Dynamic behaviors of single neuron or neural network are essential to understanding complex behaviors in brain or even serious diseases in nervous system. On account of complexity of nervous system, hundreds of equations should be set up to describe neuronal behavior precisely. However, main properties like electrical activities in neurons catch the attention of most researchers, and based on which, several models are established to physically study mode selection of neurons. For example, four-variable Hodgkin-Huxley (HH) equations are usually used to investigate the main properties of neurons via membrane potential (Hodgkin and Huxley, 1990). For further simplification, a three-variable neuron model, called as Hindmarsh-Rose (HR) equation, is reduced from the four-variable HH model (Hindmarsh and Rose, 1982). Besides ordinary differential equations, discrete dynamical systems or called maps, can also be considered as valid phenomenological neuron models to govern evolution of the transmembrane voltage and the dynamics of ionic conductances (Ibarz et al., 2011).

Among all these equations, Fitzhugh-Nagumo (FHN) neuron model shows its validity on oscillatory dynamic behavior in a neuron (Fitzhugh, 1961; Nagumo et al., 1962). The model presents properties of Van der Pol oscillator which can also be analyzed by fast- slow system (Krupa et al., 1997). Many methods are applied for constructing the exact solutions of the FHN equation (Li and Guo, 2006; Dehghan et al., 2010) to study transmission of nerve impulses. FHN equation is usually considered as a non-linear ordinary differential wave equation, and its soliton solutions are detected in different ways (Abbasbandy, 2008; Triki and Wazwaz, 2013). As the most important and complex system in nature, the brain contains a tremendous number of neurons and gliocytes. Collective behaviors of a large set of neurons and dynamics properties of neuron network are of much more importance. Patterns like spiral waves and targeted waves and their breakdown show the complexity of brain and related to horrible disease in neuron system. Sported, stripe, and hexagon patterns are also discovered in a modified FHN model which are very similar to the situation in reaction diffusion system.

Among all the factors affecting dynamic behaviors and pattern formation in neuron model, noise, and magnetic flow etc. are the most important and studied by many researchers. Different dynamical regimes are observed induced by external noise (Garcaojalvo and Schimanskygeier, 1999). Wu measures the pattern transition from subexcitable to excitable media (Ying et al., 2013). Colored noise can enhance the stochastic resonance in FHN neuronal model (Nozaki and Yamamoto, 1998). Noise could also be suppressed by a strong periodic signal (Pankratova et al., 2005). Besides, magnetic flux shows its high affection on collective electrical activities and signals propagation among neurons (Lv et al., 2016). Specifically, mode transition is detected and mismatch of frequency between electromagnetic radiation and the system is found (Ma et al., 2017). Magnetic flux on membrane potential is realized by a mimristor coupling, which leads to non-linear quasi-periodic spatial-temporal patterns (Mvogo et al., 2017).

As a matter of fact, a number of researchers are working on the effects on many facets of biological phenomena of electromagnetic radiation (Ma and Tang, 2017), such as central nervous system (Hossmann and Hermann, 2003), newborn rat cerebellar granule neurons (Lisi et al., 2005), oxidative damage to mitochondrial DNA in primary cultured neurons exposed to 1,800 MHz radio frequency radiation (Xu et al., 2010), cultured hippocampal neurons of rats exposed to microwave radiation (Xu et al., 2006). As to the neuron network, electromagnetic radiation can also play an important role in regulating collective behaviors among a large number of neurons. Usually, chemical and electric synapse is considered as the main type of connection between neurons. But in Ma's view, field coupling also lights the shadow of understanding synchronization problems in neuronal network (Ma and Tang, 2017). Dynamical features of neuron model with electromagnetic radiation considered should be paid more attention.

In spite of the fact that external magnetic flux is introduced into neuron network and it triggers complex patterns, the affection on a single neuron is still worth exploring deeply. Ma (Ma et al., 2017) develops FHN model and chooses a sinusoidal function as the external magnetic flux. There also exists a sinusoidal function as a transmembrane current mapped from external forcing in original model. The fact means the neuron system is driven by a pair of plane waves which leads to abundant non-linear behaviors in system. In order to compare with the results of Ma, we choose the same improved model, even with identical parameter values in reference paper (Ma et al., 2017). Model and numerical methods are explained in section 2. We provide and analysis the main numerical results when varying frequency of external transmembrane current ω in section 3, and when varying the amplitude of external electromagnetic radiation A in section 4, respectively. The conclusions are drawn in section 5.

