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Part II: Population Models
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapters 6-9
Laboratory of Computational Neuroscience, LCN, CH 1015 LausanneSwiss Federal Institute of Technology Lausanne, EPFL
Chapter 6: Population Equations
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 6
10 000 neurons3 km wires
1mm
Signal:action potential (spike)
action potential
Spike Response Model
iuij
fjtt
Spike reception: EPSP
fjtt
Spike reception: EPSP
^itt
^itt
Spike emission: AP
fjtt ^
itt tui j f
ijw
tui Firing: tti ^
linear
threshold
Spike emission
Last spike of i All spikes, all neurons
Integrate-and-fire Model
iui
fjtt
Spike reception: EPSP
)(tRIuudtd
ii
tui Fire+reset
linear
threshold
Spike emission
resetI
j
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
Noise models
A B C
u(t)
noise
white(fast noise)
synapse(slow noise)
(Brunel et al., 2001)
t
t
dttt^
)')'(exp()( )¦( ^ttPI
: first passagetime problem
)¦( ^ttPI Interval distribution
^t ^t ^tt
Survivor functionescape rate
)(t
))(()( tuftescape rate stochastic reset
)¦( ^ttPI )( fttG
Interval distribution
Gaussian about ft
)(tRIudtdu
ii
noisy integration
ft
Homogeneous Population
populations of spiking neurons
I(t)
?
population dynamics? t
t
tNtttntA
);()(population
activity
Homogenous network (SRM)
NJwij
0
Spike reception: EPSP
Spike emission: AP
fjtt ^
itt tui j f
ijwLast spike of i All spikes, all neurons
fjtt
^itt
Synaptic coupling
potential
fullyconnected N >> 1
dsstIs )( external input
NJwij
0
fjtt ^
itt tui j f
ijwLast spike of i All spikes, all neuronspotential
dsstIs )( external input
dsstIs )( tui ^itt dsstAsJ )(0
potential
^tt ^| ttu )(thinput potential
fullyconnected
refractory potential
Homogenous network
Response to current pulse
Spike emission: AP
s
^itt
potential
^tt ^| ttu )(thinput potential
itt ˆ tui
Last spike of ipotential
dsstIsJ )(0 external input
dsstAs )( Population activity
All neurons receive the same input
Assumption of Stochastic spike arrival: network of exc. neurons, total spike arrival rate A(t)
)()(,
0 fk
fkrest tt
Cq
NJ
uuudtd
u
0u
EPSC
Synaptic current pulses
Homogeneous network (I&F)
)()( tIRuuudtd
rest
)()( 0 tAqJtI
Density equations
Stochastic spike arrival: network of exc. neurons, total spike arrival rate A(t)
)()()(,
0 f
feext
fk
fk
erest tt
Cq
JttCq
NJ
uuudtd
u
0u
EPSC
Synaptic current pulses
Density equation (stochastic spike arrival)
)()()( ttIRuuudtd
rest Langenvin equation,Ornstein Uhlenbeck process
fqJtAqJtI ext )()( 0
u
p(u)
Density equation (stochastic spike arrival)
u
Membrane potential density
)()(),()],()([),( 2
22
21 tAuutup
utupuV
utup
t r
Fokker-Planck
drift diffusionAqJuuV 0)(
kkk w22
spike arrival rate
source term at reset
A(t)=flux across threshold
utupu
tA ),()(
Integral equations
tdtAttPtA I
t
ˆ)ˆ(ˆ|)(
Population Dynamics
tdtAttSI
t
ˆ)ˆ(ˆ|1
Derived from normalization
Escape Noise (noisy threshold)
t̂
)(t
I&F with reset, constant input, exponential escape rate
Interval distribution
t̂
)ˆ(0 ttP )')ˆ'(exp()ˆ()ˆ( ̂t
t
dtttttttP
)exp())ˆ(()ˆ()ˆ(u
ttuttuftt
escape rate
tdtAttPtA I
t
ˆ)ˆ(ˆ|)(
Population Dynamics
Wilson-Cowan
population equation
escape process (fast noise)
Wilson-Cowan model
h(t)
^t t
)(t
))(()( thftescape rate
(i) noisy firing(ii) absolute refractory time
abs
))(()( thftA
population activity
t
t abs
dttA
]')'(1[
(iii) optional: temporal averaging
))(()()( thgtAtAdtd
abs
abs
ttfor
ttforthftuft
)ˆ(00
)ˆ())(())(()(
escape rate
escape process (fast noise)
Wilson-Cowan model
h(t)
^t t
)(t
(i) noisy firing(ii) absolute refractory time
abs
))(()( thftA
population activity
t
t abs
dttA
]')'(1[
abs
abs
ttfor
ttforthftuft
)ˆ(00
)ˆ())(())(()(
escape rate
tdtAttPtA I
t
ˆ)ˆ(ˆ|)(
Population activity in spiking neurons (an incomplete history)
1972 - Wilson&Cowan; Knight Amari
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.
