Signal amplification and information transmission in neural systems Benjamin Lindner Department of Biological Physics Max-Planck-Institut für Physik mpipks group komplexer Systeme Dresden Stochastic Processes in Biophysics Tuesday, January 26, 2010
Outline • Dynamics of coupled hair bundles - enhanced signal amplification by means of coupling-induced noise reduction - Intro - Numerical simulation approach - Experimental approach - Analytical approach • Effects of short-term plasticity on neural information transfer - Intro - Broadband coding of information for a simple rate-coded signal - different presynaptic populations: frequency-dependent info transfer spike trains 0 0 1 2 3 time by additional noise . -Summary Tuesday, January 26, 2010
PART 1 • HAIRBUNDLE DYNAMICS Tuesday, January 26, 2010
Range of frequencies and frequency resolution Hearing range: 20Hz - 20kHz Two neighboring piano keys Difference of 6% Perceptible difference in hearing < 1% changes in frequency Tuesday, January 26, 2010
Range of sound amplitudes Wide dynamic range (6 orders of magnitude in sound pressure) 0 dB sound pressure level (SPL) absolute hearing threshold for humans 20 ∗ 10 − 9 % of the normal air pressure 120 dB sound pressure level (SPL) Loud rock group 20 ∗ 10 − 3 % of the normal air pressure Tuesday, January 26, 2010
www.vestibular.org Tuesday, January 26, 2010
Sound elicits a traveling wave of the basilar membrane Position of maximum vibration depends on frequency “tonotopic mapping” http://www1.appstate.edu/~kms/classes/psy3203/Ear/ Neurotransmitter causes action potentials that are sent to the brain Tuesday, January 26, 2010
The response of the basilar membrane to pure tones normal air Change in pressure pressure 2p 5 0 p=200 µ Pa -5 Basilar 5 membrane p=2000 µ Pa 0 vibrations -5 [nm] 5 p=200 mPa 0 -5 time Tuesday, January 26, 2010
The response of the basilar membrane to pure tones Nonlinear compression log 10 (BM vib) 1 1/4 ~P 0.5 Output ~P 0 -0.5 1 -3/4 log 10 ( χ ) 0.5 ~P 0 Sensitivity=Output/Input -0.5 -1 -1.5 Local Exponent -1.5 -2 -1 0 1 2 log 10 (P/P 0 ) guinea pig: data from Robles & Ruggero Physiol. Rev. 2001 Tuesday, January 26, 2010
The response of the basilar membrane to pure tones Sharp tuning Basilar membran vibration [a.u.] 3 10 2 10 1 10 0 10 0 10 20 30 Frequency [kHz] guinea pig: data from Robles & Ruggero Physiol. Rev. 2001 Tuesday, January 26, 2010
The big question What is the active mechanism which underlies frequency selectivity and nonlinear compression? Tuesday, January 26, 2010
Basilar membrane vibrations are transduced by hair cells into an electric current which is signaled to the brain Neurotransmitter causes action potentials that are sent to the brain Tuesday, January 26, 2010
Hair cells are an essential part of the cochlear amplifier inner hair basilar cells membrane outer hair cells from the Cochlea homepage from Dallos et al. The Cochlea Tuesday, January 26, 2010
Experimental model system: hair bundle from the sacculus of bullfrog Martin et al. J. Neurosci. 2003 Martin et al. PNAS 2001 Tuesday, January 26, 2010
A single hair bundle shows tuning and nonlinear compression f − 2 / 3 Martin & Hudspeth PNAS 2001 Tuesday, January 26, 2010
A stochastic model of a single hair bundle reproduces these features Tuesday, January 26, 2010
Spontaneous activity of the hair bundle Tuesday, January 26, 2010
Stimulated activity of the hair bundle - analytical results vs experiment Experiment Two-state theory noisy Hopf oscillator Theory 8 Power spectrum Simulations 6 4 2 0.6 0.8 1 1.2 1.4 ω 10 Theory χ ' Simulations 5 0 0 0.5 1 1.5 2 6 4 2 χ " 0 -2 -4 -6 0 0.5 1 1.5 2 frequency Clausznitzer, Lindner, Jülicher & Martin Jülicher, Dierkes, Lindner, Prost, & Martin Phys. Rev. E (2008) Eur. Phys. J. E (2009) Tuesday, January 26, 2010
A single hair bundle shows tuning and nonlinear compression f − 2 / 3 ... but only precursors (compared with the cochlea!) Martin & Hudspeth PNAS 2001 Tuesday, January 26, 2010
Coupling by membranes tectorial membrane cochlea Tuesday, January 26, 2010
Numerical approach X i,j = f X ( X i,j , X i,j λ ˙ a ) + F ext ( t ) + η i,j ( t ) 1 � � ∂U ( X i,j , X i + k,j + l ) /∂X i,j − k,l = − 1 λ a ˙ X i,j f X a ( X i,j , X i,j a ) + η i,j = a ( t ) , a Tuesday, January 26, 2010
