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

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://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

The response of the basilar membrane to pure tones

Change in pressure

-505 Basilar

membranevibrations [nm]

time-505

normal air pressure

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

log 10

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

log 10

~P-3/4

-1.5-1

log 10

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

g 10(B

~P-3/4

Robles & Ruggero Physiol. Rev. 2001

Sharp tuning

0 10 20 30Frequency [kHz]

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

χ' Theory

Simulations

0 0.5 1 1.5 2frequency

-6-4-20246

0.6 0.8 1 1.2 1.4ω

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)

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]

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

-2 0 20

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]

~ 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]

~ F -0.88

10-2 10-1 100 101 102 103

F [pN]

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

1 FINT

Real-timesimulation

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

No coupling

K = 0.4 pN/nm 100 ms

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

I0(fρd/D)

I1(fρd/D)−

I1(fρd/D)

I0(fρd/D)

! 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]

~ F -0.88

10-2 10-1 100 101 102 103

F [pN]

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)

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〉|

z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt

ψ = ∆ω −

ρsin(ψ) +

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 −

sin(ψ) +

ψ(t) = φ(t) + ωt

z = −(r + iω0)z − B|z|2z − C|z|4z +√

2Dξ(t) + fe−iωt

Δω ψ−(f/ρd)cos(ψ)

ψ = ∆ω −

sin(ψ) +

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)

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

I0(fρd/D)

I1(fρd/D)−

I1(fρd/D)

I0(fρd/D)

! 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

D = 10-4

D = 10-3

D = 10-2

D = 10-1 10-2

10-6 10-4 10-2 100 102

-4/5-1

(a)SUPERCRITICAL

(b)SUBCRITICAL

~f-1 ~f-1

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

Noisy normal form

10-4 10-2 100 102

f [pN]

-1-0.8-0.6-0.4-0.2

10-4 10-2 100 102

f [pN]

-1-0.8-0.6-0.4-0.2

(a)OP 1

(b)OP 2

# = 10-4

= 10-3

= 10-2

= 10-1

= 10 0

Stochastic Hair bundle model

Comparison to the hair bundle model

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

LRSNC SNC

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

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

time 0

0 1 2 3

input spike trains

Synaptic facilitation and

depression

0 1 2 3

time 0

0 1 2 3

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)

1mV100ms

Aj = FjDj

Postsynaptic amplitude

Dynamics for facilitationand depression

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

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

Power spectra

100 101

Frequency0

dominating depressiondominating facilitationTheory

constant amplitude

Power spectra

DDRFDRtheory

00.00050.0010.0015

0204060

100 101

frequency0

100 101

frequency0

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∗〉

|Sgs|2

SggSss

dt e2πift

x(t)Fourier transform

SXs = 〈Xs∗〉

Cross spectra of synaptic input/voltage and input signal

|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

|SXs|2

SssSXX

Coherence functions for various parameter sets

Cross spectra

broadband coding

|SXs|2

SssSXX

Coherence functions for various parameter sets

=N − 1

C〈xi〉,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 ?

|2 SimulationTheory

0.1 1 10frequency [Hz]

SimulationTheory

0.1 1 10frequency [Hz]

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

Coherence for a single synapse

simulation value(theoretical value)

Coherence for a single synapse

=N − 1

C〈xi〉,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

e C RX

SimulationTheory

N=1000

N=10000

Coherence-dependence on the number N of synapses

Extension I

Postsynaptic spiking

if V = −65mV then ti = t & V = −70mV

LIF output spike train

10-2 10-1 100 101 1020

γ 2 /c

PIFLIFQIF

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

10-2 10-1 100 1010

γ 2 /c

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

10-2 10-1 100 101

γ 2 /c

10-2 10-1 100 101

f10-3 10-2 10-1 100 101

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

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)

tra [H

z] Sηη

CRX (Simulation)CRX (Theory)

Facilitating synapses for signalDepressing synapses for noise

tra [H

z] Sηη

CRX (Theory)CRX (Simulation)

Depressing synapses for signalFacilitating synapses for noise

Merkel & Lindner in preparation (2009)

Extra noise

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

Signal ampliﬁcation and information transmission in neural

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