As the important components of neuronal system, neurons have a pivotal role in regulating the dynamical behaviors of neuronal system. Therefore, the complex behaviors of neurons have been investigated extensively (Ozer and Ekmekci, 2005; Postnov et al., 2009; Volman et al., 2011, 2012; Buschman et al., 2012; Lv et al., 2016; Wang et al., 2017). For this reason, some neuron models have been established and analyzed theoretically (Hodgkin and Huxley, 1952; Morris and Lecar, 1981; Hindmarsh and Rose, 1982; Izhikevich, 2004; Herz et al., 2006; Ibarz et al., 2011), some of which are beneficial for understanding the mode transition in electric activities of the neuron. For example, Hodgkin–Huxley (HH) model (Hodgkin and Huxley, 1952) and Morris–Lecar (ML) model (Morris and Lecar, 1981) are used to depict the effect of ion channels on the membrane potential of neurons. A three variable Hindmarsh–Rose (HR) neuron model is obtained (Hindmarsh and Rose, 1982), which is simplified from HH neuron model and is used to reproduce the main properties of neuronal activities. It is also effective for bifurcation analysis (Pinto et al., 2000; Storace et al., 2008; Gu et al., 2014).

To our knowledge, as a mathematical neuron model, the three-variable HR neuron model could not describe the effect of ion channel. According to this, some researchers proposed four-variable HR neuron models (Moujahid et al., 2011; Rech, 2012), which can better describe the dynamical behaviors of neurons and can be verified by experimental results (Gu and Pan, 2015). Indeed, because the neuron in neuronal system is in a complex situation, the electrical activity of neurons is too complex and more factors should be considered. For example, according to the Faraday's law of induction, the change of action potentials in neurons can generate magnet field in the media, while the magnet field could affect the electrical activity of neurons as a result of feedback effect. It means that the distribution of electromagnetic field inner and external of neurons could be altered owing to the change of membrane potentials of neurons. Therefore, the magnetic flux across membrane should be considered when exploring the dynamical behaviors of neurons. To probe the effect of electromagnetic induction on membrane potential of neurons, Lv et al. put forward a four-variable neuron model by introducing a new variable considering the magnetic flux and discussed the electric activities of it (Lv et al., 2016).

It is often thought that neuronal system can be with good memory to keep normal activities, while time-delay is often used to describe the memory effect. As we all know, magnetic field or magnet flux storage may be associated with the memory effect. In fact, when signals are propagated between neurons even within one neuron, there often exists time-delay. But, until now, the effect of time-delay on the electric activities of neurons has been seldom explored. To better describe the behaviors of neurons, more possible factors should be considered in investigating the electric activities of neurons.

Based on above, a time-delay four-variable neuron model under magnetic flow is proposed and to be investigated via numerical simulations. Other parts of this paper are arranged as follows. In section Model Description, the neuron model to be discussed is given. In section Numerical Simulation Results and Discussions, the electric activities of the addressed neuron model are discussed with the change of external forcing current and time-delay under the effect of magnetic flow. Finally, conclusions are drawn in section Conclusions.

Hindmarsh–Rose neuron model is a three-variable model (Hindmarsh and Rose, 1982), which can be described as

where x is membrane potential, y is recovery variable for slow current and z is the adaption current. I_ext is the external forcing current. a, b, c, d, r, S, k are system parameters and k is used to adjust the resting state.

Based on HR neuron model, by introducing a variable for magnetic flow, a four-variable HR neuron model (Gu and Pan, 2015) is designed to describe the effect of electromagnetic induction on neuronal activities, which can be rewritten as

where ρ(w) = (α + 3βw²) is the memory conductance of a magnetic flux-controlled memristor and is used to describe the coupling between magnetic flux as well as membrane potential of neurons. The fourth variable w can describe the magnetic flux across membrane. k₁, k₂, and k₃ represents the interaction parameters between the cell potential and the magnetic flux. Some dynamical behaviors of the membrane potential in this model have been discussed (Gu and Pan, 2015). Model (2) could be helpful for further investigating the effect of electromagnetic radiation on biological tissue. On the other hand, HR system is a slow-fast system. When the slow oscillation of z drives the fast subsystem (x, y) through periods of oscillatory and quiescent behavior, there may be a time lag. Therefore, it is necessary to investigate the dynamical behaviors of the neuron model (2) with time-delay, which can be described as

where τ is the time-delay.

In this section, considering the external forcing current and time-delay, which play an important role in neural system, the dynamical behaviors of neuron system (3) is investigated with the change of external forcing current I_ext or the time-delay τ. Numerical simulations are given to analyze the relation between the dynamics and the external forcing current I_ext or the time-delay τ. In the simulations, the system parameters are taken as a = 1.0, b = 3.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, k₁ = 0.01, k₂ = 1.0, k₃ = 6.2, α = 0.4, β = 0.01 and the initial values of system (3) is chosen as (x(0), y(0), z(0), w(0)) = (0.5, 0.2, 0.8, 0.1). The time step is selected as h = 0.01 and the time series of membrane potential x are calculated via fourth-order Runge-Kutta for different parameters.

