## 4 Structure of Local CPG Network
\tag{29}\]  
The original neuron monomer model is described with the structure shown in Figure 6. We can see that the excitatory connection from the input neurons to the output neurons in Figure 9 is equivalent to the input signal \(S\) of the original model. The inhibition from the Renshaw Cell to the output neurons is equivalent to the fatigue characteristic in the original model; thus, the input and output characteristics of signal transmission of the structure shown in Figure 9 are essentially the same as those in the Matsuoka neuron monomer model. According to Equation (29), the modified neuron monomer model only contains one time constant, _Tr_, and each axon contains a connection weight \(a\) to the postsynaptic neuron, which is consistent with the simplified model of self-growing neurons established in Section 3. Moreover, the connection structure shown in Figure 6 has the anatomical basis of the biological CPG network. Known as an inhibitory intermediate neuron directly connected to motor neurons, the Renshaw Cell is involved in the coordination of most extensor and flexor muscles as part of the ipsilateral inhibitory network [14; 25]. Moreover, the structure in Figure 6 can be seen in many models of spinal motor nerve circuits [26; 27]. A local CPG network for the control of a quadruped robot's hip joint can be formed with four identical structures, as shown in Figure 10. In Figure 10, N1, N2, N3 and N4 are the output neurons, respectively corresponding to the four hips of the quadruped robot. N5, N6, N7 and N8 are input neurons, and S1, S2, S3 and S4 are step input signals of the same amplitude. N9, N10, N11 and N12 are four output neurons connected with Renshaw Cells, _Tri_ is the time constant of each neuron and \(a_{\text{i,j}}\) is the connection weight of the synapse between the axon of neuron \(i\) and the neuron \(j\); the connection weights between peripheral neurons are not shown in Figure 10. The network structure is the growth target of the local CPG network self-growing algorithm. Figure 9: Equivalent model of the original neuron monomer model. (4) Output of local CPG network  
One of the basic features of a biological CPG network is that it can obtain a periodic output signal with a certain rhythm from a simple non-rhythmic input signal. In order to verify that the network growth model proposed in this paper can obtain growth results with biological CPG network characteristics, a CPG network growth experiment was carried out and is presented in this section. Assume that the radius \(r\) of a single neuron in the growth model is 0.1 units of length and the growth space is a cube with a side length of 5; 12 neurons were randomly distributed in the growth space as shown in Figure 11.  
Figure 11: Twelve randomly distributed neurons and the connection structure arising from the network. Figure 10: Local Central Pattern Generator (CPG) network connection structure for the hip joint control of a quadruped robot. In Figure 11, a circle represents a neuron, and the number (1, 2,..., 12) next to it shows the number of neurons. The neurons numbered 1, 2, 3 and 4 are the output neurons, represented by N1, N2, N3 and N4 (as shown by the green arrow in Figure 11); the neurons numbered 5-12 are peripheral neurons, represented by N5-N12. Numbers N5, N6, N7 and N8 are input neurons (as shown by the red arrow in Figure 11). The solid blue lines are the growing connections, as shown in Figure 11.  
For simplicity, take \(l_{1}=r=0.1\), \(\tau=10\). Putting the length of the blue solid line in Figure 11 into Equation (26), the value of Tr can be obtained as shown in Table 1.  
Step input signals with amplitude of 1 were input to the four input neurons. Observe the output signals of the network; a representative growth result was taken for discussion. The connection structure of the network growth is shown in Figure 11. In Figure 11, each neuron axon contains two parameters: the time constant Tr and the connection weight a.