A recent study repeated Standing

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A recent study repeated Standings (73) findings
-

• Subjects were presented with 2500 novel real
  world pictures, over a few hours
• Each picture was shown once, for 3 seconds
• Immediately afterwards, a 2AFC recognition
  memory test (which is familiar?)

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• Conclusions
• High memory capacity
• High fidelity

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Binary Synapses Slower Than Expected

• Work with Peter Latham

Amit Miller
Research Talk
Oct. 2008
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Outline

• Quick review
• When its good
• When its bad
• Correlations in the noise

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One shot, serial learning
The brain is doing serial learning with no
repetitions
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One shot, serial learning
External stimulus clamp
neuronal activity
stimulus A
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One shot, serial learning
External stimulus clamp
neuronal activity
stimulus A
Activity-dependent plasticity follows
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One shot, serial learning
External stimulus clamp
neuronal activity
stimulus A
stimulus B
Activity-dependent plasticity follows
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One shot, serial learning
External stimulus clamp
neuronal activity
stimulus A
stimulus B
stimulus C . . .
Activity-dependent plasticity follows
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One shot, serial learning
External stimulus clamp
neuronal activity
stimulus A
stimulus B
New memories overwrite old memories memories get
degraded and decay
stimulus C . . .
Activity-dependent plasticity follows
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Assumption discrete synapses only two
efficacies are allowed
Synapses are bound, and discrete
Strong (potentiated)
Strong (potentiated)
state switches
Weak (depressed)
Weak (depressed)
Fortunately,- we have many synapses (N of
them)- synapses are stochastic
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The learning rule

• External stimuli dictate plasticity by randomly
  choosing neurons
• Plasticity is Hebbian

(neurons and synapses, under external stimulus)
Potentiation pre post f02
Depression pre post f0
f1 Indifferent otherwise
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but synapses do not switch deterministically

• External stimuli only dictate candidates for
  potentiation and depression
• Actual state switches are performed
  stochastically
• Synapses are stochastic state machines
• State transitions are described by stochastic
  matrices
• Upon potentiation M
• Upon depression M-

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Assumption synapses are i.i.d.

• Corollary I
• We can forget about neurons for now. Care only
  for synapses
• We have a large population N synapses of
  variables

f fraction of synapses, candidates for
plasticity (sparseness) f fraction of
the candidate synapses, destined for potentiation
(balance) f - (1- f )
fraction of the candidate synapses, destined for
depression
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Assumption synapses are i.i.d.

• Corollary II
• We focus on the distribution of synaptic states -
  P(t) - over a large population of synapses
• We use a Mean-Field approach, and derive
  everything from the state distribution

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What happens when we learn a new stimulus
Space of synapses
The synaptic state distribution is at equilibrium
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What happens when we learn a new stimulus
Stimulus A
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What happens when we learn a new stimulus
Stimulus A
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What happens when we learn a new stimulus
Stimulus A
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What happens when we learn a new stimulus
Potentiated (M) sub-population
Stimulus A
Depressed (M-) sub-population
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What happens when we learn a new stimulus
Potentiated (M) sub-population
Stimulus A
Depressed (M-) sub-population
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What happens when we learn a new stimulus
Potentiated (M) sub-population
Stimulus A
Depressed (M-) sub-population
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What happens when we learn a new stimulus
Ideal Observer
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Forgetting
Stimulus A
Stimulus B
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Forgetting
Stimulus A
Stimulus B
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Forgetting
Stimulus A
Stimulus B
Stimulus C
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• Mean Signal
• S(t) w ? ( P(t) - Peq )
• Initial distribution
• P(0) M Peq
• Evolution in time Markov Chain
• P(t) Ht P(0)

Reference level - and - starting point
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Markov Chain

• The stochastic matrix H describes the mean effect
  of new memoriesH (1- f )I f f M
  (1-f )M-
• Corollaries
• P(t) converges to Peq
• The signal decays to zero

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Memory lifetime
0
The width of the distributions is given by the
noise at equilibrium - seq
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Summary

• Construct matrices
• Calculate the equilibrium distribution
• Calculate the initial distributions
• Iterate Markov Chain until stopping criterion
• Optimize w.r.t. to model parameters

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Two main models

1. The 2-State synapse (Binary synapse)
2. The Multi-State synapse

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The 2-State synapse
State transitions

