Filippov Hindmarsh-Rose Neuronal Model With Threshold Policy Control

Area of Science:

• Computational Neuroscience
• Nonlinear Dynamics
• Control Theory

Background:

• The Hindmarsh-Rose (HR) model is a foundational mathematical model for neuronal activity.
• Understanding neuronal firing patterns and their control is crucial in neuroscience.
• Existing models may not fully capture complex dynamics like sliding modes.

Purpose of the Study:

• To propose a Filippov system for the HR neuronal model with threshold policy control.
• To analyze the existence, stability, and sliding dynamics of equilibria in the 2-D and 3-D HR models.
• To investigate sliding bifurcations and their impact on neuronal activity generation.

Main Methods:

• Formulation of a Filippov system for the HR model using membrane potential as the threshold.
• Analysis of equilibria and stability for the 2-D HR subsystems.
• Study of sliding dynamics, including segments, regions, and equilibria.
• Investigation of sliding bifurcation sets and phenomena.
• Phase diagram analysis of the 3-D HR Filippov system.

Main Results:

• Established existence and stability of equilibria for 2-D HR subsystems.
• Characterized sliding dynamics, including bistable equilibria and various sliding bifurcations (boundary-node, pseudosaddle-node).
• Observed emergence and disappearance of limit cycles on the sliding line.
• Generated sliding spiking and sliding bursting in the 3-D HR model via phase diagram analysis.

Conclusions:

• The proposed Filippov system effectively models complex neuronal dynamics in the HR model.
• Sliding bifurcations play a significant role in generating diverse spiking and bursting patterns.
• This framework offers insights into the control and generation of neuronal electrical activity.