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Signal amplification and information transmission in neural systems Benjamin Lindner Department of Biological Physics Max-Planck-Institut fr Physik mpipks group komplexer Systeme Dresden Stochastic Processes in Biophysics Tuesday,

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SLIDE 1

Signal amplification and information transmission in neural systems

Stochastic Processes in Biophysics

mpipks group

Benjamin Lindner Department of Biological Physics Max-Planck-Institut für Physik komplexer Systeme Dresden

Tuesday, January 26, 2010

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SLIDE 2

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 by additional noise

  • Summary

1 2 3

time

spike trains

.

Tuesday, January 26, 2010

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SLIDE 3

PART 1

  • HAIRBUNDLE DYNAMICS

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SLIDE 4

Range of frequencies and frequency resolution

Perceptible difference in hearing < 1% changes in frequency Hearing range: 20Hz - 20kHz Two neighboring piano keys Difference of 6%

Tuesday, January 26, 2010

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SLIDE 5

Range of sound amplitudes

Wide dynamic range (6 orders

  • f magnitude in sound pressure)

0 dB sound pressure level (SPL) % of the normal air pressure

20 ∗ 10−9

120 dB sound pressure level (SPL) % of the normal air pressure

20 ∗ 10−3

absolute hearing threshold for humans Loud rock group

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SLIDE 6

www.vestibular.org

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SLIDE 7

http://www1.appstate.edu/~kms/classes/psy3203/Ear/

Sound elicits a traveling wave of the basilar membrane

Position of maximum vibration depends on frequency

“tonotopic mapping”

Neurotransmitter causes action potentials that are sent to the brain

Tuesday, January 26, 2010

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SLIDE 8

The response of the basilar membrane to pure tones

Change in pressure

  • 5

5

  • 5

5

Basilar membrane vibrations [nm]

time

  • 5

5

normal air pressure 2p

p=200 µPa p=2000 µPa p=200 mPa

Tuesday, January 26, 2010

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SLIDE 9

Sensitivity=Output/Input Robles & Ruggero Physiol. Rev. 2001

Nonlinear compression

guinea pig: data from

Output

  • 1.5
  • 1
  • 0.5

0.5 1

log10(χ) Local Exponent

  • 0.5

0.5 1

log10(BM vib)

~P

1/4

~P

  • 3/4

~P

  • 2
  • 1

1 2

log10(P/P0)

  • 1.5

The response of the basilar membrane to pure tones

Tuesday, January 26, 2010

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SLIDE 10

Robles & Ruggero Physiol. Rev. 2001

Sharp tuning

10 20 30

Frequency [kHz]

10 10

1

10

2

10

3

Basilar membran vibration [a.u.]

guinea pig: data from

The response of the basilar membrane to pure tones

Tuesday, January 26, 2010

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SLIDE 11

The big question

What is the active mechanism which underlies frequency selectivity and nonlinear compression?

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SLIDE 12

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

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SLIDE 13

Hair cells are an essential part of the cochlear amplifier

  • uter hair cells

inner hair cells basilar membrane

from Dallos et al. The Cochlea from the Cochlea homepage

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SLIDE 14

Experimental model system: hair bundle from the sacculus of bullfrog

Martin et al. PNAS 2001 Martin et al. J. Neurosci. 2003

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SLIDE 15

A single hair bundle shows tuning and nonlinear compression

Martin & Hudspeth PNAS 2001

f −2/3

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SLIDE 16

A stochastic model of a single hair bundle reproduces these features

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SLIDE 17

Spontaneous activity of the hair bundle

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SLIDE 18

Stimulated activity of the hair bundle - analytical results vs experiment

0.5 1 1.5 2 5 10

χ'

Theory Simulations 0.5 1 1.5 2

frequency

  • 6
  • 4
  • 2

2 4 6

χ"

0.6 0.8 1 1.2 1.4

ω

2 4 6 8

Power spectrum

Theory Simulations

Experiment Two-state theory noisy Hopf oscillator

Clausznitzer, Lindner, Jülicher & Martin

  • Phys. Rev. E (2008)

Jülicher, Dierkes, Lindner, Prost, & Martin

  • Eur. Phys. J. E (2009)

Tuesday, January 26, 2010

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SLIDE 19

A single hair bundle shows tuning and nonlinear compression

... but only precursors (compared with the cochlea!)

