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Swiss Federal Institute of Technology Lausanne, EPFL. Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne. Part III: Models of synaptic plasticity. BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 10-12.

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Swiss Federal Institute of Technology Lausanne, EPFL

Laboratoryof Computational Neuroscience, LCN, CH 1015 Lausanne

Part III: Models of synaptic plasticity

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapters 10-12


Chapter 10: Hebbian Models

  • -Hebb rules

  • STDP

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 10


Hebbian Learning

pre j

i

k

post

When an axon of celljrepeatedly or persistently

takes part in firing cell i, then j’s efficiency as one

of the cells firing i is increased

Hebb, 1949

- local rule

- simultaneously active (correlations)


pre j

u

i

spikes of i

Hebbian Learning in experiments (schematic)

pre j

u

no spike of i

EPSP

i

post

post


pre j

Both neurons

simultaneously active

i

post

pre j

no spike of i

EPSP

i

Increased amplitude

post

Hebbian Learning in experiments (schematic)

pre j

u

no spike of i

EPSP

i

post



Hebbian Learning

item memorized


Hebbian Learning

Recall:

Partial info

item recalled


Hebbian Learning

pre j

i

k

post

When an axon of celljrepeatedly or persistently

takes part in firing cell i, then j’s efficiency as one

of the cells firing i is increased

Hebb, 1949

- local rule

- simultaneously active (correlations)


activity (rate)

Hebbian Learning: rate model

pre j

i

k

post

- local rule

- simultaneously active (correlations)


+

0

0

0

-

-

-

+

0

-

+

0

Hebbian Learning: rate model

pre j

i

k

post

on

on

off

off

pre

post

on

off

on

off

+

-

-

+


Rate-based Hebbian Learning

pre j

i

k

post

- local rule

- simultaneously active (correlations)

Taylor expansion


Rate-based Hebbian Learning

pre j

i

post

a = a(wij)

a(wij)

wij


Oja’s rule

Rate-based Hebbian Learning

pre j

i

k

post



0

Pre

before post

Spike-based Hebbian Learning

pre j

i

k

post

- local rule

- simultaneously active (correlations)


Spike-based Hebbian Learning

pre j

EPSP

i

k

post

0

Pre

before post

causal rule

‘neuron j takes part in firing neuron’

Hebb, 1949


Spike-time dependent learning window

pre j

i

post

0

0

0

Pre

before post

Temporal contrast filter


Spike-time dependent learning window

pre j

i

post

Zhang et al, 1998

review:

Bi and Poo, 2001

Pre

before post


Spike-time dependent learning: phenomenol. model

pre j

i

post

0

Pre

before post



Translation invariance

W(tif-tjk )

Learning window

spike-based Hebbian Learning

pre j

BPAP

post

i


Detailed models

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 10


Detailed models of Hebbian learning

pre j

post

i

i at resting

potential


NMDA

channel

i at resting

potential

Detailed models of Hebbian learning

pre j

post

i


i at high

potential

Detailed models of Hebbian learning

pre j

BPAP

post

i

NMDA channel :

- glutamate binding after presynaptic spike

- unblocked after postsynaptic spike

elementary correlation detector


a

pre

b

post

w

Mechanistic models of Hebbian learning

pre j

BPAP

post

i


0

Pre

before post

sophisticated 2-factor

Mechanistic models of Hebbian learning

pre j

BPAP

post

i

pre

4-factor model

Gerstner et al. 1998

Buonomano 2001

post

Abarbanel et al. 2002


a

pre

b

post

w

Mechanistic models of Hebbian learning

pre j

BPAP

post

i

1 pre, 1 post


0

Pre

before post

Mechanistic models of Hebbian learning

pre j

Dynamics of NMDA

receptor (Senn et al., 2001)

BPAP

post

i


Which kind of model
Which kind of model?

