## 4 Structure of Local CPG Network  
(1) Study of neuron model  
In 1987, [PERSON] proposed a mathematical neuron monomer model and studied the \"oscillator\" network output with different numbers of neurons and different connection structures based on it [24]. The mathematical model of the neuron monomer is  
\[\left\{ \begin{array}{l}T_{\frac{d\tau}{dt}}\frac{dx}{dt}+x=s-bf\\ y=g(x-\theta),\ \ \ \ \ (g(x)=\max\{0,x\})\\ T_{a}\frac{df}{dt}+f=y\end{array}   
As can be seen from Figure 6, first, the neuron is activated and the membrane potential rises; then, the membrane potential declines slowly due to the fatigue effect and remains at a low level. Although the step signal continues, the neuron will not be activated again. We can see that, under appropriate parameters, the mathematical model of the neuron monomer conforms to the signal transmission characteristics of biological neurons. (2) Double neural oscillator model  
Connect two neuron monomer models and make the two inhibit each other as shown in Figure 7. If N1 and N2 are the neurons, the hollow endpoint connection presents the excitatory connection and the solid endpoint connection presents the inhibitory connection; S1 and S2 are the step input signals with an amplitude of 1 with slightly different step times. If S1 and S2 have the same step time, the two neurons will have the same output and will not produce an oscillation because of the symmetrical structure of the network. The mathematical model of each neuron monomer is  
\[\left\{ \begin{array}{l}T_{ri}\frac{dx_{i}}{dt}+x_{i}=s_{i}-b_{i}f_{i}-\sum \limits_{j=1}^{n}a_{ji}y_{j}\\ y_{i}=\max\{0,x_{i}\}\\ T_{ai}\frac{df_{i}}{dt}+f_{i}=y_{i}\end{array} \right. \tag{28}\]  
where \(\sum\limits_{j}a_{ji}y_{j}\) are the inputs from other neurons. If the input is excitatory, the operator preceding it has a plus sign, with a minus sign for the inhibitory input. Assume \(T_{r1}=T_{r2}=1\), \(b_{1}=b_{2}=2.5\), \(a_{21}=a_{12}=1.5\) and \(T_{a1}=T_{a2}=12\) and calculate the variation of the output signal of the oscillator model with time. The results are shown in Figure 8.  
In the spinal CPG network of mammals, the interactivity of neurons responsible for the coordination of the left and right sides of the body and the coordination of the extensor and flexor muscles of each joint inhibits the output [25]. It can be seen in Figure 5 that the reciprocal inhibition outputs of the two neurons have the same form as the reciprocal inhibition outputs of the neurons in the biological CPG network, indicating that the neuron monomer model in Equation (26) can reflect the characteristics of the reciprocal inhibition output caused by the reciprocal inhibition structure in the biological CPG network. Figure 8: Reciprocal output of two neurons. Figure 7: Network of two neurons that inhibit each other. (3) Connection structure of local CPG network  
As can be seen from the analysis above, as long as the neuron model represented by Equation (27) is composed of inhibitory connections among the neurons and appropriate parameters are taken, the interactive inhibitory output between the neurons will be obtained. Since each neuron monomer in the neuron model represented in Equation (27) has two time constants, _Tr_ and _Ta_, and each neuron in the simplified model of self-growing neurons established in Section 3 has only one time constant, _Tr_, we cannot make the growth algorithm of single neuron grow into a local CPG network which has the function of controlling a robot joint. In order to solve this contradiction, we modify the original neuron monomer model, as shown in Equation (29), according to the characteristics of the neuron connections in a biological CPG network. \[\left\{ \begin{array}{l}T_{ri}\frac{dx_{i}}{dt}+x_{i}=s_{i}-\sum\limits_{j=1} ^{n}a_{ji}y_{j}\\ y_{i}=\max\{0,x_{i}\}\end{array} \right.