Martin & Hudspeth PNAS 2001

f −2/3

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SLIDE 20

Coupling by membranes

cochlea tectorial membrane

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SLIDE 21

Numerical approach

λ ˙ Xi,j = fX(Xi,j, Xi,j

a ) + Fext(t) + ηi,j(t)

−

1

  • k,l=−1

∂U(Xi,j, Xi+k,j+l)/∂Xi,j

λa ˙ Xi,j

a

= fXa(Xi,j, Xi,j

a ) + ηi,j a (t),

Tuesday, January 26, 2010

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SLIDE 22

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SLIDE 23
  • 2
  • 1

1 2

Frequency mismatch [Hz]

10 100 1000

Sensitivity [nm/pN] 1 x 1 HBs 3 x 3 HBs 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs

  • 2

2 0.5 1

1 x 1 HBs 3 x 3 HBs 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs

Dierkes, Lindner & Jülicher PNAS (2008)

Coupling among hair cells results in refined frequency tuning...

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SLIDE 24

Dierkes, Lindner & Jülicher PNAS (2008) 10

  • 2

10

  • 1

10 10

1

10

2

10

3

F [pN]

10 10

1

10

2

10

3

Sensitivity [nm/pN]

1 x 1 HBs 3 x 3 HBs 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs

~ F

  • 0.88

Coupling among hair cells results in refined frequency tuning and enhanced signal compression

Tuesday, January 26, 2010

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SLIDE 25

10

  • 2

10

  • 1

10 10

1

10

2

10

3

F [pN]

10 10

1

10

2

10

3

Sensitivity [nm/pN]

1 x 1 HBs 3 x 3 HBs 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs

~ F

  • 0.88

10

  • 2

10

  • 1

10 10

1

10

2

10

3

F [pN]

10 10

1

10

2

10

3

Sensitivity [nm/pN]

decrease of intrinsic noise by 1/N

Coupling among hair cells results in refined frequency tuning and enhanced signal compression through noise reduction!

coupled system single hair bundle with reduced noise

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SLIDE 26

Experimental approach

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SLIDE 27

Experimental confirmation: coupling a hair bundle to two cyber clones

Cyber bundle 1 Hair bundle Cyber bundle 2 FEXT FEXT FEXT F1 FINT F2 Δ X Real-time simulation X1 X X2

Experiments by Jérémie Barral & Kai Dierkes in the lab of Pascal Martin (Paris)

No coupling K = 0.4 pN/nm

100 ms 20 nm

Hair bundle

Cyber clone 1 Cyber clone 2

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SLIDE 28

Experimental confirmation: coupling enhances response to periodic stimulus

Experiments by Jérémie Barral & Kai Dierkes in the lab of Pascal Martin (Paris)

coupled hair bundle isolated hair bundle

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SLIDE 29

Analytical approach

fρd/D 1 ⇒ α ≈ 0

α = d ln(|χ|) d ln(f) = f ρd + ρ

df

D I0(fρd/D) I1(fρd/D) − I1(fρd/D) I0(fρd/D)

  • − 2

D ρd f ρd(5Cρ4

d + 3Bρ2 d + r) ⇒ α ≈ −1

f ≥ ρd(5Cρ4

d + 3Bρ2 d + r) ⇒ α ≈

−2/3 : supercritical −4/5 : subcritical

Tuesday, January 26, 2010

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SLIDE 30

Coupled system equivalent to a single

  • scillator with reduced noise

10

  • 2

10

  • 1

10 10

1

10

2

10

3

F [pN]

