Dynamical Analysis of the Hindmarsh-Rose Neuron With Time Delays

This brief is mainly concerned with a series of dynamical analyses of the Hindmarsh-Rose (HR) neuron with state-dependent time delays. The dynamical analyses focus on stability, Hopf bifurcation, as well as chaos and chaos control. Through the stability and bifurcation analysis, we determine that in...

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Bibliographic Details
Published in:IEEE transaction on neural networks and learning systems 2017-08, Vol.28 (8), p.1953-1958
Main Authors: Lakshmanan, S., Lim, C. P., Nahavandi, S., Prakash, M., Balasubramaniam, P.
Format: Article
Language:English
Subjects:
Asymptotic properties
Bifurcation
Bursting
Chaos
Chaos theory
Computer simulation
Control stability
Delay effects
Dynamic stability
Exponents
Externality
Feedback control
Hopf bifurcation
Liapunov exponents
Linear matrix inequalities
linear matrix inequality (LMI)
Mathematical analysis
Mathematical model
Mathematical models
Matrix methods
Neurons
Nonlinear feedback
Numerical stability
stability
Stability analysis
Synchronism
Synchronization
time delay
Time lag
Time series
Time synchronization
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Summary:This brief is mainly concerned with a series of dynamical analyses of the Hindmarsh-Rose (HR) neuron with state-dependent time delays. The dynamical analyses focus on stability, Hopf bifurcation, as well as chaos and chaos control. Through the stability and bifurcation analysis, we determine that increasing the external current causes the excitable HR neuron to exhibit periodic or chaotic bursting/spiking behaviors and emit subcritical Hopf bifurcation. Furthermore, by choosing a fixed external current and varying the time delay, the stability of the HR neuron is affected. We analyze the chaotic behaviors of the HR neuron under a fixed external current through time series, bifurcation diagram, Lyapunov exponents, and Lyapunov dimension. We also analyze the synchronization of the chaotic time-delayed HR neuron through nonlinear control. Based on an appropriate Lyapunov-Krasovskii functional with triple integral terms, a nonlinear feedback control scheme is designed to achieve synchronization between the uncontrolled and controlled models. The proposed synchronization criteria are derived in terms of linear matrix inequalities to achieve the global asymptotical stability of the considered error model under the designed control scheme. Finally, numerical simulations pertaining to stability, Hopf bifurcation, periodic, chaotic, and synchronized models are provided to demonstrate the effectiveness of the derived theoretical results.
ISSN:2162-237X
2162-2388
DOI:10.1109/TNNLS.2016.2557845