  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.
