1 On Bubbles and Drifts Continuous attractor networks
1 On Bubbles and Drifts: Continuous attractor networks and their relation to working memory, path integration, population decoding, attention, and motor functions Thomas Trappenberg Dalhousie University, Canada
CANNs can implement motor functions State nodes Motor nodes Stringer, Rolls, Trappenberg, de Araujo, Self-organizing continuous attractor networks and motor functions Neural Networks 16 (2003). Movement selector nodes 2
3 My plans for this talk n Basic CANN model n Idiothetic CANN updates (path-integtration) n CANN & motor functions n Limits on NMDA stabilization
4 Once upon a time. . . n n n n (my CANN shortlist) Wilson & Cowan (1973) Grossberg (1973) Wilshaw & van der Malsburg (1976) Amari (1977) … Droulez & Berthos (1988) Sampolinsky & Hansel (1996) Redish, Touretzky, Skaggs, etc Zhang (1997) … Stringer et al (2002)
5 Basic CANN: It’s just a `Hopfield’ net … Recurrent architecture Synaptic weights Nodes can be scrambled!
6 In mathematical terms … Updating network states (network dynamics) Gain function Weight kernel
Network can form bubbles of persistent activity (in Oxford English: activity packets) End states 7
Space is represented with activity packets in the hippocampal system From Samsonovich & Mc. Naughton Path integration and cognitive mapping in a continuous attractor neural J. Neurosci. 17 (1997) 8
9 Various gain functions are used End states
Superior colliculus intergrates exogenous and endogenous inputs 10 L IP SEF FEF C N Tha l S N pr SC Cerebellum R F
11 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)
Weights 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) 12
There are phase transitions in the weightparameter space 13
14 CANNs can be trained with Hebb: Training pattern:
Normalization is important to have convergent method 15 • Random initial states • Weight normalization w(x, y) w(x, 50) x Training time x y
Gradient-decent learning is also possible (Kechen Zhang) Gradient decent with regularization = Hebb + weight decay 16
17 CANNs have a continuum of point attractors Point attractors and basin of attraction Can be mixed: Rolls, Stringer, Trappenberg A unified model of spatial and episodic memory Proceedings B of the Royal Society 269: 1087 -1093 (2002) Line of point attractors
18 CANNs work with spiking neurons Xiao-Jing Wang, Trends in Neurosci. 24 (2001)
19 Node Shutting-off works also in rate model Time
20 CANN (integrators) are stiff
… and can drift and jump 21 Trappenberg, Dynamic cooperation and competition in a network of spiking neurons ICONIP'98
22 Neuroscience applications of CANNs Persistent activity (memory) and winner-takes-all (competition) b a s i c C A N N • 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) P I • Place and head direction cells (e. g. Zhang, Redish, Touretzky, Samsonovitch, Mc. Naughton, Skaggs, Stringer et al. ) • Motor control (Stringer et al. ) Path-integration
Modified CANN solves path-integration 23
CANNs can implement motor functions State nodes Motor nodes Stringer, Rolls, Trappenberg, de Araujo, Self-organizing continuous attractor networks and motor functions Neural Networks 16 (2003). Movement selector nodes 24
25 . . . learning motor sequences (e. g. speaking a work) Experiment 1 Movement selector cells motor cells state cells
26 … from noisy examples … Experiment 2 state cells: learning from noisy examples
27 … and reaching from different initial states Experiment 3 Stringer, Rolls, Trappenberg, de Araujo, Self-organizing continuous attractor networks and motor function Neural Networks 16 (2003).
28 Drift is caused by asymmetries NMDA stabilization
29 CANN can support multiple packets Stringer, Rolls & Trappenberg, Self-organising continuous attractor networks with multiple Activity packets, and the representation of space Neural Networks 17 (2004)
30 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)
31 Stabilization can be too strong Trappenberg & Standage, Multi-packet regions in stabilized continuous attractor networks, submitted to CNS’ 04
32 Conclusion n CANN are widespread in neuroscience models (brain) n Short term memory, feature selectivity (WTA) n `Path-integration’ is an elegant mechanisms to generate dynamic sequences (self-organized)
33 With thanks to n Cognitive Neuroscience, Oxford Univ. ¡ ¡ ¡ n Psychology, Dalhousie Univ. ¡ n Ray Klein Physiology, Queen’s Univ. ¡ ¡ n Edmund Rolls Simon Stringer Ivan Araujo Doug Munoz Mike Dorris Computer Science, Dalhousie ¡ Dominic Standage
34 CANN can discover dimensionality
CANN with adaptive input strength explains express saccades 35
CANN are great for population decoding (fast pattern matching implementation) 36
37 John Lisman’s hippocampus
38 The model equations: Continuous dynamic (leaky integrator): : activity of node i : firing rate : synaptic efficacy matrix : global inhibition : visual input : time constant NMDA-style stabilization: : scaling factor : #connections per node Hebbian learning: : slope : threshold
- Slides: 38