基于PDE的神经元生长输入延迟补偿
Input Delay Compensation for Neuron Growth by PDE Backstepping
摘要：
神经学研究表明，受损的神经元可以通过软骨素酶ABC（ChABC）等治疗药物恢复功能。这些治疗剂通过操纵损伤的神经元及其细胞间隙以改变微管蛋白浓度来促进轴突伸长。这种基础蛋白是轴突伸长的来源，其空间分布是轴突生长动力学的状态。由于生物过程，这种动态通常包含时间延迟。本文提出了一种基于状态反馈控制的神经网络输入延迟补偿方法，通过调节微管蛋白的浓度来实现轴突的伸长。具有输入延迟的轴突生长动力学被建模为耦合的抛物型扩散-反应-平流偏微分方程（PDE），其边界由与传输PDE相关联的常微分方程（ODE）控制。提出了一种新的反馈控制律，利用反推方法对输入延迟进行补偿。在将互连的PDE-ODE-PDE系统变换为目标系统之后提供增益核。通过对目标系统在空间H1-范数下进行李雅普诺夫稳定性分析，利用范数等价性证明了原误差系统的局部指数稳定性.
英文摘要：
Neurological studies show that injured neurons can regain their functionality with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics promote axon elongation by manipulating the injured neuron and its intercellular space to modify tubulin protein concentration. This fundamental protein is the source of axon elongation, and its spatial distribution is the state of the axon growth dynamics. Such dynamics often contain time delays because of biological processes. This work introduces an input delay compensation with state-feedback control law for axon elongation by regulating tubulin concentration. Axon growth dynamics with input delay is modeled as coupled parabolic diffusion-reaction-advection Partial Differential Equations (PDE) with a boundary governed by Ordinary Differential Equations (ODE), associated with a transport PDE. A novel feedback law is proposed by using backstepping method for input-delay compensation. The gain kernels are provided after transforming the interconnected PDE-ODE-PDE system to a target system. The stability analysis is presented by applying Lyapunov analysis to the target system in the spatial H1-norm, thereby the local exponential stability of the original error system is proved by using norm equivalence.
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