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Received Dec 27, 2017; Accepted Jan 24, 2018
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1. Introduction
In the past three decades, numerous simplified neuron models had been fantastically extended from the classical Hodgkin–Huxley model [1] to reconstruct the main dynamical characteristics of neuronal electrical activities [2–8], among which the two- and three-dimensional Hindmarsh–Rose (HR) neuron models are effective and available for dynamical analysis in electrical activities of biological neurons [9, 10]. In the last few years, a wide variety of the HR neuron models, such as original three-dimensional HR models [10–16], extended or nonlinear feedback coupled HR models [17–20], time delayed HR models [20–22], fractional-order HR models [23, 24], and memristor based HR models under electromagnetic radiations [9, 25–27], have been proposed and further studied by bifurcation analysis methods for understanding the dynamics of electrical activities among neurons [8]. For this reason, bifurcation analysis theory plays an essential role in describing mode transitions between spiking and bursting in the neuronal electrical activities [9–27].
Inspired by the constructing approach of the three-dimensional HR neuron model [3, 24], a novel three-dimensional memristive HR neuron model is presented in this paper, which could be used to better describe complex dynamical characteristics of neuronal electrical activities with electromagnetic induction or further exhibit some undiscovered complex dynamical behaviors in neuronal electrical activities. Interestingly, our proposed memristive HR neuron model has no equilibrium point, which can be classified as a particular dynamical system with hidden oscillating patterns [28–31]. Furthermore, coexisting asymmetric attractors’ behavior can also be observed in such memristive HR neuron model as well, indicating the emergence of bistability dynamics, which has been found in some specified neuron models [32–35]. However, the phenomenon of hidden coexisting asymmetric attractors has not been previously reported for the HR neuron model.
This paper is organized as follows. In Section 2, based on the brief reviews on the HR neuron model, a three-dimensional memristive HR neuron model is presented, upon which hidden coexisting asymmetric attractors are numerically revealed by phase portraits and time series and its bistability dynamics are confirmed by the attraction basins related to the initial values. In...