Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays

Abstract

In this paper, we construct an inertial two-neuron system with multiple delays, which is described by three first-order delayed differential equations. The neural system presents dynamical coexistence with equilibria, periodic orbits, and even quasi-periodic behavior by employing multiple types of bifurcations. To this end, the pitchfork bifurcation of trivial equilibrium is analyzed firstly by using center manifold reduction and normal form method. The system presents different sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from trivial equilibrium presents a secondary pitchfork bifurcation. The system exhibits stable coexistence of multiple equilibria. Using the pitchfork bifurcation curves, we divide the parameter plane into different regions, corresponding to different number of equilibria. To obtain the effect of time delays on system dynamical behaviors, we analyze equilibrium stability employing characteristic equation of the system. By the Hopf bifurcation, the system illustrates a periodic orbit near the trivial equilibrium. We give the stability regions in the delayed plane to illustrate stability switching. The neural system is illustrated to have Hopf–Hopf bifurcation points. The coexistence with two periodic orbits is presented near these bifurcation points. Finally, we present some mixed dynamical coexistence. The system has a stable coexistence with periodic orbit and equilibrium near the pitchfork–Hopf bifurcation point. Moreover, multiple frequencies of the system induce the presentation of quasi-periodic behavior. The system presents stable coexistence with two periodic orbits and one quasi-periodic behavior.

Introduction

In recent decades, dynamical behaviors such as stability and bifurcation of neural networks have received much more attention since Hopfield (1984) proposed a simplified neural model. Most of mathematical models of neurons are presented by first-order differential equation such as Hopfield neural system, Cohen–Grossberg neural system (Ozcan 2019), and cellular neural system (Mani et al. 2019), which have been applied in many research fields such as associative memory, secure communication, and signal processing. In fact, artificial neural networks are usually constructed by simulating biological neural systems, for example Hodgkin–Huxley model (Wang et al. 2018; Wu et al. 2013), Chay model (Jia et al. 2017; Zhu et al. 2019) and Hindmarsh–Rose model (Hu and Cao 2016; Mondal et al. 2019). The mathematical model of neural node is very important to dynamical behavior of whole network system (Ma et al. 2019; Qu and Wang 2017). To further approximate biological neuron, some researchers introduced inductance term into artificial neural model (Badcock and Westervelt 1987). In reality, the membrane activity of a hair cell can be represented by the equivalent circuit having an inductance (Angelaki and Correia 1991). The squid axon was described as containing a phenomenological inductance (Mauro et al. 1970). The neuron system was modeled by a circuit system containing inertial term. The output of a neuron with inertial term is connected to its input via an RLC (resistance–inductance–capacitance) circuit. Further, the aim of the presented paper is to show how the dynamical behavior might be affected when an inertial term is added to the standard Hopfield equation.

Moreover, time delay is very important in real neural systems because of their finite switching speed. Some delayed inertial neural systems were presented to analyze their dynamical behaviors (Gu and Zhao 2015). For example, the delayed inertial one-neural system (Liu et al. 2009; He et al. 2012) was considered to study its stability, bifurcation, and even chaotic behavior (Li et al. 2004). For the delayed inertial two-neuron system, (Liu et al. 2009) focused on the equilibrium stability and Hopf bifurcation. Dong et al. (2012) illustrated a two-equilibrium coexistence, periodic orbit, and even quasi-periodic behavior by analyzing the Hopf–pitchfork bifurcation. Song and Xu (2014) presented a delayed inertial system with two neurons to analyze its stability switching and Bogdanov–Takens bifurcation. Further, Ge and Xu (2018) constructed a delayed inertial system with four neurons to study the fold-Hopf bifurcation (Ge and Xu 2015) and Hopf–Hopf bifurcation (Ge and Xu 2013). Recently, multistability coexistence was illustrated in a delayed inertial two-neuron system (Song et al. 2016). The neural system presented some stable coexistences with multi-type activity patterns such as two equilibria, multiple periodic orbits, and even quasi-periodic spiking by different bifurcation routes (Song et al. 2015a, b). Further, when the neural activation function was chosen as the Crespi function, the delayed inertial two-neuron system exhibited three chaotic attractors through the period-doubling and quasi-periodic bifurcations (Yao et al. 2019).

However, most works focused on analyzing the dynamics of the delayed inertial neural system described by two-, four-, and even eight-order differential equations. In fact, Wheeler and Schieve (1997) constructed a two-neuron system by adding inertial term into one of neurons. The neural system was presented by three first-order differential equations. Zhao et al. (2012) introduced time delay into the inertial two-neuron system and discussed in detail the equilibrium stability and Hopf bifurcation. Following the mentioned references, in this paper, we introduce self- and cross-interaction delays into the inertial two-neuron system. The mathematical model is presented by three first-order delayed differential equations. We focus on analyzing the stable coexistence with multiple equilibria, periodic orbits, and even quasi-periodic behavior. In fact, multiple coexistence is one of the most important dynamical behaviors in biological and artificial neural networks (Song et al. 2015a, b). It should be noticed that the similar two-neuron system without inertial term has been proposed in Song et al. (2019). The system has the same number of equilibrium points, but the different static bifurcation sequences. Further, in this paper, we will find that the inertial term induces a mixed dynamics coexistence including the coexistence of one periodic orbit and two equilibria, the coexistence of two periodic orbits and one quasi-periodic behavior. On the other hand, two inertial terms are introduced in two-neuron coupling system. The system just exhibits the coexistence of two equilibria and one periodic orbit by the pitchfork–Hopf bifurcation (Song et al. 2016).

The delayed inertial two-neuron system considered in the paper is described by the following equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1(t)=x_2(t),\hfill \\ {\dot{x}}_2(t)=-k{x}_2(t)-x_1(t)+c_{1}f(x_1(t-{\tau }_1))+c_{2}f(y_1(t-{\tau }_2)),\hfill \\ {\dot{y}}_1(t)=-y_1(t)+c_{1}f(y_1(t-{\tau }_1))+c_{2}f(x_1(t-{\tau }_2)),\hfill \end{array}\right)\end{array}$$ 1

where $x_1(t)$ and $y_1(t)$ denote neural activities at time t, $x_2(t)$ is derivative of $x_1(t)$, $k>0$ is a damping factor, ${\tau }_1$, ${\tau }_2>0$ describe time delays of self- and cross-interaction, $c_1$ and $c_2$ are coupled weighs for inhibition ($c_i<0$) and excitation ($c_i>0$), the activation function is $f(x)=tanh(x)$.