At first, the effect of external forcing current on the dynamical behaviors of neuron model (3) is investigated. For this end, the time series for membrane potential are calculated for different external forcing currents I_ext while the time-delay is taken as τ = 1, which are drawn in Figure 1. Figure 1 indicates that neuron system (3) possesses various discharge modes. For small value of external forcing current, for example I_ext = 0.01, 1.2, the neuron system (3) gradually tend to quiescent (see Figures 1A,B). But with the increasing of external forcing current, the electrical activities in neuron system (3) can show different modes, for example, when I_ext = 1.5, 1.9, 2.3, 2.7, 3.0, 3.3, the time series of system (3) appears period-1 bursting, period-2 bursting, period-3 bursting, period-4 bursting, period-5 bursting and chaotic bursting, respectively (see Figures 1C–H), while I_ext = 3.5, 4.5, neuron system (3) behaves spiking and period-1 bursting, respectively (see Figures 1I,J). From Figure 1, we can obtain that neuron system (3) shows various electrical activities with the change of external forcing current I_ext, such as stable state, period bursting with different period, chaotic bursting, spiking.

To further depict the effect of external forcing current on the electrical activities of membrane potential in neuron system (3), the external forcing current I_ext is taken as the bifurcation parameter and the bifurcation diagram of x verses I_ext is plotted in Figure 2. Figure 2 represents the Poncare section of membrane potential x at y = 0 with external forcing current I_ext. From Figure 2, it is also known that the electrical activity in neuron system (3) varies with the change of external forcing current and shows diversity, which can also confirm that multiple modes in electrical activities of neuron system (3) can be arrived by selecting appropriate external forcing current.

Secondly, the effect of time-delay on the dynamics of neuron system (3) is studied. The system parameters are taken as above. The external forcing current is fixed as I_ext = 1.2 and time-delay τ is chosen as different values. The time series of the membrane potential in neuron system (3) are calculated and given in Figure 3. From Figures 1B, 3, it can be obtained that, neuron system (3) tends to be stable when I_ext = 1.2 and time-delay τ = 1, but with the increasing of time-delayτ, even if the external forcing current I_ext keeps as a constant, the dynamical behaviors of membrane potential in neuron system (3) gradually shows multi-period bursting. Furthermore, Figures 3B–G demonstrate that, with the increase of τ, the time of quiescent between two adjacent bursting discharges becomes longer and longer, while the spiking frequency in one bursting discharge gets bigger and bigger.

To further explore the rhythm of electrical activity, I_ext = 1.9 and I_ext = 3.2 are chosen respectively. The time series of membrane potential in neuron system (3) are calculated and depicted in Figures 4, 5, respectively. Figure 1D means that neuron system (3) shows dynamics of 2-period bursting when I_ext = 1.9 and time-delay τ = 1, while Figures 4A–G indicates that, if I_ext is fixed at 1.9, neuron system (3) can be provided with multi-period bursting for different values of time-delay τ, such as period-3 bursting for τ = 4, period-4 bursting for τ = 12, period-5 bursting for τ = 17, period-6 bursting for τ = 25, period-8 bursting for τ = 35, period-12 bursting for τ = 50, period-19 bursting for τ = 75. From these phenomena, it is obvious to see that, with increasing of time-delay τ, bursting period becomes longer and longer as well as the spiking frequency gets larger and larger. It means that, when neuron system (3) is in periodic state, by selecting the time-delay and external forcing current, certain bursting frequency can be obtained.

Similarly, from Figures 1H, 5A–F, it is known that, when neuron system (3) is in chaotic, the dynamical behaviors of it can also show diversity with the change of time-delay for fixed value of external forcing current, such as period-5 bursting for τ = 2, period-6 bursting for τ = 5, period-7 bursting for τ = 10, period-12 bursting for τ = 30, period-18 bursting for τ = 50, period-28 bursting for τ = 80. It means that the larger the time-delay is, the larger the bursting period of the electrical activity in neuron system (3) becomes and the larger the spiking frequency in one bursting is.

In this paper, the electrical activity of a time-delay 4-D neuron system under electromagnetic induction is proposed and investigated via numerical simulations. The effect of the time-delay and external forcing current on the dynamical behaviors of the addressed neuron system is discussed. Main conclusions are given as following aspects.

The results suggest that the addressed neuron model can expand the parameter region to generate complex modes of electrical activity. Furthermore, the network of this model could be used to investigate the collective behaviors of neurons of brain and central nervous system, and the potential mechanism for disease induced by electromagnetic radiation could be explained.

KT carried out the numerical simulations. ZW analyzed the experimental results and wrote the manuscript. XS polished the language of the paper.