Transcript Potts Networks – Latching – Correlated patterns

Statistical and dynamical properties of
large cortical network models: insights
into semantic memory and language
Emilio Kropff
Thesis presentation
September 19, 2007
Potts Networks – Latching – Correlated patterns
Cerebral cortex – Braitenberg & Schüz, 1991
# of neurons >> # of
input fibers
Modifiable synapses
No prefered direction
in the connections
Mostly excitatory
synapses
Two-level
associative
memory with
formation of cell
assemblies
Great convergence &
divergence
Connections are very
weak
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Auto-associative memories
- No activity
- Pattern #1 active
- Pattern #2 active
- Pattern #3 active
Emilio Kropff, LIMBO-CNS-SISSA
Hebbian
Learning !!
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Testing the
memory
- Pattern #2 active
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Testing the
memory
Emilio Kropff, LIMBO-CNS-SISSA
Network
damage
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Two-level
associative
memory with
formation of cell
assemblies
• Anatomical studies –
Braitenberg & Schüz
• Embodied theories of
semantic memory
• Feature representation
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
The model: the trick
• S: number of alternative
states
of of
a unit
features
a unit
• a (global sparseness):
average number of units
that are active
in a global
features
describing
a
memory
concept
state 1
state 2
state 0
…
state S
Emilio Kropff, LIMBO-CNS-SISSA
state 3
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Testing Kanter’s Potts network (Kanter,1988)
J 
kl
ij
p
1
NS 2
( S 


1

i
k
N
s
h   J ijkl (S j l  1)
 1)( S   l  1)
k
i
j
j 1 l 0
• The patterns ξ are constructed randomly and stored in the network by
modifying J.
• The state of the network is set to some initial value (e.g: random or
some stored memory if we want to test its stability).
• A unit i is picked randomly and the fields hik are calculated.
• The S states of unit i are updated
following:
 i  arg max (hik )
k
 
k
i
exp(  hik )
S
 exp( h )
l 0
Emilio Kropff, LIMBO-CNS-SISSA
l
i
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Reviewing and extending the results of Kanter, 1988
• Kanter’s result for low S is

p
N
pmax  S (S  1)
• We find high S behaviour is
2
pmax
max
S ( S  1) erf(  2 )
 S ( S  1)

log(2 ) 
1
2
 S)2 erf (1  S 1 ) 
log(
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Adding a zero state and sparseness a
h  mv  
qPk
S (1 a~ )(1 a~C )
2
zk vk  (1   0 )( U  2 S 2(aC
)
1 aˆC )
k
Sparse Hopfield (Buhmann,
1989, Tsodyks, 1988)
αc
~ 1/a
Emilio Kropff, LIMBO-CNS-SISSA
Potts without
sparseness (Kanter,
1988)
~ 0.14 S(S-1)
Sparse Potts (Kropff &
Treves, 2005)
S2/a
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Highly diluted approximation
h  mv  
qPk
S (1 a~ )(1 a~C )
2
zk vk  (1   0 )( U  2 S 2(aC
)
1 aˆC )
k
• Two units speak to each other with probability cM/N
• Two states speak to each other with probability e
0.08
z k vk  (1    0 )( U )
0.05
Normalized storage capacity - ca/S
0.07
2
Normalized storage capacity - ca/S
 eff qPk
highly diluted
S (1 a~ )
fully connected
2
h  mv  


k
0.06
0.05
0.04
0.03
0.02
0.01
0.00
 eff  c p e
highly diluted
fully connected M
0.04
0.03
0.02
0.01
0.00
1E-5
1E-4
1E-3
0.01
sparseness - a
Emilio Kropff, LIMBO-CNS-SISSA
0.1
1
10
100
1000
10000
100000
Number of states in a unit - S
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
1
4
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
• The Kanter result for Potts networks has a logarithmic
correction for high S.
• If well defined, a sparse Potts network reaches an
S2


