Study on synchronization of coupled oscillators using the

Fokker-Planck equationH.Sakaguchi

Kyushu University, Japan

Fokker-Planck equation: important equation in statistical physics Synchronization in coupled oscillators

Langevin equation

Stochastic differential equation

Time evolution in a noisy environment

)()( txFdt

dx

Langevin equationX(t): a stochastic variable such as a position or a membrane potentialξ(t): 　 a random force, Gaussian white noise

Random forcesGaussian:

The probability distribution function is Gaussian with average 0 and variance 2D.

White: There is no time correlation.

)'(2)'()( ttDtt Consider a large number of independent stochastic variables 　 which obey the Langevin eq.

The stochastic variables are randomly distributed, since the random forces are different.

Fokker-Planck equation

Consider the probability density P(x,t) that the stochastic variable takes a value x.

P obeys the Fokker-Planck equation

2

2)(

x

PD

x

FP

t

P

Drift term Diffusion term

The probability density drifts with velocity F(x) 　 and diffuses owing to the random force

Random walkNo drift force

t

dttxtxtdt

dx

xF

0 )()0()( ),(

,0)(

Langevin equation

Dt

xx

DtP

x

PD

t

P

4

))0((exp

4

1

,

2

2

2

Fokker-Planck equation

Ornstein-Ulenbeck process

Linear force ( ｖ : velocity,- k ｖ viscous force)

)(vv

v)v(

tkdt

d

kF

Fokker-Planck equation

Maxwell distribution

Stationary distribution

Brownian motion of a small particle such as a pollen on water surface

Tk

m

D

k

D

kP

PD

Pk

t

P

B2

vexp

2

vexp

2

,vv

)v(

22

2

2

Thermal equilibrium distribution

)(

)(

tx

U

dt

dxx

UxF

Potential force

Fokker-Planck equation

Tk

xUP

x

PD

x

xPU

t

P

B

)(exp

,)/(

2

2

D=kBT; T: Temperature

Thermal equilibrium distributionNo probability flow: Detailed balance

Fokker-Planck equation for coupled Langevin equations

Coupled Langevin equations for two variables

)'(2)'()( ),(),(

)'(2)'()( ),(),(

2222

1111

ttDtttyxGdt

dy

ttDtttyxFdt

dx

Fokker-Planck equation for P(x,y)

2

2

22

2

1

)),(()),((

y

PD

x

PD

y

PyxG

x

PyxF

t

P

Synchronization of coupled biological oscillators

Synchronization of flashing of fireflies.

Synchronization of cell activity in suprachiasmaticnucleus which control the circadian rhythms

Sleep spindle waves are brain waves which appear in the second stage of sleep. Spindle waves are created by synchronous firing

of inhibitory neurons in thalamus. (Steriade et al.)

Human EEG

Phase oscillatorsLimit cycle oscillation

Phase description:

phase variables

Two-coupled phase oscillators

K

KKdt

d

Kdt

d

Kdt

d

Kdt

d

2 for 0

,2 for )2() () (

)sin(2 )(

), sin(

), sin(

2 1

2 122

2 12 1

2 1 212 1

2 122

1211

Mutual Entrainment

Globally coupled noisy phase oscillators

2

2

1

,

1

)sin()(

),()sin(

,)cos(/)cos(

),'( 2)'()(

),() sin(

PDPK

dt

dP

tKdt

d

N

ttDtt

tN

K

dt

d

iii

N

jj

jiji

i

N

jij

i

Uniform state σ ＝０， P=1/2πUnstable 　 for 　Ｋ＞２Ｄ

P(φ): The probability that the phase takes φ

Mean field coupling:

Each oscillator interactswith all other oscillatorsby the same coupling.

Order parameter

＜ sin(φ) ＞ =0 　 by the symmetry

Self-Consistent MethodOrder parameter

12

16 ,

162

by expansionTaylor

)cos(

exp)(

)sin(

)cos()( )cos(

3

33

2

2

2

0

D

K

K

D

D

K

D

K

D

KP

PDPK

t

P

dP

Order-Disorder transitionWeak interaction, Large noise

Phases are randomly distributed.

Disordered phase.

Strong interaction, small noise

Phases gather together.

＜ cos(φ ）＞　 is nonzero.

Ordered state

Order-Disorder transition

Phase transition from ferromagnetism to paramagentism

If a magnet is heated above a critical temperature,

the magnetism disappears.

