Dynamics of Delay Equations, Theory and Applications - Abstract
Visser, Sid
Neural field equations are a commonly used to describe the spatio-temporal properties of the brain?s electrical activity at a much coarser level than detailed neural networks. The spatial resolution of these model, combined with the limited number of parameters, make them appealing for studying physiological or pathological phenomena observed in electro-encephalography (EEG), such as epileptic seizures, or other clinical imaging modalities. Already since their earliest incarnation in the 1970s, neural field equations have included time delays to account for the transmission delay due to the electro-chemical nature of signal propagation between neurons. For ease of analysis, however, these delays have been ignored rather persistently by the community, part due to ignorance, part due to a lacking functional analytical setting. The latter problem has been addressed detailedly only recently, using the theory of dual semigroups (sometimes referred to as sun-star calculus). In this presentation, I will highlight some key results of this theory exemplified with a 2-dimensional neural field on the surface of a sphere including the transmission delay. The results obtained from center manifold reduction are validated numerically using model evaluations. These pose several computational challenges that I will discuss as well.