# Synchronization of Chemical Synaptic Coupling of the Chay Neuron System under Time Delay
## 1 Introduction  
Synchronization is a common phenomenon in nonlinear dynamics [1]. In recent years, this phenomenon has received considerable attention due to its huge application prospects in chemistry, biology, physics, and information processing [2; 3; 4; 5; 6]. Several theoretical and experimental studies have shown that the synchronization of the nervous system may play an important role both in revealing the brain's cognitive and sensory processes and in information processing mechanisms. It was discovered that neurological diseases such as Parkinson's, trembling of the human hands, and epilepsy are caused by some morph synchronization or non-synchronization [7; 8; 9]. Therefore, the presence, the absence, or the degree of synchronization can be important parts of the nervous system function or dysfunction. In order to study the nonlinear dynamic behavior and synchronization of single neurons and neural networks, researchers have established neuron mathematical models such as the Hodgkin-Huxley (HH), Morris-Lecar (ML), Hindmarsh-Rose (HR), FitzHugh-Nagumo (FHM), and the Chay representations. The Chay model was established in 1985 as a theoretical formulation with a high degree of uniformity based on different types of excitable cells playing an important role in Ca\({}^{2+}\)-related K\({}^{+}\) channels, such as neurons, cardiomyocytes, and nerve endings. The model is able to simulate various excitation modes of excitable cells with high biological and physiological characteristics. The study of synchronous control of biological systems became an important topic having in mind the control and the treatment of some diseases. It is generally believed that there are two ways for synchronizing the system: self-synchronization caused by natural coupling with feedback from various synapses between neurons, and control synchronization, caused by the coupling effect generated by an artificial system applying explicit feedback control [10]. In the first case, synchronization of the coupling neurons of the same type only occurs when a certain critical value of the coupling strength is reached, and this value exceeds the physiological condition of time [11]. [PERSON] [12] designed an adaptive radial basis function (RBF) neural network and an integer-order parameter adaptive law based on the Lyapunov stability theory. [PERSON] [13] proposeda Proportional and Integral based iterative learning control algorithm for the synchronization control problem of HH neurons. The master-slave neuron synchronization control under four different conditions was simulated. The results showed that after applying the control, HH neurons could quickly track the dynamic behavior of the rest of the neurons. [PERSON] [14] took hysteretic chaotic neurons and networks as their research object and proposed a control strategy based on filter tracking error to achieve neuron/network stability control. Under the control strategy, the neuron/network gradually stabilized to the extreme point of the optimization function so as to solve the optimization problem. [PERSON] [15] adopted auto-disturbance rejection control to study the chaotic synchronization between two neurons coupled through gaps. By adding controllers, it was verified that the neurons were very robust to external disturbances. In biological experiments, the synchronization of two coupled neurons could be achieved by applying an external direct current to the depolarization [16]. With the continuous development of the theory of nonlinear dynamic systems, several methods for achieving synchronization of neurons have been proposed, including feedback control, inversion design, nonlinear control, adaptive control, sliding mode control, and other techniques [17; 18; 19; 20; 21]. The time delay has a considerable influence on the dynamic properties of a coupled system.