Part III: Models of synaptic plasticity
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Transcript of Part III: Models of synaptic plasticity
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Part III: Models of synaptic plasticity
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapters 10-12
Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne
Swiss Federal Institute of Technology Lausanne, EPFL
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Chapter 10: Hebbian Models
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 10
-Hebb rules-STDP
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Hebbian Learning
pre jpost
iijw
When an axon of cell j repeatedly or persistently takes part in firing cell i, then j’s efficiency as oneof the cells firing i is increased
Hebb, 1949
k
- local rule- simultaneously active (correlations)
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Hebbian Learning in experiments (schematic)
posti
ijwEPSP
pre j
no spike of i
post
u
spikes of i
u
pre j i
ijw
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Hebbian Learning in experiments (schematic)
posti
ijwEPSP
pre j
no spike of i
EPSP
pre j
posti
ijw no spike of i
pre j
posti
ijwBoth neuronssimultaneously active
Increased amplitude 0 ijw
u
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Hebbian Learning
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Hebbian Learning
item memorized
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Hebbian Learning
item recalled
Recall:Partial info
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Hebbian Learning
pre jpost
iijw
When an axon of cell j repeatedly or persistently takes part in firing cell i, then j’s efficiency as oneof the cells firing i is increased
Hebb, 1949
k
- local rule- simultaneously active (correlations)
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Hebbian Learning: rate model
pre jpost
iijw
k
- local rule- simultaneously active (correlations)
posti
prej
corrij aw
dt
d 2
activity (rate)
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Hebbian Learning: rate modelpre jpost
iijw
k
posti
prej
corrij aw
dt
d 2
cawdt
d posti
prej
corrij 2
)(2 posti
prej
corrij aw
dt
d
))((2 posti
prej
corrij aw
dt
d
pre
poston off on offon offon off
+ +
+ 0 0 0
- -
+ 00 -
+ - - -
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Rate-based Hebbian Learning
pre jpost
iijw
k
- local rule- simultaneously active (correlations)
),;( posti
prejijij wFw
dt
d
...2110 posti
prej
corrposti
postprej
preij aaaaw
dt
d
Taylor expansion
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Rate-based Hebbian Learning
pre jpost
iijw
),;( posti
prejijij wFw
dt
d
...2110 posti
prej
corrposti
postprej
preij aaaaw
dt
d
a = a(wij)
a(wij) wij
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Rate-based Hebbian Learningpre jpost
iijw
k
...2110 posti
prej
corrposti
postprej
preij aaaaw
dt
d
222 ][ post
ipostpost
iprej
corrij aaw
dt
d
0][ 22 post
iposta
Oja’s rule
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Spike based model
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Spike-based Hebbian Learning
pre jpost
iijw
k
- local rule- simultaneously active (correlations)
0 Prebefore post
posti
prej tt
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Spike-based Hebbian Learning
pre jpost
iijw
k
causal rule‘neuron j takes part in firing neuron’ Hebb, 1949
0 Prebefore post
posti
prej tt
prejt
EPSP
)( posti
prejij ttWw
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Spike-time dependent learning window
pre jpost
iijw
0
Prebefore post
posti
prej tt
prejt
posti
prej tt
posti
prej tt 0
0
Temporal contrast filter
)( posti
prejij ttWw
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Spike-time dependent learning window
pre jpost
iijw
prejt
postit
Prebefore post
Zhang et al, 1998review:Bi and Poo, 2001
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Spike-time dependent learning: phenomenol. model
pre jpost
iijw
Prebefore post
prejt
posti
prej tt 0
o
post
t
pre
t
posti
prej
tt
ij bbbttWwprej
prej
posti
prej
)(,
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spike-based Hebbian Learning
pre j
post i
ijw),;( post
iprejijij uuwFw
dt
d
dsstusbdsstusbbwdt
d posti
postprej
preij )()()()( 110
...')'()()',(2 dsdsstustussb posti
prej
corr
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spike-based Hebbian Learningpre j
