Complexity

Volume 2019, Article ID 4835379, 10 pages

https://doi.org/10.1155/2019/4835379

## Bursting and Synchronization of Coupled Neurons under Electromagnetic Radiation

^{1}School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China^{2}School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Correspondence should be addressed to Xiaoyu Hu; moc.361@loocuyoaixuh

Received 12 May 2019; Revised 11 July 2019; Accepted 6 August 2019; Published 4 December 2019

Guest Editor: Lazaros Moysis

Copyright © 2019 Xiaoyu Hu and Chongxin Liu. 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

Bursting is an important firing activity of neurons, which is caused by a slow process that modulates fast spiking activity. Based on the original second-order Morris-Lecar neuron model, an improved third-order Morris-Lecar neuron model can produce bursting activity is proposed, in which the effect of electromagnetic radiation is considered as a slow process and the original equation of Morris-Lecar neuron model as a fast process. Extensive numerical simulation results show that the improved neuron model can produce different types of bursting, and bursting activity shows a deep dependence on system parameters and electromagnetic radiation parameters. In addition, synchronization transitions of identical as well as no-identical coupled third-order Morris-Lecar neurons are studied, the results show that identical coupled neurons experience a complex synchronization process and reach complete synchronization finally with the increase of coupling intensity. For no-identical coupled neurons, only anti-phase synchronization and in-phase synchronization can be reached. The studies of bursting activity of single neuron and synchronization transition of coupled neurons have important guiding significance for further understanding the information processing of neurons and collective behaviors in neuronal network under electromagnetic radiation environment.

#### 1. Introduction

The biological or human neural system is usually composed of millions of neurons, which can generate, transmit, receive, and process information by firing various types of electrical activities. Since the pioneering work of Hodgkin and Huxley [1], many models have been proposed for modelling and simulating the electrical activities of a neuron [2–9]. For a neuron, spiking and bursting are two major categories, and they may be periodic motion or chaotic motion. Indeed, bursting is considered as neuron activity alternates between a quiescent state and repetitive spiking, and it is a dynamical consequence of fast/slow dynamics. Bursting is an important firing pattern, and it has been confirmed that neurons in different regions of brain produce bursting activities [10].

Neurons are sensitivity to many external factors, and electrical activities of a neuron and collective behaviours in neuronal network will be changed under certain conditions. Time delay [11–17], noise [18, 19], and network topology [20–23] are common factors being considered to investigate firing behaviours of neuron and collective behaviours in neuronal network. It is worth noting that electromagnetic radiation is another one cannot be ignored. With the development of modern industry, wide utilizations of electric equipment make neural system are exposed to an environment full of electromagnetic radiation, which has a great influence on the dynamics of a single neuron and network of neurons. In Ref. [24], Wang et al. suggested that the strong external electromagnetic field facilitates the neuron firing action potentials and enhances the mean firing rate of the network, but disrupts the synchronicity of the activities of the neural network. In Ref. [25], Li et al. developed a mathematical model to describe the effect of electromagnetic radiation, the results show that electrical activities of a single neuron can be suppressed by electromagnetic radiation, and spatiotemporal patterns in neuronal network are also suppressed from the stable propagating wave state to a homogeneous resting state. Rebertson et al. [26] argued that low-frequency pulsed electromagnetic field exposure can alter neuroprocessing in humans. In Ref. [27], a small Hopfield neural network with the electromagnetic radiation being considered is constructed, in which the previous steady neural network can present abundant chaotic dynamics, and hidden attractors can be observed.

Memory is a natural characteristic of neuron, and it has been considered in studies from neuron models to collective behaviours in neuronal network recently. For example, references [28–30] have proved that the ionic channels of neuron models, e.g., Hodgkin–Huxley and Morris-Lecar neuron model, have memory effect and they can be substituted by first-order or second-order memristors. Moreover, memristive relation is also used to stress the memory effect in some memristor-based neural network [31–33]. Indeed, memristor is an effective element to characteristic the memory effect in neuron and network of neurons. From this point of view, the effect of electromagnetic radiation on neuron can be considered as a variation of magnetic flux, and the flux-controlled memristor is available to represent the memory effect of magnetic flux. As a result, Ma et al. [34–36] proposed several models to describe the effect of electromagnetic radiation on the electrical activities of neuron by using magnetic flux, in which a memristor-like feedback is employed to realize coupling between magnetic flux and membrane potential. It is found that multiple modes of spiking activities can be observed. Moreover, synchronization, noise effect, and spatiotemporal dynamics in neuron and neural networks under electromagnetic radiation were also investigated [37–41]. The effect of electromagnetic radiation can be described by time-varying magnetic flux, the coupling of electromagnetic field between neurons can be described by exchange of magnetic flux as well, which results in another effective way for coupling between neurons, i.e., field coupling. In Refs. [42–44], field coupling rather than synaptic coupling is considered as a coupling mode between neurons and neural networks, it is found that multiple modes of synchronization can be observed from coupled neurons or neural networks.

In this paper, we propose an improved Morris-Lecar neuron model with electromagnetic radiation being considered, in which the fluctuation of electromagnetic radiation is described by using magnetic flux and considered as a slow subsystem. Unlike previous models, multiple modes of bursting activities are observed. Furthermore, synchronization transitions in coupled identical bursting neurons as well as no-identical bursting neurons are studied. The organization of this paper is as follows. In Section 2, the model setting and description is introduced. In Section 3, numerical results are discussed and analysed carefully. Section 4 summarizes and concludes this paper.

#### 2. Model Setting and Description

Bursting activities are results of fast/slow dynamics, and they cannot be observed in original two-dimensional Morris-Lecar neuron model under constant external forcing current. Researchers [5, 45–47] have explored several improved Morris-Lecar neuron model, in which external forcing current is considered as a varied state variable with a very slow rate, and several types of bursting are obtained. Inspired by Refs. [35, 48–50], electromagnetic radiation is considered when improved Morris-Lecar neuron model is constructed, in which the effect of electromagnetic radiation is regarded as a slow subsystem and two-dimensional Morris-Lecar neuron model as a fast subsystem. As a result, the improved Morris-Lecar neuron model is described as follows.

where

In this model, represents membrane potential, represents recovery variable, is magnetic flux which is a slower variable than and . ,, and are steady state potentials for calcium, potassium, and leak ion channels, respectively. and define the stable values of opening probability for calcium and potassium, where , , , and are parameters of steady states. is external forcing current and . The term defines the feedback current on membrane potential when magnet flux is changed in media, and is the feedback gain. is a factor which describes the contribution of varied magnetic flux on the formation of membrane potential. is considered as leakage magnet flux, and is chosen as a constant value 0.2. The rest system parameter are selected as , , , , , , , , , , , and .

#### 3. Numerical Results and Discussion

##### 3.1. Bursting in Improved ML Neuron Model under Electromagnetic Radiation

In this section, fourth order Runge–Kutta algorithm is used with time step . The initial values are set as . At first, sampled time series for membrane potential are detected with different intensity of external forcing current, and system parameters and angular frequency are selected as , , , and . The results are shown in Figure 1.