optimal storage capacity
c
4 a log( S )
a
•In highly diluted networks this result applies, with
 = p/(cM e) .
•These results are in line with the conjecture of a limit in
the amount of information per synapse that a network can
store.
Potts Networks – Latching – Correlated patterns
latching!
Overlaps between the state and the patterns
retrieval
+ adaptation
+ correlation
Overall activity
Dynamics of the network
5
4
3
2
1
1
0.8
0.6
0.4
0.2
0
100
Emilio Kropff, LIMBO-CNS-SISSA
200
300
Timetime
cycles
400
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
• In addition, the transition matrix is not
symmetric
•
•
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Dynamics: latching of 2 patterns
Overlaps between the state and the patterns
1
0.8
0.6
0.4
0.2
0
time
Emilio Kropff, LIMBO-CNS-SISSA
200
400
Time cycles
600
800
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
100
1
Dynamics: latching of 2 patterns
80
1
60
0.8
0.8
‘alternative features’
0.6
•Units active in both
patterns
in different
0.4
states
0.6
40
0.4
0.2
0.2
20
100
200
300
400
500
600
• Units active in one
of the two patterns
1
-0.2
0
0
100
0.8
0
20
40
60
80
First Pattern M aximum at t1
100
• Unitsactive in both
in the same state
0.6
0.4
100
0.2
1
‘pathological case’
c: shared units
100
100
200
300
400
500
600
-0.2
80
0.8
60
0.6
• Weakly: units active
in both patternsin
different states
1
0.4
40
‘shared
features’
0.8
0.6
0.2
20
0.4
• Units active in one
of the two patterns
0.2
0
0
100
0
20
40
60
80
First Pattern M aximum at t1
200
300
400
500
600
• Unitactive in both in
the same state
100
-0.2
d: shared features
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Does latching present a natural rudimentary grammar?
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
• Correlation seems to be at least one of the main
properties determining latching.
• However, the transition matrix is not symmetric, which
means that there are other important factors.
• The equilibrium between global inhibition and local self
excitation can control the complexity of symbolic chains.
• Three types of latching transition exist, each in a restricted
region of parameters. Do they organize in time?
Potts Networks – Latching – Correlated patterns
Category specific deficits
• Patients were found with a significant impairment in their knowledge
about living things (animals + foodstuffs) as opposed to manmade
artifacts (Warrington & Shallice, 1984).
• Impairment for nonliving has also been reported → double dissociation.
Current ratio: 23% vs 77% (Capitani, 03)
Theoretical accounts
• The selective impairments respond to differences in the networks
representing different categories: sensory/functional theory (Warrington
& Shallice, 84), domain-specific hypothesis (Caramazza & Shelton, 98).
• The network sustaining semantic memory is quite homogeneous but
different categories have different typical correlation properties (McRee
et al, 97; Tyler & Moss, 01; Sartori & Lombardi, 04).
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Hopfield memories
• If patterns are randomly correlated (Tsodyks,88),
• However, if patterns have a non-trivial structure of correlations, the
storage capacity colapses.
• Solution #1: Orthogonalize the patterns before feeding the network.
(i.e: Dentate Gyrus in Hippocampus)
• In semantic memory correlation between stored patterns seems to play
a major role.
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Solution #2 ??
• We assume that pattern 1 is being retrieved
• We split hi into the contribution of pattern 1
(signal) and the rest (noise)
• We minimize the noise
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Classical result: hebbian learning
supports uncorrelated memories
Classical result: catastrophe
associated to correlated memories
Jij= Σμ (ξiμ – a ).(ξjμ – a )
Jij= Σμ (ξiμ - ai).(ξjμ - aj)
popularity: ak= 1/p Σμ ξkμ
New result: a modification that supports correlated memories
New result: the performance is the same with uncorrelated memories
Potts Networks – Latching – Correlated patterns
Propeties with finite α, C ≈ ln(N)
GAUSSIAN noise (If there is
independence between
neurons i and j).
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Propeties with finite α, C ≈ ln(N)
GAUSSIAN noise (If there is
independence between
neurons i and j).
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Propeties with finite α, C ≈ ln(N)
F(x)
...
(uncorrelated patterns)
If F(x) decays fast enough
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Propeties with finite α, C ≈ ln(N)
F(x)
If F(x) decays exponentially
If F(x) decays fast enough
