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Event-Triggered Control of Neuron Growth with Actuation at Soma

Cenk Demir, Shumon Koga, Miroslav Krstic

Abstract

We introduce a dynamic event-triggering mechanism for regulating the axonal growth of a neuron. We apply boundary actuation at the soma (the part of a neuron that contains the nucleus) and regulate the dynamics of tubulin concentration and axon length. The control law is formulated by applying a Zero-Order Hold (ZOH) to a continuous-time controller which guides the axon to reach the desired length. The proposed dynamic event-triggering mechanism determines the specific time instants at which control inputs are sampled from the continuous-time control law. We establish the existence of a minimum dwell-time between two triggering times that ensures avoidance of Zeno behavior. Through employing the Lyapunov analysis with PDE backstepping, we prove the local stability of the closed-loop system in $L_2$-norm, initially for the target system, and subsequently for the original system. The effectiveness of the proposed method is showcased through numerical simulations.

Event-Triggered Control of Neuron Growth with Actuation at Soma

Abstract

We introduce a dynamic event-triggering mechanism for regulating the axonal growth of a neuron. We apply boundary actuation at the soma (the part of a neuron that contains the nucleus) and regulate the dynamics of tubulin concentration and axon length. The control law is formulated by applying a Zero-Order Hold (ZOH) to a continuous-time controller which guides the axon to reach the desired length. The proposed dynamic event-triggering mechanism determines the specific time instants at which control inputs are sampled from the continuous-time control law. We establish the existence of a minimum dwell-time between two triggering times that ensures avoidance of Zeno behavior. Through employing the Lyapunov analysis with PDE backstepping, we prove the local stability of the closed-loop system in -norm, initially for the target system, and subsequently for the original system. The effectiveness of the proposed method is showcased through numerical simulations.
Paper Structure (14 sections, 7 theorems, 45 equations, 3 figures, 1 table)

This paper contains 14 sections, 7 theorems, 45 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modeling of Axon Growth
  3. Axon growth model by a moving boundary PDE
  4. Steady-state solution
  5. Reference error system
  6. Continous-time and Sample-based Control Design
  7. Control law
  8. Sample-based control law
  9. Event-triggered based boundary control
  10. Main Results
  11. Avoidance of Zeno Behavior
  12. Stability Analysis
  13. Numerical Simulations
  14. Conclusion

Key Result

Lemma 1

Under the definition of the state feedback event-triggered boundary control, it holds that $d^2(t)\leq -\gamma m(t)$ and $m(t)>0$ for $t\in [0,F)$, where $F=\sup(I)$.

Figures (3)

  • Figure 1: Schematic of neuron and state variables
  • Figure 2: The closed-loop response of the continuous-time and event-triggered control law for $l_s=12\mu m$
  • Figure 3: The closed-loop response of the designed full-state feedback control system for continuous-time and event-triggered control law.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • ...and 4 more