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Part II: Population Models
BOOK: Spiking Neuron Models,
W. Gerstner and W. Kistler
Cambridge University Press, 2002
Chapters 6-9
Swiss Federal Institute of Technology Lausanne, EPFL
Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne
Chapter 6: Population Equations
BOOK: Spiking Neuron Models,
W. Gerstner and W. Kistler
Cambridge University Press, 2002
Chapter 6
10 000 neurons
3 km wires
1mm
Signal:
action potential (spike)
action
potential
Spike Response Model
Spike emission
t  ti^ 
j
i
Spike reception: EPSP
 t  t
f
j
ui

Spike reception: EPSP
 t  t
Spike emission: AP
t  t
^
i

Last spike of i
ui t   t  t 
^
i
ui t    

All spikes, all neurons
 w  t  t 
ij
j
Firing:
f
j

f
j
linear
f
t t
^
i
threshold
Integrate-and-fire Model
Spike emission
j
i
ui
I

d
  ui  ui  RI (t )
dt
ui t     Fire+reset
Spike reception: EPSP
f reset
 t  t j
linear
threshold


Noise models
A
escape process
(fast noise)
 (t )
B
C


t^

t^
tf
stochastic reset
escape rate
noisy integration
 (t )  f (u(t )   )
Interval distribution
PI (t ¦ t ^ ) t
  (t )  exp(   (t ' )dt' )
escape
rate
stochastic spike arrival
(diffusive noise)
u(t)
t
t^
parameter changes
(slow noise)
t^
Survivor function
dui

 ui  RI   (t )
dt
Interval distribution
PI (t ¦ t ) 
noise
^
 G (t  t )
f
Gaussian about
t
f
white
synapse
^ : first passage
(fast noise) (slow noise)
I
time problem
(Brunel et al., 2001)
P (t ¦ t )
Homogeneous Population
populations of spiking neurons
t
?
I(t)
population dynamics?
t
population
activity
n(t ; t  t )
A(t ) 
N t
Homogenous network (SRM)
t  ti^ 
Spike emission: AP
 t  t
f
j

Spike reception: EPSP
Synaptic coupling
J0
wij 
N
potential Last spike of i
All spikes, all neurons
fully
connected
N >> 1
external input
f




s
I
(
t

s
)
ds



t

t
ui t   t  t   wij

j
^
i
j
f
potential
input potential
 
^



t

t
u t |t 
^
h(t )
refractory potential
ui t   t  t  J 0   s A(t  s ) ds   s I (t  s )ds
^
i
wij 
potential
J0
N
All spikes, all neurons
Last spike of i
fully
connected
external input
f




s
I
(
t

s
)
ds



t

t
ui t   t  t   wij

j
^
i
j
f
Homogenous network
t  ti^ 
Spike emission: AP
 s 
Response to current pulse
potential Last spike of i
ui t   t  tˆi

external input
 J 0  s I (t  s)ds

potential
 
^



t

t
u t |t 
^
Population activity
  s A(t  s)ds
input potential
h(t )
All neurons receive the same input
Homogeneous network (I&F)
Assumption of Stochastic spike arrival:
network of exc. neurons,
total spike arrival rate A(t)
Synaptic current pulses
J0 q
d

u  (u  urest ) 
 (t  t kf )
N C
dt
k, f
EPSC
u
u0
d
 u  (u  urest )
dt
 R I (t )
I (t )  J 0 q A(t )
Density equations
Density equation (stochastic spike arrival)
Stochastic spike arrival:
network of exc. neurons,
total spike arrival rate A(t)

Synaptic current pulses
J 0 qe
qe
f
 (t  t k )  J ext  (t  t f )

d

u  (u  urest ) 
dt
N C
k, f
f
C
EPSC
u
u0
d
 u  (u  urest )
dt
 R I (t )   (t )
I (t )  J 0 q A (t )  J extq f
Langenvin equation,
Ornstein Uhlenbeck process
Density equation (stochastic spike arrival)
Membrane potential density
u
u

A(t)=flux across
threshold
A(t ) 
p(u)

p(u, t ) u 
u
Fokker-Planck
2



  p(u, t )   [V (u ) p(u, t )]  12  2 2 p(u, t )   (u  u r ) A(t )
t
u
u
drift
V (u )  u  J 0 q A
diffusion
 2    k wk2
k
spike arrival rate
source term
at reset
Integral equations
Population Dynamics
t
A(t ) 

