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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. - PowerPoint PPT Presentation
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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
Chapter 10: Hebbian Models
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 10
-Hebb rules-STDP
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)
Hebbian Learning in experiments (schematic)
posti
ijwEPSP
pre j
no spike of i
post
u
spikes of i
u
pre j i
ijw
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
Hebbian Learning
Hebbian Learning
item memorized
Hebbian Learning
item recalled
Recall:Partial info
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)
Hebbian Learning: rate model
pre jpost
iijw
k
- local rule- simultaneously active (correlations)
posti
prej
corrij aw
dt
d 2
activity (rate)
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 -
+ - - -
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
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
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
Spike based model
Spike-based Hebbian Learning
pre jpost
iijw
k
- local rule- simultaneously active (correlations)
0 Prebefore post
posti
prej tt
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
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
Spike-time dependent learning window
pre jpost
iijw
prejt
postit
Prebefore post
Zhang et al, 1998review:Bi and Poo, 2001
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
)(,
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
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
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
ijw
i at restingpotential
Detailed models of Hebbian learning
pre j
post i
ijw
i at restingpotential
NMDAchannel
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
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
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
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
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
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
Chapter 11: Learning Equations
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 11
-rate based Hebbian learning-STDP
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
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
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
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
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
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
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
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
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 )(
Analysis of spike-based Hebbian Learning
)( posti
posti tt
.)()( dsssW
(iii) extra spike-spike correlations
)( fjtt
pre j
)( s
)(sW
spike-spike correlations
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
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
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
Learning spike patterns
BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002
Chapter 12
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
Spike-based Hebbian Learning: sequence learning
pre
jpost
iijw
0 Prebefore post
posti
prej tt
prejt
EPSP
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
suppressestemporal 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
Jeffress model
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
Delay tuning in barn owl auditory system
Delay tuning in barn owl auditory system
ca. 150 ca. 150
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 jpost
iijw
Prebefore post
prejt
posti
prej tt 0
)()(,
prej
pre
t
posti
prej
tt
ij ttbttWwprej
posti
prej
1ms
Delay tuning in barn owl auditory system
Conclusions (chapter 12)
-STDP is spiking version of Hebb’s rule
-shifts postsynaptic firing earlier in time
-allows to learn temporal codes
