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The influence of the disorder in the Kuramoto model
Eric Luon
Universit Ren Descartes - Paris 5
Partial joint works with Giambattista Giacomin and Christophe Poquet and with Wilhelm Stannat
Random Dynamical Systems, Bielefeld, Nov. 2, 2013
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 1 / 33
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1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 2 / 33
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Synchronization of individuals
Emergence of synchrony is widely encountered in complex systems of
individuals in interaction (networks of neurons, collective behavior of social
insects, chemical interactions between cells, planets orbiting, . . . ).
Three main ingredients for interacting individuals:
1 a dynamics for each individual (e.g. FitzHugh-Nagumo or Hodgkin-Huxley
for neurons)
2 a network of interactions (possibly heterogeneous and delayed)
3 absence or presence of a thermal noise (possibly correlated).
Under these conditions, for a sufficiently strong interaction between individuals
and for a sufficiently large population, synchronization should occur
(individuals exhibit similar simultaneous behavior).
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Outline
1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 4 / 33
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General framework: mean-field interacting diffusions in Rm
Here, each individual is a diffusion in Rp.
For T > 0, N > 1, consider t [0,T ] 7 (1(t), . . . ,N(t)) solution to
di(t) = c(i )dt +1
N
N
j=1
(i ,j)dt +dBi(t), i = 1, ,N,
c(): local dynamics of one individual(, ): interaction kernelBi : i.i.d. Brownian motions (thermal noise).
Exchangeability
If at t = 0, the vector (1(0), . . . ,N(0)) is exchangeable, then, at all timet > 0, the law of the vector (1(t), . . . ,N(t)) is also exchangeable.
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An example
Interesting examples include the granular media system:
di (t) =V (i)dt 1
N
N
j=1
W (i j)dt +dBi(t),
for V and W having convexity properties.
V ()
[ Carillo, McCann, Villani, Malrieu, Guillin, Cattiaux, Berglund, Gentz, Tugaut, etc.]
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Interacting diffusions in a random environment
Exchangeability may not be a suitable property: one needs to encode the fact
that the dynamics may not be the same for each individual (e.g., inhibition or
excitation for a neuron).
Idea: set a sequence of i.i.d. random variables (i ) encoding the intrinsicbehavior of the individual i . For this choice of disorder (1,2, . . . ,N), wemodify the dynamics and the interaction in the following way:
di(t) = c(i ,i)dt +1
N
N
j=1
(i ,j ,i ,j)dt +dBi(t), i = 1, ,N.
Question
What is the (quenched) influence of the disorder on the long-time/large N
behavior of the system in comparison with the case without disorder?
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Disordered mean-field models in neuroscience
Consider FitzHugh-Nagumo dynamics for the spiking activity of one neuron
= (v ,W ) R2 i.e.{
V = V V 3/3+W + IW = aW +bV ,
where V is the membrane potential and W is a recovery variable. The disorder
= (a,b) encodes the state (inhibited/excited) of one neuron.
di(t) = c(i ,i)dt +1
N
N
j=1
(i ,j ,i ,j)dt +dBi(t), i = 1, ,N,
Here models synaptic connections between neurons.[ O. Faugeras, J. Touboul et al.: similar systems with delay, two-scale of population, disorder,
etc.]
Difficulty: absence of reversibility.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 8 / 33
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A simpler model: the Kuramoto model
Kuramoto [75], Strogatz, Giacomin, Pakdaman, Pellegrin, Poquet, Bertini, L.
The state space here is the one-dimensional circle: S := R/2:
di (t) = i dt +K
N
N
j=1
sin(j i)dt +dBi(t), i = 1, . . . ,N,
Intuition: competition between
idt : random intrinsic speed of rotation for each rotator i ,
K sin()dt : synchronizing kernel between rotators.
Absence of disorder = reversibility
If i = 1, . . . ,N,i = 0, the dynamics is reversible.
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Example: The disorder is chosen by a toss of coins = 12(1 +1).
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
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Example: The disorder is chosen by a toss of coins = 12(1 +1).
1
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
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Example: The disorder is chosen by a toss of coins = 12(1 +1).
1 +1
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
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Example: The disorder is chosen by a toss of coins = 12(1 +1).
1 +1
Questions: what is the influence of the disorder on the system? Does it
depend only on its law (centered, symmetric or not) or on a typical realization
(1, . . . ,N)?