PI t | tˆ  A(tˆ) dtˆ

Derived from normalization
t
1


SI t | tˆ  A(tˆ) dtˆ
Escape Noise (noisy threshold)
I&F with reset, constant input, exponential escape rate

escape rate
 (t )
 (t tˆ)  f (u (t tˆ))   exp(
u (t tˆ ) 
u
)
tˆ
Interval distribution
P0 (t tˆ)
tˆ
t

P(t tˆ)   (t tˆ) exp(  (t ' tˆ)dt' )
tˆ
Population Dynamics
t
A(t ) 


PI t | tˆ  A(tˆ) dtˆ
Wilson-Cowan
population equation
Wilson-Cowan model
escape process
(fast noise)

 (t )
h(t)
t^
t
 abs
escape rate
 (t )  f (h(t )   )
(i) noisy firing
(ii) absolute refractory time
escape rate
abs

 f (h(t )  ) for(t  tˆ)  
 (t )  f (u(t )   )  
abs

0 for0  (t  tˆ)  
population activity
t
A(t )  [1   A(t ' )dt ' ] f (h(t )   )
t  abs
(iii) optional: temporal averaging
d
A(t )
A(t )  
 g (h(t ))
dt

Wilson-Cowan model
escape process
(fast noise)

 (t )
h(t)
t^
(i) noisy firing
(ii) absolute refractory time
t
 abs
escape rate
abs

 f (h(t )  ) for(t  tˆ)  
 (t )  f (u(t )   )  
abs

0 for0  (t  tˆ)  
population activity
t
A(t )  [1   A(t ' )dt ' ] f (h(t )   )
t  abs
t
A(t ) 
 P t | tˆ A(tˆ) dtˆ
I

Population activity in spiking neurons (an incomplete history)
1972 - Wilson&Cowan; Knight
Integral equation
Amari (Heterogeneous, non-spiking)
1992/93 - Abbott&vanVreeswijk
Gerstner&vanHemmen
Treves et al.; Tsodyks et al.
Bauer&Pawelzik
1997/98 - vanVreeswijk&Sompoolinsky
Amit&Brunel
Pham et al.; Senn et al.
1999/00 - Brunel&Hakim; Fusi&Mattia
Nykamp&Tranchina
Omurtag et al.
Mean field equations
density (voltage, phase)
Heterogeneous nets
stochastic connectivity
Fast transients
Knight (1972), Treves (1992,1997), Tsodyks&Sejnowski (1995)
Gerstner (1998,2000), Brunel et al. (2001), Bethge et al. (2001)
Chapter 7: Signal Transmission
and Neuronal Coding
BOOK: Spiking Neuron Models,
W. Gerstner and W. Kistler
Cambridge University Press, 2002
Chapter 7
Coding Properties of Spiking Neuron Models
Course (Neural Networks and Biological Modeling)
session 7 and 8
reverse
correlation
Probability
of
PSTH(t) I(t)
output
?
forwardspike
correlation
I(t)
500 trials
I(t)
fluctuating input
A(t)?
t0
Swiss Federal Institute of Technology Lausanne, EPFL
Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne
Theoretical Approach
PSTH(t)
I(t)
500 trials
A(t)
I(t)
500 neurons
- population dynamics
- response to single input spike (forward correlation)
- reverse correlations
t
Population of neurons
I(t)
?
t0
h(t)
potential
A(t)
A(t)
A(t)
A(t)
population
activity
A(t ) 
n(t ; t  t )
N t
d
A(t )
A(t )  
 g (h(t ))
dt

N neurons, A(t )  g (h(t ))
- voltage threshold, (e.g. IF neurons)
(t ) excitatory)
g ( I (t ))
- same typeA
(e.g.,
---> population response ?
A(t )  g ( I (t ), I ' (t ))
Coding Properties of Spiking Neurons:
1. Transients in Population Dynamics
- rapid transmission
2. Coding Properties
- forward correlations
- reverse correlations
Swiss Federal Institute of Technology Lausanne, EPFL
Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne
Population Dynamics
t
A(t ) 


Example: noise-free
A(t ) 
PI t | tˆ  A(tˆ) dtˆ
PI t | tˆ   (t  tˆ  T (tˆ))
 h' 
A
(
t

