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9. Continuous attractor and
competitive networks
Fundamentals of Computational Neuroscience, T. P. Trappenberg, 2002.
Lecture Notes on Brain and Computation
Byoung-Tak Zhang
Biointelligence Laboratory
School of Computer Science and Engineering
Graduate Programs in Cognitive Science, Brain Science and Bioinformatics
Brain-Mind-Behavior Concentration Program
Seoul National University
E-mail: [email protected]
This material is available online at http://bi.snu.ac.kr/
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Outline
9.1
9.2
9.3
9.4
Spatial representations and the sense of direction
Learning with continuous pattern representations
Asymptotic states and the dynamics of neural fields
‘Path’ integration, Hebbian trace rule, and sequence
learning
9.5 Competitive networks and self-organizing maps
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9.1 Spatial representations and the sense of
direction


Auto-associative attractor models
General memory states in mind
 The shape of objects, their smell, texture, or color

Point attractor neural networks (PANNs)
 Memory represented by independent vectors


Continuous attractor neural networks (CANNs)
The training patterns represent continuous
 Spatial location of an object
 Topographic maps
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9.1.1 Head direction

The sense of direction
 Representation of body or head direction
 A mechanism to update this information without visual cues
Fig. 9.1 (A) Experimental response of a neuron in the
subiculum of a rodent when the rodent is heading in
different directions in a familiar maze. The dashed
line represents the new head properties of the same
neuron when the rodent is placed in the new
unfamiliar maze. The new response properties will
normally be similar to the previous one, that is, head
direction cells try to maintain approximately their
response properties to specific head directions.
However, the results shown were produced in
experiments with a rodent that had cortical lesions
that weakened the ability to maintain the response
properties after the rodent was transferred into a new
environment. (B) Neuronal response from many
hippocampal neurons in a rodent that responded to
the subject’s location (places) in a maze. The figure
shows the firing rates of the neurons in response to a
particular place. whereby the neurons were placed in
the figure so that neurons with similar response
properties were placed adjacent to each other.
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9.1.2 Place fields


Head direction representations
The spatial representation of a one-dimensional feature space
in the brain
 Apply equally to higher-dimensional representations

Neurons in the hippocampus of rats
 Fire in relation to specific locations within a maze
 A specific topography of neurons within the hippocampal tissue
with respect to their maximal response to a particular place has
not been found
 The rearrangement
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9.1.3 Spatial representations in network
models

A possible solution to representing head directions
Fig. 9.2 A proposal as to how the activity of
nodes, for clarity arranged into a circle, can
represent head directions. With the 20
nodes of this model we can represent head
directions with a resolution of 18 degree
when using a single binary node as a
representation of a head direction. The
single active node in the figure, represented
as a solid circle, indicates a head direction
of 72 degrees in this example.
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9.1.4 Graded winner-take-all models

Winner-take-all
 Only one node or one activity packet of nodes

The dynamic equation for the networks
1 dhi (t )
1
 hi (t ) 
 dt
N

ext
i
(t )
j
(9.1)
h(x, t )
 hi (x, t )     w(x, y )r (y, t )dy1  yd  I ext (x, t )
y1
yd
(9.2)
t
1-dimension


ij j
Neural field equation


 w r (t )  I
h( x, t )
 hi ( x, t )   w( x, y )r ( y, t )dy   I ext ( x, t )
y
t
(9.3)
The discretization rules
x  i x
(9.4)
 dx x
h(ix, t )  hi (t )
(9.5)
x  1 / N
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9.2 Learning with continuous pattern
representations

Recurrent neural networks
 Represent a continuous set of patterns
 Hebbian learning

Hebbian rule for the excitatory weights
 In the neural field representation
w E ( x, y )   r  ( x, t ) r  ( y , t )

(9.6)
 The firing rate r μ is the firing rate of the neural field while
dominated by the training example of a pattern μ presented to
the network

The inhibition from inhibitory interneurons
w  wE  c
(9.7)
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9.2.1 Learning Gaussian head direction
patterns
ri  e  i
2



/ 2 2
(9.8)
The external input to a node i,
Gaussian profile around a preferred direction
The displacement between
 The head direction αHD provided by the external input
 The optimal firing direction of the cell αi
 i  min(|  i   HD |,360 |  i   HD |) (9.9)

(9.10)
A contribution to each weight component
w  ri rj  e
E
ij
 ( i2  2j ) / 2 2
E
HD
 i )  e
(9.11) wij (
 ( i  j ) 2 / 2 2
E
HD
 i  n )  e
(9.12) wij (
(9.13) w  Ne
E
ij
(9.14)
[( i  j ) 2 ( i  j )( n ) 2 ] / 2 2
for a node with a preferred direction different
from the direction of the training example
For infinite resolution of the model, i.e. Δ α → 0:


for a node with a preferred direction equal to that of training example
 (  i  j )2 / 2  2
The weight matrix has a Gaussian shape and the same width as the receptive fields
of the nodes
The continuous notations
wE ( x, y)  wE (| x, y |) : weight matrix depends only on the distance between nodes
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9.2.2 Gaussian interaction profiles in the
brain

