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.

通过神经网络方法实现闭环深度脑刺激 (DBS)。以找到最佳神经刺激策略为目标，将问题转化为控制问题。我们通过离线使用神经网络近似价值函数，结合 Pontryagin 的最大值原理和 Hamilton-Jacobi-Bellman 方程更新值函数估计，实时依据患者的神经活动调整 DBS 系统参数，从而优化治疗效果。

霍奇金 - 休克利神经元动力学的学习控制策略

Learning Control Policies of Hodgkin-Huxley Neuronal Dynamics

Simulation-based inference (SBI) enables amortized Bayesian inference for
simulators with implicit likelihoods. But when we are primarily interested in
the quality of predictive simulations, or when the model cannot exactly
reproduce the observed data (i.e., is misspecified), targeting the Bayesian
posterior may be overly restrictive. Generalized Bayesian Inference (GBI) aims
to robustify inference for (misspecified) simulator models, replacing the
likelihood-function with a cost function that evaluates the goodness of
parameters relative to data. However, GBI methods generally require running
multiple simulations to estimate the cost function at each parameter value
during inference, making the approach computationally infeasible for even
moderately complex simulators. Here, we propose amortized cost estimation (ACE)
for GBI to address this challenge: We train a neural network to approximate the
cost function, which we define as the expected distance between simulations
produced by a parameter and observed data. The trained network can then be used
with MCMC to infer GBI posteriors for any observation without running
additional simulations. We show that, on several benchmark tasks, ACE
accurately predicts cost and provides predictive simulations that are closer to
synthetic observations than other SBI methods, especially for misspecified
simulators. Finally, we apply ACE to infer parameters of the Hodgkin-Huxley
model given real intracellular recordings from the Allen Cell Types Database.
ACE identifies better data-matching parameters while being an order of
magnitude more simulation-efficient than a standard SBI method. In summary, ACE
combines the strengths of SBI methods and GBI to perform robust and
simulation-amortized inference for scientific simulators.

本研究提出了一种基于神经网络的 ACE 方法，通过代替可能过于严格的贝叶斯后验分布，使用适应数据的成本函数来进行科学模拟器的仿真推断，从而同时减少了计算成本和提高了精度。