en om he equilib ium co ela ion unc ion in an
un-d i en sys em.
A non-mono onic beha io o he SNR wi h he coupling cons an θ o he sinusoidal d i ing
also exis s. The quali a i e explana ion gi en be o e o he ec angula d i ing s ill s ands.
No e, none heless, ha he ocking o he mul i-dimensional ene gy su ace b ough up by he
sinusoidal d i ing is less d as ic han he one p oduced by he ec angula one. E en hough
he o ces ha e he same ampli ude and undamen al equency, he sinusoidal o ce in oduces
a bias in he ene gy elie wi h espec o ha in he ze o o ce case which is con inuously
changing wi h ime. On he o he hand, he ec angula signal keeps he su ace biased mos o
he ime, excep du ing i s ins an aneous changes o alue. The sys em has ample ime o elax
Will be inse ed by he edi o 7
0 0.2 0.4
D
0
0.2
0.4
Rou
θ=0
θ=0.2
θ=0.5
θ=1
θ=1.5
0
0.5
1
G
θ=0
θ=0.2
θ=0.5
θ=1
θ=1.5
Fig. 5. The beha io o Rou and Gwi h D o se e al alues o he coupling s eng h o an a ay o
N= 10, d i en by a sinusoidal o ce wi h A= 0.1 and equency Ω= 0.01.
0 50 100 150
0
0.05
0.1
0.15
0.2
0.25
0.3
Lincoh( )
θ=0; D=0.1
θ=0.5; D=0.16
θ=1; D=0.24
θ=1.5; D=0.28
0 500 1000 1500
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
Lcoh( )
Fig. 6. The ime beha io o Lcoh and Lincoh o se e al alues o he coupling s eng h o an a ay
o N= 10, d i en by a sinusoidal o ce wi h A= 0.1 and equency Ω= 0.01.
0 50 100 150 200
−0.02
0.08
0.18
0.28
0.38
0.48
0.58
Lincoh( )
θ=0.5; equil; D=0.16
θ=0.5; A=0.1; D=0.16
θ=0.5; A=0.1; D=0.16 (sinusoidal)
Fig. 7. Compa ison o he ime beha io o Lincoh in an a ay o N= 10 a equilib ium o d i en by
ei he a sinusoidal o ce wi h A= 0.1 and equency Ω= 0.01, o a ec angula one wi h he same
ampli ude and undamen al equency.
du ing hose biased in e als and his explains why he luc ua ions a e so d as ically educed
in he case o a ec angula d i ing.
8 Will be inse ed by he edi o
4 Conclusion
In his wo k, we ha e explo ed he phenomenon o SR in ini e a ays o noisy bis able uni s
wi h nea es neighbo coupling, d i en by ime pe iodic o ces o weak ampli ude. Ra he han
analyzing he modi ica ion o a single uni beha io wi h espec o he one in he absence o
any in e ac ion, we ha e ocused ou a en ion in a collec i e a iable desc ibing he dynamics
o he a ay as a whole.
A s ong enhancemen in he SR e ec s associa ed wi h he collec i e a iable wi h espec o
he one ound in a sys em o med by a single uni has been demons a ed by means o nume ical
simula ions. In pa icula , we ha e shown ha , e en o a he weak inpu ampli udes, he SNR
has a non-mono onic beha io wi h he noise s eng h. The di e en alues o he SNR quan i ie
a e much la ge han he co esponding ones ound in a single uni sys em. Fu he mo e, he
SR gain eaches alues highe han uni y o some alues o he pa ame e s, indica ing ha he
a ay is ope a ing in a nonlinea egime well beyond he limi s o he egimes desc ibed by linea
esponse heo y. This is o some ex en su p ising as, o he d i ing ampli udes and equencies
conside ed, he esponse o a single uni sys em is well desc ibed by he linea esponse heo y.
The e a e in p inciple wo main easons o he enhanced e ec s epo ed he e. On he one
hand, based on he cen al limi heo em, one can expec a dec ease in he luc ua ion le els
wi h espec o hose ound in single uni sys ems simply because o he sys em size. On he
o he hand, as we ha e demons a ed in ou nume ical simula ions, he size e ec mechanism
is no enough o explain he epo ed beha io . The p esence o coupling e ms be ween he
subsys ems is an essen ial ing edien o ob ain gains la ge han uni y.
We belie e ha he main eason o he s ong enhancemen obse ed is due o he d as ic
educ ion o he alue o he incohe en co ela ion unc ion and on i s co ela ion ime induced
by he sys em size, he ex e nal d i ing and he coupling e m be ween he di e en subuni s.
Indeed, a compa ison wi h he decay o luc ua ions in a ini e sys em in he absence o d i ing
wi h he same size and coupling pa ame e shows he ele ance o he d i ing o ce.
An enhancemen o he SR e ec s in ini e size a ays wi h global coupling (mean ield
coupling) ha e also been epo ed by us. One o he goals o he p esen wo k is o demons a e
he obus ness o ou esul s, ega dless o he opology o he connec ions be ween he di e en
subuni s. Indeed, a compa ison o he nume ical esul s epo ed he e wi h hose in [20] indica e
ha , a leas o small size a ays (N= 10), he Rou and gain alues a e no much di e en .
We ha e also no ed ha he e exis s a non-mono onic beha io o he SNR wi h espec o
he coupling pa ame e θ, besides he usual non-mono onic beha io wi h he noise s eng h.
The e a e good easons o belie e ha non-mono onic beha io s wi h espec o he sys em
size do also exis . Indeed, sys em size esonances ha e been analyzed by o he g oups o
sys ems o globally coupled nonlinea oscilla o s using cumulan expansion echniques [29] o
nonequilib ium po en ials [30]. Wi hin a linea esponse heo y desc ip ion, hey epo sys em
size esonance e ec s. We a e p esen ly in es iga ing he issue o he dependence on he sys em
size in coupled a ays (global and local coupling) o pa ame e egimes well beyond he linea
esponse heo y limi s.
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