# Synchronization of Chemical Synaptic Coupling of the Chay Neuron System under Time Delay
## 3 Synchronization of the Chemical Synaptic-Coupled Chay Neuron System  
Synapses have strong adaptability (plasticity) that depend on various external stimuli (such as the action potential of presynaptic neurons). Moreover, chemical synapses can be divided into two types: excitatory and inhibitory [40; 41; 42; 43]. Chemical synapses, which are the most common type of cellular connections in the animal's nervous system, are highly flexible, making this topic widely studied [44; 45; 46].  
To analyze the synchronous behavior of chemical synaptic-coupled Chay neurons, this paper adopted two chemical synaptic-coupled Chay neuron models as follows:  
\[\frac{\rm dV_{1}}{\rm dt}=g_{\rm i}m_{\infty}^{3}h_{\infty}(V_{\rm I}-V_{1})+g _{\rm K,V}(V_{\rm K}-V_{1})n_{1}^{4}+g_{\rm K,C}\frac{C_{1}}{\rm T+C_{1}}(V_{ \rm K}-V_{1})+g_{\rm L}(V_{\rm L}-V_{1})+\frac{H_{\rm syn}(V_{\rm syn}-V_{1} )}{1+\exp(-\sigma(V_{2}-\theta))} \tag{8}\]  
\[\frac{\rm dC_{1}}{\rm dt}=\varrho\Big{[}m_{\infty}^{3}h_{\infty}(V_{\rm C}-V_ {1})-k_{\rm C}C_{1}\Big{]} \tag{9}\]  
\[\frac{\rm dn_{1}}{\rm dt}=\frac{n_{\infty}-n_{1}}{\tau_{\rm n}} \tag{10}\]  
\[\frac{\rm dV_{2}}{\rm dt}=g_{\rm i}m_{\infty}^{3}h_{\infty}(V_{\rm I}-V_{2})+g _{\rm K,V}(V_{\rm K}-V_{2})n_{2}^{4}+g_{\rm K,C}\frac{C_{2}}{\rm T+C_{2}}(V_{ \rm K}-V_{2})+g_{\rm L}(V_{\rm L}-V_{2})+\frac{H_{\rm syn}(V_{\rm syn}-V_{2} )}{1+\exp(-\sigma(V_{1}-\theta))} \tag{11}\]  
\[\frac{\rm dC_{2}}{\rm dt}=\varrho\Big{[}m_{\infty}^{3}h_{\infty}(V_{\rm C}-V_ {2})-k_{\rm C}C_{2}\Big{]} \tag{12}\]  
\[\frac{\rm dn_{2}}{\rm dt}=\frac{n_{\infty}-n_{2}}{\tau_{\rm n}} \tag{13}\]  
The following numerical values were adopted: synaptic reversible potential \(\rm V_{syn}=-65\), synaptic threshold \(\theta=-45\), and ratio constant \(\sigma=10\). Equation groups with subscripts 1 and 2 denote sub-systems 1 and 2, respectively. The variable \(\rm H_{syn}\) represents the coupling strength and \(\rm V_{syn}\) describes the synapse reversible potential of the neuron, which depends on the presynaptic neuron and the recipient. The parameter \(\theta\) is the synaptic threshold beyond which its presynaptic neurons begin to act on post-synaptic neurons, and \(\sigma\) denotes the rate constant for the onset of excitation or inhibition. The rest of the variables indicate that the significance and value do not change.  
In order to better study the synchronization of coupled neurons, we define the synchronization discrepancy between coupled neurons: \(\rm e_{0}=V_{2}-V_{1}\), \(\rm e_{1}=n_{2}-n_{1}\), \(\rm e_{2}=C_{2}-C_{1}\), and \(\rm e_{3}=(\,|\,e_{0}\,|\,+\,|\,e_{1}\,|\,+\,|\,e_{2}\,|)/3\).  
Figure 1: Inter-spike interval bifurcation diagram versus the parameter \(\rm V_{I}\).  
Figure 4: Locus (V1, V2) when H\({}_{\rm syn}\) = 0.8.  
Figure 3: Time history diagram and synchronization discrepancy graph when H\({}_{\rm syn}\) = 0.  
Figure 2: Locus (V\({}_{1}\), V\({}_{2}\)) when H\({}_{\rm syn}\) = 0.  
It can be concluded after the above analysis that appropriate changes in the coupling strength can induce the synchronization of the chemical synaptic-coupled Chay neuron system.