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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
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
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 )[ ]
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
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