## 3 Establishment of Neuronal Axonal Signal Transmission Model  
In order to describe the signal transmission process of the neurons listed above, a simplified model of neuron signal transmission, as shown in Figure 3, is established in this paper. If there are n neurons on a neuron's dendrite to form synaptic connections with it, the output signals of these n neurons are \(y_{1},y_{2},\ldots,y_{n}\), the connection weights of synapses are \(a_{1}\), \(a_{2},\ldots,a_{n}\), the change of membrane potential caused by the input is \(s\) and the excitation threshold is \(\theta\); the membrane potential  
Figure 3: The signal transmission model of the neuron. Figure 2: Growth of growth cone under different intensities of random motion. is \(x\) when reaching the axon's terminal after being transmitted through the whole axon, and the neuron output signal is \(y\). Then, the signal transmission process of the neuron is  
\[\left\{ \begin{array}{l}s=\sum\limits_{i=1}^{n}a_{i}y_{i}\\ x=f(s-\theta)\\ y=g(x)\end{array} \right. \tag{10}\]  
where \(f(. )\) represents the function of an axon on membrane potential and \(g(. )\) represents the nonlinear output function of neurons, which is different for different biological neurons. In this paper, the commonly used threshold function is taken; i.e., \(g(x)=\max(0,\,x)\). In 1957, [PERSON] proposed the equivalent cable model of an axon during signal transmission [22], which regarded the cable as composed of many identical resistance capacitance circuits, considered the current flow in extracellular fluid and the distribution of extracellular potential and finally obtained the distribution of membrane potential in time space \(\lambda^{2}(\partial^{2}V/\partial x^{2})=\tau(\partial V/\partial t)+V\), where \(V\) is the potential of the membrane and \(\lambda\) and \(\tau\) are the time constant and length constant of the thin film of the cable joint. This model is a second-order partial differential equation, and the solution is relatively complex. In this paper, we only consider the input and output characteristics of the neuron as a signal processing unit instead of the distribution of membrane potential on the whole axon, so we assume that the extracellular potential along the whole axon is always 0--equivalent to grounding--as can be seen in Figure 4. Thus, \(U\) is equal to the membrane potential. Given the output change of an input signal after the membrane potential is transmitted through the whole axon, we can calculate \(f(. )\) as shown in Formula (10). First, we analyze the basic resistance capacitance circuit in Figure 4 separately and calculate the transfer function between the output \(U_{0}\) and the input \(U_{i}\) of each section. In Figure 4, \(i\) is the current, \(R\) is the resistance and \(C\) is the capacitance [18]. The following equation can be obtained from [PERSON]'s law and [PERSON]'s law. \[i_{1}=C\frac{dU_{o}}{dt} \tag{11}\]  
\[U_{o}=i_{2}R_{2} \tag{12}\]  
\[U_{i}=(i_{1}+i_{2})(R_{1}+R_{3})+U_{o} \tag{13}\]  
Equations (11) and (12) are substituted into Equation (13):  
\[U_{i}=(C\frac{dU_{o}}{dt}+\frac{U_{o}}{R_{2}})(R_{1}+R_{3})+U_{o} \tag{14}\]  
The Laplace transform of Equation (14) is  
\[U_{i}(s)=(sCU_{0}(s)+\frac{U_{o}(s)}{R_{2}})(R_{1}+R_{3})+U_{o}(s) \tag{15}\]  
Figure 4: The cable model without considering the extracellular potential. Finally, we obtain the transfer function between \(U_{0}\) and \(U_{i}\):  
\[G(s)=\frac{U_{o}(s)}{U_{i}(s)}=\frac{1}{C(R_{1}+R_{3})s+\frac{R_{1}+R_{2}+R_{3}}{ R_{2}}} \tag{16}\]  
Assume \(\tau\) and \(b\) as Equation (17):  
\[\left\{ \begin{array}{l}\tau=C(R_{1}+R_{3})\\ b=\frac{R_{1}+R_{2}+R_{3}}{R_{2}}\end{array} \right. \tag{17}\]  
Then, Equation (16) is  
\[G(s)=\frac{1}{\tau s+b} \tag{18}\]  
As can be seen from Figure 4, the total transfer function of the axon to the membrane potential is \(F(S)=G^{N}(S)\). \(N\) is the number of basic links on the whole axon. Thus far, the signal transmission model of the neuron has been determined.