This paper is organized as follows. In Sect. 2, multiple pitchfork bifurcations of trivial and nontrivial equilibria are analyzed. Employing center manifold reduction and normal form method, we find that the neural system (1) illustrates multiple bifurcation sequences with supercritical and subcritical pitchfork bifurcations. Using these pitchfork bifurcation curves, we give the detail parameter regions with the different number of system’s equilibrium. In Sect. 3, we analyze time delays on equilibrium’s stability. With time delay increase, the neural system can repeatedly switches its stability by the forward and reverse Hopf bifurcation, which is called the stability switching. In Sect. 4, we give stability regions. The Hopf–Hopf bifurcation points are illustrated by the Hopf bifurcation curves. The neural system presents two-coexisting periodic orbits. In Sect. 5, through theoretical analysis and numerical simulation, we present some mixed dynamics coexistence, such as the coexistence of one periodic orbit and two equilibria, the coexistence of two periodic orbits and one quasi-periodic behavior. Conclusion is provided in Sect. 6.

Multi-pitchfork bifurcations of trivial/nontrivial equilibria

Bifurcation is one of the most important dynamics in the behavior analysis of neural network system (Guan et al. 2019; Wang and Zhu 2016; Zhang et al. 2019). The pitchfork bifurcation can lead neural system to have a new pair of stable equilibria, which can be used as the static retrievable memory. Nakajima and Ikegami (2010) proposed a dynamical model to interpret the crossed-hand deficit of temporal order judgment (TOJ) from the pitchfork bifurcation analysis. Further, the coexistence of two stable equilibria provides a great flexibility in system function. Multiple coexistence of resting state is important for applications in content-addressable memory, pattern recognition and automatic control (Huang and Cao 2010; Nie et al. 2013).

In this section, to get the number of equilibria in system (1), we will analyze the static bifurcations using center manifold reduction and normal form method. However, the center manifold reduction of the delayed dynamical system is too complicated to be understand. Further, time delay cannot affect the number of equilibria. So, in the following section, we just analyze static bifurcations of the delay-free system. The stability of the equilibrium will be proposed in the next section by the Hopf bifurcation. The results show that the neural system has different bifurcation sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from the trivial equilibrium presents a secondary pitchfork bifurcation. The system illustrates a coexistence with multiple equilibria. The delay-free neural system is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1=x_2,\hfill \\ {\dot{x}}_2=-k{x}_2-x_1+c_{1}f(x_1)+c_{2}f(y_1),\hfill \\ {\dot{y}}_1=-y_1+c_{1}f(y_1)+c_{2}f(x_1).\hfill \end{array}\right)\end{array}$$ 2

Obviously, (0, 0, 0) is the trivial equilibrium of system (2). The characteristic equation of the linearized system at (0, 0, 0) is

$$\begin{array}{c}\hfill {\lambda }^3+(1+k-c_1){\lambda }^2+(1+k)(1-c_1)\lambda +{(c_1-1)}^2-c_2^2=0.\end{array}$$ 3

System (2) presents a static bifurcation if the eigenvalue passes through the imaginary axis along real axis. Therefore, letting $\lambda =0$ in (3), we have

$$\begin{array}{c}\hfill {(c_1-1)}^2-c_2^2=0.\end{array}$$ 4

Equation (4) is called as the bifurcation set, where $c_1$ and $c_2$ can be regarded as bifurcation parameters. In the following, we analyze the bifurcation types (supercritical or subcritical) to show the system’s equilibrium by center manifold reduction and normal form method. To this end, firstly, regard $c_2$ as the bifurcation parameter for the fixed parameter $c_1$. The corresponding bifurcation points are $c_2^{\ast }=c_1-1$ and $c_2^{\ast }=1-c_1$. Let $c_2=c_2^{\ast }+\epsilon$ in system (2), where $\epsilon$ is the unfolding parameter, one has

Theorem 1

Neural system (2) presents a subcritical pitchfork bifurcation at the bifurcation point$c_2^{\ast }=c_1-1$and a supercritical pitchfork bifurcation at$c_2^{\ast }=1-c_1$. The corresponding normal form systems are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{z}}_1=-{\displaystyle \frac{2}{1+k}}\epsilon z_1-{\displaystyle \frac{2}{3(1+k)}}{z}_1^3+\cdots ,\hfill \\ \dot{\epsilon }=0,\hfill \end{array}\right)\end{array}$$ 5

for$c_2^{\ast }=c_1-1$and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{z}}_1={\displaystyle \frac{2}{1+k}}\epsilon z_1-{\displaystyle \frac{2}{3(1+k)}}{z}_1^3+\cdots ,\hfill \\ \dot{\epsilon }=0,\hfill \end{array}\right)\end{array}$$ 6

for$c_2^{\ast }=1-c_1$, respectively.

Proof

Let $c_2=c_2^{\ast }=c_1-1$ in (3). The characteristic equation is simplified to

$$\begin{array}{c}\hfill \lambda ({\lambda }^2+(1+k-c_1)\lambda +(1+k)(1-c_1))=0.\end{array}$$ 7

The eigenvalues are

$$\begin{array}{c}\hfill {\lambda }_1=0,{\lambda }_{2,3}=-{\displaystyle \frac{c_1-k-1\pm \sqrt{\mathrm{\Delta }}}{2}},\end{array}$$ 8

where $\mathrm{\Delta }=c_1^2+2(c_1+k-3)(1+k)$. The corresponding eigenvector is

$$\begin{array}{c}\hfill v_1=\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right),v_2=\left(\begin{array}{c}\frac{c_1-1}{c_1-1+k{\lambda }_2+{\lambda }_2^2}\\ \frac{(c_1-1){\lambda }_2}{c_1-1+k{\lambda }_2+{\lambda }_2^2}\\ 1\end{array}\right),v_3=\left(\begin{array}{c}\frac{c_1-1}{c_1-1+k{\lambda }_3+{\lambda }_3^2}\\ \frac{(c_1-1){\lambda }_3}{c_1-1+k{\lambda }_3+{\lambda }_3^2}\\ 1\end{array}\right).\end{array}$$ 9

