A review of models used for understanding epileptic seizures

Abstract

The development and application of mathematics, fostered by a mathematics-in-industry study group (MISG) activity, is a key outcome of the challenges posed when aiming to answer the questions raised by real-world problems. Mathematics-in-industry study groups, which now thrive internationally, bring together mathematicians from tertiary institutions and industries to provide a stimulating environment for the discussion, analysis and solutions of such problems. Mathematics-in-industry study groups (MISGs) have been actively providing a forum for the development and application of mathematics for industrial modelling projects. When considering industrial mathematical applications, it is important to recognise that industrial processes work because of their inherent robustness (de Hoog, 2009). Consequently, the formulation and analysis of the simple models which encapsulate the associated robustness are an essential part of industrial mathematical modelling. This approach to modelling allows for the uncertainty in modelling to be managed by a relatively simple model that describes the process in question and can capture an answer the industrial partners can exploit operationally. At the MISG 2010 at RMIT University, in a study of the recovery of the location and time of the initiation of an epileptic episode, the importance of the joint utilisation of electroencephalogram (EEG) and functional MRI (fMRI) data became apparent for: 1. Reducing the uncertainty about the location and time of the commencement of an epileptic seizure. 2. Providing an understanding about the neuronal mechanism causing epilepsy as a means to reduce the uncertainty around minimizing associated risks. In particular, the process of sychronisation forms a foundation for the understanding of epileptic episodes. Synchronisation occurs when two or more process couple to harmonise in a similar manner. It is a naturally occurring physical, physicochemical and biological phenomenon. Examples include superconductivity, neuronal behaviour in the brain, adjacent clocks, circadian rhythm in animals and plants and the fundamental processes in population dynamics. In this paper, the modelling of synchronisation with the simple and exactly solvable Kuramoto model is discussed. Generalisations of this model are reviewed which more correctly reflects the known behaviour of synchronisation in the brain. A strategy for the joint inversion of the EEG and fMRI data is proposed.