## 3 Two Coupled FHN Neural Systems with Time Delay
\begin{array}{l}e_{i}=a_{i}\cos(\theta_{i}),i=1,2,\\ \dot{e}_{1}=-C_{1}(1-\frac{1}{10}\cos(4\theta_{1})-\frac{2}{5}\cos(2\theta_{1 })),\\ \dot{e}_{2}=-C_{2}(1-\frac{1}{10}\cos(4\theta_{2})-\frac{2}{5}\cos(2\theta_{2 })),\end{array}  \tag{15}\]  
where  
\[C_{1}=\frac{\sqrt{5\alpha_{3}}a_{1}^{2}}{4},\ C_{2}=\frac{\sqrt{5\alpha_{3}}a _{2}^{2}}{4}, \tag{16}\]  
Employing the energy method, we have:  
\[ \begin{array}{l}\frac{dE_{1}}{dt}=-f_{1}(e_{1},\dot{e}_{1},e_{1}(t-\tau), \dot{e}_{1}(t-\tau))\dot{e}_{1}\\ \qquad=[(3(\varepsilon_{1}^{2}+\varepsilon_{2}^{2})+\alpha_{1}e_{2}+\alpha_{2 })\dot{e}_{1}+(6e_{2}+\alpha_{1})e_{1}\dot{e}_{2}\\ \qquad+(3\alpha_{3}e_{2}^{2}+2\alpha_{4}e_{2}+\alpha_{5})e_{1}-\alpha_{6}e_{1 }(t-\tau)-\dot{c}\dot{e}_{1}(t-\tau)]\dot{e}_{1}.\end{array}  \tag{17}\]  
Consider \(\theta_{1}\) and \(\theta_{2}\) have the following forms:  
\[ \begin{array}{l}\theta_{1}=\omega t+\varphi_{1},\\ \theta_{2}=\omega t+\varphi_{2},\end{array}  \tag{18}\]  
for \(\omega=\omega_{0}\) and some critical value of delay determined by Equation (24), the origin of Equation (12) will lose its stability. In such a case, the coupled neural system cannot remain at neural rest. Synchronous spikes in the coupled system will occur due to an appropriate value of time delay. The distribution of positive real roots of an algebraic equation similar to Equation (23) has been treated in other studies [40; 41]. In the following, we will employ a complete discrimination system for polynomials [42; 43] and the Descartes sign method to analyze the condition that Equation (23) has one or more positive real roots. We denote the Sylvester matrix of \(p(y)\) and \(p^{\prime}(y)\) by \(Discr(p)\), where \(p^{\prime}(y)\) is the derivative of \(p(y)\) with respect to \(y\) and denote the determinant of the submatrix of \(Discr(p)\) formed by the first \(2k\) rows and the first \(2k\) columns by \(D_{k}(p)\), \(k=1\), \(2\), \(3\), \(4\). If the number of the sign changes in the discriminant sequence:  
\[\{D_{1}(p),D_{2}(p),D_{3}(p),D_{4}(p)\} \tag{27}\]  
is \(\upsilon\), then the number of pairs of distinct conjugate imaginary roots of \(p(y)\) is equal to \(\upsilon\). Furthermore, if the number of nonvanishing members of the list is \(l\), then the number of the distinct real roots of \(p(y)\) equals to \(l-2\upsilon\). According to the procedure [42], we have:  
\[ \begin{array}{l}D_{1}=1,\\ D_{2}=3r_{1}^{2}-8r_{2},\\ D_{3}=-4r_{2}^{2}+r_{1}^{2}r_{2}^{2}+2(7r_{3}r_{1}+8r_{4})r_{2}-6r_{1}^{2}r_{4 }-18r_{3}^{2}-3r_{1}^{3}r_{3},\\ D_{4}=-27r_{1}^{4}r_{4}^{4}+18r_{3}^{3}r_{2}r_{3}-4r_{1}^{3}r_{3}^{3}-4r_{1}^{ 2}r_{3}^{2}r_{4}+r_{1}^{2}r_{2}^{2}r_{3}^{2}\\ \quad\quad+144r_{2}^{2}r_{2}^{2}-6r_{1}^{2}r_{3}^{2}r_{4}-80r_{1}^{2}r_{2}^{2}r _{3}r_{4}+18r_{1}r_{2}r_{3}^{2}+16r_{2}^{4}r_{4}\\ \quad\quad-4r_{2}^{3}r_{3}^{2}-192r_{1}r_{3}r_{4}^{2}-128r_{2}^{2}r_{4}^{2}+144 r_{2}r_{3}^{2}r_{4}-27r_{3}^{4}+256r_{4}^{3}.\end{array}  \tag{28}\]  
We can further determine the number of positive real roots of Equation (23) based on the Descartes sign method by considering the number of the sign changes in the discriminant sequence:  
\[\{1,r_{1},r_{2},r_{3},r_{4}\}. \tag{29}\]  
Then we have the following:  
* If \(r_{4}<0\), Equation (23) has at least one positive real root. * If the number of sign changes in Equation (27) is zero, Equation (23) only has real roots. The number of the positive real roots equals to the number of the sign changes in Equation (29). * If the number of the sign changes in Equation (27) is one and the number of nonvanishing members in the sequence is \(4\), Equation (23) has two different real roots, the sign of which can be determined by the Descartes sign method. * If the number of sign changes in Equation (27) is one and the number of nonvanishing members in the sequence is \(3\), Equation (23) has two identical real roots, the sign of which can be determined by the Descartes sign method. Next, we numerically show that the existence of time delay in signal transmission can make the synchronous spikes occur in Equation (12).