Synchronization of Chemical Synaptic Coupling of the Chay Neuron System under Time Delay

: This paper studies the chemical synaptic coupling of Chay neurons and the effect of adding time delay on their synchronization behavior. The results indicate that coupling strength stimuli can affect the discharge activity and the synchronization behavior. In the absence of coupling strength, the Chay neurons have chaotic discharge behavior and the system is in a nonsynchronous state. When a certain coupling strength is added, the neurons change from chaotic discharge to ordered periodic discharge, and the system state changes from asynchronous to synchronous. On the other hand, a time lag can alter the coupled system from synchronous to asynchronous. delay and coupling strength on the coupled Chay neuron system was discussed. The model simulations showed that in a chemical synaptic-coupled Chay neuron system, coupling strength can promote synchronization behavior. Moreover, adding a time delay can inhibit the occurrence of its synchronous behavior, changing from a synchronous to a non-synchronized state. The results have given a deeper insight into the dynamic characteristics of neuron models and provide a basis for further research of neuronal system synchronization.


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 morbid 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].
Lin Feifei [12] designed an adaptive radial basis function (RBF) neural network and an integer-order parameter adaptive law based on the Lyapunov stability theory. Deng Bin [13] proposed inactivation of Na + channels are based on electrophysiological properties and relevant experimental data. This description improves the HH model, which involved a considerable number of variables that made it a cumbersome task to handle. More important, the addition of changes in the Ca 2+ -sensitive K + channels explains the nonlinear characteristics of neuronal impulses, spikes, and chaotic discharges. The differential equations are as follows: Equation (1) describes the variation of neuronal cells. The four terms on the right side of (1) represent the mixed Na + -Ca 2+ channels, K + ion channels with potential-dependent conductance, the K + ion channel current, and the transmembrane leakage current having a conductance that depends on the Ca 2+ ion concentration rather than the potential. The symbol V K denotes the reversible potential of the K + ion channels, V L is the reversible potential of various ions in other channels, V I is the reversible potential of the Na + -Ca 2+ channels, and V is the equilibrium potential. The parameters g I , g K,V , g K,C , and g L represent the maximum conductance of each channel, respectively.
Equation (2) describes the variation of the Ca 2+ ion concentration C in cell membranes. The two terms on the right side of (2) indicate the Ca 2+ ion channel current in and out of the cell membrane, K C denotes the ratio constant of intracellular Ca 2+ ion efflux, ρ is the proportionality constant, and V C represents the reversible potential of Ca 2+ ions.
Equation (3) describes the variation of the opening probability for the gate of the K + ion channels with the potential.
In the sequel Equations (4)-(6) show that the variables m ∞ , n ∞ and h ∞ are the stable probability values of the opening of the Ca 2+ , K + , and Na + ion channels. In Equation (7), τ n represents a time scale factor and the smaller its value, the steeper is the spike; the parameter λ n is the time kinetic constant. The expressions can be written as follows: The parameters adopted herein are depicted in Table 1.  The inter-spike interval bifurcation plot versus the parameter V I in the Chay neuron model is shown in Figure 1. With the change in V I in the range [75,165], the Chay neuron model reveals rich dynamic characteristics. From Figure 1, we find that the value of V I varies from small to large (marked as region A) and the neuron model enters six cycles and five cycles of periodical discharge patterns. As the V I value continues to increase (region B), when V I = 98.2 mV, the neuron model performs four cycles of cluster discharge activity. As the value of V I increases further, when V I = 107.8 mV, the neuron model enters a chaotic discharge mode, producing the shaded region marked as C in Figure 1. This is the expression in the inter-spike interval bifurcation diagram. As V I increases and the value of V I is greater than 117.6 mV we enter region D. The inter-spike interval sequence is restored to the second-cycle, and the first-cycle discharge mode after an inverse-periodic bifurcation.
characteristics. From Figure 1, we find that the value of VI varies from small to large (marked as region A) and the neuron model enters six cycles and five cycles of periodical discharge patterns. As the VI value continues to increase (region B), when VI = 98.2 mV, the neuron model performs four cycles of cluster discharge activity. As the value of VI increases further, when VI = 107.8 mV, the neuron model enters a chaotic discharge mode, producing the shaded region marked as C in Figure 1. This is the expression in the inter-spike interval bifurcation diagram. As VI increases and the value of VI is greater than 117.6 mV we enter region D. The inter-spike interval sequence is restored to the second-cycle, and the first-cycle discharge mode after an inverse-periodic bifurcation.

