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Neural Networks with Short-Term Synaptic Dynamics
(Leiden, May 23 2008)
Misha Tsodyks, Weizmann Institute
• Mathematical Models of Short-Term Synaptic plasticity
• Computations in Neural Networks with Short-Term Synaptic Plasticity
Experiments: Henry Markram, Eli NelkenTheory: Klaus Pawelzik, Alex Loebel
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Multineuron Recording
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Principles of Time-Dependency of Release
AP onset
Facilitation
Depression
Synaptic Facilitation
Synaptic Depression
Three Laws of ReleaseLimited Synaptic Resources
Release Dependent Depression
Release Independent Facilitation
Four Parameters of Release
•Absolute strength
•Probability of release
•Depression time constant
•Facilitation time constant
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A Phenomenological Approach to Dynamic Synaptic Transmission
• 4 Key Synaptic Parameters– Absolute strength
– Probability of release
– Depression time constant
– Facilitation time constant
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Phenomenological model of synaptic depression (deterministic)
(1 )( )
( )
sp
I s
r
p
dR RRu
Au
t tdt
I I R t t
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.
.
.
Stochastic transmission
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Binomial model of synaptic release
, , , rP Q N
Loebel et al, submitted
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The fitting process (deterministic model)
, , rA U
,mem I
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Testing the Model
experiment
model
experiment
model
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Frequency-dependence of post-synaptic response
(1 )( )sp
r
dR Ru R t t
dt
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Frequency-dependence of post-synaptic response
(1 )( )sp
r
dR Ru R t t
dt
/ /1
/ /1 1
(1 ) 1
(1 ) (1 )
r r
r r
T Tn n
T Tn n
R R u e e
E E u e E e
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Frequency-dependence of post-synaptic response
(1 )( )sp
r
dR Ru R t t
dt
/ /1
/ /1 1
(1 ) 1
(1 ) (1 )
r r
r r
T Tn n
T Tn n
R R u e e
E E u e E e
/
1 /
1
1 (1 )
r
r
T
st T
eE E
u e
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Frequency-dependence of post-synaptic response
(1 )( )sp
r
dR Ru R t t
dt
/ /1
/ /1 1
(1 ) 1
(1 ) (1 )
r r
r r
T Tn n
T Tn n
R R u e e
E E u e E e
/1
1 /
1
1 (1 )
r
r
T
st Tr r
Ee AE E
u e u f f
Tsodyks et al, PNAS 1997
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The 1/f Law of Release
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Redistribution of Synaptic Efficacy
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RSERSE
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Shifting the Distribution of Release Probabilities
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Population signal
1( )
( )
r
I
RR Ru E t
I I AuRE t
Tsodyks et al, Neural Computation 1998
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Population signal
1( )
( ) ( )r
RR Ru E t
I t AuRE t
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Population signal
1( )
( ) ( )r
RR Ru E t
I t AuRE t
‘High-pass filtering’ of the input rate
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Steady-state signal
1
1
1
r
r
Ru E
EI Au
u E
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Tsodyks et al PNAS 1997
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Synchronous change in pre-synaptic rates
1 2E E E E
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Synchronous change in pre-synaptic rates
1 2E E E E
1 2exp( / )recI AuR E t R
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Synchronous change in pre-synaptic rates
1 2E E E E
1 2 2exp( / ) exp( / )rec rec
EI AuR E t R t R
E
Abbott et al Science 1997
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Modeling synaptic facilitation (deterministic)
(1 ) ( )
1( )
( )
sf
sr
I s
u Uu U u t t
RR uR t t
I I AuR t t
Markram, Wang & Tsodyks PNAS 1998
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FacilitationFacilitation
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Frequency-dependence of post-synaptic response
(1 ) ( )
1( )
sf
sr
u Uu U u t t
RR uR t t
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Frequency-dependence of post-synaptic response
/1
/ /1
/ /1 1
(1 )
(1 ) 1
(1 ) (1 )
f
r r
r r
Tn n
T Tn n n
T Tn n n n
u u U e U
R R u e e
E E u e Au e
(1 ) ( )
1( )
sf
sr
u Uu U u t t
RR uR t t
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Frequency-dependence of post-synaptic response
/1
/ /1
/ /1 1
(1 )
(1 ) 1
(1 ) (1 )
f
r r
r r
Tn n
T Tn n n
T Tn n n n
u u U e U
R R u e e
E E u e Au e
/
/
/
1 (1 )
1
1 (1 )
f
r
r
st T
T
st st Tst
Uu
U e
eE Au
u e
(1 ) ( )
1( )
sf
sr
u Uu U u t t
RR uR t t
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Population signal
(1 ) ( )
1( )
( ) ( )
f
r
u Uu U u E t
RR Ru E t
I t AuRE t
Tsodyks et al 1998
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Steady-state signal
1
1
1
1
1
f
f
r
r
Eu U
U E
Ru E
EI AuRE Au
u E
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A Small Circuit
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Recurrent networks with synaptic depression
Tsodyks et al, J. Neurosci. 2000Loebel & Tsodyks JCNS 2002Loebel, Nelken & Tsodyks, Frontiers in Neurosci. 2007
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Simulation of Network Activity
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Simulation of Network Activity
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Origin of Population Spikes
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Network response to stimulation
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Experimental evidence for population spikes
DeWeese & Zador 2006
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Simplified model )no inhibition, uniform connections, rate equations(
1ei
Ne
J J
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The rate model equations
There are two sets of equations representing the excitatory units firing rate, E , and their
depression factor, R :
1
[ ]N
ii j j i
j
dE JE E R e
dt N
1i i
i ir
dR RuR E
dt
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Tone
The tonic stimuli is represented by a constant shift of the {e}’s, that, when large enough, causes the network to spike and reach a new steady state
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DeWeese, …, Zador 2003
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(Nelken et al, Nature 1999)
0
Noi
se a
mpl
itude
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Extended model
Loebel, Nelken & Tsodyks, 2007
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Rotman et al, 2001
Forward suppression
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Network response to complex stimuli
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Network response to complex stimuli