Emilio Kropff, LIMBO-CNS-SISSA
If F(x) decays as a power law
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
pmax
c 
C
p
c 
C min
Emilio Kropff, LIMBO-CNS-SISSA
Storing memories
Damaging the network
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Storage capacity ~ fixed p
1
70
60
Performance
Number of neurons
Connectivity
1
c  1
aS f
50
40
30
20
10
0
0.2
0.4
Popularity ai
Cc1
C
Cc
Cc2
(# of afferent connections per neuron)
Entropy Sf= Σi ai (1-ai)
summed over active neurons in the pattern
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Category specific effects
McRae feature norms
non living
Probability Distribution
• 541 concepts described
in terms of 2526 features
living
• i=1 if feature i is
included in the
description of concept 
and i =0 otherwise
Entropy Sf of objects
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
• When memories are correlated, they have variable
degrees of ressistance to damage.
• The robustness of a memory is inverse to how
informative it is (Sf).
• In addition, popular neurons affect negatively the general
performance (decay of F(x)).
• These results show how the current trend in category
specific deficits (‘living’ weaker than ‘non living’) could
emerge even in a purely homogeneous network.
• A singel cortical network with Potts units
including addaptation and storing correlated
patterns of activity in its long range synapses,
presents all the properties studied in this thesis.
Thank you!
McRae’s feature norms
# of features
• In the semantic memory literature, auto-associative
networks are often presented as weak models. Why?
popularity
• To convince psychologists one must show an autoassociative memory that is able to store feature norms.
Performance of the network
McRae’s feature norms
Size of the subgroup of patterns
If – average
information
a – average sparseness
McRae’s feature norms
Number of patterns p in the subgroup
Performance of the network
McRae’s feature norms
Size of the subgroup of patterns
McRae’s feature norms
Why the real network performs poorly?
• Independence between features is not valid (e.g: beak
and wings). Is this effect strong enough? In case it is,
there would be a storage capacity colapse.
• The system works but the approximation of diluted
connectivity is not good.
McRae’s feature norms: the full solution
 +  2+  3+ ...
Performance of the network
McRae’s feature norms: the full solution
Size of the subgroup of patterns
McRae’s feature norms: strategies to store
more patterns
% of patterns retrieved
1- add unpopular neurons
2- eliminate popular neurons
100
100
80
80
60
60
40
40
20
20
a
0
3000
4000
5000
6000
b
2490
2500
Total number of neurons
2510
2520
McRae’s feature norms: strategies to store
more patterns
3- recombination
neurons i and j have high
popularity: their coincidence will
be less popular. If applied
massively, this principle could
change the whole distribution.
4- popularity deppendent
connectivity
The (episodic)
memory pyramid
popularity
calculation
orthogonalization
Hippocampus
final
storage
Entorhinal cortex
Perirhinal and
parahippocampal cortex
Unimodal and polymodal
association areas
Primary cortex:
sensory motor areas
Potts Networks – Latching – Correlated patterns
Propeties with α ≈ 0, C ≈ ln(N)
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Propeties with α ≈ 0, C ≈ ln(N)
ac
a c
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Propeties with α ≈ 0, C ≈ ln(N)
• If you want to be an attractor, you should pick at
least some unpopular units.
• Lowering U can make any pattern retrievable ->
ATTENTION
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007
Potts Networks – Latching – Correlated patterns
Dynamics: latching of 2 patterns
1
0.8
‘alternative features’
•Units active in both
patterns in different
states
0.6
0.4
0.2
100
200
300
400
500
600
• Units active in one
of the two patterns
1
-0.2
0.8
• Unitsactive in both
in the same state
0.6
100
100
1
0.4
80
80
0.8
60
60
0.6
c: alternative features
0.2
‘pathological case’
100
40
200
300
400
500
600
0.4
40
-0.2
20
• Weakly: units active
in both patternsin
different states
0.2
20
0
0
0
0
20
40
60
80
First Pattern M aximum at t1
100
0
20
40
60
80
First Pattern M aximum at t1
100
1
100
100
80
80
‘shared
features’
1
0.8
0.8
0.6
60
60
0.6
40
40
0.4
0.4
• Units active in one
of the two patterns
0.2
20
0.2
20
100
200
300
400
500
600
• Unitactive in both in
the same state
0
0
0
0
20
40
60
80
First Pattern M aximum at t1
100
-0.2
0
20
40
60
80
First Pattern M aximum at t1
100
d: shared features
Emilio Kropff, LIMBO-CNS-SISSA
Thesis presentation. September 19th, 2007