Tuesday, January 26, 2010
Coupling among hair cells results in refined frequency tuning... 1000 1 x 1 HBs 1 x 1 HBs 1 3 x 3 HBs 3 x 3 HBs 4 x 4 HBs 4 x 4 HBs 6 x 6 HBs 6 x 6 HBs Sensitivity [nm/pN] 0.5 9 x 9 HBs 9 x 9 HBs 100 0 -2 0 2 10 -2 -1 0 1 2 Frequency mismatch [Hz] Dierkes, Lindner & Jülicher PNAS (2008) Tuesday, January 26, 2010
Coupling among hair cells results in refined frequency tuning and enhanced signal compression 1 x 1 HBs 3 10 3 x 3 HBs Sensitivity [nm/pN] 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs 2 10 -0.88 ~ F 1 10 0 10 -2 -1 0 1 2 3 10 10 10 10 10 10 F [pN] Dierkes, Lindner & Jülicher PNAS (2008) Tuesday, January 26, 2010
Coupling among hair cells results in refined frequency tuning and enhanced signal compression through noise reduction! coupled system 1 x 1 HBs 3 3 10 10 single hair bundle 3 x 3 HBs Sensitivity [nm/pN] Sensitivity [nm/pN] 4 x 4 HBs with reduced noise 6 x 6 HBs 9 x 9 HBs 2 2 10 10 -0.88 ~ F 1 1 10 10 decrease of intrinsic noise by 1/N 0 0 10 10 -2 -1 0 1 2 3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 F [pN] F [pN] Tuesday, January 26, 2010
Experimental approach Tuesday, January 26, 2010
Experimental confirmation: coupling a hair bundle to two cyber clones X No coupling X 1 X X 2 Real-time Cyber simulation clone 1 F EXT F EXT F EXT 20 nm Hair F 1 F INT F 2 K = 0.4 pN/nm 100 ms bundle Cyber clone 2 Cyber Hair Cyber bundle 1 bundle bundle 2 Δ Experiments by Jérémie Barral & Kai Dierkes in the lab of Pascal Martin (Paris) Tuesday, January 26, 2010
Experimental confirmation: coupling enhances response to periodic stimulus coupled hair bundle isolated hair bundle Experiments by Jérémie Barral & Kai Dierkes in the lab of Pascal Martin (Paris) Tuesday, January 26, 2010
Analytical approach � I 0 ( fρ d /D ) � α = d ln( | χ | ) d ln( f ) = f ρ d + ρ � d f I 1 ( fρ d /D ) − I 1 ( fρ d /D ) − 2 D I 0 ( fρ d /D ) fρ d /D � 1 ⇒ α ≈ 0 D � f � ρ d (5 Cρ 4 d + 3 Bρ 2 d + r ) ⇒ α ≈ − 1 ρ d � − 2 / 3 : supercritical f ≥ ρ d (5 Cρ 4 d + 3 Bρ 2 d + r ) ⇒ α ≈ − 4 / 5 : subcritical Tuesday, January 26, 2010
Coupled system equivalent to a single oscillator with reduced noise 1 x 1 HBs 3 3 10 10 3 x 3 HBs Sensitivity [nm/pN] Sensitivity [nm/pN] 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs 2 2 10 10 -0.88 ~ F 1 1 10 10 decrease of intrinsic noise by 1/N 0 0 10 10 -2 -1 0 1 2 3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 F [pN] F [pN] Tuesday, January 26, 2010
A generic oscillator: Hopf normal form √ z = − ( r + iω 0 ) z − B | z | 2 z − C | z | 4 z + 2 Dξ ( t ) + fe − iωt ˙ 2 1 Im(z) 0 -1 -2 -2 -1 0 1 2 Re(z) Tuesday, January 26, 2010
Amplitude and phase dynamics √ z = − ( r + iω 0 ) z − B | z | 2 z − C | z | 4 z + 2 Dξ ( t ) + fe − iωt ˙ Polar coordinates ( � ( z ) , � ( z )) ( ρ, φ ) ⇒ Phase difference between oscillator and driving phases ψ ( t ) = φ ( t ) + ωt � z ( t ) � = � ρe iφ ( t ) � = � ρe iψ � e − iωt Mean output is | χ | = |� ρ e iψ �| Sensitivity is f Tuesday, January 26, 2010
Amplitude and phase dynamics √ z = − ( r + iω 0 ) z − B | z | 2 z − C | z | 4 z + 2 Dξ ( t ) + fe − iωt ˙ Phase difference between oscillator and driving phases ψ ( t ) = φ ( t ) + ωt √ 2 D ψ = ∆ ω − f ˙ ρ sin( ψ ) + ξ ( t ) ρ Amplitude dynamics ρ = − rρ − Bρ 3 − Cρ 5 + f cos( ψ ) + D/ρ + √ ˙ 2 Dξ ρ ( t ) Tuesday, January 26, 2010
Amplitude and phase dynamics √ z = − ( r + iω 0 ) z − B | z | 2 z − C | z | 4 z + 2 Dξ ( t ) + fe − iωt ˙ Phase difference between oscillator and driving phases ψ ( t ) = φ ( t ) + ωt √ 2 D ψ = ∆ ω − f ˙ sin( ψ ) + ξ ( t ) ρ d ρ d Amplitude dynamics for r<0 and weak noise we can approximate 0 = − rρ d − Bρ d 3 − Cρ d 5 + f � cos( ψ ) � Tuesday, January 26, 2010
Amplitude and phase dynamics √ z = − ( r + iω 0 ) z − B | z | 2 z − C | z | 4 z + 2 Dξ ( t ) + fe − iωt ˙ Phase difference between oscillator and driving phases ψ ( t ) = φ ( t ) + ωt √ 2 D ψ = ∆ ω − f ˙ sin( ψ ) + ξ ( t ) ρ d ρ d Δω ψ− (f/ ρ d )cos( ψ ) I 1+ i ∆ ωρ 2 d /D ( fρ d ( f ) /D ) � e iψ � = I i ∆ ωρ 2 d /D ( fρ d ( f ) /D ) ψ Haken et al. Z. Phys. 1967 Tuesday, January 26, 2010
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