Dynamic topological description of brainstorm during epileptic seizure

Abstract

Electroencephalograph is one of the useful and favoured instruments in diagnosing various brain disorders especially in epilepsy due to its non-invasive characteristic and ability in providing wealthy information about brain functions. At present, a large amount of quantitative methods for extracting “hidden” information which cannot be seen by “naked” eye from an electroencephalogram has been invented by scientist around the world. Among those, Flat Electroencephalography (Flat EEG) is one of the novel methods developed by Fuzzy Research Group (FRG), UTM which has been intended to localize epileptic foci of epilepsy patients. The emergence of this invention has led to the development of several Flat EEG based research (e.g., Non Polar CEEG and Fuzzy Neighborhood Clustering on Flat EEG). The verification of the method has been made via comparison with some substantial clinical results. However, in this thesis, theoretical foundation of the method is justified via the construction of a dynamic mathematical transformation called topological conjugacy whereby isomorphism between dynamics of epileptic seizure and Flat EEG is established. Firstly, these two dynamic events are composed into sets of points. Then, they are forced to be strictly linearly ordered and composed into topological spaces. Subsequently, an isomorphism is constructed between corresponding mathematical structures to show that their properties are preserved and conjugate topologically. The constructed topological conjugacy is generalized into a class of dynamical systems. Within this class of dynamical system, Flat EEG’s flow is shown to be structurally stable. Additionally, topological properties on the event of epileptic seizure and Flat EEG have also been established.