An effective interaction structure
 Short-distance excitation and long-distance inhibition
 Columnar organizations in the cortex

The superior colliculus from cell recordings in monkeys
Fig. 9.3 Data from cell recordings in
the superior colliculus in a monkey
that indicate the interaction strength ρw
between cells in this midbrain structure.
The
solid
line
displays
the
corresponding measurement from
simulations of a CANN model of this
brain structure.
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9.2.3 Self-organized interaction structures
in CANNs
Fig. 9.4 A recurrent associative attractor network model, similar to the model shown in Fig. 9.2, where
the nodes have been arbitrarily placed in the physical space on a circle. The relative connection
strength between the nodes is indicated by the thickness of the lies between the nodes. Each node
responds during learning with a Gaussian firing profile around the stimulus that excites the node
maximally. Each node is assigned a center of the receptive field randomly from a pool of centers
covering the periodic training domain. (A) Before training all nodes have the same relative weights
between them. (B) After training the relative weight structure has changed with a few strong
connections and some weaker connections. (C) The regularities of the interactions can be revealed
when reordering the nodes so that nodes with the strongest connections are adjacent to each other.
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9.3 Asymptotic states and the dynamics of
neural fields




The asymptotic states (attractors)
The weight matrix is shift invariant after training the network
on continuous Gaussian patterns
Local cooperation and global competition
Activity packet
 A collection of nodes to be active

Shift invariant
 The activity packet can be stabilized at any location in the
network depending on an initial external stimulus

Dynamic competition
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9.3.1 Attractor regimes

The different regimes in the CANN model depend on the level of inhibition c
1.
Growing activity
 The inhibition is weak compared to the excitation
2.
Decaying activity
 The inhibition is strong compared to the excitation
3.
Stable activity packet
 In an intermediate range of the strength of inhibitions relative to that of the
excitations
Fig. 9.5 (A) Time evolution of the firing rates
in a CANN model with 100 nodes. Equal
external inputs to nodes 30-70 were applied
at t = 0. This external input was removed at t
= 10τ. The inhibition was set to three times
the average firing rate of a node when driven
by a Gaussian external input like that used for
training the network. (B) The solid line
represents the firing rate profile of the
simulation shown in (A) at t = 20τ. The
dashed line corresponds to the firing rate
profile in a similar simulation with reduced
(by a factor of three) inhibition.
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9.3.2 Formal analysis of attractor states



A threshold activation function g(x) = 1/exp(0.007x)
x
h
(
x
)

The stationary state of the dynamics eqn 9.3
x w( x, y)dy (9.15)
x
h(x1)=h(x2)=0, x w( x1 , y)dy  0 (9.16)

x2  x1
E
N

erf
(
)  c( x2  x1 )
For the weighting function w = w - c, 2
2
(9.17)
2
1
2
1

Fig. 9.6 (A) Plot of the functions
(9.17) and two linear functions
with slope c = 1 and c = 0.4. The
intersection of the functions
(other than at x2 – x1 = 0) gives
the solutions we are seeking of
eqn 9.17. (B) The solution of
eqn 9.17 as a function of the
inhibition constant.
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9.3.3 Stability of the activity packet

The stability of the activity packet with respect to its
movement
 Calculate the velocity of the boundaries



h
dx
The velocity dt  x t (9.18)
The centre x (t )  12 ( x (t )  x (t )) (9.19)
The velocity of the centre of the activity packet
dxc
1

dt
2x1
c

x2
1
2
w( x1 , y )dy 
1
2x2

x2
Fig. 9.7 (A) Two Gaussian bell curves
x1
x1
(9.20) centered around two different values x1 and
x2. The striped and dotted areas are the
same due to the symmetry of the bell curve.
The integrals from x1 to x2 over the two
different curves are therefore the same. This
is not true if the two curves are not
symmetric and have different shapes. (B)
The dashed line outlines the shape of an
activity packet from a simulation. The
symmetry of this activity packet makes the
gradients of boundaries equal except for a
sign.
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w( x2 , y )dy
9.3.4 Drifting activity packets
Fig. 9.8 (A) Noisy weight matrix. Time evolution of the center of gravity of activity packets in
CANN model with 100 nodes. The model was trained with activity packets on all possible
locations. Each component of the resulting weight matrix was then convoluted with some
noise. (B) Irregular or partial learning. Partial view of the weight matrix resulting from
training the network with activity packets on only a few locations. (C) Time evolution of the
center of gravity of activity packets in CANN model with 100 nodes after training the network
on only 10 different locations.
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9.3.5 Stabilization of the activity packet


The drift in the activity packet can be stabilized by a small
increase in the excitability of neurons once they have been
recently activated
NMDA receptor
 Voltage-dependent nonlinearity

