A Mathematical Approach to the Stability of Electroencephalography Signals Model Using Integral Equations During Epileptic Seizures

In this study, the mathematical stability of EEG signal models represented by the integral equation during an epileptic seizure is investigated. The modelling of the EEG signals should be done with utmost accuracy to understand the actual nature of the brain dynamics and to enhance seizure prediction approaches. We investigate the stability of such models by applying Lyapunov's stability theorem and show that stability is powerfully dependent on parameters such as seizure duration and intensity, along with neural connectivity. We demonstrate that some of the parameter regimes yield stable behaviour, while others give rise to instability, making EEG interpretation complicated during epileptic seizures. By comparing integral equation-based models with current approaches, their advantages are highlighted in capturing the transient dynamics of epileptic activity. This work focuses on developing solid mathematical frameworks that will allow for real-time EEG monitoring. It contributes to the wider knowledge on the dynamics of seizures in the brain and will have practical implications for the development of neurological diagnostic and therapeutic tools.