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Periodic Event-Triggered Boundary Control of Neuron Growth with Actuation at Soma

Cenk Demir, Mamadou Diagne, Miroslav Krstic

TL;DR

The paper addresses control of axon growth modeled as a coupled moving-boundary PDE-ODE system for tubulin concentration and growth cone dynamics. It introduces a periodic event-triggered boundary control (PETC) built on PDE backstepping to emulate a continuous-time boundary controller, with a periodic triggering function that bounds event occurrences and prevents Zeno behavior. The authors prove local exponential convergence in the $L^2$-norm under PETC, derive explicit design of the periodic trigger and sampling period, and validate the approach through numerical simulations showing comparable performance to continuous-time and reduced actuation. The work advances practically feasible stabilization of neuron growth processes, with potential extensions to adaptive identification of unknown parameters using BaLSI techniques.

Abstract

Exploring novel strategies for the regulation of axon growth, we introduce a periodic event-triggered control (PETC) to enhance the practical implementation of the associated PDE backstepping control law. Neurological injuries may impair neuronal function, but therapies like Chondroitinase ABC (ChABC) have shown promise in improving axon elongation by influencing the extracellular matrix. This matrix, composed of extracellular macromolecules and minerals, regulates tubulin protein concentration, potentially aiding in neuronal recovery. The concentration and spatial distribution of tubulin influence axon elongation dynamics. Recent research explores feedback control strategies for this model, leading to the development of an event-triggering control (CETC) approach. In this approach, the control law updates when the monitored triggering condition is met, reducing actuation resource consumption. Through the meticulous redesign of the triggering mechanism, we introduce a periodic event-triggering control (PETC), updating control inputs at specific intervals, but evaluating the event-trigger only periodically, an ideal tool for standard time-sliced actuators like ChABC. PETC is a step forward to the design of practically feasible feedback laws for the neuron growth process. The PETC strategy establishes an upper bound on event triggers between periodic examinations, ensuring convergence and preventing Zeno behavior. Through Lyapunov analysis, we demonstrate the local exponential convergence of the system with the periodic event-triggering mechanism in the $L^2$-norm sense. Numerical examples are presented to confirm the theoretical findings.

Periodic Event-Triggered Boundary Control of Neuron Growth with Actuation at Soma

TL;DR

The paper addresses control of axon growth modeled as a coupled moving-boundary PDE-ODE system for tubulin concentration and growth cone dynamics. It introduces a periodic event-triggered boundary control (PETC) built on PDE backstepping to emulate a continuous-time boundary controller, with a periodic triggering function that bounds event occurrences and prevents Zeno behavior. The authors prove local exponential convergence in the -norm under PETC, derive explicit design of the periodic trigger and sampling period, and validate the approach through numerical simulations showing comparable performance to continuous-time and reduced actuation. The work advances practically feasible stabilization of neuron growth processes, with potential extensions to adaptive identification of unknown parameters using BaLSI techniques.

Abstract

Exploring novel strategies for the regulation of axon growth, we introduce a periodic event-triggered control (PETC) to enhance the practical implementation of the associated PDE backstepping control law. Neurological injuries may impair neuronal function, but therapies like Chondroitinase ABC (ChABC) have shown promise in improving axon elongation by influencing the extracellular matrix. This matrix, composed of extracellular macromolecules and minerals, regulates tubulin protein concentration, potentially aiding in neuronal recovery. The concentration and spatial distribution of tubulin influence axon elongation dynamics. Recent research explores feedback control strategies for this model, leading to the development of an event-triggering control (CETC) approach. In this approach, the control law updates when the monitored triggering condition is met, reducing actuation resource consumption. Through the meticulous redesign of the triggering mechanism, we introduce a periodic event-triggering control (PETC), updating control inputs at specific intervals, but evaluating the event-trigger only periodically, an ideal tool for standard time-sliced actuators like ChABC. PETC is a step forward to the design of practically feasible feedback laws for the neuron growth process. The PETC strategy establishes an upper bound on event triggers between periodic examinations, ensuring convergence and preventing Zeno behavior. Through Lyapunov analysis, we demonstrate the local exponential convergence of the system with the periodic event-triggering mechanism in the -norm sense. Numerical examples are presented to confirm the theoretical findings.
Paper Structure (14 sections, 4 theorems, 65 equations, 3 figures, 1 table)

This paper contains 14 sections, 4 theorems, 65 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tubulin-Driven Axon Growth Modeling and Control
  3. Understanding axon growth
  4. A model with a moving boundary PDE
  5. The steady-state solution and reference error system
  6. Control Law Design
  7. Linearization of the finite-dimensional part of the cascade system
  8. Continuous-time and sampled-data control law
  9. Continuous-time Event Triggered control
  10. Periodic Event Triggering Mechanism
  11. Design of the periodic event triggering function $\Gamma^p(t)$
  12. Local exponential convergence under PETC
  13. Numerical Simulations
  14. Conclusion

Key Result

Theorem 1

demir2023event For the event-triggered mechanism described in eqn:def-Utj-eqn:def-S, the set of event-times $\{t_k^c\}_{k \in \mathbb{N}}$ ensures that the function $\Gamma^c(t) := d(t)^2 - \gamma m(t)$ remains non-positive for all $t \in [t_k^c, t_{k+1}^c)$, where $k \in \mathbb{N}$.

Figures (3)

  • Figure 1: Schematic of neuron and state variables
  • Figure 2: Comparison between periodic-event triggering control input $U_k^p(t)$, continuous time event triggering control input $U_k^c(t)$ and the continuous control law $U(t)$
  • Figure 3: The tubulin concentration governed by \ref{['sys1']}-\ref{['sys5']}, $c(x,t)$, converges to the steady-state tubulin concentration, $c_{\rm eq}(t)$ by about $t=4.5$min for both continuous control input, CETC and PETC. The axon length, $l(t)$, also converges to the desired axon length, $l_s$, by about $t=4$min for all sampling mechanisms.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more