Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis. (English) Zbl 1170.92005
Summary: We review modern nonlinear dynamical methods used in neuroscience and complex data analysis. We start with the general description of nonlinear dynamics, its geometrical (and topological) picture, as well as its extreme case, deterministic chaos, including its most popular models and methods: Lorenz attractor, Lyapunov exponents, and Kolmogorov-Sinai entropy. Then we review the most important nonlinear models in modern computational neuroscience:
(i) Spiking and bursting neurons, including: integrate-and-fire neurons (linear and quadratic, without and with adaptation as well as bursting), complex-valued resonate-and-fire neurons, FitzHugh-Nagumo neuron, Hindmarsh-Rose thalamic neuron, Morris-Lecar neuron, Wilson-Cowan model of interacting neural populations, as well as classical (more general) Hodgkin-Huxley and FitzHugh-Nagumo neural models; (ii) Synchronization in oscillatory neural networks, based on oscillatory phase neurodynamics of the famous Kuramoto network model; and (iii) Neural attractor dynamics, based on Amari’s neural field theory and its application to behavioral and motivational autonomous robot dynamics.
Similarly, we review nonlinear and chaos methods suitable for complex data analysis, including: (i) Basic nonlinear analysis of the heart interbeat time series (with phase plots of delayed data and its approximate entropy calculations); (ii) Lyapunov spectrum and the related fractal and Kaplan-Yorke dimensions; (iii) Other chaotic dimensions and entropies of the complex data structures; and (iv) Medical applications to breast cancer diagnosis and preterm birth analysis. The main applications of all presented tools are in various areas of medical diagnosis. Other applications include Grossberg-Arbib type brain modeling, autonomous robotics, nonlinear, and adaptive control, as well as nonlinear data classification.
(i) Spiking and bursting neurons, including: integrate-and-fire neurons (linear and quadratic, without and with adaptation as well as bursting), complex-valued resonate-and-fire neurons, FitzHugh-Nagumo neuron, Hindmarsh-Rose thalamic neuron, Morris-Lecar neuron, Wilson-Cowan model of interacting neural populations, as well as classical (more general) Hodgkin-Huxley and FitzHugh-Nagumo neural models; (ii) Synchronization in oscillatory neural networks, based on oscillatory phase neurodynamics of the famous Kuramoto network model; and (iii) Neural attractor dynamics, based on Amari’s neural field theory and its application to behavioral and motivational autonomous robot dynamics.
Similarly, we review nonlinear and chaos methods suitable for complex data analysis, including: (i) Basic nonlinear analysis of the heart interbeat time series (with phase plots of delayed data and its approximate entropy calculations); (ii) Lyapunov spectrum and the related fractal and Kaplan-Yorke dimensions; (iii) Other chaotic dimensions and entropies of the complex data structures; and (iv) Medical applications to breast cancer diagnosis and preterm birth analysis. The main applications of all presented tools are in various areas of medical diagnosis. Other applications include Grossberg-Arbib type brain modeling, autonomous robotics, nonlinear, and adaptive control, as well as nonlinear data classification.
MSC:
| 92C20 | Neural biology |
| 92C50 | Medical applications (general) |
| 37N25 | Dynamical systems in biology |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
Software:
TISEANReferences:
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