New Eigenvalue-Based Analysis for Precise Limit Cycle Stability Assessment in a Two-State Epileptor Model

The Epileptor model is a mathematical framework utilized for simulating the transition from interictal to ictal local field potential (LFP) activity in the brain, with the aim of predicting and preventing epileptic seizures. This article introduces a novel approach integrating Lyapunov and Poincaré–Bendixson methods to analyze the stability of limit cycles in nonlinear systems, specifically focusing on Epileptors with a two-state dynamic. Our method accurately delineates the limit cycle boundary through eigenvalue-based analysis, facilitating precise assessment of stability properties and identification of critical regions linked to seizure initiation and termination. Through the investigation of the two-state dynamics of Epileptors, we gain deeper insights into the transition between low activity and seizure states, consequently improving our understanding of epileptic seizures. Our approach can be employed to establish stability conditions and determine the existence of limit cycles in Epileptor models, which can further aid in predicting and preventing epileptic seizures by identifying critical regions associated with seizure initiation and termination. The simulations conducted in this study demonstrate that the model under investigation exhibits stable limit cycle behavior and manifests bifurcation, with significant implications for the development of targeted interventions and more effective prediction and treatments for epilepsy. The findings indicate that the suggested approach establishes that external stimulation should not surpass 10.8 mA. Moreover, the initial normal state lies within the range of $-$ 1.6 to $-$ 0.1 ictal LFP. On the other hand, the LaSalle and eigenvalue methods individually cannot precisely determine the limit cycle region.