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**On Bubbles and Drifts: Continuous attractor networks in brain models**

Thomas Trappenberg Dalhousie University, Canada

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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)

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**It’s just a `Hopfield’ net …**

Recurrent architecture Synaptic weights

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**In mathematical terms …**

Updating network states (network dynamics) Gain function Weight kernel

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**Weights describe the effective interaction profile in Superior Colliculus**

TT, Dorris, Klein & Munoz, J. Cog. Neuro. 13 (2001)

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

End states

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**Space is represented with activity packets in the hippocampal system**

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

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**There are phase transitions in the weight-parameter space**

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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**

Node Time

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**Various gain functions are used**

End states

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**CANNs can be trained with Hebb**

Training pattern:

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**Normalization is important to have convergent method**

Random initial states Weight normalization w(x,y) w(x,50) x x y Training time

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**Gradient-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 memory Proceedings B of the Royal Society 269: (2002)

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**Neuroscience applications of CANNs**

Persistent activity (memory) and winner-takes-all (competition) Working memory (e.g. Compte, Wang, Brunel etc) Place and head direction cells (e.g. Zhang, Redish, Touretzky, Samsonovitch, McNaughton, Skaggs, Stringer et al.) Attention (e.g. Olshausen, Salinas & Abbot, etc) Population decoding (e.g. Wu et al, Pouget, Zhang, Deneve, etc ) Oculomotor programming (e.g. Kopecz & Schoener, Trappenberg) etc

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**Superior colliculus intergrates exogenous and endogenous inputs**

h a l E F L I P R Cerebellum

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**Superior Colliculus is a CANN**

TT, Dorris, Klein & Munoz, J. Cog. Neuro. 13 (2001)

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**CANN with adaptive input strength explains express saccades**

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

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**CANN (integrators) are stiff**

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… and drift and jump TT, ICONIP'98

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**Modified CANN solves path-integration**

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**CANNs can learn dynamic motor primitives**

Stringer, Rolls, TT, de Araujo, Neural Networks 16 (2003).

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**Drift is caused by asymmetries**

NMDA stabilization

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**CANN can support multiple packets**

Stringer, Rolls & TT, Neural Networks 17 (2004)

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**How many activity packets can be stable?**

T.T., Neural Information Processing-Letters and Reviews, Vol. 1 (2003)

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**Stabilization can be too strong**

TT & Standage, CNS’04

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**CANN can discover dimensionality**

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**Continuous dynamic (leaky integrator):**

The model equations: Continuous dynamic (leaky integrator): : activity of node i : firing rate : synaptic efficacy matrix : global inhibition : visual input : time constant : scaling factor : #connections per node : slope : threshold NMDA-style stabilization: Hebbian learning:

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