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Edited by: Jun Ma, Lanzhou University of Technology, China

Reviewed by: Xia Shi, Beijing University of Posts and Telecommunications, China; Veli Baysal, Bülent Ecevit University, Turkey

*Correspondence: Shenquan Liu

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Electrical activities are ubiquitous neuronal bioelectric phenomena, which have many different modes to encode the expression of biological information, and constitute the whole process of signal propagation between neurons. Therefore, we focus on the electrical activities of neurons, which is also causing widespread concern among neuroscientists. In this paper, we mainly investigate the electrical activities of the Morris-Lecar (M-L) model with electromagnetic radiation or Gaussian white noise, which can restore the authenticity of neurons in realistic neural network. First, we explore dynamical response of the whole system with electromagnetic induction (EMI) and Gaussian white noise. We find that there are slight differences in the discharge behaviors via comparing the response of original system with that of improved system, and electromagnetic induction can transform bursting or spiking state to quiescent state and vice versa. Furthermore, we research bursting transition mode and the corresponding periodic solution mechanism for the isolated neuron model with electromagnetic induction by using one-parameter and bi-parameters bifurcation analysis. Finally, we analyze the effects of Gaussian white noise on the original system and coupled system, which is conducive to understand the actual discharge properties of realistic neurons.

Neural network is composed of a large number of neurons and the connection between neural networks is through signal propagation between neurons such as chemical or electrical signal. Neurodynamics researchers really pay much attention to dynamical properties of electrical activity in neurons or neural networks starting from the establishment of a reliable Hodgkin-Huxley (Hodgkin and Huxley,

With the development of neural dynamics, investigators propose some ways to deeply research mathematical mechanism of neuron model. For example, Izhikevich (

As reported in Lv and Ma (

We use an improved M-L neuron model, which is reported in the previous investigation and it is a real biological neuron model which describes the giant barnacle muscle fiber. As we known, its dynamical behavior is greatly abundant although the model contains only calcium ion channel and potassium ion channel. Two variables are membrane voltage

Where _{Ca}, _{K}, _{L}, _{D}, and _{1}, _{1} are angular frequency and amplitude for forcing currents, _{Ca}, _{K}, _{L}, reversal potential _{Ca}, _{K}, _{L}, the other kinetics parameter ϕ, ε, _{1}, _{2}, _{3}, _{4}. Some parameters of electromagnetic radiation are α, β, _{1}, _{2}. Readers can refer to Morris and Lecar (_{couple} = _{c}(_{1} − _{2}), where _{c} represents the coupling coefficient and _{1(2)} denote the membrane potential of one(another) of neuron.

In fact, we are familiar with the M-L model, but we still have a novel understanding and discovery for the improved M-L model via adjusting electromagnetic induction and phase noise. In this paper, we adopt fourth-order Runge-Kutta algorithm to exhibit numerical solution of the neuronal system with time step

In this section, we discuss that amplitude _{1} is how to adjust electricity activity via changing the range of forcing amplitude, and we also examine firing pattern of improved M-L neuron model by adjusting magnetic flux parameter

As shown in Figure _{1} = 1 and the firing occurs when we increase _{1} to 3. We find that the number of spiking is gradually increased when we continue to increase amplitude from 3 to 5. It is described that the amplitude of phase noise makes a positive response to the system although the range of amplitude is small. Certainly, this process is reversible. That is to say, we can better adjust the spiking of neurons by regulating the amplitude so that neuronal model is more reasonable in the simulation of realistic neural network.

Time series of membrane potential under different amplitudes. _{1} = 1, _{1} = 3, _{1} = 5, _{1} = 10, the noise intensity is selected as _{1} = 0.5.

Indeed, multiple patterns of electrical activity are detected in the improved neuron model by modifying the feedback coefficient _{1}) plane and presents a detailed suprathreshold change.

Time series of membrane potential under different feedback coefficient. _{1} = 6, ω_{1} = 0.5.

Bi-parameter bifurcation diagram of suprathreshold spiking. The number of suprathreshold spiking is presented at the right sides by the colorful belt and 0 indicates the relatively resting state.

The phase noise is induced by differential equations with Gaussian white noise and it can be grasped by noise intensity

Time series of membrane potential under different noise intensity _{1} = 6, ϕ = 0.22, ω_{1} = 0.5.