10 10

1

10

2

10

3

Sensitivity [nm/pN]

1 x 1 HBs 3 x 3 HBs 4 x 4 HBs 6 x 6 HBs 9 x 9 HBs

~ F

  • 0.88

10

  • 2

10

  • 1

10 10

1

10

2

10

3

F [pN]

10 10

1

10

2

10

3

Sensitivity [nm/pN]

decrease of intrinsic noise by 1/N

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SLIDE 31

A generic oscillator: Hopf normal form

˙ z = −(r + iω0)z − B|z|2z − C|z|4z + √ 2Dξ(t) + fe−iωt

  • 2
  • 1

1 2

Re(z)

  • 2
  • 1

1 2

Im(z)

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SLIDE 32

Amplitude and phase dynamics

˙ z = −(r + iω0)z − B|z|2z − C|z|4z + √ 2Dξ(t) + fe−iωt

Mean output is

z(t) = ρeiφ(t) = ρeiψe−iωt

Polar coordinates

((z), (z)) ⇒ (ρ, φ)

Phase difference between oscillator and driving phases

ψ(t) = φ(t) + ωt

Sensitivity is

|χ| = |ρeiψ| f

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SLIDE 33

Amplitude and phase dynamics

˙ z = −(r + iω0)z − B|z|2z − C|z|4z + √ 2Dξ(t) + fe−iωt

Phase difference between oscillator and driving phases

˙ ψ = ∆ω − f ρ sin(ψ) + √ 2D ρ ξ(t)

Amplitude dynamics

˙ ρ = −rρ − Bρ3 − Cρ5 + f cos(ψ) + D/ρ + √ 2Dξρ(t)

ψ(t) = φ(t) + ωt

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SLIDE 34

Amplitude and phase dynamics

˙ z = −(r + iω0)z − B|z|2z − C|z|4z + √ 2Dξ(t) + fe−iωt

Phase difference between oscillator and driving phases Amplitude dynamics

0 = −rρd − Bρd3 − Cρd5 + f cos(ψ)

for r<0 and weak noise we can approximate

˙ ψ = ∆ω − f ρd sin(ψ) + √ 2D ρd ξ(t)

ψ(t) = φ(t) + ωt

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SLIDE 35

Amplitude and phase dynamics

˙ z = −(r + iω0)z − B|z|2z − C|z|4z + √ 2Dξ(t) + fe−iωt

Phase difference between oscillator and driving phases

ψ

Δω ψ−(f/ρd)cos(ψ)

˙ ψ = ∆ω − f ρd sin(ψ) + √ 2D ρd ξ(t)

ψ(t) = φ(t) + ωt

Haken et al. Z. Phys. 1967

eiψ = I1+i∆ωρ2

d/D(fρd(f)/D)

Ii∆ωρ2

d/D(fρd(f)/D)

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SLIDE 36

Solution for the sensitivity

0 = −rρd − Bρ3

d − Cρ5 d + fe−iψ

e−iψ = I1+i∆ωρ2

d/D(fρd(f)/D)

Ii∆ωρ2

d/D(fρd(f)/D)

|χ| = ρd(f) f

  • I1+i∆ωρ2

d/D(fρd(f)/D)

Ii∆ωρ2

d/D(fρd(f)/D)

  • Lindner, Dierkes & Jülicher Phys.Rev.Lett. (2009)

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SLIDE 37

Instead of fitting power laws ...

∆ω = 0 fρd/D 1 ⇒ α ≈ 0

... let’s calculate the local exponent ( )

α = d ln(|χ|) d ln(f) = f ρd + ρ

df

D I0(fρd/D) I1(fρd/D) − I1(fρd/D) I0(fρd/D)

  • − 2

D ρd f ρd(5Cρ4

d + 3Bρ2 d + r) ⇒ α ≈ −1

f ≥ ρd(5Cρ4

d + 3Bρ2 d + r) ⇒ α ≈

  • −2/3

: supercritical −4/5 : subcritical

Lindner, Dierkes & Jülicher Phys.Rev.Lett. (2009)

Tuesday, January 26, 2010

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SLIDE 38

Exponents for ...