Fast transientsKnight (1972), Treves (1992,1997), Tsodyks&Sejnowski (1995)Gerstner (1998,2000), Brunel et al. (2001), Bethge et al. (2001)
Integral equation
Mean field equationsdensity (voltage, phase)
Heterogeneous netsstochastic connectivity
(Heterogeneous, non-spiking)
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 ModelsCourse (Neural Networks and Biological Modeling) session 7 and 8
Laboratory of Computational Neuroscience, LCN, CH 1015 LausanneSwiss Federal Institute of Technology Lausanne, EPFL
PSTH(t)
500 trials
I(t)
forward correlationfluctuating input
I(t)reverse correlationProbability of
output spike ?
I(t) A(t)?0t
Theoretical Approach
- population dynamics
- response to single input spike (forward correlation)
- reverse correlations
A(t)
500 neurons
PSTH(t)
500 trials
I(t) I(t)
Population of neurons
h(t)
I(t) ?0t
))(()( thgtA
A(t)
A(t)
A(t)
))(()( tIgtA
))('),(()( tItIgtA
potential
A(t) ))(()()( thgtAtAdtd
t
tNtttntA
);()(population
activity
N neurons,- voltage threshold, (e.g. IF neurons)- same type (e.g., excitatory) ---> population response ?
Coding Properties of Spiking Neurons:
Laboratory of Computational Neuroscience, LCN, CH 1015 LausanneSwiss Federal Institute of Technology Lausanne, EPFL
- forward correlations- reverse correlations
1. Transients in Population Dynamics - rapid transmission2. Coding Properties
Example: noise-free
tdtAttPtA I
t
ˆ)ˆ(ˆ|)(
))ˆ(ˆ(ˆ| tTttttPI
)(tA )( TtA
''1
uh
Population Dynamics
I(t) h’>0h(t)
T(t^)
higher activity
noise-free
Theory of transients
)(tA )( TtA
''1
uh
I(t)h(t)
I(t) ?0t
potential dsstIs )( ^tt ^| ttu
)(thinput potential 0)(' ttth
)()( 00 ttAAtA A(t)
External input.No lateral coupling
Theory of transients A(t)
no noise
I(t)h(t)
noise-free
noise model B
slow noise
I(t)h(t)
(reset noise)
u
p(u)
u
Membrane potential density
Hypothetical experiment: voltage step
u
p(u)
Immediate responseVanishes linearly
Transients with noise
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
Noise models
A B C
u(t)
noise
white(fast noise)
synapse(slow noise)
(Brunel et al., 2001)
t
t
dttt^
)')'(exp()(
)¦( ^ttPI Interval distribution
^t ^t ^tt
Survivor functionescape rate
)(t
))(()( tuftescape rate stochastic reset
)¦( ^ttPI )( fttG
Interval distribution
Gaussian about ft
)(tRIudtdu
ii
noisy integration
ft
Transients with noise:Escape noise (noisy threshold)
linearize
tdtAttPtA I
t
ˆ)ˆ(ˆ|)(
)()( 0 tAAtA
Theory with noise A(t)
)()( 0 thhth
I(t)h(t)
0A
dsstIsth )()()(
sA 1
0 inverse mean interval
I
Llow noiselow noise: transient prop to h’
high noise: transient prop to h
h: input potential
high noise
Theory of transients A(t)
low noise
I(t)h(t)
noise-free
(escape noise/fast noise) noise model A
low noise
fast
noise model A
I(t)h(t)
(escape noise/fast noise)
high noise
slow
Transients with noise:Diffusive noise (stochastic spike arrival)
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
Noise models
A B C
u(t)
noise
white(fast noise)
synapse(slow noise)
(Brunel et al., 2001)
t
t
dttt^
)')'(exp()(
)¦( ^ttPI Interval distribution
^t ^t ^tt
Survivor functionescape rate
)(t
))(()( tuftescape rate stochastic reset
)¦( ^ttPI )( fttG
Interval distribution
Gaussian about ft
)(tRIudtdu
ii
noisy integration
ft
u
p(u)
Diffusive noiseu
Membrane potential density
p(u)
Hypothetical experiment: voltage step
Immediate responsevanishes quadratically
),(
)],()([
),(
2
22
21 tup
u
tupuAu
tupt
Fokker-Planck
u
p(u)
SLOW Diffusive noiseu
Membrane potential density
Hypothetical experiment: voltage step
Immediate responsevanishes linearly
p(u)
Signal transmission in populations of neurons
Connections4000 external4000 within excitatory1000 within inhibitory
Population- 50 000 neurons- 20 percent inhibitory- randomly connected
-low rate-high rate
input
Population- 50 000 neurons- 20 percent inhibitory- randomly connected
Signal transmission in populations of neurons
100 200time [ms]
Neuron # 32374
50
u [mV]
100
0
10A [Hz]
Neu
ron
#
32340
32440
100 200time [ms]50
-low rate-high rate
input
Signal transmission - theory
- no noise
- slow noise (noise in parameters)
- strong stimulus
- fast noise (escape noise) prop. h(t) (potential)
prop. h’(t) (current)
See also: Knight (1972), Brunel et al. (2001)
fast
slow
Transients with noise: relation to experiments
Experiments to transients A(t)
V1 - transient response
V4 - transient response
Marsalek et al., 1997
delayed by 64 ms
delayed by 90 ms
V1 - single neuron PSTH
stimulus switched on
Experiments
input A(t)
A(t)
A(t)
A(t)
See also: Diesmann et al.
