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One principle of DLM is that the links between two layers can be cleaned up and structured on the basis of correlations between pairs of neurons (see Figure 5 and Movie 1). The correlations result from the layer synchronization described in the previous section. The link dynamics typically consists of a growing term and a normalization term. The former lets the weights grow according to the correlation between the connected neurons. The latter prevents the links from growing infinitely and induces competition such that only one link per neuron survives which suppresses all others. The corresponding equations are (cf. Equations 6):
Links are initialized by the similarity between the jets of connected nodes .
The parameter guarantees a minimal positive synaptic weight, permitting each link to suppress others, even if the similarity between the connected neurons is small. This can be useful to obtain a continuous mapping if a link has a neighborhood of strong links inducing high correlations between the pre- and postsynaptic neurons of the weak link. The synaptic weights grow exponentially, controlled by the correlation between connected neurons defined as the product of their activities . The learning rate is additionally controlled by . Due to the Heavyside-function , normalization takes place only if links grow beyond their initial value. Then, the link dynamics is dominated by the normalization term, with a common negative contribution for all links converging to the same neuron. Notice that the growth term, based on the correlation, is different for different links. Thus the link with the highest average correlation will eventually suppress all others converging to the same neuron. Since the similarities cannot be larger than 1, the synaptic weights are restricted to the interval .
Figure 5: (click on the image to view a larger version) Connectivity and correlations developing in time. It can be seen how the correlations develop faster and are cleaner than the connectivity. Both are iteratively refined on the basis of the other.
Movie 1: Connectivity and correlations developing in time as in Figure 5 (350 kB).