Neural noise and neural signals - spontaneous...
Transcript of Neural noise and neural signals - spontaneous...
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Neural noise and neural signals - spontaneous activity and signal transmission
in models of single nerve cells
Benjamin Lindner
Theory of Complex Systems and Neurophysics Institut für Physik Humboldt-Universität Berlin
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Formal matter
Weekly tutorial on Mondays 13:15-15:45 starting on May 2nd
Problems are handed out two weeks before the tutorial; required for passing: 70% of the points
Oral examination before the new semester starts
ECTS: 6 (both Computational Neuroscience & Physics)
URL of the lecture: http://people.physik.hu-berlin.de/~neurophys/neusig/
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Motivation
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Sensory signals are represented by sequences of electrical discharges -
neural spike trains
time [s]
membrane voltage [mV]
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Neurons fire randomly
Nawrot et al. Neurocomp. (2007)
Spontaneous firing (Neocortex of rat)
Different responses to the same stimulus (auditory neuron of locust)
Karin Fisch & Jan Benda, LMU
Interspike interval (ISI) statistics ?
Aim: reproduce and understand the spike statistics Statistics shapes information transmission and processing! 5
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Aim of the course
Analytical calculation of the statistics of single-neuron activity, modeled by dynamical systems in a stochastic framework
Spontaneous activity
Evoked activity
Noise
Spike trainStimulus
Noise
Spike train
neuron
neuron
Statistics: Rate, Cv, ISI density & correlations; spike train spectrum & Fano factor
Statistics: Rate modulation, phase lag; signal-to-noise ratio, coherence, mutual information rate
neuron
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What can be achieved by stochastic models of
single-neuron activity?
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p(I) = r0 exp[�r0I]
Spiking statistics of a Poisson process ...
0 50 100 150Frequency [Hz]
10-1
100
101
102
103
Spike trainpower spectrum
Poisson process
2 4 6 8 10 12 14 16 18 20lag
-0.5
0
0.5
1
Intervalcorrelationcoefficient
Poisson process
0 20 40 60 80ISI [ms]
0
0.02
0.04
0.06
0.08
0.1
ISIprobability density
Poisson process
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Spiking statistics of a Poisson process ...... and of a sensory neuron
0 50 100 150Frequency [Hz]
10-1
100
101
102
103
Spike trainpower spectrum
Poisson processelectro-sensory neuron
2 4 6 8 10 12 14 16 18 20lag
-0.5
0
0.5
1
Intervalcorrelationcoefficient
Poisson processelectro-sensory neuron
0 20 40 60 80ISI [ms]
0
0.02
0.04
0.06
0.08
0.1
ISIprobability density
Poisson processelectro-sensory neuron
data: Neiman & Russell, Ohio University9
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Spiking statistics of a Poisson process ...... and of a sensory neuron and the right theory!
data: Neiman & Russell, Ohio University
0 50 100 150Frequency [Hz]
10-1
100
101
102
103
Spike trainpower spectrum
theory
Poisson process
electro-sensory neuron
2 4 6 8 10 12 14 16 18 20lag
-0.5
0
0.5
1
Intervalcorrelationcoefficient
theory
Poisson process
electro-sensory neuron
0 20 40 60 80ISI [ms]
0
0.02
0.04
0.06
0.08
0.1
ISIprobability density
theory
Poisson process
electro-sensory neuron
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11What can be achieved by stochastic models of
single-neuron activity?
• understand neural spontaneous activity(single neurons are complex non-equilibrium systems)
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How much info does the spike train carry about the sensory signal?
time
Information theory of neural spiking
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Do neurons encode more information on slow or on fast components of a sensory stimulus?
Chacron et al. Nature 2003
encoding of fast stimulus
components
encoding of slow stimulus
components
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Visual System • Reinagel et al. J Neurophisiol (1999) • Passaglia et al. J Neurophisiol (2004) ... Auditory System • Rieke et al. Proc. R. Soc. Lond. (1995) • Marsat et al. . J Neurophisiol (2004) ... Vestibular System • Sadeghi et al. J Neurosci (2007) • Massot et al. J Neurophisiol (2011) ... Electrosensory System • Oswald et al. J Neurosci (2004) • Middleton et al. PNAS (2006) ...
Information filtering is ubiquitous
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15What can be achieved by stochastic models of
single-neuron activity?
• understand neural spontaneous activity(single neurons are complex non-equilibrium systems)
• understand the neuron’s capability to transmit and filter information about time-dependent stimuli
(equivalent to populations of uncoupled neurons)
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16Methods are useful for problem of recurrent neural nets
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17Methods are useful for problem of recurrent neural nets
Network of model neurons:
Equation for the probability density of the membrane voltage:
Brunel J. Comp. Neurosci (2000)
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Brunel J. Comp. Neurosci (2000)
Collective behavior of neurons in recurrent networks18
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19What can be achieved by stochastic models of
single-neuron activity?
