Dynamics of Excitation and SelfOrganization in Neural Fields

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Dynamics of Excitation and Self-Organization in
Neural Fields

• Shun-ichi Amari
• Laboratory for Mathematical Neuroscience
• RIKEN Brain Science Institute

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Dynamics of Neural Field
Wilson-Cowan Amari Ermentrout-Cowan .
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one-dimension one layer
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Boundary of Excitation
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Dynamics of Boundary
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multi-dimension
maximum detector
travelling waves
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oscillation and chaos
general activation function
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Input from outsidesearch for maximum
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general activation function
existence of bump solution stability
Wiener-Krasnoselskii theory
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2-dimensional field
1. symmetry
2. stability
curvature
stripes chaos
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Oscillation
Travelling wave
exc inh
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Learning and Self-Organization
information source
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generalized Hebbian learning
information source
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averaged learning equation
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Dynamics of Self-Organizing Neurofield
presynaptic field
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Self-organization of topological map
signal space
neural field

• amplification
• topology
• patch structure (colummar structure)

dynamical instability
Willshaw-Malsburg Takeuchi-Amari Kohonen
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dynamics of neural excitation
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dynamics of learning
adiabatic hypothesis
average learning equation
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equilibrium
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special case 1-dim.
amplification sizes of receptive field
conformal map
inverse law
Amari, 1980 Bull. Math. Biol. 42, 339-364
J. Zhang S. Filho
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Topological properties
emergence of block structure
patch structure
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dynamics of receptive field
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Equilibrium solution
Stabilityvariational equation
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stability
unstable
stable
correlation narrow
unstable
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two types of stability
neural fields
Kohonen cactus
emergence of patch or column structure
Kurata
Information Geometry
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Conclusions Dynamics of Neural Fields
Oscillations in Neural Fields Synchronous
Firing Self-Organization of Neural Fields
Beyond Kohonen Topological Map