Normalization of weights in Kohonen algorithm

[Ken Miller]

re point 3 of recent posting about Kohonen algorithm: 

"3	In Kohonen's book "Self Organization and Associative Memory", Ch 5
	the algorithm for weight adaptation does not produce normalized
	weights."

the algorithm

du_{ij}/dt = a(t)[e_j(t) - u_{ij}(t)], i in N_c

where u = weights, e is input pattern, N_c is topological neighborhood of
maximally responding neighborhood, should I believe be written

du_{ij}/dt = a(t)[ e_j(t)/\sum_k(e_k(t)) - u_{ij}(t)/\sum_k(u_{ik}(t)) ], 
i in N_c.

That is, the change should be such as to move the jth synaptic weight on the
ith cell, as a PROPORTION of all the synaptic weights on the ith cell, in the
direction of matching the PROPORTION of input which was incoming on the jth
line.  Note that in this case \sum_j du_{ij}(t)/dt = 0, so weights remain
normalized in the sense that sum over each cell remains constant.

If inputs are normalize to sum to 1 (\sum_k(e_k(t)) = 1) then the first
denominator can be omitted.  If weights begin normalized to sum to 1 on each
cell ( \sum_k(u_{ik}(t)) = 1 for all i) then weights will remain normalized
to sum to 1, hence the second denominator can be omitted.  Perhaps Kohonen
was assuming these normalizations and hence dispensing with the denominators?

ken miller (ken at phyb.ucsf.edu)