T
)
1


 u ' 
higher activity

I(t)
h(t)
T(t^)
h’>0
Theory of transients
noise-free
A(t ) 
potential

 h' 
1  u '   A(t  T )

u t | t ^  t  t ^    s  I (t  s ) ds
input potential
A(t)
h(t )
A(t )  A0  A  (t  t0 )
I(t)
h(t)
?
I(t)
t0
External input.
No lateral coupling
h' (t )   t  t0 
Theory of transients
noise model B
no noise
(reset noise)
noise-free
I(t)
A(t)
slow noise
h(t)
I(t)
h(t)
Membrane potential density
u
u

p(u)
Hypothetical experiment: voltage step
u

p(u)
Immediate response
Vanishes linearly
Transients with noise
Noise models
A
escape process
(fast noise)
 (t )
B
C


t^

t^
tf
stochastic reset
escape rate
noisy integration
 (t )  f (u(t )   )
Interval distribution
PI (t ¦ t ^ ) t
  (t )  exp(   (t ' )dt' )
escape
rate
stochastic spike arrival
(diffusive noise)
u(t)
t
t^
parameter changes
(slow noise)
t^
Survivor function
dui

 ui  RI   (t )
dt
Interval distribution
PI (t ¦ t ) 
noise
^
 G (t  t )
f
Gaussian about
tf
white
(fast noise)
synapse
(slow noise)
(Brunel et al., 2001)
Transients with noise:
Escape noise (noisy threshold)
Theory with noise
t
A(t ) 

A(t)
PI t | tˆ  A(tˆ) dtˆ
A0
I(t)
I
h(t)

linearize
A(t )  A0  A(t )
h(t )  h0  h(t )
h: input potential
A0 
1
inverse mean interval
 s
h(t )    ( s) I (t  s) ds
low noise: transient prop to h’
high noise: transient prop to h
low noise
L
high noise
Theory of transients
noise model A
(escape noise/fast noise)
low
noise
low
noise
noise-free
I(t)
fast
A(t)
noise model A
(escape noise/fast noise)
high noise
h(t)
I(t)
slow
h(t)
Transients with noise:
Diffusive noise (stochastic spike arrival)
Noise models
A
escape process
(fast noise)
 (t )
B
C


t^

t^
tf
stochastic reset
escape rate
noisy integration
 (t )  f (u(t )   )
Interval distribution
PI (t ¦ t ^ ) t
  (t )  exp(   (t ' )dt' )
escape
rate
stochastic spike arrival
(diffusive noise)
u(t)
t
t^
parameter changes
(slow noise)
t^
Survivor function
dui

 ui  RI   (t )
dt
Interval distribution
PI (t ¦ t ) 
noise
^
 G (t  t )
f
Gaussian about
tf
white
(fast noise)
synapse
(slow noise)
(Brunel et al., 2001)
Membrane potential density
u
Fokker-Planck
Diffusive noise
u

p(u)

  p(u, t ) 
t

 [ A(u ) p(u, t )]
u
2

 12  2 2 p(u, t )
u
Hypothetical experiment: voltage step

p(u)
Immediate response
vanishes quadratically
Membrane potential density
u
u
SLOW Diffusive noise

p(u)
Hypothetical experiment: voltage step

p(u)
Immediate response
vanishes linearly
Signal transmission in populations of neurons
Population
- 50 000 neurons
- 20 percent inhibitory
- randomly connected
Connections
input -low rate
-high rate
4000 external
4000 within excitatory
1000 within inhibitory
Signal transmission in populations of neurons
A [Hz]
Neuron #
10
32440
input -low rate
-high rate
32340
50
100
Population
- 50 000 neurons
- 20 percent inhibitory
- randomly connected
100
time [ms]
200
Neuron # 32374
u [mV]
0
50
100
time [ms]
200
Signal transmission - theory
- no noise
fast
- slow noise (noise in parameters)
prop. h’(t)
(current)
- strong stimulus
slow
- fast noise (escape noise)
See also: Knight (1972), Brunel et al. (2001)
prop. h(t)
(potential)
Transients with noise:
relation to experiments
Experiments to transients A(t)
Experiments
V1 - single neuron
PSTH
Marsalek et al., 1997
V1 - transient
response
delayed by 64 ms
stimulus switched on
V4 - transient
response
delayed by 90 ms
input
A(t)
A(t)
A(t)
A(t)
See also: Diesmann et al.
How fast is neuronal signal processing?
animal -- no animal
Simon Thorpe
Nature, 1996
Reaction time experiment
Visual processing
eye
Memory/association
Output/movement
How fast is neuronal signal processing?
Simon Thorpe
Nature, 1996
animal -- no animal
# of
images
Reaction time
400 ms
Visual processing
Reaction time
Memory/association
eye
Recognition time 150ms
Output/movement
Coding properties of spiking neurons
Coding properties of spiking neurons
A(t)
I(t)
500 neurons
- response to single input spike
(forward correlations)
PSTH(t)
I(t)
500 trials
Coding properties of spiking neurons
I(t)
Spike ?
Two simple arguments
1)
PSTH=EPSP
(Moore et al., 1970)
- response to single input spike
(forward correlations)
Experiments:
Fetz and Gustafsson, 1983
Poliakov et al. 1997
2)
PSTH=EPSP’
(Kirkwood and Sears, 1978)
Forward-Correlation Experiments
A(t)
Poliakov et al., 1997
I(t)
PSTH(t)
noise
1000 repetitions
high noise
prop. EPSP
low noise
prop.
d
EPSP
dt
Population Dynamics
 P t | t  A(t
t
A(t ) 
^
I
^
) dt
^
full theory

