Signal ampliﬁcation and information transmission in neural

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

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

0 1 2 3

time 0

0 1 2 3

time0

spik

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PART 1

• HAIRBUNDLE DYNAMICS


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%


Range of sound amplitudes

Wide dynamic range (6 orders of magnitude in sound pressure)

0 dB sound pressure level (SPL)

% of the normal air pressure20 ∗ 10−9

120 dB sound pressure level (SPL)

% of the normal air pressure20 ∗ 10−3

absolute hearing threshold for humans

Loud rock group


www.vestibular.orgTuesday, January 26, 2010

http://www.vestibular.org

http://www.vestibular.org

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


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

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

The response of the basilar membrane to pure tones

Change in pressure

-505

-505 Basilar

membranevibrations [nm]

time-505

normal air pressure

2p

p=200 µPa

p=2000 µPa

p=200 mPa


Sensitivity=Output/Input

Robles & Ruggero Physiol. Rev. 2001

Nonlinear compression

guinea pig: data from

Output

-1.5-1

-0.50

0.51

log 10

(χ)

-2 -1 0 1 2log10(P/P0)

-1.5

-1

-0.5

0

Loca

l Exp

onen

t

-0.5

0

0.5

1

log 10

(BM

vib

)

~P1/4

~P-3/4

~P

-1.5-1

-0.50

0.51

log 10

(χ)

-2 -1 0 1 2log10(P/P0)

-1.5

-1

-0.5

0

Loca

l Exp

onen

t

-0.5

0

0.5

1lo

g 10(B

M v

ib)

~P1/4

~P-3/4

~P



Robles & Ruggero Physiol. Rev. 2001

Sharp tuning

0 10 20 30Frequency [kHz]

100

101

102

103

Basil

ar m

embr

an v

ibra

tion

[a.u

.]

guinea pig: data from



The big question

What is the active mechanism which underlies frequency selectivity and

nonlinear compression?


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


Hair cells are an essential part of the cochlear amplifier

outer hair cells

inner hair cells

basilar membrane

from Dallos et al. The Cochlea from the Cochlea homepage


Experimental model system: hair bundle from the sacculus of bullfrog

Martin et al. PNAS 2001Martin et al. J. Neurosci. 2003Tuesday, January 26, 2010

A single hair bundle shows tuning and nonlinear compression

Martin & Hudspeth PNAS 2001

f−2/3


A stochastic model of a single hair bundle reproduces these features


Spontaneous activity of the hair bundle


Stimulated activity of the hair bundle - analytical results vs experiment

0 0.5 1 1.5 20

5

10

χ' Theory

Simulations

0 0.5 1 1.5 2frequency

-6-4-20246

χ"

0.6 0.8 1 1.2 1.4ω

2

4

6

8

Pow

er sp

ectru

m TheorySimulations

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)


A single hair bundle shows tuning and nonlinear compression

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

Martin & Hudspeth PNAS 2001

f−2/3


Coupling by membranes

cochleatectorial membrane


Numerical approach

λXi,j = fX(Xi,j , Xi,ja ) + Fext(t) + ηi,j(t)

−

1∑

k,l=−1

′∂U(Xi,j , Xi+k,j+l)/∂Xi,j

λaXi,ja = fXa

(Xi,j , Xi,ja ) + ηi,j

a (t),



-2 -1 0 1 2Frequency mismatch [Hz]

10

100

1000

Sen

siti

vit

y [

nm

/pN

]

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

-2 0 20

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...


Dierkes, Lindner & Jülicher PNAS (2008)

10-2 10-1 100 101 102 103

F [pN]

100

101

102

103

Sens

itivi

ty [n

m/p

N]


~ F -0.88

Coupling among hair cells results in refined frequency tuningand enhanced signal compression


10-2 10-1 100 101 102 103

F [pN]

100

101

102

103

Sens

itivi

ty [n

m/p

N]


~ F -0.88

10-2 10-1 100 101 102 103

F [pN]

100

101

102

103

Sens

itivi

ty [n

m/p

N]

decrease of intrinsic noise by 1/N

Coupling among hair cells results in refined frequency tuningand enhanced signal compression

through noise reduction!

coupled systemsingle hair bundle

with reduced noise


Experimental approach


Experimental confirmation: coupling a hair bundle to two cyber clones

Cyberbundle 1

Hairbundle

Cyberbundle 2

FEXT

FEXT

FEXTF

1 FINT

F2

Δ

X

Real-timesimulation

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 n

m Hair bundle

Cyberclone 1

Cyberclone 2


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


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


Coupled system equivalent to a single oscillator with reduced noise

10-2 10-1 100 101 102 103

F [pN]