　　　

Globally coupled oscillators with different frequencies

rotation: , ||For

ent Entrainm)/(sin , ||For

sin

)cos(

)g(-)g( , of ondistributi :)g(

timeinconstant but random :

,) sin(

1

1

ii

iii

iii

N

jiji

i

K

KK

Kdt

d

N

K

dt

d

Phase oscillator model

Kuramoto model

Self-Consistent analysis

KK

Kdtd

gK

KKg

gK

gKKg

dg

c

c

c

||for )(

||for 0/

,|)0("|

))(0(8

)0(

2

,)0("16

)0(2

expansionTaylor

)())(cos(

22

3

33

ω

Globally coupled oscillators with different frequencies and external noises

2

2

,

1

)sin(),(

),() sin(

,)cos(

),'( 2)'()(

),() sin(

PDPK

dt

dP

tKdt

d

ttDtt

tN

K

dt

d

iiii

jiji

i

N

jiji

i

g(ω): Distribution of the natural frequency ω

Stationary solution of the Fokker-Planck equation for nonzero ω

2

0

0

))cos(1())cos(1(

2

2

)0()2( :ity Periodic,1)( :ionNormalizat

)0(),(

,0)sin(),(

PPdP

deD

cPeP

PDPK

t

P

D

K

D

K

　Flow of probability: average circulation of phase Stationary but non-equilibrium distribution

Phase transition via synchronization

Complete entrainment is impossible owing to noises

22

2/1

22

2

0

2

1

,)(

)/()(

1

for unstable is state Uniform

)cos(),()(

)cos(

D

K

KK

DdDgKK

ddPg

c

c

Integrate-and-fire modelHodgkin-Huxley equation

Detailed dynamics of membrane potential

and several ion channels

IF model

simplest model of the neural firing

0. reset to is ,1 If

1for ,

xx

,xbxIdt

dx

x:membrane potential

Synchronization of two IF neurons

fires.neuron other when theby shifts variableThe

fires.neuron second) (thefirst when thejth time theis )(

,)(1

,)(1

21

122

211

gx

tt

ttgbxdt

dx

ttgbxdt

dx

jj

j

j

Instantaneous interactionResponse time 0

Complete synchronization for t>80

01.0 ,8.0 gb

δｘ＝ｘ１－ｘ２

Noisy integrate-and-fire model and the Fokker-Planck equation

,10for ,

1)0(

,0for )0(

ondistributi Stationary

)/(

,)()(

xfor equationPlanck -Fokker

)'(2)'()( ),(

1

0

/}2/1 ({

0

/)}2/1 ({

/)2/1 (

/)2/1 (

10

02

2

2

2

2

2

xdze

dzeeP

xePP

xPDJ

Jxx

PDPbxI

xt

P

ttDtttbxIdt

dx

DbzzI

x DbzzI

DbxxI

DbxxI

x

reset process

Stochastic resonance in the noisy IF model

Stochastic resonanceResponse of excitable systems to periodic force + noises

Response is maximum for intermediate strength of noise

.0for 1:solution Stationary

system, Excitable

1 ),'(2)'()(

),()sin(1

,

eD/bx

bttDtt

ttebxdt

dx

jiji

iii

Direct simulation of the Fokker-Planck equation

ly.periodical changes )/( rate Firing

,)())sin(1(

equationPlanck -Fokker

10

02

2

xxPDJ

Jxx

PDPbxte

xt

P

Firing rateOscillation of P

Oscillation of J0

D=0.005,0.0015 b=1.1,e=0.05

Phase transition in a globally coupled IF models

)(

),()(}))(1{(

neuron kth theof timefiring jth : ,)(

)()(1

0

02

2

1

tgJIdt

dI

tJxx

PDPtIbx

xt

P

tttN

gI

dt

dI

ttIbxdt

dx

jk

N

k j

jk

iii

Oscillation amplitude vs. D

Disorder

Order

b=0.8,D=0.215,g=0.6,τ=0.01

τ ： response time

Phase transition in a nonlocally coupled IF model

|)'|exp(48.0|)'|4exp(8.1)'(

,')'()'(),(

),,()()(

),,()(})),(1{(),(

0

02

2

yyyyyyg

dyyJyygtyJ

tyJyIdt

ydI

tyJxx

PDPytIbx

xt

yxP

Nonlocal interactionMaxican-hat type

Excitatory in the neighborhoodInhibitory in far regions

Synaptic coupling is nonlocal.Synaptic current at y is determined by the firing rate at y’ by the integral.

