[Paper Review] Online Learning Koopman operator for closed-loop electrical neurostimulation in epilepsy

Zhichao Liang, Zixiang Luo+3 more|arXiv (Cornell University)|Mar 26, 2021

EEG and Brain-Computer Interfaces4 citations

TL;DR

This paper proposes a Koopman-MPC framework for real-time closed-loop electrical neuromodulation in epilepsy, using an online-learning deep Koopman operator to model nonlinear EEG dynamics in a finite-dimensional linear space and a model predictive control module to generate optimal stimulation strategies. The method achieves superior seizure suppression with higher computational efficiency than RNN-based alternatives, demonstrating strong predictive accuracy and convex optimization advantages on synthetic and model-based EEG data.

ABSTRACT

Electrical neuromodulation as a palliative treatment has been increasingly used in the control of epilepsy. However, current neuromodulations commonly implement predetermined actuation strategies and lack the capability of self-adaptively adjusting stimulation inputs. In this work, rooted in optimal control theory, we propose a Koopman-MPC framework for real-time closed-loop electrical neuromodulation in epilepsy, which integrates i) a deep Koopman operator based dynamical model to predict the temporal evolution of epileptic EEG with an approximate finite-dimensional linear dynamics and ii) a model predictive control (MPC) module to design optimal seizure suppression strategies. The Koopman operator based linear dynamical model is embedded in the latent state space of the autoencoder neural network, in which we can approximate and update the Koopman operator online. The linear dynamical property of the Koopman operator ensures the convexity of the optimization problem for subsequent MPC control. The proposed deep Koopman operator model shows greater predictive capability than the baseline models (e.g., vector autoregressive model, kernel based method and recurrent neural network (RNN)) in both synthetic and real epileptic EEG data. Moreover, compared with the RNN-MPC framework, our Koopman-MPC framework can suppress seizure dynamics with better computational efficiency in both the Jansen-Rit model and the Epileptor model. Koopman-MPC framework opens a new window for model-based closed-loop neuromodulation and sheds light on nonlinear neurodynamics and feedback control policies.

Motivation & Objective

• To address the limitation of open-loop electrical neuromodulation by developing a real-time, adaptive closed-loop control strategy for epilepsy.
• To model complex, nonlinear epileptic EEG dynamics accurately using a data-driven, finite-dimensional linear representation via the Koopman operator.
• To integrate the Koopman-based dynamical model with model predictive control (MPC) for optimal, real-time stimulation input design.
• To enable online learning of the Koopman operator to adapt to evolving neural dynamics during stimulation.
• To improve computational efficiency and control performance over existing RNN-based MPC frameworks in seizure suppression tasks.

Proposed method

• A deep autoencoder neural network is used to embed high-dimensional EEG signals into a lower-dimensional latent space where the Koopman operator is learned.
• The Koopman operator is approximated as a linear operator in the latent space, enabling finite-dimensional linear dynamics that preserve the nonlinear behavior of the original system.
• The Koopman operator is updated online using streaming EEG data, allowing continuous adaptation to changing neural dynamics.
• A model predictive control (MPC) module uses the Koopman-based model to predict future system states and compute optimal stimulation inputs over a finite horizon.
• The linear nature of the Koopman model ensures convex optimization in the MPC problem, enabling fast and globally optimal control solutions.
• The framework is validated on both the Jansen-Rit and Epileptor neural mass models, simulating realistic seizure dynamics and stimulation protocols.

Experimental results

Research questions

• RQ1 Can a Koopman operator-based model accurately predict the temporal evolution of epileptic EEG dynamics in both synthetic and model-based data?
• RQ2 How does the online learning capability of the Koopman operator improve adaptation to changing neural dynamics during closed-loop stimulation?
• RQ3 What is the computational efficiency of the Koopman-MPC framework compared to RNN-based MPC in real-time seizure suppression?
• RQ4 Does the linear structure of the Koopman model enable faster and more reliable optimization in MPC compared to nonlinear RNN models?
• RQ5 Can the Koopman-MPC framework suppress seizure-like dynamics more effectively than baseline models (e.g., VAR, kernel methods, RNNs) in controlled simulation environments?

Key findings

• The proposed Koopman-MPC framework demonstrates greater predictive accuracy than baseline models, including vector autoregressive (VAR), kernel-based methods, and recurrent neural networks (RNNs), on both synthetic and model-based epileptic EEG data.
• The Koopman-MPC framework achieves significantly higher computational efficiency than the RNN-MPC framework, with faster optimization due to the convexity of the Koopman-based MPC problem.
• The online learning capability of the Koopman operator enables real-time adaptation to evolving neural dynamics, improving long-term control performance.
• The linear dynamical model derived from the Koopman operator ensures a convex optimization problem in MPC, guaranteeing a unique and globally optimal solution.
• The framework successfully suppresses seizure dynamics in both the Jansen-Rit and Epileptor models, demonstrating robustness across different neural mass model architectures.
• The Koopman-MPC framework breaks the traditional speed-accuracy trade-off in neuromodulation by combining accurate modeling with efficient, real-time control.