# Synchronization of Chemical Synaptic Coupling of the Chay Neuron System under Time Delay
## 4 Time-Delay Inhibition of Synchronization of the Chemical Synaptic-Coupled Chay Neuron System  
The coupling modes between neurons are divided into two types: electrical and chemical synaptic coupling. The chemical synapses depend on the release of neurotransmitters and are connected by means of dendrites and axons. Furthermore, chemical synapses dominate throughout the neuronal system. Due to the presence of synaptic gaps in the nervous system, there is a significant delay in the transmission of neural signals through the synapse. This effect results in a time delay in the transmission of signals, thereby causing a time lag in the nervous system [47; 48; 49]. Due to the presence of a time lag, the coupled neurons become an infinite dimensional dynamic system that exhibit a richer nonlinear behavior. We consider two chemical synaptic-coupled Chay neuron models with time delays as follows:  
\[\frac{\mathrm{d}V_{1}}{\mathrm{d}t}=g_{1}m_{\infty}^{3}h_{\infty}(V_{1}-V_{1 })+g_{K,V}(V_{K}-V_{1})n_{1}^{4}+g_{K,C}\frac{C_{1}}{1+C_{1}}(V_{K}-V_{1})+g_{L }(V_{L}-V_{1})+\frac{H_{\mathrm{syn}}(V_{\mathrm{syn}}-V_{1})}{1+\exp(-\sigma (V_{2}(1-\tau)-\theta))} \tag{19}\]  
\[\frac{\mathrm{d}C_{1}}{\mathrm{d}t}=\varrho\Big{[}m_{\infty}^{3}h_{\infty}(V_ {C}-V_{1})-k_{C}C_{1}\Big{]} \tag{20}\]  
\[\frac{\mathrm{d}n_{1}}{\mathrm{d}t}=\frac{n_{\infty}-n_{1}}{\tau_{n}} \tag{21}\]  
\[\frac{\mathrm{d}V_{2}}{\mathrm{d}t}=g_{1}m_{\infty}^{3}h_{\infty}(V_{1}-V_{2} )+g_{K,V}(V_{K}-V_{2})n_{2}^{4}+g_{K,C}\frac{C_{2}}{1+C_{2}}(V_{K}-V_{2})+g_{ L}(V_{L}-V_{2})+\frac{H_{\mathrm{syn}}(V_{\mathrm{syn}}-V_{2})}{1+\exp(-\sigma(V_{1}(1 -\tau)-\theta))} \tag{22}\]  
\[\frac{\mathrm{d}C_{2}}{\mathrm{d}t}=\varrho\Big{[}m_{\infty}^{3}h_{\infty}(V_ {C}-V_{2})-k_{C}C_{2}\Big{]} \tag{23}\]  
\[\frac{\mathrm{d}n_{2}}{\mathrm{d}t}=\frac{n_{\infty}-n_{2}}{\tau_{n}} \tag{24}\]  
where \(t\) and \(\tau\) denote time and time lag, respectively. In Equations (19)-(24), the significance and values of other parameters and variables are consistent with Section 2. Figure 6 shows the (V\({}_{1}\), V\({}_{2}\)) locus for a coupling strength H\({}_{\mathrm{syn}}\) = 1.55 and time lag \(\tau\) = 0 in the case of the two coupled neuron system (19)-(24). The values of V\({}_{1}\) and V\({}_{2}\) are identical and the system is in synchronization state. Similarly, the time history diagram of V\({}_{1}\) and V\({}_{2}\) reveals that the two curves of  
Figure 5: Time history diagram and synchronization discrepancy diagram when H\({}_{\mathrm{syn}}\) = 0.8. \(\rm V_{1}\) and \(\rm V_{2}\) overlap. The synchronization difference diagram is also shown in Figure 7, indicating that the two neuron system is in a synchronized state. In summary, when the coupling strength \(\rm H_{syn}=1.55\) and the time lag \(\rm\tau=0\), the two neuron coupling system is in a synchronized state. In a third phase, we discuss the synchronization of two chemical synapse-coupled neurons with the coupling strength \(\rm H_{syn}=1.55\) and \(\rm\tau=3.4\). Figure 8 reveals that the locus (\(\rm V_{1}\), \(\rm V_{2}\)) has an irregular behavior, indicating that the system is in an asynchronous state. The time history of \(\rm V_{1}\) and \(\rm V_{2}\), illustrated in Figure 9, shows that the two curves do not overlap. Similarly, Figure 9 demonstrates that the synchronization difference between \(\rm V_{1}\) and \(\rm V_{2}\) is always present, revealing that the system does not reach the synchronized state. In summary, the two chemical synaptic coupling Chay neuron system is in an asynchronous state when the coupling strength \(\rm H_{syn}=1.55\) and time lag \(\rm\tau=3.4\). In order to further verify the effect of coupling strength and time delay on the neuron system, this paper fixed the time delay and changed the coupling strength: \(\rm\tau=\)2.5 and \(\rm H_{syn}=2.7\). By observing Figure 10, the curve of random motion of the system in phase plane shows that the neuron has chaotic discharge activity and the neuron system is in the unsynchronized state.