  Epilepsy is one of the most common neurological disorders globally, affecting
millions of individuals. Despite significant advancements, the precise
mechanisms underlying this condition remain largely unknown, making accurately
predicting and preventing epileptic seizures challenging. In this paper, we
employ a bistable model, where a stable equilibrium and a stable limit cycle
coexist, to describe epileptic dynamics. The equilibrium captures normal
steady-state neural activity, while the stable limit cycle signifies
seizure-like oscillations. The noise-driven switch from the equilibrium to the
limit cycle characterizes the onset of seizures. The differences in the regions
of attraction of these two stable states distinguish epileptic brain dynamics
from healthy ones. We analytically construct the regions of attraction for both
states. Further, using the notion of input-to-state stability, we theoretically
show how the regions of attraction influence the stability of the system
subject to external perturbations. Generalizing the bistable system into
coupled networks, we also find the role of network parameters in shaping the
regions of attraction. Our findings shed light on the intricate interplay
between brain networks and epileptic activity, offering mechanistic insights
into potential avenues for more predictable treatments.
