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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 Time Node

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Various gain functions are used End states

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CANNs can be trained with Hebb Hebb: Training pattern:

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Normalization is important to have convergent method Random initial states Weight normalization w(x,50) Training time x xy w(x,y)

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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

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

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