From spiking neurons to brain waves

No single model would be able to capture all processes in the brain at once, since its interactions are too numerous and too complex. Therefore, it is common practice to simplify the parts of the system. Typically, the goal is to describe the collective action of many underlying processes, without studying each individually. The work presented here analyzes a particular class of models which has revealed itself as being challenging for such simplifications: neural networks. As no generic procedure exists for reducing an arbitrary network of spiking neurons, an ad hoc approach is often chosen, which emphasizes particular features of the network. In the first part of thesis, it is illustrated that such approaches can be convincing by showing that the reduced model matches the original complicated model. Being based on imprecise assumptions, it is essential to routinely validate the reduced model with the original network, reducing the effectiveness. Therefore it is desirable to develop a more general framework, to facilitate the lumping of networks of spiking neurons. The second half of this thesis focuses on the establishment of a functional analytic setting for neural fields with transmission delay, showing that their behavior can be characterized in full detail. Consequently, attention is given to the formulation of such models from a neural network. Explicitly stating all assumptions made in the reduction, the framework is shown to facilitate the inclusion of complex cell dynamics. Although the majority of the presented results relates to conceptual and simplistic net- works of spiking neurons, it is proposed that a clinical application towards epilepsy is within reach. During seizures the majority of the neurons behaves very coherently, such that elementary networks would be able to captures these dynamics adequately. In this setting, the mathematical framework offers an effective setting for a complete characterization of the insult.