Input Delay Compensation for Neuron Growth by PDE Backstepping

TL;DR

This paper develops a PDE backstepping control method to compensate for input delays in neuron growth models, enhancing axon elongation regulation through tubulin concentration control.

Contribution

It introduces a novel backstepping-based feedback law for PDE-ODE coupled systems with input delay, specifically applied to neuron axon growth dynamics.

Findings

Proposes a PDE backstepping control law for delay compensation.

Proves local exponential stability of the controlled system.

Provides explicit gain kernels for the control law.

Abstract

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.