100

101

102

103

Sens

itivi

ty [n

m/p

N]


~ F -0.88

10-2 10-1 100 101 102 103

F [pN]

100

101

102

103

Sens

itivi

ty [n

m/p

N]

decrease of intrinsic noise by 1/N


A generic oscillator: Hopf normal form

z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt

-2 -1 0 1 2Re(z)

-2

-1

0

1

2

Im(z)


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)〉 = 〈ρeiψ〉e−iωt

Polar coordinates (!(z),"(z)) ⇒ (ρ, φ)

Phase difference between oscillator and driving phases

ψ(t) = φ(t) + ωt

Sensitivity is |χ| =

|〈ρeiψ〉|

f



z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt


ψ = ∆ω −

f

ρsin(ψ) +

√

2D

ρξ(t)

Amplitude dynamics

ρ = −rρ − Bρ3− Cρ5 + f cos(ψ) + D/ρ +

√

2Dξρ(t)

ψ(t) = φ(t) + ωt



z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt


Amplitude dynamics

0 = −rρd − Bρd3 − Cρd

5 + 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



z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt


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)


Solution for the sensitivity

0 = −rρd − Bρ3

d − Cρ5

d + f"〈e−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)Tuesday, January 26, 2010

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

Exponents for ... ∆ω = 0

10-2

100

102

104

|χ|

D = 10-4

D = 10-3

D = 10-2

D = 10-1 10-2

100

102

104

|χ|

10-6 10-4 10-2 100 102

f

-1

-0.8

-0.6

-0.4

-0.2

0

α

10-6 10-4 10-2 100 102

f

-1

-0.8

-0.6

-0.4

-0.2

0α

0

-2/3

-1

0

-4/5-1

(a)SUPERCRITICAL

(b)SUBCRITICAL

~f-1 ~f-1

~f-4/5~f-2/3

-1

Noisy normal form

100

102

104

|!|

[nm

|/pN

]

100

102

104

10-4 10-2 100 102

f [pN]

-1-0.8-0.6-0.4-0.2

0

"

10-4 10-2 100 102

f [pN]

-1-0.8-0.6-0.4-0.2

0

"

0

-2/3

-1

0

-1

(a)OP 1

(b)OP 2

# = 10-4

= 10-3

= 10-2

= 10-1

= 10 0

####

|!|

[nm

/pN

]

Stochastic Hair bundle model


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

LRSNC SNC

LR

f f

D DNC

HB modelnumerically from sensitivity curves

subcritical Hopf oscillatorfrom formula

Exponents of nonlinear compression


•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 bundlesas an essential part of the cochlear amplifier

Summary


PART 2

• SHORT-TERM PLASTICITY AND INFORMATION TRANSFER


output spikes

synaptic background

+signals

Central questionHow do dynamic synapses affect

the transfer of time-dependent signals and noise?

dynamic synapses(short-term plasticity)

Setting


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)


0 1 2 3

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0 1 2 3

time0

spik

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input spike trains

F-D

F-D

Synaptic facilitation and

depression

.

.

.

0 1 2 3

time 0

0 1 2 3

time0

spik

e tra

ins

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.

.

Synapticinput

∑δ(t− ti,j)

∑Ai,jδ(t− ti,j)

Facilitation & depression add an amplitude to each spike


•shift in response times to population bursts Richardson et al. (2005)

•network oscillationsMarinazzo et al. Neural Comp. 2007

•self-organized criticalityLevina et al. Nature Physics 2007

•working memoryMongillo et al. Science 2008

Network levelSingle 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 aboutthe presynaptic spike train seen so far (e.g. Fuhrmann et al. 2001)

Here:information transmissionacross dynamic synapse

Known effects of dynamic synapses


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

Model


1mV100ms

Aj = FjDj

Postsynaptic amplitude

Dynamics for facilitationand depression

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

Model


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


output spikes

dynamic synapses

synaptic input,postsynaptic conductance

Power spectra

Poissonian spike trains

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


Correlation function or power spectra?

0 1 2 3

time 0

0 1 2 3

time0

spik

e tra

ins

.

.

.

