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Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis

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Abstract

In this paper, 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 neuron (linear and quadratic, without and with adaptation as well as bursting), complex-valued resonate-and-fire neuron, 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 application of all presented tools is 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.

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Authors and Affiliations

  1. School of Electrical and Information Engineering, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes Campus, Mawson Lakes, South Australia, 5095, Australia

    Tijana Ivancevic, Lakhmi Jain, John Pattison & Alex Hariz

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  1. Tijana Ivancevic
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  2. Lakhmi Jain
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Correspondence to Tijana Ivancevic.

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Ivancevic, T., Jain, L., Pattison, J. et al. Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis. Nonlinear Dyn 56, 23–44 (2009). https://doi.org/10.1007/s11071-008-9376-9

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