Submitting $c_2=c_2^{\ast }+\epsilon$ into (2) and applying Taylor expansion, one has the following system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1=x_2,\hfill \\ {\dot{x}}_2=(c_1-1)x_1-k{x}_2+(c_1-1+\epsilon )y_1-{\displaystyle \frac{c_1}{3}}{x}_1^3-{\displaystyle \frac{c_1-1+\epsilon }{3}}{y}_1^3+\cdots ,\hfill \\ {\dot{y}}_1=(c_1-1+\epsilon )x_1+(c_1-1)y_1-{\displaystyle \frac{c_1-1+\epsilon }{3}}{x}_1^3-{\displaystyle \frac{c_1}{3}}{y}_1^3+\cdots ,\hfill \\ \dot{\epsilon }=0.\hfill \end{array}\right)\end{array}$$ 10

Let

$$\begin{array}{c}\hfill \left(\begin{array}{c}{x}_1\\ x_2\\ y_1\end{array}\right)=T\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\end{array}\right),\end{array}$$

where

$$\begin{array}{c}\hfill T=\left(\begin{array}{ccc}-1& \frac{c_1-1}{c_1-1+k{\lambda }_2+{\lambda }_2^2}& \frac{c_1-1}{c_1-1+k{\lambda }_3+{\lambda }_3^2}\\ 0& \frac{(c_1-1){\lambda }_2}{c_1-1+k{\lambda }_2+{\lambda }_2^2}& \frac{(c_1-1){\lambda }_3}{c_1-1+k{\lambda }_3+{\lambda }_3^2}\\ 1& 1& 1\end{array}\right).\end{array}$$

System (10) is transformed into the following standard form, which is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{z}}_1=0z_1+f_1,\hfill \\ {\dot{z}}_2={\lambda }_2{z}_2+f_2,\hfill \\ {\dot{z}}_3={\lambda }_3{z}_3+f_3,\hfill \\ \dot{\epsilon }=0,\hfill \end{array}\right)\end{array}$$ 11

where

$$\begin{array}{cc}\hfill f_1& =m_1\epsilon z_1+m_2\epsilon z_2+m_3\epsilon z_3+\frac{m_1}{3}{z}_1^3\hfill \\ \hfill & +m_2{z}_1^2{z}_2+m_3{z}_1^2{z}_3+m_4{z}_1{z}_2^2\hfill \\ \hfill & +m_5{z}_1{z}_2{z}_3+m_6{z}_1{z}_3^2+m_7{z}_2^3\hfill \\ \hfill & +m_8{z}_2^2{z}_3+m_9{z}_2{z}_3^2+m_{10}{z}_3^3+\cdots ,\hfill \\ \hfill f_2& =n_1\epsilon z_1+n_2\epsilon z_2+n_3\epsilon z_3+\frac{n_1}{3}{z}_1^3\hfill \\ \hfill & +n_4{z}_1^2{z}_2+n_5{z}_1^2{z}_3+n_6{z}_1{z}_2^2\hfill \\ \hfill & +n_7{z}_1{z}_2{z}_3+n_8{z}_1{z}_3^2+n_9{z}_2^3+n_{10}{z}_2^2{z}_3\hfill \\ \hfill & +n_{11}{z}_2{z}_3^2+n_{12}{z}_3^3+\cdots ,\hfill \\ \hfill f_3& =q_1\epsilon z_1+q_2\epsilon z_2+q_3\epsilon z_3+\frac{q_1}{3}{z}_1^3\hfill \\ \hfill & +q_4{z}_1^2{z}_2+q_5{z}_1^2{z}_3+q_6{z}_1{z}_2^2\hfill \\ \hfill & +q_7{z}_1{z}_2{z}_3+q_8{z}_1{z}_3^2+q_9{z}_2^3\hfill \\ \hfill & +q_{10}{z}_2^2{z}_3+q_{11}{z}_2{z}_3^2+q_{12}{z}_3^3+\cdots ,\hfill \\ \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill m_1& =-\frac{2}{1+k},m_2=\frac{3-3c_1-k-\sqrt{\mathrm{\Delta }}}{2(c_1-1)(1+k)},\hfill \\ \hfill m_3& =\frac{3-3c_1-k+\sqrt{\mathrm{\Delta }}}{2(c_1-1)(1+k)},\hfill \\ \hfill m_4& =\frac{-2(1-2k+{(c_1+k)}^2)+2(c_1-1+k)\sqrt{\mathrm{\Delta }}}{(1+k){(c_1-1+k-\sqrt{\mathrm{\Delta }})}^2},\hfill \\ \hfill m_5& =\frac{2(2-c_1)}{(c_1-1)(1+k)},\hfill \\ \hfill m_6& =\frac{-2(1-2k+(c_1+k)(c_1+k+\sqrt{\delta }))-\sqrt{\delta }}{(1+k)(c_1-1+k+\sqrt{\mathrm{\Delta }})}.\hfill \end{array}$$

The expressions of other coefficients (i.e., $m_i,i=7,\dots ,10$ and $n_j,q_j,j=1,\dots ,12$) are too complicated to be presented. Assume the center manifold of system (11) as

$$\begin{array}{c}\hfill \begin{array}{c}{W}^c(0)=\left\{(z_1,z_2,z_3,\epsilon ),\in ,R^4,|,z_2,=,h_2,(z_1,\epsilon ),,,z_3,=,h_3,(z_1,\epsilon ),,\right)\hfill \\ \left(|,z_1,|<,{\delta }_1,,,|\epsilon |,<,{\delta }_2,,,h_i,(0,0),=,0,,,D,h_i,(0,0),=,0,,,i,=,2,,,3\right\},\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{c}{z}_2=h_2(z_1,\epsilon )=r_1{z}_1^2+r_2{z}_1\epsilon +r_3{\epsilon }^2+\cdots ,\hfill \\ z_3=h_3(z_1,\epsilon )=s_1{z}_1^2+s_2{z}_1\epsilon +s_3{\epsilon }^2+\cdots ,\hfill \end{array}\end{array}$$ 12

and ${\delta }_1$ and ${\delta }_2$ are small. We let

$$\begin{array}{c}\hfill DH(0·z_1+f_1)-BH-L=0,\end{array}$$ 13

where D is the differential operator, and

$$\begin{array}{c}\hfill B=\left(\begin{array}{cc}{\lambda }_2& 0\\ 0& {\lambda }_3\end{array}\right),H=\left(\begin{array}{c}{h}_2\\ h_3\end{array}\right),L=\left(\begin{array}{c}{f}_2\\ f_3\end{array}\right).\end{array}$$

