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In this paper we show that the quantization error for Dynamic Cell Structures (DCS) Neural Networks (NN) as defined by Bruske and Sommer provides a measure of the Lyapunov stability of the weight centers of the neural net. We also show, however, that this error is insufficient in itself to verify that DCS neural networks provide stable topological representation of a given fixed input feature manifold. While it is true that DCS generates a topology preserving feature map, it is unclear when and under what circumstances DCS will have achieved an accurate representation. This is especially important in safety critical systems where it is necessary to understand when the topological representation is complete and accurate. The stability analysis here shows that there exists a Lyapunov function for the weight adaptation of the DCS NN system applied to a fixed feature manifold. The Lyapunov function works in parallel during DCS learning, and is able to provide a measure of the effective placement of neural units during the NN's approximation. It dnes not, however, guarantee the formation of an accurate representation of the feature manifold. Simulation studies from a selected CMU-Benchmark involving the use of the constructed Lyapunov function indicate the existence of a Globally Asymplotically Stable (GAS) state for the placement of neural units, hut an example is given where the topology of the constructed network fails to mirror that of the input manifold even though the quantization error continues to decrease monotonically. is no guarantee for the existence of a solution to a non-linear differential equation at all times, let alone the complicated task of solving them. In practice however, there exists a variety of techniques and tools that analyze the existence of stable states in non-linear systems, without actually the need for solving any differential equation [l], [4], [SI, [6]. Almost all stability definitions for linear systems lead to a similar stability criterion (R-locus, eigenvalues, etc). Therefore defining a precise stability definition for linear systems can be relaxed. However, eigenvalue methods are not applicable to non-linear systems and so one needs to be careful while dealing with systems that have non-linear properties. It's often seen that if such systems are stable under one definition of stability they may tend to become unstable under other definitions [SI. This difficulty in imposing strong stability restrictions for non-linear systems was realized as early as a century ago by a Russian mathematician A.M.Lyapunov. Details on Lyapunov's stability analysis technique for non-linear discrete systems can be found in [I], [Z], [ 6 ] , [41. The fact that Lyupunov's Direct Method or Non-linear Theory of Stability Analysis can be readily applied to validate the existence (or non-existence) of stable states in non-linear systems, intrigued us in using Lvapunov function --, . in this stability analysis as a means to answer the question posed earlier. Dynamic Cell Structures (DCS) NN represents a fa...

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