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: The following numerical values were adopted: synaptic reversible potential Vsyn = −65, synaptic threshold θ = −45, and ratio constant σ = 10. Equation groups with subscripts 1 and 2 denote sub-systems 1 and 2, respectively. The variable Hsyn represents the coupling strength and Vsyn describes the synapse reversible potential of the neuron, which depends on the presynaptic neuron and the recipient. The parameter θ is the synaptic threshold beyond which its presynaptic neurons begin to act on post-synaptic

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: The following numerical values were adopted: synaptic reversible potential V syn = −65, synaptic threshold θ = −45, and ratio constant σ = 10. Equation groups with subscripts 1 and 2 denote sub-systems 1 and 2, respectively. The variable H syn represents the coupling strength and V syn describes the synapse reversible potential of the neuron, which depends on the presynaptic neuron and the recipient. The parameter θ is the synaptic threshold beyond which its presynaptic neurons begin to act on post-synaptic neurons, and σ 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: e 0 = V 2 − V 1 , e 1 = n 2 − n 1 , e 2 = C 2 − C 1 , and e 3 = (|e 0 | + |e 1 | + |e 2 |)/3. Constructing Lyapunov function as then the derivative of L is: The type of substitution (15) can be obtained as follows: The substitution of e 0 = V 2 − V 1 , e 1 = n 2 − n 1 and e 2 = C 2 − C 1 in (17) gives the following: When Equation (18) is set up, dL dt < 0, then we can get the existence and limit of lim t → ∞ t 0 e i (τ)dτ, i = 0, 1, 2. According to the Lyapunov stability theory, by adjusting the parameters, the dL dt < 0, the error state e 0 , e 1 , e 2 and e 3 are asymptotically stable at the origin, namely, lim t → ∞ e i (t) = 0, i = 0, 1, 2, so the system has reached sync.
In a first phase, let us consider the discharge activities of two Chay neuron models without the coupling strength. When the coupling strength is H syn = 0, solving the two sets of equations leads to the (V 1 , V 2 ) locus of the two synaptic-coupled Chay neurons shown in Figure 2. The irregular movements indicate that the system is in an asynchronous state. The time history of V 1 and V 2 are represented in Figure 3. We verify that the values of V 1 and V 2 are different and that the discharge of the neurons is in a state of chaos; in other words, the system is in a chaotic state. In addition, the synchronization difference map of the coupled neurons is depicted in Figure 3. We observe that the synchronization discrepancy for all time instants indicates that the two neuron system is not in synchronization. In summary, it can be concluded that when H syn = 0, the two neuron systems are in a state of chaotic discharge and are in an asynchronous state.
In a second phase, let us consider the coupling strength H syn = 0.8 that leads to the (V 1 , V 2 ) diagram of the chemical synaptic-coupled Chay neuron system shown in Figure 4. The plot is a straight line with a slope of 1, indicating that the system is in a synchronized state. The time evolution of V 1 and V 2 depicted in Figure 5 shows also that the curves of V 1 and V 2 coincide. Similarly, it can also be seen from the synchronization discrepancy diagram of Figure 5 that the difference of the synchronization discrepancy with time is always zero, indicating that the coupled neuron system is in a synchronized state. In summary, when the coupling strength is H syn = 0.8, the system is in a synchronized state.           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.

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: where t and τ 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 (V1, V2) locus for a coupling strength Hsyn = 1.55 and time lag τ = 0 in the case of the two coupled neuron system (19)- (24). The values of V1 and V2 are identical and the system is in synchronization state. Similarly, the time history diagram of V1 and V2 reveals that the two curves of V1 and V2 overlap. The synchronization difference diagram is also shown in Figure 7, indicating that the two 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.

