## 3 Establishment of Neuronal Axonal Signal Transmission Model
Finally, the amplitude-frequency and phase-frequency characteristics of \(G(S)\) are analyzed to determine the physical characteristics of the neuron self-growing model:  
\[G(j\omega)=\frac{1}{\tau j\omega+b} \tag{19}\]  
where \(\omega\) is the angular frequency. The amplitude-frequency characteristic of Equation (19) is  
\[|G(j\omega)|=\frac{1}{\sqrt{\tau^{2}\omega^{2}+b^{2}}} \tag{20}\]  
The phase-frequency characteristic of Equation (19) is  
\[\angle G(j\omega)=-\arctan(\frac{\tau\omega}{b}) \tag{21}\]  
(1) Influence of axon length on amplitude and phase lag  
Suppose an axon cable is composed of \(N\) basic links as shown in Figure 4; then, the amplitude of the input signal is attenuated to \((1/\sqrt{\tau^{2}\omega^{2}+b^{2}})^{N}\) after being transmitted through the axon, and the phase lag is -\(N\)arctan(\(\tau\omega\)/\(b\)). The longer the axon is, the more basic links in the corresponding cable model there should be; in other words, the larger \(N\) is. We can see that \(b>1\) from Equation (17), and so the longer the axon is, the larger \(N\) is and the larger the amplitude attenuation and phase lag are. The length of the axon can be obtained by the iteration in Equation (8). If the length of the axon corresponding to the basic link of a resistance capacitance loop is \(l_{1}\) and the total length of the axon is \(L\)[23], then the amplitude of the signal will decay to \((1/\sqrt{\tau^{2}\omega^{2}+b^{2}})^{U_{1}}\) and the phase lag will be -\(N\)arctan(\(\tau\omega\)/\(b\)) after passing through the whole axon. (2) Diameter of the axon  
As can be seen from Equations (20) and (21), the amplitude-frequency and phase-frequency characteristics of the neuron signal transmission are determined by the two parameters \(\tau\) and \(b\). In Equations (16) and (17), \(R_{2}\) is the electrical leakage resistance between the inside and outside of the cell within a basic link, whose resistance value is much higher than \(R_{1}\) and \(R_{3}\), which is equivalent to the myelin sheath of the outer layer of the axon of biological neurons and plays an insulating role. \(C\) is the leakage capacitance between the inside and outside of the cell within a basic link. \(R_{3}\) is the conductive resistance of the extracellular fluid within a basic link. \(R_{2}\), \(R_{3}\) and C are the same for every neuron and do not change with the gradual growth of axons. \(R_{1}\) represents the resistance of the axon core within a basic link, which can be obtained according to the resistance law:  
\[R_{1}=\rho\frac{l_{1}}{s} \tag{22}\]  
where \(\rho\) is the resistivity of the inner core of the axon, \(s\) is the cross-sectional area of the axon--\(s=\pi^{2}d/4\) since the cable model is cylindrical--and \(d\) is the diameter of the axon. The axon diameters of different neurons determine the amplitude-frequency and phase-frequency characteristics of the basic link in the cable model. (3) Connection weight to the postsynaptic neuron  
Since \(|G(j\omega)|<1\), the cable model shown in Figure 4 must cause the attenuation of signal amplitude during signal transmission. However, we occasionally want a single neuron to transmit the signal without changing its amplitude when controlling, only changing its phase. For example, in the CPG control network of the joint of a legged robot, each joint is controlled by a motor neuron with joint angular displacement output, and the output signals of the motor neuron in the symmetrical position of the body need to have the same amplitude, because the joint angle of these joints has the same range. If all the intermediate neurons in the network do not change the amplitude of the signal during signal transmission, the output of each joint with the same amplitude can be obtained from the input with the same amplitude.