I(t)
A(t)
h(t )  h0  h(t )
h: input potential
I(t)
h(t )    ( s) I (t  s) ds
PSTH(t)
linear theory
Forward-Correlation Experiments
A(t)
Theory: Herrmann and Gerstner, 2001
high noise
red: linearized theory
blue: full theory
high noise Poliakov et al., 1997
low noise
red: linearized theory
blue: full theory
low noise
Forward-Correlation Experiments
A(t)
Poliakov et al., 1997
I(t)
PSTH(t)
noise
prop. EPSP
prop.
1000 repetitions
high noise
prop. EPSP
d
EPSP
dt
low noise
prop.
d
EPSP
dt
Reverse Correlations
I(t)
fluctuating input
Swiss Federal Institute of Technology Lausanne, EPFL
Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne
Reverse-Correlation Experiments
I (t )
after 1000 spikes
Linear Theory
h(t )  h0  h(t )
h: input potential
h(t )    ( s) I (t  s) ds
Fourier Transform
~
~
~
A( )  G( ) I ( )
Inverse Fourier Transform

A(t )   G( s) I (t  s) ds
0
~
~
i A0 L ( ) ~( )
G( ) 
~
1  P ( )
Signal transmission
A( f )
G( f ) 
I( f )
noise model A
(escape noise/fast noise)
low noise
high noise
I(t)
A(t)
T=1/f
noise model B
(reset noise/slow noise)
low noise
high noise
no cut-off
Reverse-Correlation Experiments
(simulations)
I (t )
after 1000
25000spikes
spikes

A(t )   G( s) I (t  s) ds
0
theory:
G(-s)
Coding Properties
of spiking neurons
?
I(t)
- spike dynamics -> population dynamics
- noise is important
- fast neurons for slow noise
- slow neurons for fast noise
- implications for
- role of spontaneous activity
- rapid signal transmission
- neural coding
- Hebbian learning
Laboratory of Computational Neuroscience, EPFL, CH 1015 Lausanne
Chapter 8: Oscillations
and Synchrony
BOOK: Spiking Neuron Models,
W. Gerstner and W. Kistler
Cambridge University Press, 2002
Chapter 8
Stability of Asynchronous State
Stability of Asynchronous State
 P t | t  A(t
A(t)
t
A(t ) 
^
I
^
) dt
^

fully connected
coupling J/N
linearize
A(t )  A0  A(t )
h(t )  h0  h(t )
h(t )  J   ( s) A(t  s) ds
h: input potential
A(t )  A1eit t
Search for bifurcation points   0
Stability of Asynchronous State
A(t)
 (s) 
noise

stable
0

0 for s  
(s  ) e( s)
 (s)
s
20
 delay
T period
30
Stability of Asynchronous State

  0.4ms
2
0 
T
  1.0ms
60
  1.2ms
50
  1 .4
40
  2.0ms
30
  3.0ms
20
 (s)
s
Chapter 9: Spatially structured
networks
BOOK: Spiking Neuron Models,
W. Gerstner and W. Kistler
Cambridge University Press, 2002
Chapter 9
Continuous Networks
Several populations
Ai (t )
Continuum
i
k

Continuum: stationary profile
A( , t )  A( )