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: where t and τ 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 syn = 1.55 and time lag τ = 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 V 1 and 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 H syn = 1.55 and the time lag τ = 0, the two neuron coupling system is in a synchronized state. neuron system is in a synchronized state. In summary, when the coupling strength Hsyn = 1.55 and the time lag τ = 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 Hsyn = 1.55 and τ = 3.4. Figure 8 reveals that the locus (V1, V2) has an irregular behavior, indicating that the system is in an asynchronous state. The time history of V1 and V2, illustrated in Figure 9, shows that the two curves do not overlap. Similarly, Figure 9 demonstrates that the synchronization difference between V1 and V2 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 Hsyn = 1.55 and time lag τ = 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: τ =2.5 and Hsyn = 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. In Figure 11, the time history diagram does not overlap the motion curve of the neurons and the difference of the synchronous difference map is not zero, which indicates that the system is in an asynchronous state. neuron system is in a synchronized state. In summary, when the coupling strength Hsyn = 1.55 and the time lag τ = 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 Hsyn = 1.55 and τ = 3.4. Figure 8 reveals that the locus (V1, V2) has an irregular behavior, indicating that the system is in an asynchronous state. The time history of V1 and V2, illustrated in Figure 9, shows that the two curves do not overlap. Similarly, Figure 9 demonstrates that the synchronization difference between V1 and V2 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 Hsyn = 1.55 and time lag τ = 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: τ =2.5 and Hsyn = 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. In Figure 11, the time history diagram does not overlap the motion curve of the neurons and the difference of the synchronous difference map is not zero, which indicates that the system is in an asynchronous state. In a third phase, we discuss the synchronization of two chemical synapse-coupled neurons with the coupling strength H syn = 1.55 and τ = 3.4. Figure 8 reveals that the locus (V 1 , V 2 ) has an irregular behavior, indicating that the system is in an asynchronous state. The time history of V 1 and V 2 , illustrated in Figure 9, shows that the two curves do not overlap. Similarly, Figure 9 demonstrates that the synchronization difference between V 1 and 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 H syn = 1.55 and time lag τ = 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: τ =2.5 and 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. In Figure 11, the time history diagram does not overlap the motion curve of the neurons and the difference of the synchronous difference map is not zero, which indicates that the system is in an asynchronous state.
Finally, when the coupling strength increases to H syn = 5.3 with τ = 2.5, the phase plane diagram of the coupled neuron system (Figure 12) shows that the trajectory of the system is still an irregular random curve, indicating that the system does not reach the synchronous state. The time history diagram and the synchronous difference diagram in Figure 13 also verify that the system is in an asynchronous state. Finally, when the coupling strength increases to Hsyn = 5.3 with τ = 2.5, the phase plane diagram of the coupled neuron system (Figure 12) shows that the trajectory of the system is still an irregular random curve, indicating that the system does not reach the synchronous state. The time history diagram and the synchronous difference diagram in Figure 13 also verify that the system is in an asynchronous state.   Finally, when the coupling strength increases to Hsyn = 5.3 with τ = 2.5, the phase plane diagram of the coupled neuron system (Figure 12) shows that the trajectory of the system is still an irregular random curve, indicating that the system does not reach the synchronous state. The time history diagram and the synchronous difference diagram in Figure 13 also verify that the system is in an asynchronous state.           From the above discussion, we conclude that the addition of time lags in the two chemical synapticcoupled Chay neuron system yields a system change from periodic to chaotic discharge activity, leading to the modification from synchronous to non-synchronized state and destroying the original synchronization state.

Conclusions
In this paper, the chemical synaptic coupling of a Chay neuron system was studied. The discharge characteristics of the Chay neuron model were analyzed through the state locus, synchronization discrepancy plot, and inter-spike interval bifurcation diagram. The effect of time delay and coupling strength on the coupled Chay neuron system was discussed. The model simulations showed that in a chemical synaptic-coupled Chay neuron system, coupling strength can promote synchronization behavior. Moreover, adding a time delay can inhibit the occurrence of its synchronous behavior, changing from a synchronous to a non-synchronized state. The results have given a deeper insight into the dynamic characteristics of neuron models and provide a basis for further research of neuronal system synchronization. From the above discussion, we conclude that the addition of time lags in the two chemical synaptic-coupled Chay neuron system yields a system change from periodic to chaotic discharge activity, leading to the modification from synchronous to non-synchronized state and destroying the original synchronization state.

Conclusions
In this paper, the chemical synaptic coupling of a Chay neuron system was studied. The discharge characteristics of the Chay neuron model were analyzed through the state locus, synchronization discrepancy plot, and inter-spike interval bifurcation diagram. The effect of time delay and coupling strength on the coupled Chay neuron system was discussed. The model simulations showed that in a chemical synaptic-coupled Chay neuron system, coupling strength can promote synchronization behavior. Moreover, adding a time delay can inhibit the occurrence of its synchronous behavior, changing from a synchronous to a non-synchronized state. The results have given a deeper insight into the dynamic characteristics of neuron models and provide a basis for further research of neuronal system synchronization.
Author Contributions: Author (1) Kaijun Wu designed the main idea and methods of the paper. Author (2) Dicong Wang implemented the methods of the paper. Author (3) Chao Yu added included extra simulations according to the opinions of the reviewers. Author (4) J. Tenreiro Machado revised the study and perfected the language of the paper.