Descriptive Models

Gerstner et al. 1996

Song et al. 2000

Gütig et al. 2003

Mechanistic Models

Senn et al. 2000

Abarbanel et al. 2002

Shouval et al. 2000

Optimal Models

Chechik, 2003

Hopfield/Brody, 2004

Dayan/London, 2004


Chapter 11: Learning Equations

  • -rate based Hebbian learning

  • STDP

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 11


Rate-based Hebbian Learning

pre j

i

post

a = a(wij)

a(wij)

wij


Analysis of rate-based Hebbian Learning

x1

x2

xk

xk

t

Linear model

Analysis - separation of time scales, expected evolution

Correlations

in the input


supress index i

eigenvectors

Analysis of rate-based Hebbian Learning

x1

x2

xk

xk

t

Linear model

Correlations

in the input


moves towards data cloud

w

Analysis of rate-based Hebbian Learning

x1

x2

xk

xk

t

x1


becomes aligned

with principal axis

w

Analysis of rate-based Hebbian Learning

x1

x2

xk

xk

t

x1



Translation invariance

W(tif-tjk )

Learning window

spike-based Hebbian Learning

pre j

BPAP

post

i


Analysis - separation of time scales, expected evolution

Average over doubly

stochastic process

Correlations

pre/post

Analysis of spike-based Hebbian Learning

vjk

vj1

Point process

vk

Linear model


Stable if

Rate stabilization

(ii) input covariance

(plus extra terms)

Average over

ensemble of rates

Covariance of input

Analysis of spike-based Hebbian Learning

Rewrite equ.

(i) fixed point equation for postsyn. rate


Analysis of spike-based Hebbian Learning

(iii) extra spike-spike correlations

pre j

spike-spike correlations


Spike-based Hebbian Learning

- picks up spatio-temporal correlations on the time scale

of the learning window W(s)

- non-trivial spike-spike correlations

- rate stabilization yields competition of synapses

Synapses grow at the expense of others

Neuron stays in sensitive regime


Chapter 12: Plasticity and Coding

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 12


Learning to be fast: prediction

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 12


Derivative filter and prediction

pre j

Mehta et al. 2000,2002

Song et al. 2000

+

-


+

-

Derivative filter and prediction

pre j

Mehta et al. 2000,2002

Song et al. 2000

Postsynaptic firing shifts, becomes earlier


Derivative filter and prediction

pre j

Mehta et al. 2000,2002

Song et al. 2000

+

-

derivative of postsyn. rate

Roberts et al. 1999

Rao/Sejnowski, 2001

Seung


Learning spike patterns

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 12


Spike-based Hebbian Learning

pre j

EPSP

i

k

post

0

Pre

before post

causal rule

‘neuron j takes part in firing neuron’

Hebb, 1949


pre j

i

post

Spike-based Hebbian Learning: sequence learning

EPSP

0

Pre

before post

Strengthen the connection

with the desired timing


Subtraction of expectations:

electric fish

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 12


Spike-based Hebbian Learning

suppresses

temporal structure

experiment

model

C.C. Bell et al.,

Roberts and Bell

Novelty detector

(subtracts expectation)


Learning a temporal code:

barn owl auditory system

BOOK: Spiking Neuron Models,

W. Gerstner and W. Kistler

Cambridge University Press, 2002

Chapter 12


Delay tuning in barn owl auditory system

Accuracy 1 degree

Temporal precision <5us


Jeffress model

Accuracy 1 degree

Temporal precision <5us



Tuning of delay lines

Delay tuning in barn owl auditory system

Sound source

Jeffress, 1948

Carr and Konishi, 1990

Gerstner et al., 1996




Delay tuning in barn owl auditory system

Problem: 5kHz signal (period 0.2 ms)

but distribution of delays 2-3 ms


Spike-timing dependent plasticity: phenomenol. model

pre j

i

1ms

post

0

Pre

before post



Conclusions (chapter 12)

-STDP is spiking version of Hebb’s rule

-shifts postsynaptic firing earlier in time

-allows to learn temporal codes


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