• understand neural spontaneous activity(single neurons are complex non-equilibrium systems)
• understand the neuron’s capability to transmit and filter information about time-dependent stimuli
(equivalent to populations of uncoupled neurons)
• advanced theory of recurrent networks of spiking neurons rely on accurate theory of single-cell activity
(methods are an extension of those for the single cell)
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Schedule (approximate)
20.4. Motivation; spike-train statistics 27.4. Spike-train statistics II 4.5. How much information does a spike train contain? 11.5. Mutual information - direct estimates and lower bound 18.5. Beyond statistical models: Sources of neural noise and their incorporation into dynamical neuron models 25.5. Effective time-constant approximation & IF models:Fokker-Planck analysis (intro) 1.6. IF models:Fokker-Planck analysis of spontaneous activity 8.6. IF models:Fokker-Planck analysis of driven activity I15.6. IF models:Fokker-Planck analysis of driven activity II 22.6. Stochastic resonance in driven IF models 29.6. Multi-dimensional IF models: colored noise, spike- frequency adaptation, subthreshold oscillations 6.7. Properties of spontaneous firing & signal transmission 13.7. Signal transmission in neural populations of uncoupled neurons 20.7. Outlook: recurrent networks
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•Stochastic Processes in general ...
- Risken The Fokker-Planck Equation Springer (1984)
- Gardiner Handbook of Stochastic Methods Springer (1985)
- Van Kampen Stochastic Processes in Physics and Chemistry North-Holland (1992)
- Cox & Isham Point Processes Chapman and Hall (1980)
•... and in neuroscience:
- Holden Models of the Stochastic Activity of Neurones Springer-Verlag (1976)
- Tuckwell Stochastic Processes in the Neuroscience SIAM (1987)
- Gerstner & Kistler Spiking Neuron Models Cambridge University Press (2002)
- Gabbiani & Cox Mathematics for Neuroscientists Elsevier (2010)
- Dayan & Abbott Theoretical Neuroscience MIT Press (2001)
General references21
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Spike train statistics
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x(t) =�
�(t� ti)
From the membrane dynamics to the point process
Spike train:
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A stationary process - time-independent firing rate
Spike train:
time
.
.
.
Spike trains
r(t) =p(t, t + dt)
dt
firing rate per unit time
Estimate for the firing probability in dt
p(t, t + dt) =# of neurons that fire in(t, t + dt)
total # neurons
Steveninck et al. Science 1997
spontaneous firing of a motion-sensitive neuron
in the blow fly
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A non-stationary process - time-dependent firing rate
.
.
.
Spike trains
r(t) =p(t, t + dt)
dt
firing rate per unit time
Estimate for the firing probability in dt
p(t, t + dt) =# of neurons that fire in(t, t + dt)
total # neurons
Steveninck et al. Science 1997
evoked firing of a motion-sensitive neuron
in the blow fly
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Simple example: Poissonian spike train (homogeneous)
x(t) =�
�(t� ti)
1. draw N points from a uniform density along the t -axis
tiDefining property: all are independent!
Three ways to simulate a Poisson process} If
Poisson in
with rate
Only one parameter: the firing rate
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Simple example: Poissonian spike train (homogeneous)
x(t) =�
�(t� ti)
2. discretize t -axis; draw independent random number for each bin
tiDefining property: all are independent!Only one parameter: the firing rate
Three ways to simulate a Poisson process
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Simple example: Poissonian spike train (homogeneous)
x(t) =�
�(t� ti)
3. draw time interval to the next spike (Gillespie method)
tiDefining property: all are independent!Only one parameter: the firing rate
Three ways to simulate a Poisson process
0 2 4 6 8 10ISI
0
0.2
0.4
0.6
0.8
1
ISI density
Prob. density
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C(�) = ⇥x(t)x(t + �)⇤ � ⇥x(t)⇤2
Spike train correlation function
For a stationary process
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C(⇥) = r0[�(⇥) + m(⇥)] � r20
S(f) = r0
C(⇥) = r0�(⇥)
Spike train correlation function
For a Poisson process
corresponding to a flat spike train power spectrum
Generally, for any stationary stochastic process
Wiener-Khinchin theorem:
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x̃(f) =T�
0
dt x(t) e2�ift
Spectral measures
Fourier transform of the process
a new random function (of frequency)!
