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BRAIN biological science information science

OCNC---2004 St atistical Approach to Neural Learning and Population Coding ---- Introduction to Mathematical Neuroscience Shun-ichi Amari Laboratory for Mathematical Neuroscience RIKEN Brain Science Institute.

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BRAIN biological science information science

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  1. OCNC---2004Statistical Approach toNeural Learning and Population Coding---- Introduction to Mathematical NeuroscienceShun-ichi AmariLaboratory for Mathematical Neuroscience RIKEN Brain Science Institute

  2. BRAINbiological science information science Computational neuroscience Neurocomputing Mathematical Neuroscience

  3. I. Mathematical Neuroscience • ----classical theories II. Population Coding ---- modern topics III. Bayesian Inference ---- its merits and critique

  4. 1. Mathematical Neurons • Dynamics of Neuro-Ensembles • Dynamics of Neuro-Fields • Learning and Self-Organization • 5. Self-Organization of Neuro-Fields

  5. I Mathematical Neurons Simple model

  6. output function u

  7. spiking neuron integration-and-fire neuron rate coding

  8. synchrony : spatial correlations firing probability

  9. rate coding ensemble coding

  10. 1-layer network

  11. macroscopic law Ensemble of networks macroscopic state

  12. stability = =

  13. Associative memory m pairs

  14. Randomly generated Random matrix

  15. II Dynamics of Neuro-Ensembles spiking neurons : stochastic point process synchronization Ensemble coding : macrodynamics

  16. S Simple examples Bistable S Multi-stable

  17. oscillation Amari (1971); Wilson-Cowan (1972)

  18. competitive model (winner-take-all) ・・・ (winner-share-some)

  19. multistable associative memory (Anderson, Amari, Nakano, Kohonen Hopfield) decision process (Hopfield) travelling salesman problem

  20. General Theory Transient Attractors • stable state • limit cycle • chaos (strange attractors)

  21. Chaotic behavior random stable states chaos Chaotic memory search

  22. random attractor Associative memory (content-addressable memory) dynamics

  23. Theory 1 =

  24. =

  25. Theory 2 …..

  26. Macroscopic state Amari & Maginu, 1998

  27. Dynamics of recalling processes Direction cosine Correct pattern 1 0 time simulations

  28. Direction cosine 1 0 theory time

  29. simulation Threshold of recalling Spurious memory

  30. Dynamics of temporal sequence (Amari, 1972) non-monotonic output function Morita model

  31. Nonmonotonic model non-monotonic

  32. memory capacity : sparse exact : no spurious memories chaotic oscillation inhibitory connection

  33. Biology hippocumpus, Rolls et al Tonegawa et al CA3 Chaotic associative memory Aihara et al Chaotic search

  34. Associative Memory Dynamics of a Chaotic Neural Networks Each neuron model shows chaotic dynamics Synaptic weights are determined by an auto-correlation matrix of the stored patterns Stored Patterns t=0 t=1 t=2 t=3 t=4

  35. t=5 t=8 t=9 t=6 t=7 t=10 t=11 t=12 t=13 t=14 t=15 t=16 t=17 t=18 t=19

  36. t=20 t=21 t=22 t=23 t=24 t=25 t=26 t=28 t=29 t=27 t

  37. III Field Dynamics of Neural Excitation timing local excitations: travelling wave: oscillatory: memory decision Amari, Biol. Cybern,1978

  38. Dynamics of Neural Fields

  39. unstable stable

  40. excitatory and inhibitory fields traveling wave oscillation

  41. Neural Learning (Hebbian) classic theory ……… Information source I

  42. Hebbian correlation generalized inverse principal component analyzer Perceptron Amari, Biol,Cybern,1978 ….

  43. ……. …. Neural learning (STDP) Spike-time dependent plasticity emergence of synchrony LTP LTD

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