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Dynamical Analysis of the Hindmarsh-Rose Neuron With Time Delays.

S Lakshmanan, C P Lim, S Nahavandi

    IEEE Transactions on Neural Networks and Learning Systems
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    This study analyzes the Hindmarsh-Rose neuron model with time delays, revealing how external current and delays influence its stability and lead to complex behaviors like chaos. It also demonstrates effective chaos control for synchronization.

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    Area of Science:

    • Computational Neuroscience
    • Nonlinear Dynamics
    • Dynamical Systems Theory

    Background:

    • The Hindmarsh-Rose (HR) neuron model is a fundamental mathematical model for simulating the electrical activity of neurons.
    • Time delays in neural systems can significantly alter neuronal dynamics, leading to complex behaviors such as bursting and chaos.
    • Understanding these dynamics is crucial for comprehending neural network function and dysfunction.

    Purpose of the Study:

    • To conduct a comprehensive dynamical analysis of the Hindmarsh-Rose neuron model incorporating state-dependent time delays.
    • To investigate the effects of external current and time delays on neuronal stability, bifurcation, and chaotic behavior.
    • To develop and validate a nonlinear control strategy for achieving synchronization in chaotic time-delayed HR neuron models.

    Main Methods:

    • Stability and Hopf bifurcation analysis were performed to understand the transition to different firing patterns.
    • Time series, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions were used to characterize chaotic dynamics.
    • A nonlinear feedback control scheme based on a Lyapunov-Krasovskii functional was designed for chaos control and synchronization.
    • Linear matrix inequalities were employed to derive synchronization criteria ensuring global asymptotic stability.

    Main Results:

    • Increasing external current induces periodic or chaotic bursting/spiking behaviors and subcritical Hopf bifurcation in the HR neuron.
    • Varying time delays affects the stability of the HR neuron under a fixed external current.
    • Chaotic behaviors were analyzed and quantified using standard dynamical system tools.
    • The proposed nonlinear control scheme effectively achieved synchronization between uncontrolled and controlled chaotic HR neuron models.

    Conclusions:

    • The study provides a thorough dynamical characterization of the time-delayed Hindmarsh-Rose neuron model.
    • External current and time delays are critical parameters influencing neuronal excitability and dynamics.
    • The developed nonlinear control method is effective for synchronizing chaotic neural systems, offering potential applications in understanding and treating neurological disorders.