post i
ijw
)()( fjf
prej tttu
)()( 110f
ipost
f
fj
pre
fij ttbttbbw
dt
d
...),(2,
kj
fi
corr
kj
ttttb
BPAP
)()( fif
posti tttu
Translation invariance
W(tif-tj
k )
Learning window
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Detailed models
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 10
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Detailed models of Hebbian learning
pre j
post i
ijw
i at restingpotential
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Detailed models of Hebbian learning
pre j
post i
ijw
i at restingpotential
NMDAchannel
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Detailed models of Hebbian learning
pre j
post i
ijw
i at highpotential
BPAP
NMDA channel : - glutamate binding after presynaptic spike - unblocked after postsynaptic spike
elementary correlation detector
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Mechanistic models of Hebbian learningpre j
post i
ijw
BPAP
prejt
badt
dwij
)( prej
a
tta
dt
da
prea
)( posti
a
ttb
dt
db
post
b
w
)( postj
prejij ttWw
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Mechanistic models of Hebbian learningpre j
post i
ijw
BPAP
prejt
][ dcbadt
dwij
4-factor modelpre
post
sophisticated 2-factor
][ babadt
dw qqij
)( postj
prejij ttWw
Prebefore post
posti
prej tt 0
Abarbanel et al. 2002
Gerstner et al. 1998Buonomano 2001
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Mechanistic models of Hebbian learningpre j
post i
ijw
BPAP
prejt
badt
dwij
)( prej
a
tta
dt
da
prea
)( posti
a
ttb
dt
db
post
b
w
)( postj
prejij ttWw
1 pre, 1 post
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Mechanistic models of Hebbian learning
pre j
post i
ijw
BPAP
Dynamics of NMDA receptor (Senn et al., 2001)
Prebefore post
posti
prej tt 0
prejt
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Which kind of model?
exp , if
exp , if
j ij i
j ij i
t tA t t
wt t
A t t
Descriptive Models
Optimal Models
Mechanistic Models
( ) ( ) ( ) ( )w a t b t a t b t
Song et al. 2000Gerstner et al. 1996
Abarbanel et al. 2002Senn et al. 2000
Chechik, 2003Hopfield/Brody, 2004Dayan/London, 2004
Shouval et al. 2000
Gütig et al. 2003
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Chapter 11: Learning Equations
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 11
-rate based Hebbian learning-STDP
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Rate-based Hebbian Learning
pre jpost
iijw
),;( posti
prejijij wFw
dt
d
...2110 posti
prej
corrposti
postprej
preij aaaaw
dt
d
a = a(wij)
a(wij) wij
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Analysis of rate-based Hebbian Learning
ijw
prejij
posti w
txk
x1 xkx2
.2post
iprej
corrij aw
dt
d
Linear model
Analysis - separation of time scales, expected evolution
.2prek
prejik
corrij waw
dt
d
Correlationsin the input
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Analysis of rate-based Hebbian Learning
ijw
prejij
posti w
txk
x1 xkx2
Linear model
.2prek
prejik
corrij waw
dt
d
Correlationsin the input
.2 jkkcorr
j Cwawdt
d
supress index i
wCawdt
d corr 2
eigenvectors
nnn eeC
)exp()( 2 taetw ncorr
nn
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Analysis of rate-based Hebbian Learning
ijw
txk
x1 xkx2
wCawdt
d corr 2
)exp()( 2 taetw ncorr
nn
x1
12 xaw
dt
d postcorr
prej
posti
corrij aw
dt
d 2
)(tw
moves towards data cloud
w
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Analysis of rate-based Hebbian Learning
ijw
txk
x1 xkx2
wCawdt
d corr 2
)exp()( 2 taetw ncorr
nn
x1
12 xaw
dt
d postcorr
)(tw
becomes aligned with principal axis
w
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spike-based Hebbian Learning
pre j
post i
ijw),;( post
iprejijij uuwFw
dt
d
dsstusbdsstusbbwdt
d posti
postprej
preij )()()()( 110
...')'()()',(2 dsdsstustussb posti
prej
corr
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spike-based Hebbian Learningpre j
post i
ijw
)()( fjf
prej tttu
)()( 110f
ipost
f
fj
pre
fij ttbttbbw
dt
d
...),(2,
kj
fi
corr
kj
ttttb
BPAP
)()( fif
posti tttu
Translation invariance
W(tif-tj
k )
Learning window
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Analysis of spike-based Hebbian Learning
ijwvk
Linear model
vj1 vj
k
Analysis - separation of time scales, expected evolution
Correlationspre/post
.)()()(
)()( 110
dsstStSsW
tSatSaawdt
d
posti
prej
prej
preposti
postij
)()( fj
f
