# Dynamic Pathway Selection Mechanisms of Brain Networks
## 2 Materials and Methods
### Dynamic Synaptic Model  
The synapse is an important part of the nervous system, and neurons transmit signals through synaptic structures. In the process of information transmission, mixed synapses have a certain effect on signal transmission, and the number of synapses between neurons will also affect the response ability of the nervous system. Therefore, taking into account the respective advantages of chemical synapses and electrical synapses, this section constructs a bionic dynamic synaptic model according to the proportion of chemical synapses and electrical synapses and the regulation of \(ECM\) concentration on the number of synapses. (1) Electrical synaptic coupling takes into account the fast channel of ion exchange between neurons. \(I_{i,e}^{syn}\) represents the current of the neuron after the electric synapse, as shown in its Equation (2):  
\[I_{i,e}^{syn}=\sum_{j}g_{e}^{syn}A(i,j)\big{(}V_{i}-V_{j}\big{)} \tag{2}\]  
where \(g_{e}^{syn}\) is the electrical conductivity of the electrical synaptic channel, \(A(i,j)\) represents the coupling relationship between neuron \(i\) and neuron \(j\), \(V_{i}\) and \(V_{j}\) represent the membrane potential of neuron \(i\) and neuron \(j\), \(A(i,j)=A(j,i)=1\), indicating that there is a coupling between the neuron \(i\) and the neuron \(j\). \(A(i,j)=A(j,i)=0\) means that there is no coupling between neurons. (2) Chemical synaptic coupling takes into account the direction of transmission of pre-and post-synaptic neurons and the release process of neurotransmitters. \(I_{i,c}^{syn}\) represents the post-synaptic current of neurons \(i\), as shown in (3):  
\[I_{i,c}^{syn}=\sum_{j}g_{c}^{syn}A(i,j)S_{j}\big{(}V_{i}-V_{syn}\big{)} \tag{3}\]  
where \(g_{c}^{syn}\) is the coupling strength of chemical synapses, \(A(i,j)\) is expressed as the coupling relationship between neuron \(i\) and neuron \(j\) and \(V_{syn}\) is the reverse potential of neurons which determines the type of synapse: \(V_{syn}=0\) means excitatory postsynaptic potential (\(EPSP\)); \(V_{syn}=-1.2\) represents inhibitory postsynaptic potential (\(IPSP\)). The release process of chemical synaptic vesicles is measured by \(S_{j}\), which is determined by the membrane voltage \(V_{j}\), as shown in (4):  
\[\frac{dS_{j}}{dt}=\frac{\alpha\big{(}V_{j}\big{)}\big{(}1-S_{j}\big{)}}{ \varepsilon}-\frac{S_{j}}{\tau_{syn}} \tag{4}\]  
where \(\tau_{syn}=1/\delta\) is the synaptic decay rate. When the neuron is in the resting state (\(V_{j}<0\)), \(S_{j}\) decreases slowly and \(dS_{j}/dt=-S_{j}/\tau_{syn}\); when \(V_{j}>0\), \(S_{j}\to 1\) jumps rapidly to 1. Presynaptic neurons act on postsynaptic neurons only when \(S_{j}\) changes rapidly, which is different from the rapid coupling of electrical synapses. \(\alpha\big{(}V_{j}\big{)}\) is the synaptic recovery equation, such as Equation (5):  
\[\alpha\big{(}V_{j}\big{)}=\frac{\alpha_{0}}{1+e^{\left(-V_{j}/V_{\rm{sip}} \right)}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Izhikevich Neuronal Pulse Type** & \(\boldsymbol{a/\rm{ms}^{-1}}\) & \(\boldsymbol{b/\rm{10}^{-9}\Omega^{-1}}\) & \(\boldsymbol{c/\rm{mV}}\) & \(\boldsymbol{d}\) \\ \hline Regular Spiking & \(0.02\) & \(0.2\) & \(-65\) & \(8\) \\ \hline Intrinsically Bursting & \(0.02\) & \(0.2\) & \(-55\) & \(4\) \\ \hline Fast Spiking & \(0.1\) & \(0.2\) & \(-65\) & \(2\) \\ \hline Low-threshold Spiking & \(0.02\) & \(0.25\) & \(-65\) & \(2\) \\ \hline Resonator & \(0.1\) & \(0.26\) & \(-65\) & \(2\) \\ \hline Chattering & \(0.02\) & \(0.2\) & \(-50\) & \(2\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter tables of six typical Izhikevich neurons. where \(\alpha(V_{j})\) is the Heaviside equation that models the mechanism of chemical synaptic vesicle release. (3) According to the latest microbiological research [16; 17], the number of synapses will be affected by the concentration of \(ECM\), thereby regulating the synaptic efficiency. Therefore, we will build the synaptic number model \(N[C]\) under the influence of concentration.