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
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Example: The disorder is chosen by a toss of coins = p1 +(1p)1.
1
Questions: what is the influence of the disorder on the system? Does it
depend only on its law (centered, symmetric or not) or on a typical realization
(1, . . . ,N)?
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
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Simulation I: N = 500, K = 3, = 1, no disorder
di (t) =K
N
N
j=1
sin(j i)dt +dBi(t), i = 1, . . . ,N,
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 11 / 33
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Simulation II: N = 600, K = 6, = 1, = 12(1+1)
di (t) = i dt +K
N
N
j=1
sin(j i)dt +dBi(t), i = 1, . . . ,N,
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 12 / 33
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Outline
1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 13 / 33
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The empirical measure
di(t) = c(i ,i)dt +1
N
N
j=1
(i ,j ,i ,j)dt +dBi(t), i = 1, ,N.
We want to understand the (quenched vs annealed) behavior as N of theempirical measure
N,t :=1
N
N
j=1
(j(t),j).
Law of Large Numbers?
Does the continuous limit say anything on the particle system?
Central Limit Theorem? Large Deviations?
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Quenched convergence of the empirical measure N
Under Lipschitz regularity on c and , moment condition on and
convergence of the initial condition ()N,0
N 0,
Proposition (Quenched LLN - L. 2011)
For a.e. (i)i > 1, the empirical measure ()N converges in law, as a process to
t 7 t(d, d)
that is the unique weak solution to the following McKean-Vlasov equation:
tt =1
2div
(
T t)
div[
t
(
c(,)+
(,, , )dt)]
.
Self-averaging phenomenon
At the level of the LLN, the dependence in the disorder lies in its law, not a
typical realization.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 15 / 33
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Outline
1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 16 / 33
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McKean-Vlasov equation in the Kuramoto model
In the limit of an infinite population: qt(,), density at time t of oscillators withphase and frequency solves
tqt(,) =2
2qt(,)K
[
qt(,)(
sinqt()+)]
.
What makes the Kuramoto tractable is that the nonlinearity is nice (it only
concerns the first Fourier coefficients of q). In particular, if there is no disorder,
one can show that
the microscopic system is reversible,
there exists a Lyapounov functional for the continuous model.
From this, one can derive many things in the case of small disorder, by
perturbation arguments.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 17 / 33
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Non-symmetric disorder: existence of traveling waves
1 The law of the disorder is not centered (E() 6= 0): we can go back tothe centered case by the change of variables i(t) := i(t) tE()(existence of traveling waves).
2 The law of the disorder is centered (E() = 0) but not symmetric:
Theorem (Giacomin, L., Poquet, 2012 )
If the disorder is small, there exist solutions to the McKean-Vlasov equation of
the following type
q(,) := q( c()t )and the family q() is stable by perturbation.
Question
What if the law of the disorder is centered and symmetric?
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 18 / 33
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Symmetric disorder: synchronization
In this case, the McKean-Vlasov admits stationary solutions that can be
explicitly computed:
0 = tqt(,) =2
2qt(,)K
[
qt(,)(
sinqt()+)]
,
Theorem (Giacomin, L., Poquet, 2012 )
For small disorder, there exists Kc > 0 such that
if K 6 Kc , qi 12 is the only stationary solution (incoherence),if K > Kc,
12 coexists with a circle (rotation invariance) of nontrivial
stationary solutions {(,) 7 qs(+0,); 0 S} (synchronization).Moreover, such a circle of synchronized solutions is stable under perturbations.
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K 1
qi(, ) 12
qs(+ 0, )
Figure : The incoherent
solution qi 12
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 20 / 33
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K > 1
qi(, ) 12
qs(+ 0, )
Figure : The incoherent
solution qi 12Figure : One synchronized
solution qs
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 20 / 33
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Outline
1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 21 / 33
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Fluctuations of N around its McKean-Vlasov limit
For fixed t [0,T ], fixed disorder (), consider the random tempereddistribution
()N,t :=
N
(
()N,t t
)
S .
Semi-martingale representation of ()N : for all regular, t 6 T :
()N,t ,
=
()N,0 ,
+ t
0
()N,s , LN()
ds+M()N,t (),
where LN is a linear operator and M()N,t () a martingale.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 22 / 33
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Some negative answer
Remark
There cannot be any real quenched Central Limit Theorem, in the sense that
for fixed disorder (), the process ()N may not converge.
