From single cell to neural field - dynamics on different scales
This thesis deals with the development and analysis of mathematical models of brain activity, with a focus on pathological conditions. The models we discuss span a wide range of both spatial and temporal scales, ranging from picometer-sized ion channels to centimeter-sized neural networks, and from spiking dynamics on the order of milliseconds to cell volume dynamics on the order of hours.
We first focus on swelling of brain cells due to energy shortage, as observed in stroke or brain trauma. By analyzing a biophysical model of a single cell, we identify both a critical point of energy supply below which the cell volume significantly increases, and an intervention that could reverse cell swelling.
Secondly, we derive a phenomenological rate-based neuron model that can mimic a wide variety of experimentally observed spiking behaviors. This enables us to incorporate non-trivial spiking patterns in so-called neural field models, which describe the averaged activity of large populations of neurons.
Thirdly, we model epileptic seizure dynamics by coupling two different types of neural field models that correspond to distinct spatial scales. This approach is motivated by recent recordings of epilepsy patients that reveal a clear separation between high-frequency activity at a localized wavefront, and global low-frequency oscillations in the surrounding areas.
Finally, in a more mathematically driven contribution, we analyze neural field models with transmission delays. We illustrate how qualitative changes in the system dynamics can be detected and classified, which permits a full characterization of the models’ behavior