Symbolic Methods in Simulation

The goal of the project is the anticipation of full body motions. Combining purely data-driven with physics-based modeling approaches, anticipation of full body motions on the basis of very sparse sensor signals of different nature, e.g., inertial measurement units, EMG sensors, or ground-contact sensors, will be realized. Extending the information on the physics-based layer by model-based anticipation components (including information from balance control, physical constraints) is another important objective to allow robust extrapolations from the range of motions similar to ones recorded in an existing knowledge base to new motion ranges, especially those relatedto disabilities. The anticipated whole-body motions can be used to determine the optimal robot placement for collaborative tasks and for direct entrainment and modulation of ongoing motor behavior. Symbolic labels and trajectories from affordances obtained from other projects of the research unit will be incorporated as additional a priori knowledge on motions, which will reduce computations times, will stabilize short term predictions and even open the door for the method to long-term anticipations.

SYMBIONT is an interdisciplinary project ranging from mathematics via computer science to systems biology and systems medicine. The project has a clear focus on fundamental research on mathematical methods, and prototypes in software, which is in turn benchmarked against models from computational biology databases. Computational models in systems biology are built from molecular interaction networks and rate parameters resulting in large systems of differential equations. These networks are foundational for systems medicine. The currently prevailing numerical approaches shall be complemented with our novel algorithmic symbolic methods, which will address fundamental problems in this area. One important problem is that statistical estimation of model parameters is computationally expensive and many parameters are not identifiable from experimental data. In addition, there is typically a considerable uncertainty about the exact form of the mathematical model itself. The parametric uncertainty (with wide potential variations of parameters by several orders of magnitudes) leads to severe limitations of numerical approaches even for rather small and low dimensional models. Furthermore, extant model inference and analysis methods suffer from the curse of dimensionality that sets an upper limit of about ten variables to the tractable models. For those reasons, the formal deduction of principle properties of large and very large models has a very high relevance. The main goal of SYMBIONT is to combine symbolic methods with model reduction methods for the analysis of biological networks. We propose new methods for symbolic analysis, which overcome the above mentioned obstacles and therefore can be applied to large networks. In order to cope more effectively with the parameter uncertainty problem, we impose an entirely new paradigm replacing thinking about single instances with thinking about orders of magnitude. Our computational methods are diverse and involve various branches of mathematics such as tropical geometry, real algebraic geometry, theories of singular perturbations, invariant manifolds and symmetries of differential systems. The foundations and validity of our methods will be carefully secured by mathematical investigation. Corresponding computer algebra problems are NP-hard, but experiments point at their feasibility for biological networks. We have already shown that complexity parameters such as tree-width or number of distinct metastable regimes grow only slowly with size for models available in existing biological databases. We will exploit this observation to solve challenging problems in network analysis including determination of parameter regions for the existence and stability of attractors, model reduction, and characterization of qualitative dynamics of nonlinear networks. The methods developed in this project will be benchmarked against existing biological models and also against more challenging models, closer to the needs of systems and precision medicine that will be generated using biological pathways databases.

We will analyse bifurcations and singularities of algebraic systems of ordinary differential equations with particular emphasis on questions concerning the existence of oscillations.