Coefficient variability of ISIs series of membrane potential. The abscissa is noise intensity _{1} = 6, ϕ = 0.22, ω_{1} = 0.5.

The above discussion is based on adding noise and electromagnetic field at the same time and then changing amplitude, feedback coefficient or noise intensity to obtain some firing behaviors of dynamical system. Next, we will discern the oscillating mode only adding electromagnetic radiation or adding Gaussian white noise. The result in Figure

Interspike intervals for time series of membrane potential with the increasing of feedback coefficient

We calculate bi-parameter bifurcation diagram and show it in Figure _{K} whose value is taken from −96 to −80, and the ordinate denotes the feedback coefficient _{Ca}) plane, where the abscissa indicates the maximal conductance _{Ca} whose value is taken from 3.3 to 5.3, and the ordinate represents the feedback coefficient

Spiking-counting diagram as the change of bi-parameter. The number of spiking per bursting is exhibited at the right sides by the colorful belt. And no Gaussian white noise is considered. _{K}) plane; the number 0 indicates quiescent condition and the number 1 represents tonic spiking whilst the number 2–45 denotes regular bursting; _{Ca}) plane; similarly, the number 0 indicates quiescent condition and the number 1 represents tonic spiking; the number 2–69 denotes regular bursting whilst the number 70 indicates chaotic states.

By comparing Figure _{K} at the same time, we will get a continuous spike-adding mode. But if we transform the maximal conductance _{Ca}, we will see that electrical activity will return to period bursting via chaotic bursting. The principle is hidden behind a large number of patterns, and it can be well discerned by simulating the neuronal system.

In this section, we will discuss dynamical response of an isolated neuron and two coupled neurons. These two systems are exposed to Gaussian white noise, and they have many different responses to the noise system. In addition, we examine how the noise affects coupled neuronal system and compare the discharge activity of coupling neuronal system with that of original coupling system without noise. Furthermore, we may find some new phenomena by comparing electrical mode of isolated neurons with that of coupled neurons under noise, and we also present a new perspective to explore neurons responding to noise. They are analyzed as follows.

It is inevitable that many realistic neuron system will be exposed to noisy environment and we find that the effect has two-sided. Therefore, we will explore neuronal system by adding the noise to neuron model without adding electromagnetic radiation. Figure

Time series of membrane potential under different reversal potential _{K}. _{K} = −96, _{K} = −92, _{K} = −88, _{K} = −84, and no noise is introduced.

Time sequences of membrane potential under different reversal potential _{K}. _{K} = −96, _{K} = −92, _{K} = −88, _{K} = −84, and Gaussian white noise is considered.

In fact, we have explored multiple modes of discharge activity in coupled neuron system by changing reversal potential and adding Gaussian white noise to system and it has more patterns than in an isolated neuron. Therefore, we are interested in the coupling system with noise. Figure

Time series of membrane potential of coupled neurons under different reversal potential _{K}. _{K} = −92, _{K} = −88, _{K} = −84, _{K} = −76, and adding Gaussian white noise.

Time series of membrane potential of coupled neurons under different reversal potential _{K}. _{K} = −92, _{K} = −88, _{K} = −84, _{K} = −76, and Gaussian white noise is considered.

Tonic spiking and bursting are effective coding of signals between neurons and abundant discharge activity patterns are accompanied by complex signal propagation. The effects of electromagnetic radiation and noise on discharge activity of system are two-sided and it has been investigated by some researchers and their co-workers (Lv et al.,

The idea of this article is proposed by SL and the specific writting is done by FZ.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11172103 and 11572127.

^{3}

Fixed parameter values are used in the calculation.

Reversal potentials (mv) | _{Ca} = 120, _{K} = −84, _{L} = −60 |

Maximal conductance (ms/^{2}) |
_{Ca} = 1.0, _{K} = 8.0, _{L} = 2.0 |

Gating variable parameters (mv) | _{1} = −1.2, _{2} = 18, _{3} = 12, _{4} = 17.4, _{0} = −26 |

Other dynamical parameters | ^{2}, ϕ = 0.23, ε = 0.001 |

Electromagnetic induction parameters | α = 0.1, β = 0.02, _{1} = 0.9, |

_{2} = 0.5, ω_{1} = 0.5, _{1} = 6, |