�x̃(f)⇥ = 0 For a stationary process x(t):
S(f) = limT⇥⇤
�x̃x̃�⇥T
.... and the power spectrum indicates possible stochastic
periodicity by finite peaks
hair bundle from a sensory hair cell
of bullfrog
sensory neuron in paddlefish
Neiman & Russell J. Neurophysiol. 200431
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Spike train power spectra are not flat in general
Neiman & Russell J. Neurophysiol. 2004
fa
fe
f −a fe
f +a fe
Frequency
100
0
Bair et al. J. Neuroscie. (1994)
Power spectrum
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N(t) =⇥ t
0dt�x(t�) =
�
0<ti<t
�(t� ti)
spik
e tra
ins
0 1 2 3time
0
10
20
30
40
spik
e co
unts
Spike train -> spike count
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�N(t)⇥ =� t
0dt��x(t�)⇥ =
� t
0dt�r(t�)
Spike train -> spike count -> diffusion process with drift
spik
e tra
ins
0 1 2 3time
0
10
20
30
40
spik
e co
unts
�N(t)⇥ = r0T
For a stationary process
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⇥�N2(t)⇤ =� t
0
� t
0dt1dt2⇥x(t1)x(t2)⇤ � ⇥x(t1)⇤⇥x(t2)⇤
⇤�N2(t)⌅ � 2De�t, for t �⇥
F (t) =��N2(t)⇥�N(t)⇥
F (t)� 2De�
r0for t�⇥
Spike train -> spike count -> diffusion process with drift
Spike count variance
Often, spike count shows normal diffusion
Temporal structure of variability is described in the Fano factor
For a Poisson process at all times F (t) = 135
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F (t) =��N2(t)⇥�N(t)⇥
Fano factor does not always saturate ...
Lowen & Teich J. Acoust. Soc. Am. 1992
cat auditory nerve unit A
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Interval statistics
IntervalBrenner et al. Phys. Rev E (2003)
Probability density of the interspike interval (ISI)
��I2⇥Variance of the ISI
Mean ISI �I⇥
CV =�
��I2⇥�I⇥
Coefficient of variation
CV = 1For a Poisson process
p(I)p(I)
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Examples for ISI densities
Brenner et al. Phys. Rev E (2003)
p(I) = r0 exp[�r0I]Poisson process
0 2 4 6 8 10ISI
0
0.2
0.4
0.6
0.8
1
ISI density
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The moments do not determine the probability density
Example:
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interspike intervals (ISI)
with density
and
n-th order intervalswith densities
tti ti+1 ti+2
Ii Ii+1 Ii+2
ti+3ti-1
Ii-1 Ii+3
tti ti+1 ti+2
T1,i
T2,i
T3,i
ti+3ti-1
N-th order intervals
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ρ1 > 0
ρ1 < 0
time
short short short
long long
time
short short short
long long long
long
Interval correlations (in nonrenewal spike trains)
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Chacron et al. Phys. Rev. Lett. 2001
Neiman & Russel Phys. Rev. Lett. 2001P-units in weakly
electric fish
Electroreceptors in paddle fish
long-range positive correlations for auditory
neurons
Lowen & Teich J. Acoust. Soc. Am. Rev. 1992
ISI correla*ons reviewed in Farkhooi et al. Phys. Rev. E 2009 Akerberg & Chacron 2011
Interval correlations (in nonrenewal spike trains)
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Renewal spike trains
All intervals are mutually independent!
Consequence for the Fourier transforms of the n-th order interval density
Example: Poisson process
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Relation between different spike train statistics
Time-averaged firing rate =inverse mean ISI
(stationary processes)
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Power spectrum - Fourier transforms of n-th order interval densities
Relation between different spike train statistics
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Lindner Phys. Rev. E (2004)
For a renewal spike train
��T 2n⇥ = n��T 2
1 ⇥
tti ti+1 ti+2
T1,i
T2,i
T3,i
ti+3ti-1⇥�T 2
n⇤ �= n⇥�T 21 ⇤
For a nonrenewal spike train
�k = 0, k > 0
Relation between different spike train statistics
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Lindner Phys. Rev. E (2004)
Relation between different spike train statistics
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Last time ... For a stationary process:
1.Firing rate
2.n-th order interval variance
3. Interval correlation coefficient
4. Power spectrum
--> renewal process: 5. Fano factor
renewal:
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For a stationary process:
1.Firing rate
2.n-th order interval variance
3. Interval correlation coefficient
4. Power spectrum
--> renewal process: 5. Fano factor
nonrenewal: renewal:
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Negative ISI correlations cause a drop in spectral power at low frequency
Chacron et al. Proc. SPIE 2005
electroreceptor (P-units) in the weakly electric fish
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Negative ISI correlations cause a drop in spectral power at low frequency
Shuffled spike train
Shuffled spike train
Original spike trainOriginal spike train
Chacron et al. Proc. SPIE 2005
electroreceptor (P-units) in the weakly electric fish
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Gamma process - ISI density and spike train power spectrum
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Summary: spike-train statistics
-Point processes can be characterized •by spike train (rate) and count statistics (Fano factor) •by interval statistics (ISI and nth-order intervals, interval correlations) -ISI densities often unimodal; spectra non-flat --- not well described by Poisson statistics!
-Spike train and interval statistics are related in a nontrivial manner
- Cox & Lewis The Statistical Analysis of Series of Events Chapman and Hall (1966)
- Cox & Isham Point Processes Chapman and Hall (1980)
- Holden Models of the Stochastic Activity of Neurones Springer-Verlag (1976)
- Gerstner & Kistler Spiking Neuron Models Cambridge University Press (2002)
References
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