Self-Organisation, Metastable States and the ODE Method in the Kohonen Neural Network

Metastable states are analysed for a simple Kohonen Neural Network (KNN) using the ODE method. Normally the ODEs are determined for each possible configuration of the neuron weights. By comparing the trajectories of the ODEs of the neuron weights and those obtained from simulation we are forced to rethink how the ODEs should be formed for the KNN before a complete analysis of the self-organisation process in the KNN can be analysed with the ODE method. It is shown that the ODEs predict better what happens in practise if the general ODEs are formed by using a weighted average of the ODEs for each possible configuration of the neuron weights. The weight accorded to each configuration depends amongst other things on the type of gain function used.