An increases of the voltage-dependent nonlinearity would
make more states stable
Fig. 9.8 (D) ‘NMDA’-stabilization. The
network trained o the 10 locations was
augmented a stabilization mechanism that
reduces the firing threshold of active
neurons.
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9.4 ‘Path’ integration, Hebbian trace rule,
and sequence learning
The possibility of ‘updating’ the state
 A subject might not have an absolute value available

 Rotate a subject with closed eyes

Path integration
 Calculate the new position from the old position
9.4.1 Path integration with asymmetric
weighting functions


The path integration problem involves using such
asymmetries in a systematic way
The strength of the asymmetry to the velocity of the
movement

Idiothetic cues
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9.4.2 Idiothetic update of head direction
representations
Fig. 9.9 Model for path integration in
CANNs. The central nodes are part of the
network with collateral connections as used
to represent head directions (Fig. 9.2). The
rotation nodes represent collections of
neurons that signal rotation velocities
proportional to their activity. The afferents
of these rotation cells can modulate the
collateral connections within the head
direction network. We symbolized this with
synapses close to the synapses of the
collateral connections. Each rotation cell can
synapse on to each synapse in the head
direction network. The separation of the
connections, as indicated by the solid and
dashed lines in the figure, is self-organized
during learning
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9.4.3 Self-organization of a rotation
network

Biologically realistic model
 Self-organize the network

Clockwise rotation
 Clockwise synapse

Short-term memory
 Trace term
ri (t  1)  (1  )ri (t )  ri (t ) (9.21)

The weight between rotation nodes with Hebbian rule
rot
wijk
 kri rj rkrot

(9.22)
The rule strengthens the weights between the rotation node
and the appropriate synapses in the recurrent network
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9.4.4 Updating the network after learning

The dynamics of the model
h( x, t )
 h( x, t )   weff ( x, y, r rot )r ( y, t )ri rot (t )dy (9.23)
y
t
rot rot
(9.24)
wijeff  ( wij  c)(1  wijk
rk )

Fig. 9. 10 (A) Simulation of a CANN
model with idiothetic updating
mechanisms. The acitivity packet can
be moved with idiothetic inputs in
either clockwise or anti-clockwise
directions, depending on the firing
rates of the corresponding rotation cells.
(B) The different weighting functions
from node 50 to the other nodes in the
network after learning. w, solid line;
wrot, dashed line, weff, dotted line.
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9.4.5 Sequence learning

Sequence learning
 Apply the generic mechanisms of asymmetric weighting
functions
 Including a trace term (in pattern space) in the canonical
learning rule
wij 

1
N
(      


i
j
i
 j )
1
(9.25)
For a sufficient strength of the asymmetric component
 Using the strength parameter
 The network is able to jump between the patterns in sequences
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9.5 Competitive networks and self-organizing maps
9.5.1 Two-dimensional SOM

Two-dimensional feature vectors
 r1in 
r   in 
 r2  (9.26)
in
Fig. 9.11 Architecture of a twodimensional self-organizing map.
Each of the two input values rin1
and rin2 , each representing one of
two feature components, is
mapped on the map network with
individual weight values win. The
nodes in the map network are
arranged in a two-dimensional
sheet with collateral connections
(not shown) corresponding to the
distances between nodes in this
two-dimensional sheet.
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9.5.2 Simplifying winner-take-all
description

The response of the network
 A Gaussian firing rate around the node that receives the
strongest input
 Wining node and label it with ‘*’
 (( i i )  ( j  j ) ) / 2
(9.27)
 The firing rate of other nodes, rij  e
 Hebbian learning rule, wxini  krx (riin  wxini ) (9.28)
* 2

* 2
2
w inx*
The weight vector of the wining node
is closest to the
corresponding input vector, w inx    w inx   for all x (9.29)
*
Fig. 9.12 Experiment with two-dimensional
self-organizing feature maps. (A) Initial map
with random weight values. (B) and (C) Two
examples of the resulting feature map after
1000 random training examples with different
random initial conditions. These simulations
are discussed further in Chapter 12.
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9.5.3 Other competitive networks (1)
Fig. 9.13 Another example of a twodimensional self-organizing feature
map. In this example we trained the
network on 1000 random training
examples from the lower left
quadrants. The new training examples
were then chosen randomly from the
lower-left and upper-right quadrant.
The parameter t specifies how many
training examples have been presented
to the network.
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9.5.3 Other competitive networks (2)
Fig. 9. 14 Example of categorization (vector quantization) of two-dimensional input
data (two-dimensional training vectors). The training data are represented as dots,
and the input vector that would best evoke a response of one of the three output
nodes is represented by a cross. (A) Before training there is no correspondence
between the group of input data and the output node representing category. (B) After
training we have a ‘preferred vector’ for each node that corresponds to each of the
clusters in the training data set.
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Conclusion

The continuous attractor model
 Spatial representation

Winner-take-all models
 Hebbian learning with continuous pattern
 Gaussian interaction
 Self-organization models
 Attractor regimes
 Path integration and Sequence learning
 Competitive networks
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