Complexity

Volume 2018, Article ID 3012743, 13 pages

https://doi.org/10.1155/2018/3012743

## Dynamic Behaviors in Coupled Neuron System with the Excitatory and Inhibitory Autapse under Electromagnetic Induction

^{1}Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China^{2}College of Electronics and Information Engineering, South-Central University for Nationalities, Wuhan 430074, China^{3}School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China

Correspondence should be addressed to Ya Jia; nc.ude.uncc.liam@yaij

Received 4 April 2018; Revised 3 June 2018; Accepted 25 June 2018; Published 26 July 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Ying Xu 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

The induced current produced by electromagnetic induction can adjust the membrane potential of neuron through the feedback of a magnetic flux-controlled memristor. We adopt the numerical simulation method with the aim of investigating the synchronous behavior in the neuronal system that is coupled by chemical and electrical synapses under electromagnetic induction. Within the improved model, the effects of electromagnetic induction on neurons are described with additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The simulation results show that the two coupling modes play an important role in the synchronization of the system. By increasing the chemical synaptic feedback gain, we observe a transition from mixed oscillatory to periodic state at a critical value. In addition, two Hopf bifurcation points are found with the change of the external stimuli, and the state of neuron discharge is influenced by initial values. Furthermore, there is a domain of coupling strength and feedback gain values, in which the two coupled neuron system is synchronized and longer time lag is not conducive to the system synchronization.

#### 1. Introduction

A neural system, which is made up of a large number of neurons, is a complex information network. Different types of discharge patterns can be switched under the control of external stimulation or bifurcation parameter. In order to understand the regulating function of the nervous system, many models of neuronal electrical activity have been proposed. Commonly used models include the FitzHugh-Naguma model [1], Morris-Lecar neuron model [2], Hindmarsh-Rose model [3, 4], Nagumo-Sato neuron model [5], and Wilson-Cowan neuron model [6]. These models that describe neuron dynamics with a set of differential equations are almost derived from the Hodgkin-Huxley [7] model or the simplified version. Some results from biological experiments [8–10] can be explained by theoretical neuron models, such as the Morris-Lecar neuron model. In this model, the membrane potential exhibits quiescent, spiking, or bursting state by changing the external forced current [11]. Neurons do not work in isolation, but they interact to affect the processing of information. There are two forms of synaptic coupling found in the real nervous system, namely, electrical synapse and chemical synapse. The synchronization phenomena are a typical manifestation of the rhythms of group movement; that is, all neurons in the system have a certain connection at the same time or rhythm [12–14]. Bazhenov et al. [15] designed a coupled linear chain of Hindmarsh-Rose model neurons with reciprocal inhibition between neighboring neurons that exhibited synchronous oscillations. Zhang et al. [16] proposed a class of synchronization problems of nonlinear time-delay dynamic networks with a nonuniform impulse effect. Burić et al. [17] studied the synchronization of Hindmarsh-Rose neurons with a time-delayed fast threshold modulation synapse. Xu et al. [18] analyzed the synchronization behavior and mode selection in neural networks under the coupling of chemical or electrical synapses. Yao et al. [19] investigated the influence of coupling strength, time delay, and network topology on synchronization behavior in delay-coupled networks of chaotic pendulums. Gokul and Kapitaniak [20] studied the synchrony of coupling multistable systems which have hidden attractors with each other. In coupled oscillators or coupled neurons, synchronization may occur because of the appropriate coupling effect [21–24]. Interestingly, the stochastic and coherence resonance [25–27] of the nervous system is induced by appropriate noise intensity and external periodic stimulus. The synchronization of the coupling system is an interesting research filed. It is challenging to analyze the dynamic mechanism caused by the variation of the coupling parameters and modes of the system. The synchronization phenomena in Hindmarsh-Rose (HR) neurons that are connected by electrical coupling and chemical coupling, moreover, complete synchronization, phase synchrony, and antisynchrony of neurons are realized [28, 29]. The neural electrical activity has also been widely studied and verified in the circuit [30–35]. For example, Vaidyanathan et al. [36–38] designed electronic circuits to study the feasibility of the 3D novel jerk chaotic system with hyperbolic sinusoidal nonlinearity. Conti and Turchetti [39] performed a circuit to realize approximate identity neural network for the analog synthesis nonlinear dynamical system. Pham et al. [40] proved the existence of chaotic behavior in a three-dimensional autonomous chaotic system with a circular equilibrium by using OrCAD PSpice software and experimental.

It is necessary to study the effects of electromagnetic induction on neuronal cells [41–44]. The changes of membrane potential can induce electromagnetic induction between neurons. As reviewed in [45, 46], the effects of electromagnetic radiation in *Homo sapiens* include electrical activity of neurons, energy metabolism, genomic responses, neurotransmitter balance, blood-brain barrier permeability, cognitive function, sleep, brain tumors, and other encephalopathy. Lu et al. [47–49] investigated the effects of high- and low-frequency signal stimulus on neural activity under electromagnetic radiation. According to Faraday’s law of induction, the magnetic field is a result of fluctuations in the action potential. That is, the distribution of electromagnetic field both inside and outside neurons can be changed by the fluctuation of the membrane potential. Therefore, a new three-variable ML neuron model is established by introducing an additional variable as magnetic flux which adjusts the membrane potential via a memristor [50, 51].

