## 4 GS between Two HR Neurons  
A HR neuron is described by the following equation of motion:  
\[ \begin{split}\dot{u}&=v-au^{3}+bu^{2}-w+L_{xt},\\ \dot{v}&=c-du^{2}-v,\\ \dot{w}&=rs_{0}(u-x_{0})-rw,\end{split}  \tag{25}\]  
where \(u\) is the membrane potential, \(v\) is a recovery variable associated with fast current, and \(w\) represents a slowly changing adaptation current. \(a\), \(b\), \(c\), \(d\), \(s_{0}\), \(r\), \(x_{0}\) are parameters,and \(l_{ext}\) is the external current input. For simplicity, by letting \(x=u-u_{0}\), \(y=v-v_{0}\), and \(z=w-w_{0}\), where \((u_{0},v_{0},w_{0})\) is the equilibrium of System (25), System (25) becomes  
\[\dot{x} =-ax^{3}+\rho_{1}x^{2}+\rho_{2}x+y-z, \tag{26}\] \[\dot{y} =-dx^{2}+\rho_{3}x-y,\] \[\dot{z} =r(s_{0}x-z),\]  
where \(\rho_{1}=-3 au_{0}+b\), \(\rho_{2}=-3 au_{0}^{2}+2 bu_{0}\), \(\rho_{3}=-2 du_{0}\), \(u_{0}\) is the unique real root to equation \(au_{0}^{3}+(d-b)u_{0}^{2}+s_{0}u_{0}-(s_{0}x_{0}+c+l_{ext})=0\). Consider two HR neurons described by System (26) have the following coupling form:  
\[\dot{x}_{1} =-ax_{1}^{3}+\rho_{1}x_{1}^{2}+\rho_{2}x_{1}+y_{1}-z_{1}, \tag{27}\] \[\dot{y}_{1} =-dx_{1}^{2}+\rho_{3}x_{1}-y_{1},\] \[\dot{z}_{1} =r(s_{0}x_{1}-z_{1}),\] \[\dot{x}_{2} =-ax_{2}^{3}+\rho_{1}x_{2}^{2}+\rho_{2}x_{2}+y_{2}-z_{2}+\alpha( x_{2}-J(x_{1})),\] \[\dot{y}_{2} =-dx_{2}^{2}+\rho_{3}x_{2}-y_{2},\] \[\dot{z}_{2} =r(s_{0}x_{2}-z_{2}),\]  
where \(J(x_{1})\) is a continuous function of \(x_{1}\), and \(\alpha\) represents the coupling strength. Similarly, we consider that System (27) has the property  
\[\lim_{t\to\infty}||x_{2}-J(x_{1})||=0. \tag{28}\]  
Clearly, \(J(x_{1})\) can have the form of \(J(x_{1})=x_{1}(t-\tau)\), in which \(\tau\geq 0\) is a delay. From the auxiliary system approach, the auxiliary system corresponding to System (27) is given by  
\[\dot{y} =-dx^{2}+\rho_{3}x-y,\]0 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]1 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]2  
According to the analysis in the last section, Condition (28) will be satisfied if  
\[\dot{y} =-dx^{2}+\rho_{3}x-y,\]3  
By letting  
\[\dot{y} =-dx^{2}+\rho_{3}x-y,\]4  
the response system and the auxiliary system in Systems (27) and (29) can be rewritten as  
\[\dot{y} =-dx^{2}+\rho_{3}x-y,\]5 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]6 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]7 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]8 \[\dot{y} =-dx^{2}+\rho_{3}x-y,\]9 \[\dot{z} =r(s_{0}x-z),\]0  
Then Condition (30) is equivalent to  
\[\dot{z} =r(s_{0}x-z),\]1From the Laplace transform method used in the last section, the Laplace transform of \(E_{1}\) can be given by  
\[\dot{z} =r(s_{0}x-z),\]2  
where  
\[\dot{z} =r(s_{0}x-z),\]3 \[\dot{z} =r(s_{0}x-z),\]4 \[\dot{z} =r(s_{0}x-z),\]5 \[\dot{z} =r(s_{0}x-z),\]6  
According to the analysis in the last section based on the theory of Volterra integral equations (Equations (17)-(19)), the condition that \(E_{1}=0\) means the following polynomial (the denominator in the right-hand side of Equation (34))  
\[\dot{z} =r(s_{0}x-z),\]7  
has no roots with positive real parts. That is,  
\[\dot{z} =r(s_{0}x-z),\]8  
Next, numerical simulations are performed to prove that the method to investigate GS proposed in this paper is valid even if the drive-response system has chaotic attractor. Here, \(a=1.0\), \(b=3.0\), \(c=1.0\), \(d=5.0\), \(s_{0}=4.0\), \(r=0.013\), \(x_{0}=-1.6\), and \(I_{ext}=3.0\). The two HR neurons in System (27) with \(\alpha=0\) exhibit burst-spike chaotic behavior. From Condition (36), the functional relation \(x_{2}=x_{1}(t-\tau)\) appears between two HR neurons when \(\alpha<-1.129\). To demonstrate the validity of the condition, \(\alpha=-0.9\) and \(\alpha=-1.3\) are taken to perform the numerical simulations of System (27), respectively. The initial conditions are given by \(x_{1}(t)=0.2\), \(t\in(-\tau,0]\), \(y_{1}(0)=0.1\), \(z_{1}(0)=0.2\), \(x_{2}(0)=0.1\), \(y_{2}(0)=0.1\), \(z_{2}(0)=0.1\), \(x_{3}(0)=0.2\), \(y_{3}(0)=0.2\), and \(z_{3}(0)=0.1\).