1. 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
This material is available online at

2. Outline
9.1
9.2
9.3
9.4
9.5
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
Competitive networks and self-organizing maps

3. 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

4. 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.

5. 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

6. 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.

7. 9.1.4 Graded winner-take-all models
Only one node or one activity packet of nodes
The dynamic equation for the networks
Neural field equation
1-dimension
The discretization rules
(9.1)
(9.2)
(9.3)
(9.4)
(9.5)

8. 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
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
(9.6)
(9.7)

9. 9.2.1 Learning Gaussian head direction patterns
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
A contribution to each weight component
For infinite resolution of the model, i.e. Δ α → 0:
The continuous notations
(9.8)
(9.9)
(9.10)
for a node with a preferred direction equal to that of training example
(9.11)
for a node with a preferred direction different from the direction of the training example
(9.12)
The weight matrix has a Gaussian shape and the same width as the receptive fields of the nodes
(9.13)
(9.14)
: weight matrix depends only on the distance between nodes

10. 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.

11. 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.

12. 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

13. 9.3.1 Attractor regimes
The different regimes in the CANN model depend on the level of inhibition c
Growing activity
The inhibition is weak compared to the excitation
Decaying activity
The inhibition is strong compared to the excitation
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 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.

14. 9.3.2 Formal analysis of attractor states
A threshold activation function g(x) = 1/exp(0.007x)
The stationary state of the dynamics eqn 9.3
h(x1)=h(x2)=0,
For the weighting function w = wE - c,
(9.15)
(9.16)
(9.17)
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 (B) The solution of eqn 9.17 as a function of the inhibition constant.

15. 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
The velocity
The centre
The velocity of the centre of the activity packet
(9.18)
(9.19)
Fig. 9.7 (A) Two Gaussian bell curves 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.
(9.20)

16. 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.

17. 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.

18. 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

19. 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

20. 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
The weight between rotation nodes with Hebbian rule
The rule strengthens the weights between the rotation node and the appropriate synapses in the recurrent network
(9.21)
(9.22)

21. 9.4.4 Updating the network after learning
The dynamics of the model
(9.23)
(9.24)
Fig (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.

22. 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
For a sufficient strength of the asymmetric component
Using the strength parameter
The network is able to jump between the patterns in sequences
(9.25)

23. 9. 5 Competitive networks and self-organizing maps 9. 5
9.5 Competitive networks and self-organizing maps Two-dimensional SOM
Two-dimensional feature vectors
(9.26)
Fig Architecture of a two-dimensional 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.

24. 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 ‘*’
The firing rate of other nodes,
Hebbian learning rule,
The weight vector of the wining node is closest to the corresponding input vector,
(9.27)
(9.28)
(9.29)
Fig 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.

25. 9.5.3 Other competitive networks (1)
Fig Another example of a two-dimensional 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.

26. 9.5.3 Other competitive networks (2)
Fig 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.

27. Conclusion The continuous attractor model Winner-take-all models
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