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Part II Population Models
BOOK Spiking Neuron Models, W. Gerstner
and W. Kistler Cambridge University Press,
2002 Chapters 6-9
2
Chapter 6 Population Equations
BOOK Spiking Neuron Models, W. Gerstner and
W. Kistler Cambridge University Press, 2002
Chapter 6
3
10 000 neurons 3 km wires
1mm
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Spike Response Model
Spike emission
i
Spike emission AP
All spikes, all neurons
Last spike of i
linear
threshold
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Integrate-and-fire Model
Spike emission
i
reset
I
linear
Firereset
threshold
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Noise models
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
A
B
C
u(t)
t
stochastic reset
Interval distribution
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Homogeneous Population
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populations of spiking neurons
t
population dynamics?
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Homogenous network (SRM)
Spike emission AP
Spike reception EPSP
fully connected N gtgt 1
Synaptic coupling
potential
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refractory potential
potential
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Homogenous network
Spike emission AP
Response to current pulse
potential
Population activity
Last spike of i
All neurons receive the same input
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Homogeneous network (IF)
Assumption of Stochastic spike arrival network
of exc. neurons, total spike arrival rate A(t)

Synaptic current pulses
EPSC
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Density equations
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Density equation (stochastic spike arrival)
Stochastic spike arrival network of exc.
neurons, total spike arrival rate A(t)
Synaptic current pulses
EPSC
Langenvin equation, Ornstein Uhlenbeck process
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Density equation (stochastic spike arrival)
A(t)flux across threshold
u
p(u)
Fokker-Planck
diffusion
source term at reset
drift
spike arrival rate
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Integral equations
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Population Dynamics
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Escape Noise (noisy threshold)
IF with reset, constant input, exponential
escape rate
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Population Dynamics
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Wilson-Cowan population equation
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Wilson-Cowan model
escape process (fast noise)
(i) noisy firing
(ii) absolute refractory time
h(t)
t
population activity
escape rate
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Wilson-Cowan model
escape process (fast noise)
(i) noisy firing
(ii) absolute refractory time
h(t)
t
population activity
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Population activity in spiking neurons (an
incomplete history)
1972 - WilsonCowan Knight Amari
Integral equation
(Heterogeneous, non-spiking)
1992/93 - AbbottvanVreeswijk
GerstnervanHemmen Treves et al. Tsodyks et
al. BauerPawelzik 1997/98 -
vanVreeswijkSompoolinsky AmitBrunel
Pham et al. Senn et al. 1999/00 - BrunelHakim
FusiMattia NykampTranchina Omurtag
et al.
Mean field equations density (voltage, phase)
Heterogeneous nets stochastic connectivity
Fast transients Knight (1972), Treves
(1992,1997), TsodyksSejnowski (1995) Gerstner
(1998,2000), Brunel et al. (2001), Bethge et al.
(2001)
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Chapter 7 Signal Transmission and Neuronal
Coding
BOOK Spiking Neuron Models, W. Gerstner and
W. Kistler Cambridge University Press, 2002
Chapter 7
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Coding Properties of Spiking Neuron Models
Course (Neural Networks and Biological Modeling)
session 7 and 8
Probability of output spike ?
forward correlation
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Theoretical Approach
A(t)
PSTH(t)
I(t)
I(t)
500 trials
500 neurons
- population dynamics - response to single input
spike (forward correlation) - reverse
correlations
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Population of neurons
?
I(t)
potential
N neurons, - voltage threshold, (e.g. IF
neurons) - same type (e.g., excitatory) ---gt
population response ?
