Attractor neural networks Vi Tij, Tji Vj X U i = T ij V j - PowerPoint PPT Presentation

Mar 14, 2023 •41 likes •211 views

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)

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 ( α ) P ( α ) X T ij = i j α P (1) P (1) + P (2) P (2) + P (3) P (3) + ... = i j i j i j or T = P (1) P (1) T + P (2) P (2) T + P (3) P (3) T + ... Thus ( P (1) P (1) T + P (2) P (2) T + P (3) P (3) T + ... ) V U = ∼ P (1) ( P (1) · V ) + P (2) ( P (2) · V ) + P (3) ( P (3) · V ) + ... =
Capacity vs. error rate
Hopfield network with analog units
Liapunov function
From Liapunov function to dynamics u i ∝ − ∂ E � Let ˙ = T ij V j + I i − u i ∂ V i j ̸ = i Thus
State space
left Marr-Poggio right stereo algorithm (Marr & Poggio 1976) - + + -
‘Bump circuits’ and ring attractors (Zhang, Sompolinsky, Seung and others)
Head-direction neurons
Shifting the bump
2D bumps

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