∆ω = 0

10

  • 2

10 10

2

10

4

|χ|

D = 10

  • 4

D = 10

  • 3

D = 10

  • 2

D = 10

  • 1

10

  • 6

10

  • 4

10

  • 2

10 10

2

f

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

α

  • 2/3
  • 1

(a)

SUPERCRITICAL

~f

  • 1

~f

  • 2/3
  • 1

Noisy normal form

10 10

2

10

4

10

  • 4

10

  • 2

10 10

2

f [pN]

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 1

(a) (b)

OP 2

|| [nm/pN] Stochastic Hair bundle model

Tuesday, January 26, 2010

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SLIDE 39

Comparison to the hair bundle model

1e-02 1e-01 1e+00 1e+01 1e+02 1e+03 1e-03 1e-02 1e-01 1e+00

NC

  • 1.0
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.0 0.2

1e-02 1e-01 1e+00 1e+01 1e+02 1e+03 1e-03 1e-02 1e-01 1e+00

LR LR SNC SNC LR

f f

D D

NC

HB model numerically from sensitivity curves subcritical Hopf oscillator from formula

Exponents of nonlinear compression

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SLIDE 40
  • sharp tuning and high exponents of nonlinear

compression through coupling-induced noise reduction

  • numerical, experimental, and analytical results give a

unique picture of small groups of coupled hair bundles as an essential part of the cochlear amplifier

Summary

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SLIDE 41

PART 2

  • SHORT-TERM PLASTICITY AND INFORMATION TRANSFER

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SLIDE 42
  • utput spikes

synaptic background + signals

Central question How do dynamic synapses affect the transfer of time-dependent signals and noise?

dynamic synapses (short-term plasticity)

Setting

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SLIDE 43

1mV 100ms

Lewis &Maler J. Neurophysiol. (2002) Abbott & Regehr Nature. (2004)

EPSCs Field potentials

depression facilitation facilitation facilitation

Change in the released transmitter by incoming spikes Increase in efficacy = synaptic facilitation Decrease in efficacy = synaptic depression

[Markram & Tsodyks 1997, Abbott et al. 1997, Zucker & Regehr 2002]

Short-term plasticity (STP)

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SLIDE 44

1 2 3

time

1 2 3

time

spike trains

. . .

input spike trains F-D F-D Synaptic facilitation and depression

. . .

1 2 3

time

1 2 3

time

spike trains

. . .

Synaptic input

  • δ(t − ti,j)
  • Ai,jδ(t − ti,j)

Facilitation & depression add an amplitude to each spike

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SLIDE 45
  • shift in response times to population

bursts Richardson et al. (2005)

  • network oscillations

Marinazzo et al. Neural Comp. 2007

  • self-organized criticality

Levina et al. Nature Physics 2007

  • working memory

Mongillo et al. Science 2008

Network level Single neurons

  • sensory adaptation and decorrelation

(Chung et al. 2002)

  • input compression

(Tsodyks & Markram 1997, Abbott et al. 1997)

  • switching between different neural codes

(Tsodyks & Markram 1997)

  • spectral filtering

(Fortune & Rose 2001, Abbott et al. 1997)

  • synaptic amplitude can keep info about

the presynaptic spike train seen so far (e.g. Fuhrmann et al. 2001)

Here: information transmission across dynamic synapse

Known effects of dynamic synapses

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SLIDE 46

(similar to phenomenological models by Abbott et al. and Tsodyks & Markram)

