<p>This paper presents an optimization and extension program to address limitations in the FitzHugh-Nagumo (FHN) model. The traditional fixed nonlinear term is replaced with a more flexible formulation with an improved inverse <i>N</i>-shaped nonlinear reaction term. The proportional-integral-derivative (PID) algorithm with multivariate characteristics is designed and incorporated into the improved FHN model. The spatiotemporal excited state thresholds are derived for the uncontrolled and controlled models, respectively. The effects and sensitivities of the controller and nonlinear parameters on spatiotemporal dynamics are systematically examined. The stability and direction of spatiotemporal bifurcating periodic solutions are explored. Numerical simulations reveal that the theoretical results established can effectively predict the spatiotemporal bifurcation thresholds of the improved FHN model. The PID controller with multivariate characteristics is able to realize the precise regulation of the spatiotemporal excited state. Furthermore, compared to the only two states observed in the uncontrolled diffusive FHN model (the resting state and the completely excited state), the PID controller with multivariate characteristics enriches the model’s dynamical behaviors, including the resting state, the sparsely excited pulse state, the densely excited pulse state, the sparsely excited clustered state, the densely excited clustered state, and the completely excited state. The PID controller enables precise regulation of transitions between the resting and excited states. In particular, the proportional gain effectively modulates the excitation period of the controlled model while preserving its inherent excitation waveforms. Results confirm the controller’s effectiveness and underscore the advantages of the proposed control algorithm in optimizing dynamic performance.</p>

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PID algorithm with multivariate characteristics: Neuronal spatiotemporal dynamics analysis and design

  • Yunxiang Lu,
  • Min Xiao,
  • Yonghui Sun,
  • Zhengxin Wang,
  • Leszek Rutkowski,
  • Jinde Cao,
  • Tingwen Huang

摘要

This paper presents an optimization and extension program to address limitations in the FitzHugh-Nagumo (FHN) model. The traditional fixed nonlinear term is replaced with a more flexible formulation with an improved inverse N-shaped nonlinear reaction term. The proportional-integral-derivative (PID) algorithm with multivariate characteristics is designed and incorporated into the improved FHN model. The spatiotemporal excited state thresholds are derived for the uncontrolled and controlled models, respectively. The effects and sensitivities of the controller and nonlinear parameters on spatiotemporal dynamics are systematically examined. The stability and direction of spatiotemporal bifurcating periodic solutions are explored. Numerical simulations reveal that the theoretical results established can effectively predict the spatiotemporal bifurcation thresholds of the improved FHN model. The PID controller with multivariate characteristics is able to realize the precise regulation of the spatiotemporal excited state. Furthermore, compared to the only two states observed in the uncontrolled diffusive FHN model (the resting state and the completely excited state), the PID controller with multivariate characteristics enriches the model’s dynamical behaviors, including the resting state, the sparsely excited pulse state, the densely excited pulse state, the sparsely excited clustered state, the densely excited clustered state, and the completely excited state. The PID controller enables precise regulation of transitions between the resting and excited states. In particular, the proportional gain effectively modulates the excitation period of the controlled model while preserving its inherent excitation waveforms. Results confirm the controller’s effectiveness and underscore the advantages of the proposed control algorithm in optimizing dynamic performance.