Propagating pulse statesUniform state is unstable Pulse propagation

Oscillation amplitude of I(y,t) J0(y,t)

Order-disorder phase transion from a uniform state to a traveling wave stateInhibitory interaction suppresses global synchronization

D=0.01

Another IF model and inhibitory networkThalamus(thalamic reticular neurons) Synchronization occurs among inhibitory neurons. Synchronization between two inhibitory IF neurons is possible if the response time τtakes a suitable value. Another IF model including the dynamics of excited state

50 ,35' ,30

40, -35, ,2,035.0

. tojumps , todown goes If

. tojumps ,over goes If

for )'(

for )(

200

2

22

1

0

0

VVV

VVC

VVVV

VVVV

VVVVdt

dVC

VVVVdt

dVC

TT

iTi

iTi

Tiii

Tiii

V>VT Excited state

VT

V1VT2

V2

Two IF neurons with inhibitory coupling

Synchronization of two inhibitory IF neurons

0for 0 ,0for 1)(

,)}(2/{

for ,)'(

for ,)(

2

1

0

0

VVV

VKIdt

dI

VVIVVdt

dVC

VVIVVdt

dVC

jjs

s

Tisii

Tisii

-Is inhibitory synapse

V1 and V2 are synchronized

K=0.5,V0=-18

Synchronization becomes easier owing to finite duration of excited state

Phase transition in globally coupled IF models with mutual inhibition

0

2211

2

2

0

1

0

)(

),()()()(

}),,({

,)(

)(),,(

dxxPKIdt

dI

tJVxtJVxx

PDPIIVf

xt

P

VN

KI

dt

dI

tIIVfdt

dV

i

N

jj

iii

Langevin equation

Fokker-Planck equation

Oscillatory phase transition in inhibitory systems

Oscillation amplitude vs. K

τ

Phase diagram

D=0.2,τ=20 D=0.2

Finite response time is preferable for global oscillation

Vi and the average at K=10, D=0.2Vi and the averageat K=2, D=0.2

Time evolution of P at K=10,D=0.2Time evolution of P at K=2,D=0.2

Two types of oscillation

Oscillation is synchronized.Firing is not synchronized.

Fokker-Planck equation

Langevin equation of 1000 neurons

Synchronized firing

The firing of some neurons suppresses the firing of the other neurons

Integrate-and-fire-or-burst model

Low threshold Ca2+ current: IT(t) plays

important role for thalamic neurons

70,120,07.0

))(())()((

10

),()(

hHT

hHTT

hh

VVg

VtVtVVthgI

h

VVhdt

dhV

This current flows for a short time after the potential V goes over Vh .

Phase transition in globally coupled IFB models with mutual inhibition

0

1

0

2112

2

0

),(

),()()()(

]}/)([{}),,({

dxdhhxPKIdt

dI

tJVxtJVxx

PD

PxVhh

PIIIxfxt

P

R

hhT

h(t) is a stochastic variable.

Bistability of globally coupled IFB model

I0=1.6,K=40, D=0.2 and τ=20.

h≠0, in one mode (rebound mode, burst mode).h=0, and IT does not work, in another mode (tonic mode).

Average membrane potential E(t) <h(t)> vs.I0.

Ｖｈ =-70I0 is external input

Two modes are bistable for 0.55<I0<2.2

Summary

1 Phase transition via mutual synchronization

2 Direct simulation of Fokker-Planck equation

3 Phase oscillator model and IF models

4 Transition to traveling wave states

5 Mutual synchronization in inhibitory systems 　　 Intermittent firing in strongly inhibited systems

Discussions and ProblemsGood points of the Fokker-Planck equation:

1. Stationary distribution might be solved.

2. Numerical results are clear, since it is

a deterministic equation.

Weak points of the Fokker-Planck equation:

If the number of stochastic variables is not one

or two, numerical simulations are rather hard.

Langevin simulations may be efficient for realistic equations such as noisy Hodgkin-Huxley equations.

References

Y.Kuramoto, “Chemical Oscillations, Waves

and Turbulence” Springer (Berlin, 1984).

H.Sakaguchi, Prog.Theor.Phys. Vol.79(1988) 39

S.H.Strogatz, Physica D Vol. 143 (2000) 1.

H.Sakaguchi, Phys.Rev. E Vol.70(2004) 022901.

M.Steriade and R.R.Llinas, Phsiol.Rev. Vol.68

(1988) 649