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 ( 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