Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis

摘要

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.