Control of Seizure Dynamics in a Bistable System with Pulsatile Inputs

Bistable neural population models describe transitions associated with epilepsy through the coexistence of a stable equilibrium representing normal neural activity and a stable limit cycle representing seizure-like oscillations. Perturbations that drive the system outside the equilibrium's basin of attraction can trigger transitions to seizure-like activity. Enlarging this basin enables the system to recover normal activity from a larger set of perturbed states, providing a model-based mechanism for reducing frequency of seizure initiation. Motivated by electrical brain stimulation, this paper investigates basin enlargement using pulsatile feedback control. Combining higher-order averaging with periodic near-identity transformations, we establish conditions under which nonzero-mean and zero-mean pulsatile feedback gains enlarge the basin of attraction and derive guidelines for pulse design. Numerical simulations validate the theoretical enlargement.