In biology and the medical sciences, complexity and detail prevail. Many branches of mathematics may be employed to great detail to provide biological insight. The biomathematics group is active in the following areas.
The research interests in dynamical systems encompass various domains:
- Mathematical Neuroscience: Our research interests are in modeling, simulation, and statistical analysis of neuronal activity. We analyze deterministic and stochastic models of brain activity at various scales, from single neurons to networks of coupled oscillators, neural masses, and continuum neural fields to gain insights into neural dynamics. We develop statistical methods and perform statistical analysis of brain signals obtained by functional magnetic resonance imaging (fMRI), electroencephalography (EEG), and magnetoencephalography (MEG). We also develop numerical methods with provable accuracy that predict activity, quantify uncertainties in models with random data, and infer model inputs from data.
- Statistics for Big Biomedical Data: We work on statistical methodology, ranging from parameter estimation, inference, prediction to class discovery, to learn from high-throughput experiments on the cell. This methodology aids, for instance, in the analysis of oncogenomic studies. Such studies characterize the expression levels of many genes simultaneously in a limited set of cancer patients. The methodology then enables the identification of genes predictively associated with the clinical outcome of the patients. Alternatively, we develop methods for the reconstruction of the cancer cell's gene-gene interaction network from longitudinal in vitro studies.
- Dynamics of Cellular Processes: The department also has a long-standing collaboration with systems biologists. Here, we try to develop dynamical models of cellular processes and behaviour. How does a single cell sense its environment and deal with stochastic information? How does it decide whether to invest into growth processes or instead mount stress responses? Understanding such broad questions, relevant for all of unicellular life, requires both stochastic and deterministic modeling, formal analysis of ODE and PDE models, and optimization and control theory.
- Networks and Topology: Graphs and hypergraphs are common topological models to understand a variety of biological systems, including gene expression, protein-protein interaction, phylogeny, neural networks and metabolic networks. We are interested in developing the mathematical foundations of network theory and extending topological methods while exploring applications to biology. For example, how do topological properties of the network of interactions between neural oscillators shape emergent synchronization and, thus, neural function?