• Simple
• Analytically solvable
• Exponential decay

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• Fast learning means fast forgetting
• Tradeoff initial Signal-to-Noise vs. memory
  lifetime

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The Multi-State synapse

• The Cascade model Fusi, Drew, Abbott. Neuron,
  2005
• Allow both strong initial S/N and long memory
  lifetime
• Power-law behavior

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• A model may have n internal states
  for n 8

Transition probabilities in/out of a state are
falling off exponentially For state of depth d,
xd
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• A model may have n internal states
  for n 8

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Results

• Comparison of the Multi-State and 2-State synapses

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Enhanced
Standard
Perfect balance f 1/2
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Enhanced
Modified
Standard
f 0.9
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Modified
Enhanced
Standard
N 109, f 0.01
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Modified
Enhanced
Standard
N 109, f 0.01
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Outline

• Quick review
• When its good
• When its bad
• Correlations in the noise

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Eigenvalue decomposition of H

• And therefore

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Finally
The signal is written as a sum of weighted
exponentials
S(t) ?k ßk exp(?k t)
w vk vkP(0)
, k ? 0 ßk w vk
vkP(0) wPeq 0 , k 0
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n -1 exponentials
Last one to survive
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At long times, a Multi-State model behaves as its
slowest exponential
S(t) ß1 exp(?1 t)
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Memory lifetime depends on ß1 and ?1
?n-1 and ßn-1
ß1
Better
Worse
Binary model
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Memory lifetime depends on ß1 and ?1
?n-1 and ßn-1
Multi-State ?
ß1
Binary model
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Optimizing ß1 and ?1 for memory
lifetimeyields an optimal memory lifetimewhich
is independent of the number of states. Thus,
all models 2-State included are
equivalent. The only condition the eigenvalues
must be well separated
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d
0

e ltlt d
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Outline

• Quick review
• When its good
• When its bad
• Correlations in the noise

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Its all about the Equilibrium Distribution
exponential fall-off in transition probabilities

sensitivity to the balance between potentiation
and depression
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Standard model, x 1/2
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• The noise is lower
• But the signal is hit much more badly

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Solution 1 the Modified model

• fine-tune w.r.t. f
• Uniform distribution among negative/positive
  states
• Comes at a price
  x 1- f

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Enhanced
Modified
Standard
f 0.9
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Solution 2 the Enhanced model

• Optimize both x and
• The constraint now is
  x
• goes very low, x approaches 1
• No exponential fall-off, transition probabilities
  are of the same magnitude

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Enhanced
Modified
Standard
f 0.9
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Outline

• Quick review
• When its good
• When its bad
• Correlations in the noise

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Dependence on the Signal-to-Noise ratio

• Memory lifetime is extracted by solving
• We assumed that synapses are i.i.d., hence
  Noiseeq was proportional to the variance of a
  single synapse
• But, in reality, different synapses may share
  pre- and post-synaptic neurons

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Correlations 2nd order only
Pairs of synapses may co-vary their efficacy
depending on the learning rule
post
post
pre
pre
pre
post
post
Type 1
Type
3 Type 2

Type 4
pre
post/pre
pre/post
post
pre
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• Nsynapses Mneurons x Kconnections
  K in 1 .. M-1
• Random connectivity, every neurons making
  exactly K pre- post-synaptic connections

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Signal
Mean Signal
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And the noise (at equilibrium)
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• We use K 1500
• Still, M gtgt K

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How to calculate the joint P(Ja,Jb)?
Peq is extracted from the transition matrix
(principal eigen vector)
- -
--
H n x n
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Approximation

• Collapse all positive and negative states
  together
• We are interested in the net negative / positive
  probability p(), p(-)
• At equilibrium and not the convergence
  towards equilibrium
• Synaptic pairs, at equilibrium, still maintain
  the same marginals

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A necessary condition for equilibrium
Equal transition mass between the two sets of
states
H2 4 x 4
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Some results
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N 1010 synapses, f 0.1
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N 1010 synapses, f 0.1
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So

• Disclaimers
• I might be wrong
• Reasonable choice of K
• Reasonable learning rule

• Conclusions with a barrel of salt
• Multi-State synapses cant really deliver
• Strong initial S/N ? large covariance ? stronger
  noise
• You cant beat the trade-off