How fast is neuronal signal processing?
animal -- no animalSimon ThorpeNature, 1996
Visual processing Memory/association Output/movement
eye
Reaction time experiment
How fast is neuronal signal processing?
animal -- no animalSimon ThorpeNature, 1996
Reaction time
Reaction time
# ofimages
400 msVisual processing Memory/association Output/movement
Recognition time 150ms
eye
Coding properties of spiking neurons
Coding properties of spiking neurons
- response to single input spike
(forward correlations)
A(t)
500 neurons
PSTH(t)
500 trials
I(t)I(t)
Coding properties of spiking neurons
- response to single input spike
(forward correlations)
I(t) Spike ?Two simple arguments1)
2)
Experiments: Fetz and Gustafsson, 1983 Poliakov et al. 1997
(Moore et al., 1970)
PSTH=EPSP
(Kirkwood and Sears, 1978)
PSTH=EPSP’
Forward-Correlation Experiments A(t)
Poliakov et al., 1997
I(t) PSTH(t)
1000 repetitionsnoise
high noise low noiseprop. EPSP prop. EPSP
ddt
^^^ )(|)( dttAttPtA I
t
Population Dynamics
)()( 0 thhth h: input potential dsstIsth )()()(
A(t) PSTH(t)I(t)I(t)
full theory
linear theory
Forward-Correlation Experiments A(t)
Theory: Herrmann and Gerstner, 2001
high noise low noisePoliakov et al., 1997
high noise low noise
blue: full theoryred: linearized theory
blue: full theoryred: linearized theory
Forward-Correlation Experiments A(t)
Poliakov et al., 1997
I(t) PSTH(t)
1000 repetitionsnoise
high noise low noiseprop. EPSP prop. EPSP
ddt
prop. EPSP
prop. EPSPddt
Reverse Correlations
Laboratory of Computational Neuroscience, LCN, CH 1015 LausanneSwiss Federal Institute of Technology Lausanne, EPFL
fluctuating input
I(t)
Reverse-Correlation Experiments
after 1000 spikes
)(tI
)()( 0 thhth h: input potential dsstIsth )()()(
Linear Theory
Fourier Transform
)(~)(~)(~ IGA
0
)()()( dsstIsGtA
Inverse Fourier Transform
)(~1)(~)(~
)(~ 0
PLAiG
Signal transmissionI(t) A(t)
)()()(
fIfAfG T=1/f
(escape noise/fast noise) noise model A
low noise
high noise
noise model B (reset noise/slow noise)
high noiseno cut-off
low noise
Reverse-Correlation Experiments (simulations)
after 1000 spikes
0
)()()( dsstIsGtA
theory:G(-s)
)(tI
after 25000 spikes
Laboratory of Computational Neuroscience, EPFL, CH 1015 Lausanne
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
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
Search for bifurcation points
linearize
^^^ )(|)( dttAttPtA I
t
)()( 0 tAAtA )()( 0 thhth dsstAsJth )()()(
h: input potential
A(t)
ttieAtA 1)(0
fully connected coupling J/N
Stability of Asynchronous State A(t)
delayperiod
)()( sess0 for
stable0
03
02
noise
T
)(s
s
)(s
Stability of Asynchronous State s
)(s
ms0.1
ms2.1
4.1
ms0.2
ms0.3
ms4.0
06
05
04
03
02
T 2
0
Chapter 9: Spatially structured networks
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 9
Continuous Networks
)(tAi
Several populations
i k
Continuum
)(),( AtA
Continuum: stationary profile