Ch10 : Self-Organizing Feature Maps 。 Cortex: 。Ordered Feature Map e.g. Auditory cortex -- tonotopic maps Hippocampal cortex -- geographic maps Somatosensory cortex -- somatic maps Visual cortex -- retinotectal maps ------------------------- ------------------------ i) 6 layers of neurons ii) Size: iii) Thick:

。 Macroscopic scale: Consistently uniform structure. Microscopic scale: Located consistently relative to one another logical ordering of functionality Tissue level – inherence Mental level – learning

◎ Architecture of SOFM Neural Networks Primary mechanisms: (1) Lateral feedback (2) Clustering (3) Topology preserving projection

Objective : formation of localized responses ○ Lateral Feedback Objective : formation of localized responses Lateral interaction function : input, : output

Lateral interaction function ＋: excitation －: inhibition 。Output: primary input lateral input a: transfer function

○ Clustering e.g., Clustering of activity in a 1-D array: input signal Clustering of activity in a 1-D array: Clustering of activity in a 2-D array:

。 Nonsmooth input signal or Nonlinear transfer function a Arbitrary lateral feedback function Irregular activity bubbles

○ Topology Preserving Projection 。 3-D Physical space Weight space 。 2-D

。 1-D

Mathematical treatment of self – organization ○ 1-D case : a scalar input signal to a neuron neurons : 1, 2, …. , l weights :

。 The best match 。 Neighborhood neurons (first-order neighbors)

。 Weight update : learning step, g : Gaussian ○ 2-D System Input scalar signals: Output response: : weight

Find the best match between x and 2 ways 。 Topological neighborhood -- To be engaged in learning but with different degrees (approximated by Gaussian that is reduced with time)

◎ Applications – Geometrical Modeling S. W. Chen, G. C. Stockman, and K. E. Change, “SO Dynamic Deformation for Building of 3D Models,” IEEE Trans. on Neural Networks, Vol. 7, No. 2, pp. 374-387, 1996.

○ Data Acquisition