Substituting (12) into (13), we have the coefficients of (12). Then neural system (2) can be transformed into (5), which is the normal form of the pitchfork bifurcation at $c_2^{\ast }=c_1-1$. By the same computation process, the normal form system for $c_2=1-c_1$ is shown as (6). This completes the proof. $\square$

On the other hand, if $c_1$ is regarded as the bifurcation parameter for the fixed $c_2$ , we obtain the bifurcation points $c_1^{\ast }=1-c_2$ and $c_1^{\ast }=1+c_2$, respectively. Similarly, let $c_1=c_1^{\ast }+\epsilon$, where $\epsilon$ is the unfolding parameter, we have

Theorem 2

Neural system (2) presents two supercritical pitchfork bifurcations at the bifurcation points$c_1^{\ast }=1-c_2$and$c_1^{\ast }=1+c_2$, respectively. The corresponding normal form systems are all

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{z}}_1={\displaystyle \frac{2}{1+k}}\epsilon z_1-{\displaystyle \frac{2}{3(1+k)}}{z}_1^3+\cdots ,\hfill \\ \dot{\epsilon }=0.\hfill \end{array}\right)\end{array}$$ 14

It follows that system (2) presents the different types of pitchfork bifurcations. Regarding $c_2$ as the bifurcation parameter, we obtain the supercritical and subcritical pitchfork bifurcations in system (2) . For example, choosing $k=1$, one has the pitchfork bifurcation points $c_2^{\ast }=c_1-1=6$ and $c_2^{\ast }=1-c_1=6$ for $c_1=-5$. By theorem 1, the pitchfork bifurcation is subcritical for $c_2^{\ast }=-6$ and supercritical for $c_2^{\ast }=6$. The bifurcation diagrams are shown in Fig. 1. It implies that neural system (2.1) firstly presents the subcritical, and then supercritical pitchfork bifurcations, as shown in Figs. 1a, b. However, if we choose $c_1=5$, the subcritical pitchfork bifurcation point is $c_2^{\ast }=c_1-1=4$ and the supercritical pitchfork bifurcation is $c_2^{\ast }=1-c_1=4$ . That is, neural system (2) firstly presents the supercritical, and then the subcritical pitchfork bifurcation, as shown in Figs.1c, d.

Fig. 1.

The bifurcation diagrams with $c_2$ variation show the subcritical and supercritical pitchfork bifurcations for a, b$c_1=-5$ and c, d$c_1=5$ in system (2)

On the other hand, if $c_1$ is regarded as the bifurcation parameter for the fixed parameter $c_2=5$, one has the pitchfork bifurcation points $c_1^{\ast }=1-c_2=-4$ and $c_1^{\ast }=1+c_2=6$. Further, it follows from theorem 2 that these bifurcation are all supercritical. The bifurcation diagrams are exhibited in Fig. 2a, b. The neural system (2) has the first supercritical pitchfork bifurcation at $c_1^{\ast }=-4$. The single trivial equilibrium (0, 0, 0) bifurcates into three equilibria. Increasing $c_1$ to pass through $c_1^{\ast }=6$, system (2) exhibits the second supercritical pitchfork bifurcation, as shown in Fig. 2b. The trivial equilibrium generates another two equilibria. Neural system (2) illustrates a bifurcation sequence with two supercritical pitchfork bifurcations.

Fig. 2.

The bifurcation diagrams with $c_1$ variation show the supercritical pitchfork bifurcation a$c_1^{\ast }=1-c_2=-4$ and b$c_1^{\ast }=1+c_2=6$ with $c_2=5$ in system (2)

Now, we will show a new pitchfork bifurcation of the nontrivial equilibria. In fact, neural system (2) presents the nontrivial equilibrium $(x_{10},0,y_{10})$ satisfied with the dynamic nullclines $x_{10}=c_{1}f(x_{10})+c_{2}f(y_{10})$ and $y_{10}=c_{1}f(y_{10})+c_{2}f(x_{10})$. By $u_1=x_1-x_{10}$, $u_2=x_2$, $u_3=y_1-y_{10}$, one has the linearizing system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{u}}_1=u_2,\hfill \\ {\dot{u}}_2=-k{u}_2-u_1+c_1{P}_1{u}_1+c_2{P}_2{u}_3,\hfill \\ {\dot{u}}_3=-u_3+c_1{P}_2{u}_3+c_2{P}_1{u}_1,\hfill \end{array}\right)\end{array}$$ 15

where $P_1=1-{tanh}^2(x_{10})$, $P_2=1-{tanh}^2(y_{10})$ The characteristic equation is

$$\begin{array}{cc}& {\lambda }^3+(1+k-c_1{P}_2){\lambda }^2+(1+k-c_1{P}_1-c_{1}k{P}_2)\lambda \hfill \\ \hfill & +(c_1{P}_1-1)(c_1{P}_2-1)-c_2^2{P}_1{P}_2=0.\hfill \end{array}$$ 16

Neural system (2) presents a pitchfork bifurcation at $(x_{10},0,y_{10})$ when the following condition is valid, which is

$$\begin{array}{c}\hfill Q(c_1)=(c_1{P}_1-1)(c_1{P}_2-1)-c_2^2{P}_1{P}_2=0,\end{array}$$ 17

where $P_1$ and $P_2$ are the function of $x_{10}$ and $y_{10}$. The pitchfork bifurcation point cannot be obtained by theoretical expression since (17) is a transcendental equation. So, we just give some numerical results for the given parameters in system (2). As the above section, we regard $c_1$ as the bifurcation parameter for the fixed $c_2$ with $k=1$. The neural system (2) presents two bifurcations when the bifurcation parameter $c_1$ passes through the second supercritical pitchfork bifurcation, i.e., $c_1>c_1^{\ast }=6$. It follows from Fig. 3a that $Q(c_1)=0$ has three roots, i.e., $c_1^{\ast }=-4$, 6 and 6.99917. In fact, the critical values $c_1^{\ast }=-4,6$ are the pitchfork bifurcation points corresponded to the equilibrium (0, 0, 0). The third root $c_1^{\ast }=6.99917$ corresponds to the bifurcation of the nontrivial equilibrium. The bifurcation diagram with $c_1$ varying is illustrated in Fig. 3b. The nontrivial equilibria bifurcated from the second pitchfork bifurcation present simultaneously two supercritical pitchfork bifurcations.