Why? Consider the example of independent Brownian motions with random
drifts (i.e. 0 and c(,) = ):
di(t) = i dt + dBi(t).
In the quenched model, the (i)i > 1 are fixed. In order to study thefluctuations of this system, one needs to understand the quantity
N
(
1
N
N
i=1
i E())
,
which does not converge for fixed (i)i > 1 (but only in law w.r.t. (i)i > 1).
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 23 / 33
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The correct set-up
Instead of looking at
()N :=
N
(
()N
)
C ([0,T ],S ),
for fixed (), one can always consider the application
() 7 HN() := law of ()N M1(C ([0,T ],S )).
The correct set-up is to say that the sequence of random variables (HN)Nconverges in law in the big space M1(C ([0,T ],S
)).
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 24 / 33
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Quenched CLT
Hypothesis
b and c are regular,
(j) are i.i.d. and
R||4(d) < for some > 0.
Theorem (L. 2011)
Let HN() be the law of the process ()N . Then (HN)N converges in law in
M1(C ([0,T ],S)) to 7 H() satisfying the following characterization: for all
, H() is the law of the solution of the SPDE:
t = X()+ t
0
Lqs s ds+Wt ,
where, Wt is explicit and for all , X() is a Gaussian process that is notcentered. W is independent with X .
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 25 / 33
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Kuramoto: asymptotic behavior of the fluctuation process
Binary disorder: = 12(0 +0), 0 > 0.
Theorem (L. 2012)
For all K > 1, there exists 0 = 0(K ) such that satisfies
, t
t
in lawt
v()q.
Moreover, as a function of , 7 v() is a Gaussian random variable withvariance
2v :=204.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 26 / 33
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Outline
1 Mean-field interacting diffusions
2 N : Law of Large Numbers
3 The example of the Kuramoto model
4 The symmetric case: fluctuations around the McKean-Vlasov equation
5 Spatially extended neurons
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 27 / 33
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The case with spatial extension
Joint work with W. Stannat.
We want to take into account the positions of the particles i : we place
one particle at each point of the lattice Zd and the interaction between
two particles depends on the distance between them.
i
j
NN Z
d
N(i, j)
0 NN
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 28 / 33
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Spatially extended weakly interacting diffusions
The system becomes
di(t)= c(i ,i)dt+1
|N | jN ,j 6=i(i ,j ,i ,j)
(
i
2N,
j
2N
)
dt+dBi(t),
where
N is a box in Zd of size N, with volume |N |,
(, ) is a spatial weight.Possible choices of weights are:
A cut-off function: (x ,y) 1|xy |6 RA power-law interaction: (x ,y) = 1|xy |
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 29 / 33
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The power-law case for < d
di (t) = c(i ,i)dt +1
|N | jN ,j 6=i(i ,j ,i ,j)
ij2N
dt +dBi(t).
Proposition (L. - Stannat, 2013 )
The empirical measure
N :=1
|N | jN ,j 6=i(i ,i , i2N )
converges, as a process, to the unique solution dt = qt(,,x)d(d)dxwhere
tqt =1
2div
(
T qt)
div[
qt
(
c(,)+
(,, , )|x | dqt
)]
.
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 30 / 33
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Precise fluctuations estimates in the case < d
For an appropriate weighted Wasserstein distance (for some p > 1 and forsome adequate domain D),
d(,) := supfD
(
E
f d
f d
p)1/p
.
Theorem (L. - Stannat, 2013 )
For any < d2
, there exists a constant C > 0 such that:
sup0 6 t 6 T
d(N,t ,t) 6 C
N(1), if [
0, d2
)
,
(lnN) N( d2 1), if = d2,
N((d)1), if (
d2,d)
.
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Perspectives
Is it possible to prove a quenched LDP in the mean field case?
What can we say about the phase transition in the spatial case? About
the central limit theorem?
What if the positions of the particles are chosen randomly?
Can we derive similar McKean-Vlasov equations for more general graphs
(small-world graphs, etc.)?
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Thank you for your attention!
Eric Luon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 33 / 33
Mean-field interacting diffusionsN: Law of Large NumbersThe example of the Kuramoto modelThe symmetric case: fluctuations around the McKean-Vlasov equationSpatially extended neurons