Model

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SLIDE 47

1mV 100ms

Aj = FjDj

Postsynaptic amplitude

Dynamics for facilitation and depression

Dittman et al. J. Neurosci. (2000), Lewis &Maler J. Neurophysiol. (2002,2004)

Model

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SLIDE 48

Synaptic inputs Conductance dynamics Membrane voltage dynamics [postsynaptic spiking with fire&reset rule (LIF)] Presynaptic spike trains

  • δ(t − ti,j)

Conductance and voltage dynamics

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SLIDE 49
  • utput spikes

dynamic synapses

synaptic input, postsynaptic conductance Power spectra Poissonian spike trains

Effect of FD dynamics on the temporal structure of the postsynaptic activity

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SLIDE 50

Correlation function or power spectra?

1 2 3

time

spike trains

.

Power spectra

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SLIDE 51

10 10

1

Frequency

20 40 60

power spectra

dominating depression dominating facilitation Theory constant amplitude

Power spectra

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SLIDE 52

2 4 6 8 10

DDR FDR theory

20 40 60

spike train power spectrum

10 10

1

frequency

50 100

r=1Hz r=10Hz r=100Hz

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

Power spectra

Tuesday, January 26, 2010

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SLIDE 53

Modulation of the input firing rate by a periodic signal

R(t) = r · [1 + εs(t)]

Model with rate modulation

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SLIDE 54

R(t) = r · [1 + εs(t)]

SNR largely independent of frequency !

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

Modulation of the input firing rate by a periodic signal

Model with rate modulation

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SLIDE 55

Modulation of the input firing rate by a band-limited Gaussian white noise (0-100Hz)

R(t) = r · [1 + εs(t)]

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

Model with rate modulation

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SLIDE 56

Sgs = ˜ g˜ s∗ Cgs = |Sgs|2 SggSss

˜ x = 1 √ T

T

  • dt e2πiftx(t)

Fourier transform

SXs = ˜ X˜ s∗

Cross spectra of synaptic input/voltage and input signal

CXs = |SXs|2 SssSXX

Coherence functions

Spectral measures

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SLIDE 57

Relation to information theoretic measures Lower bound on mutual information Error of linear reconstruction

ILB = −

  • d

f log2[1 − C(f)] ǫ =

  • d

fSss[1 − C(f)]

Why the coherence?

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SLIDE 58

broadband coding

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

CXs = |SXs|2 SssSXX

Coherence functions for various parameter sets

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SLIDE 59

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

Cross spectra

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SLIDE 60

broadband coding

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

CXs = |SXs|2 SssSXX

Coherence functions for various parameter sets

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SLIDE 61

1 CX,R = N − 1 N 1 Cxi,R + 1 N 1 Cxi,R

Coherence between rate and time-dependent mean value of the single FD modulated spike train Coherence between rate and the single FD modulated spike train

Cxi,R ≈ 1

Merkel & Lindner submitted (2009)

Why is the coherence flat ?

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SLIDE 62

0.0 0.1 0.2

~|cross spectrum|

2

Simulation Theory

0.0 0.1

~power spectrum

0.1 1 10

frequency [Hz]

0.000 0.002 0.004

coherence

A B C

0.00 0.01 0.02 0.03 0.04

~|cross spectrum|

2

Simulation Theory

0.00 0.01 0.02 0.03

~power spectrum

0.1 1 10

frequency [Hz]

0.000 0.002 0.004

coherence

A B C

pure facilitation pure depression

CRx(f) ≈ ε2rSRR(f) 1 + [1+(2πfτF )2]·∆2

linrτF /2

(F1+∆linrτF )2+(2πfτF )2·F 2

1

F1 = F0,lin + DlinrτF

with

CRx(f) ≈ ε2rSRR(f) ·

  • 1 − F 2

0 rτD

2β

  • Merkel & Lindner submitted (2009)

Coherence for a single synapse

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slide-63
SLIDE 63

simulation value (theoretical value) Merkel & Lindner submitted (2009)

Coherence for a single synapse

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SLIDE 64

1 CX,R = N − 1 N 1 Cxi,R + 1 N 1 Cxi,R

Coherence between rate and time-dependent mean value of the single FD modulated spike train Coherence between rate and the single FD modulated spike train

Cxi,R ≈ 1

Merkel & Lindner submitted (2009)

Why is the coherence flat ?

