As an application of the generalized quantum theory developed within recent years, we could formally work out how complementary (or incompatible, resp.) observables, traditionally restricted to quantum systems, can arise in classical systems as well. Such observables have been discussed for decades without an appropriate formal framework. The key to such a framework is the concept of generating partitions in the theory of nonlinear dynamical systems. A partition is generating if it divides the state space into regions prescribed by the dynamics of the system, thus permitting the definition of states that are stable under the dynamics.

We could show that complementary observables arise in classical systems whenever the partitioning of the corresponding state space is not generating. Conversely, observables are compatible if and only if they are based on generating partitions. Only in this case are the partitioned (symbolic) dynamics and the original dynamics (topologically) equivalent. Scattered observations of "quantum-like" features in classical systems could be explained in this manner. This is particularly important for relations between mental states and their underlying neurodynamics.

Lattices of coupled nonlinear maps with discretized space and time variables, but with continuous state variables, were introduced by Kaneko und utilized in many applications. Only recently it turned out that they are specifically relevant for models of neural networks with nonlinear transfer functions of individual neurons. We showed that suitably coupled map lattices permit a global stabilization of locally unstable fixed points without external control — a new and powerful alternative to broadly applied approaches of chaos control. In addition, such stabilizing mechanisms are of crucial significance for our understanding of so-called acategorial mental states.

Continuing earlier numerical work we studied the impact of causal, simultaneous, and anti-causal coupling on the stabilization properties of coupled maps. We discovered that stabilization is supported by causal coupling, while it is strongly obstructed by anti-causal coupling. Since mental states presuppose stable neural ensembles, this result suggests an interesting relationship between neuronal causality and the psychological arrow of time.

Graph theoretical spectral analysis offers the possibility to calculate the stabilization properties of coupled maps. It turned out that all established numerical results are in prefect agreement with theory. In future work we intend to develop methods for the characterization of transients close to instabilities. Moreover, numerical algorithms will be refined toward time-dependent coupling strengths (neural plasticity, Hebb's rule).

Coupled map lattices were additionally studied as examples for the failure of reproducibility. This is due to long-time transients, implying non-stationarity and problems with limit theorems and ergodicity. We extended our investigations to weakly interacting lattices of coupled maps, separated from each other by boundary regions with small coupling. We demonstrated that such weakly interacting lattices can nevertheless have unexpected striking effects on each other. The stabilization properties of the lattices can be significantly affected even by a minimal information flow through the boundary regions.

The optimization of small graphs (with less than 30 vertices with nonlinear dynamics) was studied with respect to predefined input-output-relations. The optimization is achieved by a learning process, in which successive random changes of the graph (mutations) are accepted whenever they improve its output and thus increase its fitness (selection).

To begin with, we found a significant stepwise behavior of the fitness, resembling the phenomenon of so-called "punctuated equilibrium" in biological evolution. This is remarkable since in our model these transitions are not initiated by environmental changes but by internal modifications of the graph. Moreover it is surprising that such non-trivial behavior occurs already in very small graphs.

We also found an associative, (for learning typical) non-commutative, multiplicative structure of observables corresponding to the reaction onto particular inputs. The representation space of this structure is a (complete) set of attractor states, whose number can be interpreted as a measure for the complexity of the learning process. At the start and at the end of the processes this complexity is (much) smaller than during its intermediate stages.

We plan to study in more detail whether this behavior can unveil deeper reasons for a well-known heuristic relationship between complexity and pragmatic information, a proposed measure of meaning. Moreover, the convex behavior of complexity during learning is interesting for particular aspects of the solution of insight problems.