Complexity

Volume 2018 (2018), Article ID 3872573, 11 pages

https://doi.org/10.1155/2018/3872573

## Three-Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

School of Information Science and Engineering, Changzhou University, Changzhou 213164, China

Correspondence should be addressed to Bocheng Bao; moc.621@oabnivrem

Received 27 December 2017; Accepted 24 January 2018; Published 28 February 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Bocheng Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

#### 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 Section 3, hidden coexisting asymmetric attractors’ behaviors are demonstrated by bifurcation diagrams, Lyapunov exponents, and dynamical maps, from which numerous types of coexisting asymmetric attractors are easily observed. In addition, a physical implementation circuit is fabricated and breadboard experiments are carried out to confirm the hidden coexisting asymmetric attractors in Section 4. The conclusions are summarized in Section 5.

#### 2. Three-Dimensional Memristive HR Neuron Model

##### 2.1. Brief Reviews on the HR Neuron Model

Through simplifying the classical Hodgkin–Huxley model [1], the two-dimensional Hindmarsh–Rose (HR) neuron model was proposed by Hindmarsh and Rose [2] in 1982, which is described by two first-order ordinary differential equations asin which two variables and are the membrane potential and recovery variable (also called spiking variable), respectively, and a term is the externally applied current. The parameters , , , and are four positive constants, which are often assumed as , , , and , respectively [2, 11–14].

To permit numerous dynamical behaviors, for example, chaotic dynamics, for the membrane potential, an extra third equation was introduced by Hindmarsh and Rose [3] in 1984 to improve the two-dimensional neuron model (1), which is expressed by three first-order ordinary differential equations aswhere the variable is the bursting variable and the constant is the resting potential of the model. The newly added parameters and are two positive constants but is very small. Thus, a new variable , a slowly evolving current, is coupled into the first equation of the two-dimensional model (1) to tune the externally applied current . If the three-dimensional neuron model (2) is in its firing state, the value of increases [24].

##### 2.2. Constructed Memristive HR Neuron Model

Motivated by the above constructing approach of the model (2), a three-dimensional memristive HR neuron model with electromagnetic induction effect is proposed through introducing a flux-controlled ideal memristor into the first equation of the model (1), which can be mathematically modeled bywhere the new variable is the magnetic flux indicating the time integral of the membrane potential . The newly adding term represents the externally applied electromagnetic induction and is the strength of the electromagnetic induction.

It is important to stress that the introduced memristor in (3) is ideal and flux-controlled. According to the definitions of ideal memristor by state-dependent Ohm’s laws between the terminal voltage and terminal current [36, 37], a flux-controlled ideal memristor is thereby given aswhere the memductance can be interpreted as the flux-dependent rate of change of charge. Therefore, the memductance utilized in (3) can be written byin which the coefficient is positive.

To exhibit three characteristic fingerprints of pinched hysteresis loop of the memristor modeled by (4) and (5) [38], a sinusoidal voltage source is connected at the input terminals of the memristor, where and are the amplitude and frequency, respectively. Let . When is maintained unchanged and is assigned as 0.1, 0.2, and 0.5, respectively, the plots are shown in Figure 1(a), while when is fixed and is determined as 3, 4, and 5, respectively, the plots are shown in Figure 1(b). It is seen from Figure 1 that the plots are the hysteresis loops pinched at the origin. The hysteresis loop is pinched regardless of the stimulus amplitudes but shrinks into a linear function at infinite frequency and its lobe area decreases with increasing the frequency. The numerical results in Figure 1 indicate that the memristor modeled by (4) can behave three fingerprints for distinguishing memristors [38].