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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
1
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/
2
Motivation
3
Sensory signals are represented by sequences of electrical discharges -
neural spike trains
time [s]
membrane voltage [mV]
4
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
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
6
7
What can be achieved by stochastic models of
single-neuron activity?
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
8
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
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
10
11What can be achieved by stochastic models of
single-neuron activity?
• understand neural spontaneous activity(single neurons are complex non-equilibrium systems)
How much info does the spike train carry about the sensory signal?
time
Information theory of neural spiking
12
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
13
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
14
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)
16Methods are useful for problem of recurrent neural nets
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)
Brunel J. Comp. Neurosci (2000)
Collective behavior of neurons in recurrent networks18
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)
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
20
•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
Spike train statistics
22
x(t) =�
�(t� ti)
From the membrane dynamics to the point process
Spike train:
23
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
24
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
25
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
26
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
27
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
28
C(�) = ⇥x(t)x(t + �)⇤ � ⇥x(t)⇤2
Spike train correlation function
For a stationary process
29
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:
30
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
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
32
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
33
�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
34
⇥�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
F (t) =��N2(t)⇥�N(t)⇥
Fano factor does not always saturate ...
Lowen & Teich J. Acoust. Soc. Am. 1992
cat auditory nerve unit A
36
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)
37
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
38
The moments do not determine the probability density
Example:
39
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
40
ρ1 > 0
ρ1 < 0
time
short short short
long long
time
short short short
long long long
long
Interval correlations (in nonrenewal spike trains)
41
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)
42
Renewal spike trains
All intervals are mutually independent!
Consequence for the Fourier transforms of the n-th order interval density
Example: Poisson process
43
Relation between different spike train statistics
Time-averaged firing rate =inverse mean ISI
(stationary processes)
44
Power spectrum - Fourier transforms of n-th order interval densities
Relation between different spike train statistics
45
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
46
Lindner Phys. Rev. E (2004)
Relation between different spike train statistics
47
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:
48
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:
49
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
50
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
51
Gamma process - ISI density and spike train power spectrum
52
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
53