The following study is based on the proposed Morris-Lecar neuron model with consideration of magnetic flux, in which the dynamic characteristics of the neurons are studied by using bifurcation diagrams and time series of the discharge. A preliminary synchronization analysis was conducted in the excitatory and inhibitory neural system. The study revealed that excitatory and inhibitory neurons can be synchronized under the appropriate coupling strength. The synchronization behavior of the system is also affected by the time lag when the coupling strength and the feedback gain are maintained.

#### 2. Model and Scheme

The Morris-Lecar (ML) equations were originally developed as a mathematical model of muscle fiber. For the neuron, the effect of electromagnetic induction should be considered during the discharge process of the membrane potential. The electric activity will change because of the fluctuation of electromagnetic induction and ion concentration in the process of ion exchange. We modify the basic ML model, including the impact of the electromagnetic radiation. The improved ML neuronal model [44] contains three variables, and the dynamic properties are described as follows:
with
where and denote the variables for the membrane potential (mV) and gate channel, respectively. Parameter is the capacitance of the membrane (*μ*F/cm^{2}). The , , and denote the maximum conductance (mS) of calcium ion, potassium ion, and leak ion, respectively. , , and are the reversal potential (mV) corresponding to these channels. and define the value of the opening probability for the calcium ion channel and the potassium ion channel in the steady state, where , , , and are the parameters of the steady system, and defines the rate constant for the opening of potassium ion channel. The parameter is marked as the variation between the fast and the slow scales of neurons.

As described in [52, 53], the variations of the intercellular and extracellular ion concentration can induce electromagnetic induction, which can be expressed by magnetic flux according to Faraday’s law of electromagnetic induction. The induced current produced by electromagnetic induction can adjust the membrane potential by the feedback of the memristor. The memristor in model (1) can be divided into two ways: the charge controlled and the magnetic controlled. For the potassium ion-channel memristor, the second term in the right of (1) can be rewritten as with and , and is the potassium memductance function. The fourth term in the right of (1) can be rewritten as with and , which defines another first-order memristor, and the conductance value of the memristor depends on the input current. The expression of denotes the memory conductance of a magnetic flux-controlled memristor [54], it is used to calculate the effect of feedback regulation on the membrane potential when the magnetic flux is changed, and and are fixed parameters. Therefore, as in [54], the induced current and electromagnetic induction can be described by

The variable represents induction current. The term represents the inhibitory modulation of membrane potential, and it describes the induced current induced by electromagnetic induction. The parameter is the induction coefficient, and its value depends mainly on the medium itself. is the external forcing current. The terms and in the (1) mean the influence of membrane potential on magnetic flux and leakage of magnetic flux, respectively.

For the analysis of the possibility and stability of the synchronized dynamics between two neurons under bidirectional coupling, the dynamic equations are given by where the subscripts and are a pair of coupled ML neurons under electromagnetic radiation. denotes coupling intensity between adjacent neurons.

In order to simulate the chemical synapse feedback of neurons, we shall use the so-called fast threshold modulation scheme proposed by Somers and Kopell [55] and often used by others, for example, [56, 57]. This chemically feedback form, which clearly combines the time lag of the synapse, is provided by the following functions:

The variable parameter is the feedback gain at time with itself connected at time . The symbol indicates the time lag (ms) of the signal propagation. is coupling strength between two neurons. represents the synaptic reversal potential (mV), which depends on the presynaptic neurons and receiver. The chemical coupling is characterized by the difference between the synaptic reversal potential and the synaptic potential. A positive or negative sign of the difference corresponds to an excitatory or inhibitory effect of the synapse. If the synapsis is excitatory, mV, and if the synapsis is inhibitory, mV. The parameter is a synaptic threshold. Considering that the neuron membrane potential value of the improved ML model is between −17 mV and 15 mV, mV is selected to ensure that the spike of the is over the threshold, and the quiescent state of the is less than the threshold. That is, the membrane potential of the presynaptic neuron is more than , and it can play a role in the postsynaptic neuron [58, 59]. is the ratio constant to the start of excitement or inhibition. In this paper, we focus on the collective behavior of the two coupled neuron system driven by the excitatory and inhibitory autapse, and the schematic diagram is shown in Figure 1. Parameters of the improved ML neuronal model are given as = 20 *μ*F, = 120 mV, = −84 mV, = −60 mV, = 4 mS, = 8 mS, = 2 mS, = −1.2 mV, = 18 mV, = 12 mV, = 17.4 mV, = 0. 067, = 0.1, = 0.01, and = −1.