prej tttS
)(}{obPr fj
fjij
posti ttwt )()( post
ipost
posti tttS
)( fjtt
Point process
Average over doublystochastic process
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Analysis of spike-based Hebbian Learning
][ FPpost
ipost
idt
d
Rewrite equ.(i) fixed point equation for postsyn. rate
post
pre
FP aW
aa
1
10
wQwdt
d
.)()()( dssWQ prek
prejjk Covariance of input
(ii) input covariance
(plus extra terms)
Stable if01 postaW
Average over ensemble of rates
Rate stabilization
dssWW )(
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Analysis of spike-based Hebbian Learning
)( posti
posti tt
.)()( dsssW
(iii) extra spike-spike correlations
)( fjtt
pre j
)( s
)(sW
spike-spike correlations
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Spike-based Hebbian Learning
.)()( dsssW
- picks up spatio-temporal correlations on the time scale of the learning window W(s)
)( s
)(sW
)(sW
- non-trivial spike-spike correlations
- rate stabilization yields competition of synapses
ijw Synapses grow at the expense of others
Neuron stays in sensitive regime
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Chapter 12: Plasticity and Coding
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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Learning to be fast: prediction
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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Derivative filter and prediction
pre j Mehta et al. 2000,2002
Song et al. 2000
+
-
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Derivative filter and prediction
pre j Mehta et al. 2000,2002
Song et al. 2000
+
-
Postsynaptic firing shifts, becomes earlier
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Derivative filter and prediction
posti
prej dt
dw
derivative of postsyn. rate
pre j Mehta et al. 2000,2002
Song et al. 2000
+
-
Roberts et al. 1999Rao/Sejnowski, 2001Seung
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Learning spike patterns
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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Spike-based Hebbian Learning
pre jpost
iijw
k
causal rule‘neuron j takes part in firing neuron’ Hebb, 1949
0 Prebefore post
posti
prej tt
prejt
EPSP
)( posti
prejij ttWw
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Spike-based Hebbian Learning: sequence learning
pre
jpost
iijw
0 Prebefore post
posti
prej tt
prejt
EPSP
Strengthen the connection with the desired timing
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Subtraction of expectations: electric fish
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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Spike-based Hebbian Learning
suppressestemporal structure
experiment model
C.C. Bell et al.,Roberts and Bell
Novelty detector(subtracts expectation)
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Learning a temporal code: barn owl auditory system
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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Delay tuning in barn owl auditory system
Accuracy 1 degree
Temporal precision <5us
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Jeffress model
Accuracy 1 degree
Temporal precision <5us
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Jeffress model
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Delay tuning in barn owl auditory system
1
mst 5
Sound source
k
Tuning of delay lines
Jeffress, 1948Carr and Konishi, 1990
Gerstner et al., 1996
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Delay tuning in barn owl auditory system
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Delay tuning in barn owl auditory system
ca. 150 ca. 150
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Delay tuning in barn owl auditory system
Problem: 5kHz signal (period 0.2 ms) but distribution of delays 2-3 ms
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Spike-timing dependent plasticity: phenomenol. model
pre jpost
iijw
Prebefore post
prejt
posti
prej tt 0
)()(,
prej
pre
t
posti
prej
tt
ij ttbttWwprej
posti
prej
1ms
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Delay tuning in barn owl auditory system
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Conclusions (chapter 12)
-STDP is spiking version of Hebb’s rule
-shifts postsynaptic firing earlier in time
-allows to learn temporal codes