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On Bubbles and Drifts:Continuous attractor networks and their relation to working memory, path integration, population decoding, attention, and motor functions

Thomas TrappenbergThomas Trappenberg

Dalhousie University, Canada Dalhousie University, Canada

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CANNs can implement motor functions

Stringer, Rolls, Trappenberg, de Araujo, Self-organizing continuous attractor networks and motor functions Neural Networks 16 (2003).

State nodes Motor nodes

Movement selector nodes

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My plans for this talk

Basic CANN model

Idiothetic CANN updates (path-integtration)

CANN & motor functions

Limits on NMDA stabilization

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Once upon a time ... (my CANN shortlist)

Wilson & Cowan (1973) Grossberg (1973) Amari (1977) … Sampolinsky & Hansel (1996) Zhang (1997) … Stringer et al (2002)

Wilshaw & van der Malsburg (1976)

Droulez & Berthos (1988)

Redish, Touretzky, Skaggs, etc

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Basic CANN: It’s just a `Hopfield’ net …

I ext rout

w

w

x

Recurrent architecture Synaptic weights

Nodes can be scrambled!

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In mathematical terms …

Updating network states (network dynamics)

Gain function

Weight kernel

7Network can form bubbles of persistent activity (in Oxford English: activity packets)

0 5 10 15 20

20

40

60

80

100

Time [t]

Nod

e in

dex

External stimulus

End states

8Space is represented with activity packets in the hippocampal system

From Samsonovich & McNaughtonPath integration and cognitive mapping in a continuous attractor neural J. Neurosci. 17 (1997)

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Various gain functions are used

End states

10Superior colliculus intergrates exogenous and endogenous inputs

C N

S N p r

T h a l

S E F

F E F

L IP

S C

R F

Cerebellum

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Superior Colliculus is a CANN

Trappenberg, Dorris, Klein & Munoz,A model of saccade initiation based

on the competitive integration of exogenous and endogenous inputs

J. Cog. Neuro. 13 (2001)

12Weights describe the effective interaction in Superior Colliculus

Trappenberg, Dorris, Klein & Munoz,A model of saccade initiation based on the competitive integration of exogenous and endogenous inputs J. Cog. Neuro. 13 (2001)

13There are phase transitions in the weight-parameter space

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CANNs can be trained with Hebb

Hebb:

Training pattern:

15Normalization is important to have convergent method

• Random initial states• Weight normalization

w(x,50)

Training timex

x y

w(x,y)

16Gradient-decent learning is also possible (Kechen Zhang)

Gradient decent with regularization = Hebb + weight decay

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CANNs have a continuum of point attractors

Point attractors and basin of attraction

Line of point attractors

Can be mixed: Rolls, Stringer, Trappenberg A unified model of spatial and episodic memoryProceedings B of the Royal Society 269:1087-1093 (2002)

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CANNs work with spiking neurons

Xiao-Jing Wang, Trends in Neurosci. 24 (2001)

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Shutting-off works also in rate model

Time

No

de

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CANN (integrators) are stiff

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… and can drift and jumpTrappenberg, Dynamic cooperation and competition in a network of spiking neuronsICONIP'98

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Neuroscience applications of CANNs

Persistent activity (memory) and winner-takes-all (competition)

• Cortical network (e.g. Wilson & Cowan, Sampolinsky, Grossberg)

• Working memory (e.g. Compte, Wang, Brunel, Amit (?), etc)

• Oculomotor programming (e.g. Kopecz & Schoener, Trappenberg et al.)

• Attention (e.g. Sompolinsky, Olshausen, Salinas & Abbott (?), etc)

• Population decoding (e.g. Wu et al, Pouget, Zhang, Deneve, etc )

• SOM (e.g. Wilshaw & van der Malsburg)

• Place and head direction cells (e.g. Zhang, Redish, Touretzky, Samsonovitch, McNaughton, Skaggs, Stringer et al.)

• Motor control (Stringer et al.)

basic

CANN

PI

Path-integration

23Modified CANN solves path-integration

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CANNs can implement motor functions

Stringer, Rolls, Trappenberg, de Araujo, Self-organizing continuous attractor networks and motor functions Neural Networks 16 (2003).

State nodes Motor nodes

Movement selector nodes

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... learning motor sequences (e.g. speaking a work)

Movement selector cells motor cells state cells

Experiment 1

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… from noisy examples …

state cells:learning from noisy examples

Experiment 2

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… and reaching from different initial states

Stringer, Rolls, Trappenberg, de Araujo,Self-organizing continuous attractor networks and motor function

Neural Networks 16 (2003).

Experiment 3

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Drift is caused by asymmetries

NMDA stabilization

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CANN can support multiple packets

Stringer, Rolls & Trappenberg,Self-organising continuous attractor networks with multiple

Activity packets, and the representation of spaceNeural Networks 17 (2004)

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How many activity packets can be stable?

Trappenberg, Why is our working memory capacity so large?Neural Information Processing-Letters and Reviews, Vol. 1 (2003)

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Stabilization can be too strong

Trappenberg & Standage,Multi-packet regions in stabilized continuousattractor networks, submitted to CNS’04

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Conclusion

CANN are widespread in neuroscience models (brain)

Short term memory, feature selectivity (WTA)

`Path-integration’ is an elegant mechanisms to generate dynamic sequences (self-organized)

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With thanks to

Cognitive Neuroscience, Oxford Univ. Edmund Rolls Simon Stringer Ivan Araujo

Psychology, Dalhousie Univ. Ray Klein

Physiology, Queen’s Univ. Doug Munoz Mike Dorris

Computer Science, Dalhousie Dominic Standage

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CANN can discover dimensionality

35CANN with adaptive input strength explains express saccades

36CANN are great for population decoding (fast pattern matching implementation)

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John Lisman’s hippocampus

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1 1

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dh th t Iw w r t

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: activity of node i

: firing rate

: synaptic efficacy matrix

: global inhibition

: visual input

: time constant

: scaling factor

: #connections per node

: slope

: threshold

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Continuous dynamic (leaky integrator):

The model equations:

NMDA-style stabilization:1

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if 0.5( )

elsewherei

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ij i jw kr r Hebbian learning:

c hd hd cij i jw kr r r ac hd hd acij i jw kr r r