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Neural Network Models of Working Memory
Misha Tsodyks
Weizmann Institute of Science
Barak Blumenfeld, Son Preminger & Dov Sagi
Gianluigi Mongillo & Omri Barak
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Delayed memory experiments – sample to match
Miyashita et al, Nature 1988
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Persistent activity (Fuster ’73)
Miyashita et al, Nature 1988
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Neural network models of associative memory (Hopfield ’82)
Memories are represented as attractors (stable states) of network dynamics. Attractor = internal representation (memory) of a stimulus in the form
of stable network activity pattern Synaptic modifications => Changes in attractor landscape = long-
term changes in memory Convergence to an attractor = recall of item from long-term memory
into a working form
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Associative memory model
Hopfield Model (1982):
1
( 1) ( ( ))N
i ij j
j
S t sign J S t
1iS Neurons:
Synaptic connections: ijJ
Network dynamics:
Memory patterns: 1i
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Associative memory model
Hopfield Model (1982):
1iS Neurons: Memory patterns: 1i
Synaptic modifications : ij ij i jJ J
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Associative memory model
Hopfield Model (1982):
1iS Neurons: Memory patterns: 1i
1
P
ij i jJ
Synaptic modifications : ij ij i jJ J
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Associative memory model
Hopfield Model (1982):
1iS Neurons: Memory patterns: 1i
1
P
ij i j i iJ A
Synaptic modifications : ij ij i jJ J
(for each )
Amit, Guetfriend and Sompolinsky et al 1985if 0.14P N
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Network dynamics: Convergence to an attractor
Memory#
Ove
rlap
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Network dynamics: Convergence to an attractor
Memory#
Ove
rlap
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Context-dependent representations: gradual change in the pattern
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… …
Long-term memory for faces
Non Friends
Friends
Preminger, Sagi & Tsodyks, Vision Research 2007
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Experiment – Terminology
• Basic Friend or Non-Friend task (FNF task)– Face images of faces are flashed for 200 ms – for each image the subject is asked whether the
image is a friend image (learned in advance) or not. – 50% of images are friends, 50% non-friends, in
random order; each friend is shown the same number of times. No feedback is given
F/NF
?
200ms
F/NF
?F/NF
?
200ms 200ms 200ms 200ms 200ms 200ms 200ms 200ms
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1 … 20
Source(friend)
Target(unfamiliar)
… 40 … 60 … 80 100…
Morph sequence
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Subject HL -------- (blue-green spectrum) days 1-18
Morph effect
Bin number
Nu
mb
er o
f ‘F
rien
d’
resp
on
ses
Preminger, Sagi & Tsodyks, Vision Research 2007
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Morphed patterns – Hopfield model
Continuous set of patterns 0 1
{ } (0 1)
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Morphed patterns – Hopfield model
Continuous set of patterns
0 1
ij i jJ
1
( 1) ( ( ))N
i ij j
j
S t sign J S t
0 1 { } (0 1)
* 0.5i iA
Blumenfeld et al, Neuron 2006
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Network dynamics: Convergence to the common attractor
Ove
rlap
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Network dynamics: Convergence to the common attractor
Ove
rlap
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Working memory in biologically-realistic networks (D. Amit ‘90)
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Working memory in biologically-realistic networks (D. Amit ‘90)
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Working memory in biologically-realistic networks (D. Amit ‘90)
Brunel & Wang, J. Comp. Neurosci 2001
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Working memory in biologically-realistic networks
Brunel & Wang, J. Comp. Neurosci 2001
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Multi-stability in recurrent networks
( )I t
( )E t J( )dE
E g JE Idt
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Multi-stability in recurrent networksF
iring
rat
e (H
Z)
Synaptic strength (J)
( )I t
( )E t J
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Persistent activity is time-dependent.
Rainer & Miller EJN 2002
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A Phenomenological Approach to Dynamic Synaptic Transmission
• 4 Key Synaptic Parameters– Absolute strength– Probability of release– Depression time constant– Facilitation time constant
• 2 Synaptic Variables– Resources available (x)– Release probability (u)
Markram, Wang & Tsodyks PNAS 1998
(1 ) ( )
1( )
spf
spd
du u UU u t t
dt
dx xux t t
dt
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Synaptic diversity in the pre-frontal cortex
Wang, Markram et al, Nature Neuroscience 2006
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New idea
• To hold short-term memories in synapses too, with pre-synaptic calcium level.
• Use spiking activity only when the memory is needed for processing, and/or to refresh the synapses (smth like ‘rehearsal’).
Mongillo, Barak & Tsodyks 2008
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Bifurcation diagram
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Attractors in networks with synaptic facilitation
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Integrate and fire network simulations
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Integrate and fire network simulations
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Integrate and fire network simulations
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Robustness and multi-item memory
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Robustness and multi-item memory
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Random overlapping populations
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Predictions
• Increased pre-synaptic calcium concentration during the delay period, disassociated from increased firing rate
• Resistance to temporal cessation of spiking
• Synchronous spiking activity during the delay period
• Slow oscillations during the delay period (theta rhythm?).
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Correlated spiking activity during delay period
Constantinidis et al, J. Neurosci. 2001
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Theta oscillations and working memory
Lee et al, Neuron 2005
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Barak Blumenfeld)WIS, Rehovot(
Son Preminger)WIS, Rehovot(
Dov Sagi)WIS, Rehovot(
Omri Barak)WIS, Rehovot(
Gianluigi Mongillo)ENS, Paris(