Phase Description of Oscillations in Singularly-Perturbed Systems

• Applicant: Dr. Justus Schwabedal
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2012 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 224195127

We want to study the typically irregular oscillations of neuronal activity. For such irregular oscillations, the unsteady rhythm of breathing is a classical example. Using the established method, a simplified description of oscillations builds on their division into characteristic phases. In applications, such a phase description can be used for the detection of very weak interactions between oscillations. However, the standard phase method allows only for a treatment of highly regular oscillations. Therefore, the central goal of our proposal is a sufficient extension of the methods of phase description for an application to realistic neuronal activity, which shows substantial irregularity. In particular, we want to treat two examples of neuronal oscillations. So called noise-induced oscillations (1) exist only in the presence of external fluctuations that arouse the system out of a state that actually does not oscillate. This type of oscillation cannot be described under disregard of the perturbation as it would be tried in the standard phase method. Therefore, another starting point has to be used. For this, we want to extend a recently developed method for the description of breathing. So called bursting (2) is a type of mixed mode oscillation, in a way two oscillators in one, that arises from an interplay of fast and slow time scales. In networks of coupled bursters, the standard phase description breaks down already for weak coupling strength. We, therefore, plan to design a more adequate phase description that achieves a simplified handling of the fast mode by taking an average. In the application to burster networks as they appear in the central nervous system, our goal is to describe the variety of simultaneously occurring synchronization patterns.

• DFG Programme: Research Fellowships
• International Connection: USA
• Participating Institution: Georgia State University ; Department of Mathematics and Statistics
• Host: Professor Dr. Andrey Shilnikov