Fig. 3.

a The function $Q(c_1)=0$, b bifurcation diagram shows the pitchfork bifurcations of the trivial/nontrivial equilibria with $a_{12}=1$, $a_{21}=0.6$

Lastly, we illustrate the pitchfork bifurcation curves to show the dynamical classification of equilibrium’s number in $(c_1,c_2)$-plane. The figure is shown in Fig. 4. In region $D_1$, system (2) just illustrates a stable equilibrium (0, 0, 0). When the parameters enter into region $D_2$ or $D_3$, system (2) exhibits three equilibria through the pitchfork bifurcation. Further, neural system (2) obtains a new pair of nontrivial equilibrium when the parameters enter into region $D_4$ or $D_5$, which is divided by the second pitchfork bifurcation curve. In region $D_6$, system (2) possesses nine equilibria through adding two pairs of new nontrivial equilibria, which is a new pitchfork bifurcation of the nontrivial equilibrium.

Fig. 4.

The classification of the number of system’s equilibrium in the $(c_1,c_2)$-plane by multiple pitchfork bifurcation curves of system (2) with $k=1$, where the nullcline curves $x_{10}=c_{1}f(x_{10})+c_{2}f(y_{10})$ and $y_{10}=c_{1}f(y_{10})+c_{2}f(x_{10})$ show the number of equilibria

Hopf bifurcation and periodic orbit

In what follows, we analyze the effect of time delays on the equilibrium’s stability. The neural system exhibits a periodic orbit by the Hopf bifurcation of trivial equilibrium. The linearized system near the trivial equilibrium is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1(t)=x_2(t),\hfill \\ {\dot{x}}_2(t)=-k{x}_2(t)-x_1(t)+c_1{x}_1(t-{\tau }_1)+c_2{y}_1(t-{\tau }_2),\hfill \\ {\dot{y}}_1(t)=-y_1(t)+c_1{y}_1(t-{\tau }_1)+c_2{x}_1(t-{\tau }_2),\hfill \end{array}\right)\end{array}$$ 18

The corresponding characteristic equation is

$$\begin{array}{cc}& {\left(c_1,e^{-\lambda {\tau }_1},-,1\right)}^2-c_2^2{e}^{-2\lambda {\tau }_2}+(1+k)(1-c_1{e}^{-\lambda {\tau }_1})\lambda \hfill \\ \hfill & +(1+k-c_1{e}^{-\lambda {\tau }_1}){\lambda }^2+{\lambda }^3=0.\hfill \end{array}$$ 19

The equilibrium is stable if all eigenvalues have negative real parts. So, we firstly present the basic condition for ${\tau }_1=0$ and ${\tau }_2=0$ in (19), where the trivial equilibrium is stable. The characteristic equation is simplified to

$$\begin{array}{c}\hfill {(c_1-1)}^2-c_2^2+(1+k)(1-c_1)\lambda +(1+k-c_1){\lambda }^2+{\lambda }^3=0.\end{array}$$ 20

Applying the Routh-Hurwitz criterion, we have necessary and sufficient conditions of the stable trivial equilibrium, which is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}1+k-c_1>0\text{or}(1+k)(1-c_1)>0,\hfill \\ {(c_1-1)}^2-c_2^2>0,\hfill \\ c_2^2+(1-c_1)(k+2-c_1)k>0.\hfill \end{array}\right)\end{array}$$ 21

By $k>0$ in system (1), the basic condition is simplified into

$$\begin{array}{c}\hfill 1-c_1>0\text{and}{(c_1-1)}^2>c_2^2.\end{array}$$ 22

Time delays ${\tau }_1$ and ${\tau }_2$ can induce the trivial equilibrium to instability. To analyze its stability and obtain the stability region, we regard ${\tau }_2$ as a bifurcation parameter for the fixed ${\tau }_1$. Assuming (19) has a pure imaginary root $\lambda =i\omega ,\omega >0,$ one has

$$\begin{array}{c}\hfill \begin{array}{c}1-2c_1{e}^{-i\omega {\tau }_1}+c_1^2{e}^{-2i\omega {\tau }_1}-c_2^2{e}^{-2i\omega {\tau }_2}+i\left(1,+,k,-,c_1,e^{-i\omega {\tau }_1}\right)\omega \hfill \\ -\left(1,+,k,-,c_1,e^{-i\omega {\tau }_1}\right){\omega }^2-i{\omega }^3=0,\hfill \end{array}\end{array}$$ 23

Separating real and imaginary parts, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}1-{\omega }^2-k{\omega }^2+(c_1{\omega }^2-2c_1)cos\omega {\tau }_1+c_1^{2}cos2\omega {\tau }_1\hfill \\ -(c_1\omega +k{c}_1\omega )sin\omega {\tau }_1-c_2^{2}cos2\omega {\tau }_2=0,\hfill \\ \omega +k\omega -{\omega }^3-(c_1\omega +k{c}_1\omega )cos\omega {\tau }_1-c_1^{2}sin2\omega {\tau }_1\hfill \\ +(2c_1-c_1{\omega }^2)sin\omega {\tau }_1+c_2^{2}sin2\omega {\tau }_2=0.\hfill \end{array}\right)\end{array}$$ 24

Eliminating ${\tau }_2$ from (24) produces

$$\begin{array}{c}\hfill \left\{\begin{array}{c}cos2\omega {\tau }_2=\frac{1-(1+k){\omega }^2+c_1\left[\left(-,2,+,{\omega }^2\right),cos,\omega ,{\tau }_1,+,c_1,cos,2,\omega ,{\tau }_1,-,(1+k),\omega ,sin,\omega ,{\tau }_1\right]}{c_2^2},\hfill \\ sin2\omega {\tau }_2=\frac{{\omega }^3-(1+k)\omega +(1+k)c_1\omega cos\omega {\tau }_1+c_1\left({\omega }^2,-,2,+,2,c_1,cos,\omega ,{\tau }_1\right)sin\omega {\tau }_1}{c_2^2}.\hfill \end{array}\right)\end{array}$$ 25

