Investigating Bistable Dynamics of Coupled Oscillators With Similarities to Neural Activity in Epilepsy

The dynamics of an oscillator, which exhibits a stable equilibrium and a stable limit cycle, is investigated. We refer to it as a bistable oscillator unit (BOU) and show that two coupled BOUs (CBOUs) exhibit dynamics analogous to neural activity patterns in epilepsy, including healthy, localized, and fully spread epileptic states. By treating each CBOU as an independent system influenced by the state of the other, we establish local input-to-state stability near the equilibrium and the limit cycle, and estimate the ultimate bounds of the trajectories. Our analysis identifies the domain of the initial conditions and estimates the coupling parameter critical in determining the epileptic behavior of the CBOUs. The actual value of the coupling parameter, above which the dynamics transition from localized to fully spread epileptic states, is determined through simulations; the results closely match the derived analytical estimate.