Neuron growth control and estimation by PDE backstepping

Neurological disorders, such as spinal cord injuries, Parkinson's disease and Alzheimer's disease, limit the functionality of neurons. A recent medical therapy, Chondroitinase ABC (ChABC), aims to cure these conditions by restoring neuron functionality through axon growth for damaged neurons by manipulating the extracellular matrix (ECM), the network of macromolecules and minerals that surrounds neurons and controls their activity, thereby neurons produce tubulin proteins which elongate the axon. This process is modeled as a Partial Differential Equation (PDE), which represents the behavior of the tubulin concentration along the axon, with a moving boundary governed by Ordinary Differential Equations (ODE) consisting of the dynamics of the axon length and tubulin concentration in the growth cone. This paper proposes nonlinear design methods for a state feedback control law, an observer, and an output feedback control law for a one-dimensional model of axonal elongation. First, we introduce a novel backstepping method to obtain the target system and the associated state feedback control law. Using a similar backstepping method, we design a nonlinear observer and prove the local exponential stability of the observer-error system in the spatial H1-norm. The resulting output-feedback control law ensures the local exponential stability of the closed-loop system composed of the estimation error and the reference error states. The performance of the designed control and estimation methods is demonstrated in numerical simulation for neuron elongation by up to three orders of magnitude. Robustness to parameter changes up to 40% relative to the design/analysis model is demonstrated.