Tuesday, January 26, 2010

slide-65
SLIDE 65

0.01 0.1 1 10

frequency [Hz]

0.0001 0.001 0.01 0.1 1

coherence CRX

Simulation Theory N=1 N=10 N=100 N=1000 N=10000

Merkel & Lindner submitted (2009)

Coherence-dependence on the number N

  • f synapses

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SLIDE 66

Extension I

Postsynaptic spiking

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SLIDE 67

+µ

if V = −65mV then ti = t & V = −70mV

LIF output spike train

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SLIDE 68

10

  • 2 10
  • 1 10

10

1

10

2

0.2 0.4 0.6 0.8 1

γ 2 /c

PIF LIF QIF

10

  • 2 10
  • 1 10

10

1

10

2 10

  • 2 10
  • 1 10

10

1

10

2

10

  • 2 10
  • 1

10 10

1

0.2 0.4 0.6 0.8 1

γ 2 /c

10

  • 2 10
  • 1

10 10

1

10

  • 2 10
  • 1

10 10

1

10

  • 2 10
  • 1 10

10

1

f

0.2 0.4 0.6 0.8 1

γ 2 /c

10

  • 2 10
  • 1 10

10

1

f

10

  • 3 10
  • 2 10
  • 1 10

10

1

f

A B C D E F G H I

Perfect IF Leaky IF QuadraticIF

Coherence functions always low-pass !

Vilela & Lindner

  • Phys. Rev. E (2009)

Coherence for static synapses and different I&F models

Tuesday, January 26, 2010

slide-69
SLIDE 69

Lindner, Gangloff, Longtin & Lewis J. Neurosci. (2009)

Coherence -LIF output spike train

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SLIDE 70
  • utput spikes

dynamic synapses

R(t) = r · [1 + εs(t)]

synaptic input, postsynaptic conductance,

  • utput spike train

Info about R(t) broadband coding

So far: one presynaptic population with

  • ne rate modulation

Tuesday, January 26, 2010

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SLIDE 71

Extension II

Extra Noise channel

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SLIDE 72

R(t)

facilitation-dominated synapses depression-dominated synapses

spikes with rate modulation spikes with constant rate (just noise)

  • utput spikes

Extra noise

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SLIDE 73

1 CRX(f) = 1

N · 1 CRxi(f) + N−1 N

·

1 CRxi(f)

+ 1

N · 1 CRxi(f) · Sηη(f) NSxixi(f)

0.01 0.1 1 10

frequency [Hz]

0.00 0.01 0.02

coherence CRX (Simulation) CRX (Theory)

B

Facilitating synapses for signal Depressing synapses for noise

0.01 0.1 1 10

frequency [Hz]

0.05 0.1 0.15 0.2 0.25

coherence CRX (Theory) CRX (Simulation)

Depressing synapses for signal Facilitating synapses for noise

Merkel & Lindner in preparation (2009)

Extra noise

Tuesday, January 26, 2010

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SLIDE 74

R(t)

facilitation-dominated synapses depression-dominated synapses

spikes with rate modulation spikes with constant rate (just noise)

  • utput spikes

synaptic input, postsynaptic conductance,

  • utput spike train

Info about R(t) low or highpass coding possible

Extra noise

Tuesday, January 26, 2010

slide-75
SLIDE 75

Summary

  • analytical results for FD dynamics under Poissonian stimulation
  • “information filtering” not affected by FD dynamics -

broadband coding at the level of the conductance dynamics

  • “information filtering” possible if additional noise channels

are present

Tuesday, January 26, 2010