Analytical description of the evolution of neural networks: learning rules and complexity

We present the foundation of a physical formalism that allows us to characterize the dynamics of the evolution of neural networks both in regard to the network configuration and to network performance. Model runs were performed on a simple network consisting of six neurons, allowing complete analytical description of the network's behaviour. Order parameters are characterized that allow an analytical description of critical periods in network evolution. Thus, correlations of the local dynamics and the system's global behaviour could be computed. It is shown that local learning rules are sufficient to model complex dynamical aspects of the evolution of networks. It is demonstrated in how far novel statistical formalisms, e.g. neural complexity, can be employed to evaluate the system's dynamics. The introduction of order parameters allows an analytical characterization of transient phases in the network's behaviour, correlating network connectivity with neuronal firing patterns. The relevance of this approach for the interpretation of physiological data is discussed.