Koszul connections of flat EEG bundles for description of brain signal dynamics

Abstract

The bio-electromagnetic inverse problem is the identification of electromagnetic sources based on signals recorded from Electroencephalography (EEG) or Magnetoencephalography (MEG) and physical equations with a minimum of priori information. Following initial applications in modelling epilepsy patients' electrical brain activity, extensive research has been done in processing massive amounts of signal data from patients while preserving as much information as possible. Improvements in analyzing EEG/MEG recordings have significant utility not just in epilepsy treatment and diagnosis, but neuro-cognitive research in general. The main objective of this research is to build structures holding visualized and flattened EEG data as differential geometric entities, and to extend these structures as a dynamical system dealing with the evolution of large amounts of point data about brain activity. One aspect of existing research involves EEG signal data recorded across a time interval and processed using fuzzy clustering techniques, resulting in point data sets representing areas of high electrical activity within the brain. Concepts in differential geometry are applied to these spaces as a dynamical and visualized approach to modelling the evolution of signal clusters in the brain over time. Initially, Flat EEG data sets are shown to be topological spaces, manifolds and vector spaces. Two vector bundle structures for the Flat EEG space are thus developed: one analogous to Minkowski space-time and the other based on the classical notion of spatial change over linear time. From there, Koszul connections were constructed for both vector bundles, and both are shown to have zero curvature. Having provided a continuous differential structure to Flat EEG data, the evolution of signal clusters as a discrete dynamical system is then interpolated into a continuous form, allowing an enhanced view of the brain's state changes over time.