The improved FHN model, with magnetic flux considered, is displayed as Equation (1).

The additive magnetic flux is induced by changing the distribution of ionic concentration of lectrolytes. The evolution of magnetic flux across the membrane is described by the third equation in Equation (1), which has great effects on membrane potential u in first equation in Equation (1) by a induced current denoted by the last term k₀ρ(φ)u. The memory conductance ρ(φ) of a memristor controlled by magnetic flux, which is often described by (2)ρ(φ)=α+3βφ2

The second v is slow variable for current and magnetic flux across the membrane. The physical meaning of other parameters are present in the reference (Ma et al., 2017). It is important that both the external electromagnetic radiation φ_ext and the transmembrane current I_st are chosen as trigonometric functions like, (3)φext=A cos 2πft (4)Ist=I0 sin ωt respectively. The system is driven by these two plane waves. As to simulation, we use fourth order Runge-Kutta algorithm and the time step is h = 0.01. We choose the same initial values (u, v, φ) = (0.2, 0.1, 0.8). Values of other parameters are listed here, a = 0.15, μ₁ = 0.2, μ₂ = 0.3, ε = 0.002, α = 0.1, β = 0.2, I₀ = 0.6, k₀ = −1, k₁ = 0.2, k₂ = 1.0.

External electromagnetic radiation is chosen as a cosine function φ_ext = A cos 2πft. The amplitude A is also an important control parameter, which decides the state of system when other parameters are fixed. We still use semi-Poincáre diagram to describe the scenario as varying A during 0 < A < 1.0. The situation of A > 1.0 will not shown in this paper because the the system stay in equilibrium state, which means all the variables remain constants.

Three semi-Poincáre diagrams are plotted as ω = 0.618, ω = 0.4, and ω = 0.6, corresponding to periodic, quasi-periodic, and chaotic motion when A is fixed in 0.1, shown in Figures 12–14, respectively.

In large span of 0 < A < 1.0, periodic motion is discovered, but is is interrupted by quasi-periodic motion several times, when ω = 0.618 (Figure 12). When A is fixed at 0.4, it seems that quasi-periodic motion is more easy to be captured during the interval 0 < A < 1.0 (Figure 13). Similarly, the chance we find chaotic behavior is much more than periodic and quasi-periodic motions when ω = 0.6, which can be reflected by Figure 14.

It is worth noting that Ruelle-Takens route to chaos also exists when varying the parameter A. Take the scenario shown in Figure 14 for example, there are only two closed orbits in Poincáre section when A < 0.0019, however, three closed orbits appear when A is >0.0019. The new orbit is growing lager when we increase A. The process of growth of the new orbit is stopped by a Hopf bifurcation and system enters the chaotic state when A is >0.0832. The growth process of orbit in Poincáre section is shown in Figure 15.

In this paper, we study dynamical behavior of a single neuron system driven by two plane waves. One of them is provided by external current denoted by I_st, and another comes from the external electromagnetic radiation φ_ext.

With frequency of electromagnetic radiation φ_ext fixed, system state is chosen from periodic, quasi-periodic, and chaotic motion. Specifically, system behavior switches back and forth between periodic and quasi-periodic when 0 < ω < 0.778. In the interval of 0.778 < ω < 2.208, only chaotic motion is fond. But when ω is beyond 2.208, the system alternates again between periodic and quasi-periodic motions and the chance of finding quasi-periodic motion is much more than 0 < ω < 0.778.

We also study the non-linear behavior of system varying amplitude of electromagnetic radiation A at three ω values. Very different scenarios are displayed. Periodic, quasi-periodic, or chaotic motion type characterize the scenario at different ω value, respectively.

The route to chaos, induced by Hopf bifurcation, is discovered varying both frequency ω of electromagnetic radiation φ_ext and amplitude of electromagnetic radiation A. However, it seems ω plays a much more important role in controlling the system state. In summary, system behavior is very sensitive to frequency of external force, which might be related to the ratio of frequencies of two external plane waves.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.