###### Abstract  
In this paper, a new method is proposed based on the auxiliary system approach to investigate generalized synchronization between two identical neurons with unidirectional coupling. Different from other studies, the synchronization error system between the response and auxiliary systems is converted into a set of Volterra integral equations according to the Laplace transform method and convolution theorem. By using the successive approximation method in the theory of integral equations, an analytical criterion for the detection of generalized synchronization between two identical neurons is obtained. It is found that there is a time difference between two signals of neurons when the generalized synchronization between them is achieved. Furthermore, the value of the time difference has no relation to the generalized synchronization condition but depends on the coupling function between two neurons. The study in this paper shows that one can construct a coupling function between two identical neurons using the current signal of the drive system to predict its future signal or make its past signal reappear. generalized synchronization; FHN neuron; HR neuron;  
oscillators. However, how to find an appropriate Lyapunov function is the key to using this method. In addition, the synchronization condition obtained by using the Lyapunov function method is usually sufficient and highly conserved. [PERSON] et al. [11] proposed an indirect way of analytically verifying GS between two chaotic systems with unidirectional coupling by constructing an identical copy of the response system called the auxiliary system, which is driven by the same driving signal. The auxiliary system approach proved that GS between the drive-response system will be achieved if CS between the response and auxiliary systems occurs with different initial conditions. The master stability function method [12] was recommended in the auxiliary system approach to obtain the conditions for CS between the response and auxiliary system, whereas the auxiliary system approach can only be implemented numerically in most cases. It is a very meaningful topic to develop more convenient and effective approaches to analytically detect GS in a drive-response system. In recent years, the dynamical behavior of neurons has received increasing interest due to its applications in brain science, medical technology, artificial intelligence, and so on. Synchronization in neural systems in the brain is considered to be a crucial phenomenon related to cognition and the correlation of behavior [13; 14; 15; 16] because nerve cells do not have intelligence of their own. It is feasible to understand information processing in the brain by investigating synchronization between neurons [17; 18; 19; 20; 21; 22]. The Hodgkin-Huxley (HH) Equation [23] was usually used to construct neural systems and exhibit the neural dynamic behavior. A simplified version of the HH equation, such as the FitzHugh-Nagumo (FHN) model [24] and the Hindmarsh-Rose (HR) model [25; 26; 27; 21], which can show excitability and neural bursting of a neuron, is more convenient in the investigation of neural behavior based on nonlinear dynamical theory. The study of synchronization in FHN and HR neurons makes plenty of sense in understanding their information transfer process. In this paper, GS between two identical FHN and HR neurons with unidirectional coupling is investigated based on the auxiliary system approach. Different from other studies, the synchronization error between the response and auxiliary systems is found to satisfy a Volterra integral equation from the Laplace transform method and convolution theorem. According to the successive approximation method [28] in the Volterra integral equation theory, an analytical criterion is given for the detection of GS between two identical FHN and HR neurons.