Neuron Growth Control by PDE Backstepping: Axon Length Regulation by Tubulin Flux Actuation in Soma

TL;DR

This paper introduces a novel boundary control method using PDE backstepping to regulate axon length by controlling tubulin flux, ensuring local exponential stability of neuron growth dynamics.

Contribution

It develops a new backstepping control approach for coupled PDE-ODE systems with moving boundaries in neuron growth models.

Findings

Proposed a PDE backstepping control law for axon length regulation.

Proved local exponential stability of the controlled system.

Validated the control method through Lyapunov analysis.

Abstract

In this work, stabilization of an axonal growth in a neuron associated with the dynamics of tubulin concentration is proposed by designing a boundary control. The dynamics are given by a parabolic Partial Differential Equation (PDE) of the tubulin concentration, with a spatial domain of the axon's length governed by an Ordinary Differential Equation (ODE) coupled with the tubulin concentration in the growth cone. We propose a novel backstepping method for the coupled PDE-ODE dynamics with a moving boundary, and design a control law for the tubulin concentration flux in the soma. Through employing the Lyapunov analysis to a nonlinear target system, we prove a local exponential stability of the closed-loop system under the proposed control law in the spatial $H_1$-norm.