Using ${cos}^{2}2\omega {\tau }_2+{sin}^{2}2\omega {\tau }_2=1$, one has

$$\begin{array}{cc}\hfill L(\omega )=& \left(1,-,(1+k),{\omega }^2,+,c_1,\left[\left(-,2,+,{\omega }^2\right),cos,\omega ,{\tau }_1\right)\right)\hfill \\ \hfill & {\left(\left(+,,c_1,cos,2,\omega ,{\tau }_1,-,(1+k),\omega ,sin,\omega ,{\tau }_1\right]\right)}^2\hfill \\ \hfill & +\left({\omega }^3,-,(1+k),\omega ,+,(1+k),c_1,\omega ,cos,\omega ,{\tau }_1\right)\hfill \\ \hfill & {\left(+,,c_1,\left({\omega }^2,-,2,+,2,c_1,cos,\omega ,{\tau }_1\right),sin,\omega ,{\tau }_1\right)}^2\hfill \\ \hfill & c_2^4=0,\hfill \end{array}$$ 26

Supposing (26) exhibits some positive roots ${\omega }_i,i=1,2,\dots$, one obtains some critical values of stability region, which is

$$\begin{array}{c}\hfill {\tau }_2^{i,j}={\displaystyle \frac{{\phi }_i+2j\pi }{{\omega }_i}},i=1,2,\dots ;j=0,1,2,\dots ,\end{array}$$ 27

where ${\phi }_i\in [0,2\pi )$. Define

$$\begin{array}{c}\hfill {\tau }_2^0=min\{{\tau }_2^{i,0}:i=1,2,\dots \}.\end{array}$$ 28

Differentiating $\lambda$ with ${\tau }_2$ in (19), one checks the transversality condition, i.e., the eigenvalue passes through the imaginary axis with non-zero velocity

$$\begin{array}{c}\hfill {\lambda }^{\prime }({\tau }_2)=\frac{-2c_2^2\lambda }{e^{2\lambda {\tau }_2}(1+k+2k\lambda +2\lambda +3{\lambda }^2)-2c_1^2{e}^{2\lambda ({\tau }_2-{\tau }_1)}{\tau }_1+2c_2^2{\tau }_2+c_1{e}^{\lambda (2{\tau }_2-{\tau }_1)}((2+\lambda (1+k+\lambda )){\tau }_1-1-k-2\lambda )}.\end{array}$$ 29

Therefore, employing the Hopf bifurcation theory, one has

Theorem 3

With the basic conditions$1-c_1>0$and${(c_1-1)}^2>c_2^2$, the equilibrium (0, 0, 0) is asymptotically stable for arbitrary${\tau }_2$when (26) has no positive root. Further, when (26) has at least two positive and simple roots${\omega }_i,i=1,2,\dots$, there exist a finite number of delay intervals, in which the equilibrium (0, 0, 0) is asymptotically stable. That is to say, the neural system presents stability switching with delay increase.

For example, we choose $k=1,c_1=-2,c_2=-1$. System (1) just has the equilibrium (0, 0, 0). When time delay ${\tau }_1=0.3$, the equation $L(\omega )=0$ has no positive root. The equilibrium (0, 0, 0) is locally asymptotically stable for arbitrary ${\tau }_2$. Time delay ${\tau }_2$ cannot induce the equilibrium to instability. The time histories are exhibited with small delay ${\tau }_2=1$ (Fig. 5a) and large delay ${\tau }_2=10$ (Fig. 5b). The system trajectories evolve and enter into the equilibrium (0, 0, 0). Further, if time delay ${\tau }_1$ increases to 0.5, it follows from (26) that $L(\omega )=0$ has two different roots, i.e., ${\omega }_1=1.454$ and ${\omega }_2=1.6541$. Submitting ${\omega }_1$ and ${\omega }_2$ into (27), one has two series of values of time delay, that is ${\tau }_2^{1,j}=2.0346,4.1954,6.3562,\dots$ and ${\tau }_2^{2,j}=1.0607,2.9599,4.859,\dots$. The equilibrium (0, 0, 0) presents the stability switching with delay increase. In fact, when ${\tau }_2\in (0,1.0607)$, system (1) has a stable trivial equilibrium (0, 0, 0). Time history is shown in Fig. 6a for ${\tau }_2=0.5$ . The trajectory evolves into the equilibrium (0, 0, 0). When delay passes through the critical value ${\tau }_2^{2,0}=1.0607$ , the equilibrium (0, 0, 0) loses its stability by Hopf bifurcation with frequency ${\omega }_2$. It follows form Fig. 6b that the system presents a stable periodic orbit. Due to the existence of two roots, i.e., ${\omega }_1$ and ${\omega }_2$, the stable periodic orbit will disappear by the reverse Hopf bifurcation with frequency ${\omega }_1$. The trivial equilibrium (0, 0, 0) regains its stability. Time history is shown in Fig. 6c for ${\tau }_2=2.5$. Furthermore, the stable trivial equilibrium will lose its stability if the delay passes through ${\tau }_2^{2,1}=2.9599$. The system re-exhibits a stable periodic orbit, as shown in Fig. 6d. In such way, system (1) multi-switches its stability with delay increase, which is called as the stability switching. The corresponding eigenvalues are shown in Fig. 7a for ${\tau }_1=0.3$ and Fig. 7b for ${\tau }_1=0.5$.

Fig. 5.

Time histories in system (1) for a${\tau }_2=1$ and (b) ${\tau }_2=10$ with $k=1,c_1=-2,c_2=-1$ and ${\tau }_1=0.3$

Fig. 6.

Time histories with a${\tau }_2=0.5$, b${\tau }_2=1.5$, c${\tau }_2=2.5$, and d${\tau }_2=3.5$ show the stability switching for $k=1,c_1=-2,c_2=-1$ and ${\tau }_1=0.5$

Fig. 7.

