We present a neural network approach for closed-loop deep brain stimulation (DBS). We cast the problem of finding an optimal neurostimulation strategy as a control problem. In this setting, control policies aim to optimize therapeutic outcomes by tailoring the parameters of a DBS system, typically via electrical stimulation, in real time based on the patient's ongoing neuronal activity. We approximate the value function offline using a neural network to enable generating controls (stimuli) in real time via the feedback form. The neuronal activity is characterized by a nonlinear, stiff system of differential equations as dictated by the Hodgkin-Huxley model. Our training process leverages the relationship between Pontryagin's maximum principle and Hamilton-Jacobi-Bellman equations to update the value function estimates simultaneously. Our numerical experiments illustrate the accuracy of our approach for out-of-distribution samples and the robustness to moderate shocks and disturbances in the system.

翻译：我们提出了一种用于闭环脑深部电刺激（DBS）的神经网络方法。我们将最优神经刺激策略的求解问题建模为一个控制问题。在该框架下，控制策略旨在通过根据患者实时神经元活动调整DBS系统参数（通常通过电刺激实现）来优化治疗效果。我们使用神经网络离线逼近值函数，从而能够通过反馈形式实时生成控制信号（刺激）。神经元活动由霍奇金-赫胥黎模型所描述的非线性刚性微分方程组表征。我们的训练过程利用庞特里亚金最大值原理与哈密顿-雅可比-贝尔曼方程之间的关系，同步更新值函数估计值。数值实验表明，该方法对分布外样本具有准确性，并且对系统中的适度冲击与扰动具有鲁棒性。