Passivity-Based Control of Electrographic Seizures in a Neural Mass Model of Epilepsy

Abstract

Recent advances in neurotechnologies and decades of scientific and clinical research have made closed-loop electrical neuromodulation one of the most promising avenues for the treatment of drug-resistant epilepsy (DRE), a condition that affects over 15 million individuals globally. Yet, with the existing clinical state of the art, only 18% of patients with DRE who undergo closed-loop neuromodulation become seizure-free. In a recent study, we demonstrated that a simple proportional feedback policy based on the framework of passivity-based control (PBC) can significantly outperform the clinical state of the art. However, this study was purely numerical and lacked rigorous mathematical analysis. The present study addresses this gap and provides the first rigorous analysis of PBC for the closed-loop control of epileptic seizures. Using the celebrated Epileptor neural mass model of epilepsy, we analytically demonstrate that (i) seizure dynamics are, in their standard form, neither passive nor passivatable, (ii) epileptic dynamics, despite their lack of passivity, can be stabilized by sufficiently strong passive feedback, and (iii) seizure dynamics can be passivated via proper output redesign. To our knowledge, our results provide the first rigorous passivity-based analysis of epileptic seizure dynamics, as well as a theoretically-grounded framework for sensor placement and feedback design for a new form of closed-loop neuromodulation with the potential to transform seizure management in DRE.

Figures (4)

• Figure 1: (a) Simulated state trajectories of the autonomous Epileptor model ($u = 0$). One state is selected from each subsystem for illustration. The ultraslow modulation by $z$ drives bifurcations in the remaining (faster) states between interictal and ictal regimes. (b) Phase–plane trajectories of the autonomous Epileptor model (same model as in (a)) plotted separately for the fast $(x_1,y_1)$ and slow $(x_2,y_2)$ subsystems. Trajectories converge to a stable limit cycle corresponding to ictal oscillations.
• Figure 2: The spectral abscissa of the closed-loop Jacobian matrix $\mathbf{A} - k \mathbf{g} \mathbf{c}^\top$ (cf. \ref{['eq:lin']}) as a function of the control parameters $(u^\star, k)$ of the passive feedback controller in \ref{['eq:shunt_feedback']}. Control parameters that fail to stabilize the system ($\alpha(\mathbf{A}) > 0$ are shown as white. The red dot corresponds to the nominal parameters $u^\star = -0.8, k = 1$ used in acharya2026passive and Theroem \ref{['thm:shunt_local_stab']}.
• Figure 3: Closed-loop behavior of the Epileptor under passive feedback with parameters as in Theorem \ref{['thm:shunt_local_stab']}. Left: asymptotic convergence of two representative states to steady-state values. Right: the phase plane of the slow subsystem, showing the co-existence of the asymptotically stable equilibrium point (red dot) and a stable limit cycle (black). The apparent overlap between the regions of attraction of the stable equilibrium and the limit cycle is due to the 2D projection of the 6D state space.
• Figure 4: The spectral abscissa of the closed-loop Jacobian matrix $\mathbf{A} - k \mathbf{g} \mathbf{c}^\top$, similar to that in Fig. \ref{['fig:heatmaps']}, but for new sets of controller and output parameters. (a) $y = x_1 + x_2$, $u^\star = -2$, $k = 0$ (Theorem \ref{['thm:passive-newc1c2']}), (b) $y = x_1 - 0.29 x_2$, $u^\star = -2$, $k = 0$ (Example \ref{['ex:opt']}).

Theorems & Definitions (19)

• Definition B.1
• Proposition B.2
• Proposition B.3
• proof
• Proposition B.4
• Remark C.1
• Lemma D.1
• proof
• Theorem D.2
• proof