The real parts of system eigenvalues with ${\tau }_2$ increase: a${\tau }_1=0.3$ and b${\tau }_1=0.5$. The other parameters are $k=1,c_1=-2,c_2=-1$

Hopf–Hopf bifurcation and periodic coexistence

We have analyzed the effect of time delays on stability of trivial equilibrium. It follows that the equilibrium (0, 0, 0) of the system repeatedly loses and retrieves its stability with delay increase. In this section, we will show the Hopf bifurcation curves and give some Hopf–Hopf bifurcation points. Further, some special dynamical behaviors, such as two-coexisting periodic orbits are exhibited near the Hopf–Hopf bifurcation points.

In fact, for each ${\tau }_1$, by (27), one obtains the Hopf bifurcation curves, i.e., ${\tau }_2^{1,j}$, ${\tau }_2^{2,j}(j=0,1,2,\dots )$ in the $({\tau }_1,{\tau }_2)$-plane, as shown in Fig. 8 with $k=1,c_1=-2,c_2=-1$. The plane is divided into some regions by the equilibrium’s stability criterion. It should be noticed that the Hopf bifurcation curves have some intersection points H-H in Fig. 8. The corresponding coordinates are $({\tau }_1^{\ast },{\tau }_2^{\ast })=(0.7074,4.9759),(0.5774,6.7731)$ and (0.5191, 8.653) for the first few points. In fact, two pairs of eigenvalues with maximum real part have pure imaginary, which is called as the Hopf–Hopf bifurcations. By the Hopf–Hopf bifurcation theory, we obtain that system (1) illustrates the stable coexistence with two periodic orbits due to the frequencies ${\omega }_1$ and ${\omega }_2$. Using the Matlab package tool DDE-BIFTOOL, we can further give the parameter regions of stable coexistence, as shown in Fig. 8. The enlargement is shown in Fig.9, where T denotes the appearance and disappearance of the two-coexisting periodic orbits.

Fig. 8.

Hopf bifurcation curves ${\tau }_2^{1,j}$, ${\tau }_2^{2,j}(j=0,1,2,\dots )$ presented in $({\tau }_1,{\tau }_2)$-plane illustrate the Hopf–Hopf bifurcation points (H-H) for $k=1,c_1=-2,c_2=-1$

Fig. 9.

The parameter regions of the two-coexisting periodic orbits near the first two Hopf–Hopf bifurcation points, where T denotes the appearance and disappearance curves of the regions

In what follows, we will give some numerical simulations to illustrate the corresponding dynamical behaviors. It follows from Fig. 10 that the equilibrium (0, 0, 0) shown in Fig. 10a evolves into a periodic orbit shown in Fig. 10b, c through Hopf bifurcation with frequency ${\omega }_2$ . On the other hand, the system present a long and narrow periodic orbit (as shown in Fig. 10e, f) by the Hopf bifurcation with frequency ${\omega }_1$. Furthermore, the periodic orbits generated from the different Hopf bifurcation will be encountered in a small region of time delay. The system exhibits the stable coexistence with two periodic orbits, as shown in Fig. 10d for $({\tau }_1,{\tau }_2)=(0.8,5.29)$ with initial conditions (1, 0, 1) and $(-1,0,1)$. Similarly, system (1) presents the two-periodic coexistence for other Hopf–Hopf bifurcation points, as shown in Fig. 11 for $({\tau }_1,{\tau }_2)=(0.8,7.79)$ and (0.8, 10.3) with initial conditions (1, 0, 1) and $(-1,0,1)$. It follows from Fig. 11 that two coexisting periodic orbits repeatedly change their shapes from oval to long and narrow with delay increase.

Fig. 10.

Phase portraits near $({\tau }_1^{\ast },{\tau }_2^{\ast })=(0.7074,4.9759)$ show the system dynamical behaviors for the fixed delays a (0.6, 4.8), b (0.6, 5.2), c (0.8, 5.4), d (0.8, 5.29), e (0.8, 4.6), and f (0.6, 4.4) with $k=1,c_1=-2,c_2=-1$

Fig. 11.

Phase portraits near the second and third Hopf–Hopf bifurcation points show two-coexisting periodic orbits for the fixed delays a (0.8, 7.79) and b (0.8, 10.3) with $k=1,c_1=-2,c_2=-1$

The mixed dynamical coexistence

In above section, we present the stable coexistence of two periodic orbits near the Hopf–Hopf bifurcation points. In fact, the typical dynamical behaviors of system include equilibrium, periodic orbit, and even quasi-periodic behavior. So, in what follows, we will illustrate some mixed dynamical coexistence, such as the coexistence of two equilibria and one periodic orbit, the coexistence of two periodic orbits and one quasi-periodic behavior.

For this purpose, firstly, we choose $c_2$ and ${\tau }_2$ as the variable parameters to obtain the pitchfork–Hopf bifurcation for $k=1,c_1=-1$, and ${\tau }_1=1$. It follows from theorem 1 that the pitchfork bifurcations are $c_2^{\ast }=c_1-1=-2$ and $c_2^{\ast }=1-c_1=2$. The Hopf bifurcation curve represented by $c_2$ and ${\tau }_2$ can be obtained by (27), as shown in Fig. 12a for $c_2^{\ast }=2$. There exists an intersection point coordinated as $(c_2,{\tau }_2)=(2,0.5525)$ for the pitchfork and Hopf bifurcation curves. Similar dynamical behaviors can be exhibited for $c_2^{\ast }=-2$. The eigenvalues are illustrated in Fig. 12b with $c_2$ increase. It follows that ${\omega }_2$ evolves to zero and ${\omega }_1$ maintains a positive when $c_2$ increases and passes through the pitchfork bifurcation point $c_2=2$. The result shows that the intersection is a pitchfork–Hopf bifurcation point. In the following, we will choose some fixed parameters $(c_2,{\tau }_2)$ to illustrate the mixed dynamical coexistence with two equilibria and one periodic orbit by time histories. The detail parameter values are shown in Fig. 12a.