Power spectra


100 101

Frequency0

20

40

60

pow

er sp

ectra

dominating depressiondominating facilitationTheory

constant amplitude

Power spectra


02468

10

DDRFDRtheory

00.00050.0010.0015

0204060

spik

e tra

in p

ower

spec

trum

0

0.005

0.01

100 101

frequency0

50

100

100 101

frequency0

0.002

0.004

r=1Hz r=1Hz

r=10Hzr=10Hz

r=100Hz r=100Hz

Voltage pow

er spectrum

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

Power spectra


Modulation of the input firing rate by a periodic signal R(t) = r · [1 + εs(t)]

Model with rate modulation


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



Modulation of the input firing rate by a band-limited Gaussian

white noise (0-100Hz) R(t) = r · [1 + εs(t)]




Sgs = 〈gs∗〉

Cgs =

|Sgs|2

SggSss

x =1

√

T

T∫0

dt e2πift

x(t)Fourier transform

SXs = 〈Xs∗〉

Cross spectra of synaptic input/voltage and input signal

CXs =

|SXs|2

SssSXX

Coherence functions

Spectral measures


Relation to information theoretic measures

Lower bound on mutual information

Error of linear reconstruction

ILB = −∫

df log2[1− C(f)]

ε =∫

dfSss[1− C(f)]

Why the coherence?


broadband coding


CXs =

|SXs|2

SssSXX

Coherence functions for various parameter sets



Cross spectra


broadband coding


CXs =

|SXs|2

SssSXX

Coherence functions for various parameter sets


1

CX,R

=N − 1

N

1

C〈xi〉,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

C〈xi〉,R ≈ 1

Merkel & Lindner submitted (2009)

Why is the coherence flat ?


0.0

0.1

0.2

~|cr

oss

spec

trum

|2 SimulationTheory

0.0

0.1

~po

wer

spec

trum

0.1 1 10frequency [Hz]

0.000

0.002

0.004

cohe

renc

e

A

B

C

0.00

0.01

0.02

0.03

0.04

~|cr

oss

spec

trum

|2

SimulationTheory

0.00

0.01

0.02

0.03

~po

wer

spec

trum

0.1 1 10frequency [Hz]

0.000

0.002

0.004

cohe

renc

e

A

B

C

pure facilitation pure depression

CRx(f) ≈ ε2rSRR(f)

1 + [1+(2πfτF )2]·∆2linrτF /2

(F1+∆linrτF )2+(2πfτF )2·F 21

F1 = F0,lin + DlinrτFwith

CRx(f) ≈ ε2rSRR(f) ·[1− F 2

0 rτD

2β

]


Coherence for a single synapse


simulation value(theoretical value)


Coherence for a single synapse


1

CX,R

=N − 1

N

1

C〈xi〉,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

C〈xi〉,R ≈ 1


Why is the coherence flat ?


0.01 0.1 1 10frequency [Hz]

0.0001

0.001

0.01

0.1

1

cohe

renc

e C RX

SimulationTheory

N=1

N=10

N=100

N=1000

N=10000


Coherence-dependence on the number N of synapses


Extension I

Postsynaptic spiking


+µ

if V = −65mV then ti = t & V = −70mV

LIF output spike train


10-2 10-1 100 101 1020

0.2

0.4

0.6

0.8

1

γ 2 /c

PIFLIFQIF

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 1010

0.2

0.4

0.6

0.8

1

γ 2 /c

10-2 10-1 100 101 10-2 10-1 100 101

10-2 10-1 100 101

f

0

0.2

0.4

0.6

0.8

1

γ 2 /c

10-2 10-1 100 101

f10-3 10-2 10-1 100 101

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



Coherence -LIF output spike train


output spikes

dynamic synapses

R(t) = r · [1 + εs(t)]

synaptic input,postsynaptic conductance,output spike train

Info about R(t)broadband coding

So far: one presynaptic population with one rate modulation


Extension II

Extra Noise channel


R(t)

facilitation-dominated synapses

depression-dominated synapses

spikes with rate modulation

spikes with constant rate

(just noise)

output spikes

Extra noise


1CRX(f)

= 1N · 1

CRxi(f) + N−1N · 1

CR〈xi〉(f)

+ 1N · 1

CRxi (f) · Sηη(f)NSxixi (f)

0

10

20

30

40

spec

tra [H

z] Sηη

NSxx


0.00

0.01

0.02

cohe

renc

e

CRX (Simulation)CRX (Theory)

A

B

Facilitating synapses for signalDepressing synapses for noise

0

10

20

30

40

spec

tra [H

z] Sηη

NSxx


0

0.05

0.1

0.15

0.2

0.25

cohe

renc

e

CRX (Theory)CRX (Simulation)

Depressing synapses for signalFacilitating synapses for noise

Merkel & Lindner in preparation (2009)

Extra noise


R(t)

facilitation-dominated synapses

depression-dominated synapses

spikes with rate modulation

spikes with constant rate

(just noise)

output spikes

synaptic input,postsynaptic conductance,output spike train

Info about R(t)low or highpass coding

possible

Extra noise


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 channelsare present


Documents

Signal ampliﬁcation and information transmission in neural