Synchronization transition and universality:F. Jülicher, J. ProstInstitut Curie, Paris

Institut Max Planck, Dresde

Thomas Risler

Inner ear and global coupling:A.S. Kozlov, A.J. Hudspeth

Rockefeller University, New York

Hearing organ and performances:From global coupling to

Out-of-equilibrium criticality

Summary

P. Gillespie

5 µm

Structure and mechano-transduction

Laser Interferometer

Multi-Taper Data Analysis

The Hair Cell

Noisy Coupled Oscillators

Synchronization Transition

Renormalization Group

Universal Properties

1-Inner hair cell2-Outer hair cells3-Tunnel of Corti4-Basilar membrane5-Habenula perforata 6-Tectorial membrane7-Deiters' cells8-Space of Nuel9-Hensen's cells

10-Inner spiral sulcus

Internal ear

Hudspeth, Nature (1989)

(R. Pujol,http://www.iurc.montp.inserm.fr/cric/audition)

American Bullfrog Rana catesbeiana

Bullfrog Sacculus

A.J. Hudspeth’s Laboratory

Tokay Gecko

Bullfrog Sacculus

The transduction apparatus

5 µm

100 nm

10 nm 10 nm

50 nm

1 m

Kachar et al., PNAS (2000)

P. Gillespie

Holt, Corey, PNAS (2000)

Mechano-transduction

Corey, Hudspeth, J. Neurosc. (1983)

F: Stimulusz: “gating force”

Cstz)X(NpXK)X(F o +−= ∞

1)(

exp1)(

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡ −−+=

TkXXz

XpB

offo

Martin et al.,PNAS (2000)Howard, Hudspeth, Neuron (1988)

Force-displacement curve

The Hair Bundle: A Noisy Nonlinear Oscillator

Martin et al.,PNAS (2001)

€

˜ S (ω) = X(ω)X(−ω) Fluctuation SpectrumSpontaneous Oscillations

Response Oscillations Synchronization

Preferred Frequency

Phase LagsPhase Shifts

Martin, Hudspeth, PNAS (1999)

The Hair Bundle: A Critical Oscillator

Martin, Hudspeth, PNAS (2001)

)(~

)(~

)(~ωωωχ

fX

=

Choe et al., PNAS (1998)Camalet et al., PNAS (2000)Ospeck et al., Biophys. J. (2001)

fZZiuuZirZ at ++−+−≅∂ 20 )()( ω

)(Re ZX ≅

Fef i 1 θ−Λ≅

0 ; 0 ωω ==r

Critical Point

€

X

F∝ F

−2 / 3

5 m

A.J. Hudspeth

5 µm

P. Gillespie

Parallel or series configuration ?

Structure and High-Frequency Response

Lnc

L

nc

Ltt )1(

2

/)(

wwwwaterobject −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=−=

λπωωφ

L, nDenk et al., PNAS (1989)Denk, Webb, Appl. Opt. (1990)

Interferometry

Experimental set-up

A. Kozlov, A. Hinterwirth

Measured quantities

X2X1

€

C12(τ ) = X1(t)X2(t + τ ) ; C12( f )Correlation Functions

€

γ( f ) =C12( f )

S11( f ) S22( f )

Coherency SpectrumCoherence Spectrum

€

γ( f )

Phase Spectrum

€

arg γ( f )[ ]

Folded epithelium; bullfrog sacculus

Rms amplitudes: ≈ 5 - 6 nm

Glass fiber Adjacent bundles

Statistics Across Cells

Bullfrog Sacculus

Averaged Cross-Correlation Peak (20 records): 0.95 ± 0.01 (0.97 ± 0.02 same spot)

Coherence and Phase SD: N=29 and N=38 measurements from the same 18 Cells

Kozlov*, Risler*, Hudspeth, Nat. Neur. 10, 87 (2007)

Spectral Estimators, Leakage and Confidence Intervals

∫−

Δ−−=N

N

f

f

tfNni fdZenx )()( ]2/)1([2π

Known: )(nx Wanted: )( fS

2)( )( fdZdffS =

∑−

=

Δ−=1

0

2 )()(N

n

tfni nxefy πAccessible:

€

y( f ) =sinNπ ( f − v)Δt

sinπ ( f − v)Δt− fN

fN

∫ dZ(v) = K ∗dZ( f )

)2/(1 tfN Δ=

“Multi-Taper”:

€

yk ( f ) =1

λ k (N,W )Uk (N,W ;v)

−W

W

∫ y( f + v)dv

D. Thomson,Proc. IEEE (1982)

Kozlov, Risler et al.,In preparation.