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Coding Properties of Spiking Neurons
1. Transients in Population Dynamics -
rapid transmission 2. Coding Properties
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Population Dynamics
Example noise-free
I(t)
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Theory of transients
noise-free
External input. No lateral coupling
potential
I(t)
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Theory of transients
no noise
(reset noise)
noise-free
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Hypothetical experiment voltage step
Immediate response Vanishes linearly
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Transients with noise
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Noise models
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
A
B
C
u(t)
t
stochastic reset
Interval distribution
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Transients with noise Escape noise (noisy
threshold)
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Theory with noise
A(t)
linearize
h input potential
low noise
low noise transient prop to h
high noise transient prop to h
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Theory of transients
noise model A
(escape noise/fast noise)
low noise
low noise
noise-free
fast
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Transients with noise Diffusive noise
(stochastic spike arrival)
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Noise models
escape process (fast noise)
parameter changes (slow noise)
stochastic spike arrival (diffusive noise)
A
B
C
u(t)
t
stochastic reset
Interval distribution
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Diffusive noise
u
p(u)
Hypothetical experiment voltage step
Immediate response vanishes quadratically
p(u)
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SLOW Diffusive noise
u
p(u)
Hypothetical experiment voltage step
Immediate response vanishes linearly
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Signal transmission in populations of neurons
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Signal transmission in populations of neurons
A Hz
10
32440
Neuron
32340
200
time ms
100
50
100
Neuron 32374
u mV
Population - 50 000 neurons - 20 percent
inhibitory - randomly connected
0
time ms
200
100
50
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Signal transmission - theory
- no noise - slow noise (noise in parameters) -
strong stimulus
prop. h(t) (current)
fast
prop. h(t) (potential)
slow
- fast noise (escape noise)
See also Knight (1972), Brunel et al. (2001)
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Transients with noise relation to experiments
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Experiments to transients
Experiments
V4 - transient response
Marsalek et al., 1997
delayed by 90 ms
V1 - transient response
delayed by 64 ms
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input
A(t)
A(t)
A(t)
A(t)
See also Diesmann et al.
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How fast is neuronal signal processing?
Simon Thorpe Nature, 1996
animal -- no animal
Reaction time experiment
Visual processing
Memory/association
Output/movement
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(No Transcript)
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How fast is neuronal signal processing?
Simon Thorpe Nature, 1996
animal -- no animal
of
images
Reaction time
Reaction time
400 ms
Visual processing
Memory/association
Output/movement
Recognition time 150ms
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Coding properties of spiking neurons
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Coding properties of spiking neurons
A(t)
PSTH(t)
500 trials
500 neurons
- response to single input spike (forward
correlations)
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Coding properties of spiking neurons
Two simple arguments
Spike ?
1)
- response to single input spike (forward
correlations)
2)
Experiments Fetz and Gustafsson, 1983
Poliakov et al. 1997
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Forward-Correlation Experiments
A(t)
Poliakov et al., 1997
low noise
high noise
prop. EPSP
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Population Dynamics
full theory
A(t)
PSTH(t)
I(t)
I(t)
linear theory
h input potential
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Forward-Correlation Experiments
A(t)
Theory Herrmann and Gerstner, 2001
high noise
low noise
low noise
high noise
Poliakov et al., 1997
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Forward-Correlation Experiments
A(t)
Poliakov et al., 1997
PSTH(t)
I(t)
prop. EPSP
noise
1000 repetitions
low noise
high noise
prop. EPSP
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Reverse Correlations
I(t)
fluctuating input
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Reverse-Correlation Experiments
after 1000 spikes
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Linear Theory
h input potential
Fourier Transform
Inverse Fourier Transform
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Signal transmission
A(t)
I(t)
T1/f
noise model A
noise model B
(escape noise/fast noise)
(reset noise/slow noise)
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Reverse-Correlation Experiments
(simulations)
after 1000 spikes
after 25000 spikes
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Coding Properties of spiking neurons
- spike dynamics -gt 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
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Chapter 8 Oscillations and Synchrony
BOOK Spiking Neuron Models, W. Gerstner and
W. Kistler Cambridge University Press, 2002
Chapter 8
65
Stability of Asynchronous State
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Stability of Asynchronous State
A(t)
fully connected coupling J/N
linearize
h input potential
Search for bifurcation points
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Stability of Asynchronous State
A(t)
0 for
s
delay period
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Stability of Asynchronous State
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Chapter 9 Spatially structured
networks
BOOK Spiking Neuron Models, W. Gerstner and
W. Kistler Cambridge University Press, 2002
Chapter 9
70
Continuous Networks
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Several populations
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Continuum stationary profile