Fig. 12.

a Hopf and pitchfork bifurcation curves show the pitchfork–Hopf bifurcation point, b evolution of frequencies ${\omega }_1$ and ${\omega }_2$ with $c_2$ increase for the system parameter values $k=1,c_1=-1,{\tau }_1=1$

It follows from Fig. 13a that time history for $(c_2,{\tau }_2)=(1.8,0.2)$ is converged into the trivial equilibrium. Fixed $c_2=1.8$ and increased ${\tau }_2=1.0$, time delay crosses through the Hopf bifurcation curve. The system obtains a periodic orbit near the trivial equilibrium, as shown in Fig. 13b. Further, when $c_2$ passes through the pitchfork bifurcation curve, the neural system gets two stable nontrivial equilibria bifurcated from the trivial equilibrium. The time history is illustrated in Fig. 13d for $(c_2,{\tau }_2)=(2.2,0.2)$. The trajectories for initial conditions (1, 0, 1) and $(-1,0,-1)$ evolve into the different nontrivial equilibria. Further, choosing $c_2=2.2$ and varying ${\tau }_2=0.7$, we present a stable coexistence with one periodic orbit and two equilibria, as shown in Fig. 13c for initial condition (1, 0, 1), $(-1,0,-1)$ and $(1,0,-1)$. The phase portraits in $x_1-x_2$ and $x_1-y_1$ are illustrated in Fig. 14 for a clear presentation.

Fig. 13.

Time histories for the fixed $(c_2,{\tau }_2)$a (1.8, 0.2), b (1.8, 1.0), c (2.2, 0.7), d (2.2, 0.2) show the dynamical behavior near the pitchfork–Hopf bifurcation point with $k=1,c_1=-1$ and ${\tau }_1=1$

Fig. 14.

Phase portraits show the mixed dynamical coexistence in the different plane a$x_1-y_1$ and b$x_1-x_2$ for $(c_2,{\tau }_2)=(2.2,0.7)$ with $k=1,c_1=-1$ and ${\tau }_1=1$

At the end of this paper, we present a more complex coexistence with some mixed dynamical behaviors including two periodic orbits and one quasi-periodic behavior. The system is chosen as $k=1,c_1=-2,c_2=-1$ and ${\tau }_1=10$. The phase portraits are shown in Fig. 15. Firstly, system (1) just exhibits a long and narrow periodic orbit for ${\tau }_2=1$ shown in Fig. 15a. With delay increasing to ${\tau }_2=2$, there is a new oval periodic orbit shown in Fig. 15b. The neural system exhibits a stable coexistence with two periodic orbits, where initial conditions are chosen as (1, 0, 1) and $(-1,0,1)$, respectively. Due to multiple frequencies, the long and narrow periodic orbit loses its stability and enters into a quasi-periodic behavior. The phase portrait is illustrated in Fig. 15c for delay ${\tau }_2=3$. Neural system (1) presents a mixed dynamical coexistence with one periodic orbit and one quasi-periodic behavior, where initial conditions are fixed as (1, 0, 1) and $(-1,0,1)$.

Fig. 15.

Phase portraits show the complex coexistence for a${\tau }_2=1$, b${\tau }_2=2$, c${\tau }_2=3$, d${\tau }_2=3.89$, e${\tau }_2=4.5$, and f${\tau }_2=6$ with $k=1,c_1=-1,c_2=-2$, and ${\tau }_1=10$

Further, when delay is chosen as ${\tau }_2=3.89$, system (1) presents a more complex coexistence with two periodic orbits and one quasi-periodic behavior, as shown in Fig. 15d for initial conditions (1, 0, 1), $(-1,0,1)$ and (2, 0, 2), respectively. The oval periodic orbit loses its stability and the system presents the mixed dynamical coexistence with a long and narrow periodic orbit and quasi-periodic behavior, as shown in Fig. 15e for ${\tau }_2=4.5$, where initial conditions are chosen as (1, 0, 1) and $(-1,0,1)$. Lastly, the quasi-periodic behavior evolves into a periodic orbit for ${\tau }_2=6$. The neural system retrieves two-coexisting with a long and narrow periodic orbit and an oval periodic orbit. The phase portraits are illustrated in Fig. 15f for initial conditions (1, 0, 1) and $(-1,0,1)$. It follows that the oval periodic orbit shown in Fig. 15f is just evolved from the long and narrow periodic orbit presented in Fig. 15b. To obtain a detail evolution of the dynamical behaviors, we illustrate the bifurcation diagram employing Poincáre sections ${\dot{x}}_2=0$, as shown in Fig. 16. The system presents the mixed dynamical coexistence with two periodic orbits and one quasi-periodic behavior in a long parameter region of delay ${\tau }_2$, where Fig. 16b is a Poincáre map for ${\tau }_2=4$ shown a quasi-periodic behavior.

Fig. 16.

The bifurcation diagram obtained by Poincáre sections ${\dot{x}}_2=0$ shows the complex coexistence for the fixed parameters $k=1,c_1=-1,c_2=-2$, and ${\tau }_1=10$, where (b) is a Poincáre map for ${\tau }_2=4$ denoted quasi-periodic behavior

Conclusion

Stable coexistence with different kinds of attractors is one of the most important dynamical properties in both biological and artificial neural systems. In this paper, we introduced self- and cross-interaction delays into an inertial two-neuron system presented by three first-order delayed differential equations. The system illustrated some coexistence patterns with equilibria, periodic orbits, and even quasi-periodic behavior. We firstly illustrated the coexistence with multiple equilibria through the pitchfork bifurcation sequences of the trivial/nontrivial equilibrium. By center manifold reduction and normal form method, we found the different bifurcation sequences of the supercritical and subcritical pitchfork bifurcations. The nontrivial equilibrium bifurcated from the trivial equilibrium presents the secondary pitchfork bifurcation. The system exhibits a stable coexistence with multiple equilibria. Using all pitchfork bifurcation curves, we obtained the detail regions, where the neural system has the different number of equilibrium.

We further analyzed the stability of the equilibrium (0, 0, 0) and found a periodic orbit. By the Hopf bifurcation curves, we presented the stability regions and exhibited the stability switching. The neural system lost and regained its stability through the forward and reverse Hopf bifurcation with time delay increase. Because of the multiple frequencies, the neural system illustrated a stable coexistence with two periodic orbits in the region of the Hopf–Hopf bifurcation points. Moreover, we illustrated the mixed dynamical coexistence with one periodic orbit and two equilibria in the region of the pitchfork–Hopf bifurcation point. Further, the neural system exhibited a stable coexistence including two periodic orbits and one quasi-periodic behavior.