SLIDE 1
Vi Vj
Tij, Tji
Dynamics: Energy function:
Ui = X
j
Attractor neural networks Vi Tij, Tji Vj X U i = T ij V j - - PowerPoint PPT Presentation
Attractor neural networks Vi Tij, Tji Vj X U i = T ij V j - - PowerPoint PPT Presentation
Attractor neural networks Vi Tij, Tji Vj X U i = T ij V j Dynamics: j V i = sign( U i ) Energy function: Basins of attraction input recall Outer-product (Hebb) rule P ( ) P ( ) X T ij = i j P (1) P (1) + P (2) P (2) + P (3)
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Basins of attraction
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input recall
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Outer-product (Hebb) rule
Tij = X
α
P (α)
i
P (α)
j
= P (1)
i
P (1)
j
+ P (2)
i
P (2)
j
+ P (3)
i
P (3)
j
+ ...
- r
T = P(1) P(1)T + P(2) P(2)T + P(3) P(3)T + ...
Thus
U ∼ = (P(1) P(1)T + P(2) P(2)T + P(3) P(3)T + ...) V = P(1) (P(1) · V) + P(2) (P(2) · V) + P(3) (P(3) · V) + ...
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Capacity vs. error rate
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Hopfield network with analog units
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Liapunov function
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From Liapunov function to dynamics
Thus Let
˙ ui ∝ − ∂E ∂Vi =
- j̸=i
Tij Vj + Ii − ui
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State space
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Marr-Poggio stereo algorithm
(Marr & Poggio 1976)
left right
+ +
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‘Bump circuits’ and ring attractors
(Zhang, Sompolinsky, Seung and others)
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Head-direction neurons
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Shifting the bump
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