Analytical Characterization of Epileptic Dynamics in a Bistable System

TL;DR

The paper treats epileptic seizures as noise-driven switches between a normal stable state and a seizure-like oscillatory state within a bistable oscillator. It develops an analytical framework using Lyapunov and LaSalle methods to identify regions of attraction and employs input-to-state stability to understand perturbation resilience, then extends to networks of bistable units to reveal how coupling shapes stability. By training a network on EEG data, it demonstrates a generative, data-informed approach to reproduce brain activity and seizure-like transitions, suggesting pathways for predictive and closed-loop control. Overall, the work links dynamical-systems analysis to epileptic dynamics, highlighting how network structure and perturbations influence seizure onset and offering principled targets for intervention.

Abstract

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.

Figures (4)

• Figure 1: Nullclines and vector fields of the system \ref{['main']} in different parameter regimes. In all cases, $a=1,b=1$, and $\omega = 0.4$.
• Figure 2: Illustration of bistability when $-a^2b<\sigma<0$. (a) A stable equilibrium (the origin) and a stable limit cycle (solid circle) coexist. The region of attraction for the origin is depicted by the shaded open area. Starting from inside, the system converges to the origin; starting from outside, it converges to the limit cycle. The separatrix is the dashed circle. (b) Different regions of attractions provide a possible interpretation of the discrepancy between healthy and epileptic brains.
• Figure 3: Evolution of the system \ref{['forced']} at different regimes with the same initial condition and identical continuous perturbations ((a): $\sigma = -0.8$, (b): $\sigma = -0.2$). In both cases, $a=1,b=1, \omega = 4$. Perturbations are Gaussian with mean $0.2$ and STD $0.5$ truncated at $[-0.3,0.7]$.
• Figure 4: (a) Illustration of networks of bistable oscillators as generative models. Recorded EEG signals (left) are used to train the network parameters and the readout function so that the EEG recordings can be reproduced (right). (b) A network of 64 oscillators reproduces an EEG recording. The trained model also exhibits seizure-like activity in the presence of higher levels of noise, indicating a noise-driven transition. Shaded areas show STD.

Theorems & Definitions (7)

• Lemma 1
• Definition 1
• Theorem 1: Regions of Attraction
• Theorem 2: Input-to-State Stability
• proof
• Theorem 3: Estimate of Region of Attraction
• proof