the influence of the disorder in the kuramoto modelgentz/pages/ws13/rds13/slides/... · the...

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

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


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).


Outline







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.


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.]


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?


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.


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.


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.


Example: The disorder is chosen by a toss of coins = 12(1 +1).



1



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)?


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)?


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,


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,


Outline







The empirical measure


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?


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.


Outline







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.


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?


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.


K 1

qi(, ) 12

qs(+ 0, )

Figure : The incoherent

solution qi 12


K > 1

qi(, ) 12

qs(+ 0, )

Figure : The incoherent

solution qi 12Figure : One synchronized

solution qs


Outline







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.


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).


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

)).


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 .


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.


Outline







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


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 |


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

)]

.


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)

.


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.)?


Thank you for your attention!


Mean-field interacting diffusionsN: Law of Large NumbersThe example of the Kuramoto modelThe symmetric case: fluctuations around the McKean-Vlasov equationSpatially extended neurons

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