2

0

1

00 )(

),(

1

2

1)( fy

WNNWfS k

K

k k∑−

=

=λ

- Bias- Lack of Consistency

Synchronization Transitionor

Hopf Bifurcation ofCoupled Oscllators

∞→τξ ,

Coupled Oscillators

ξ

t

)(tC

)()0()( tXXtC =

Fluctuations and spontaneous oscillations

Noisy Oscillator

CGLE: Aranson, Kramer, Rev. Mod. Phys. (2002)

ZZiuuZirZ at

2

0 )()( +−+−=∂ ω

ηθ +Λ+Δ++ − FeZicc ida

1)(

Real Case 0 ; 0 == aa cu

Feef

ZeAiti

ti

θω

ω

10

0

−Λ=

=

ξ++Δ+−−=∂ fAcAAurAA dt

2

)Re(A

V

Field Theory of Coupled Oscillators

Exact Mapping to the XY Model

)( 00 ' ' tiRtiR effeAA δωδθδω +==

ab⋅'

'

f

A

Feef

ZeAiti

ti

θω

ω

10

0

−Λ=

=a

Renormalization

Perturbation Theory

α

β

γ

σ

α=αψ

α=αψ~α β==

0

0βααβ ψψC

α β==0

0 ~βααβ ψψχ

α

γ

β

σ

( ) =+− γσαβαβ δεδ auu

€

Z =ψ1 + iψ 2( )

ε−=4dOne-Loop Order

ac

u

u

r

€

+θ(b)..

Renormalization Group: Flow Diagram

€

ω0(b)

€

b0ω

eff0ω

α β

α β

Two-Loop Order

Risler, Prost, Jülicher, PRL 93, 175702 (2004)

( )12

eff011

effeff

''12eff

''11eff

2

1 sin cos CiC

Dωωχθχθ +

Λ=+

Fixed Point Dynamical XY Model: Equilibrium Fixed Point!

( )

( )⎥⎦⎤

⎢⎣

⎡

+Λ≅= − qic

e

qq

qi

γωωχ

θ

η2eff

eff0

1

2

1),(

50/2εη ≅

( ) ( ) 221effeffeffeff ; ωωω γγβαθθ qqqqq ≅++≅

Response Function

50/ ; 5/ 221 εωεω ≅≅

Results

Risler, Prost, Jülicher, PRE 72, 016130 (2005)

Actual Synchronization Transitions ?

Cochlear nonlinear response

1/31

Ruggero et al.,J. Acoust. Soc. Am. (1997)

Yasuda et al., Biophys. J. (1996)

Sarcomeric Oscillations

http://www.ux.his.no/~ruoff/BZ_Phenomenology.html

Belousov-Zhabotinsky

Numerical Confirmation Wood et al., Phys. Rev. Lett. (2006)Wood et al., Phys. Rev. E (2006)

Summary

P. Gillespie

5 µm

A highly-specialized nonlinear structure

Its sensitivity relies on global coupling of the ion channels

Multi-Taper Data Analysis

The Hair Bundle

Synchronization TransitionHopf Bifurcation of Coupled Oscillators: An out-of-equilibrium Phase Transition

Renormalization Group and Flow

New Universal Properties

Frank JülicherJacques Prost

Acknowledgements

Hair Cells

Markus BärEdouard Brézin

Erwin FreyKay Wiese

Karsten KrusePascal Martin

Björn NadrowskiOlivier Giraud

Synchronization Transition

Andrei KozlovJim Hudspeth

Armin HinterwirthMarcelo Magnasco

Omar AhmadBrian FabellaDaniel AndorLoïc Legoff