## 2 Spatial Navigation Cell Model  
### Boundary Cell Modeling  
The boundary cells, which mainly exist in the entorhinal cortex, presubiculum, and parasubiculum tract of the rat hippocampus, are spatial navigation cells that respond to boundary information and can reflect the relative positions of rats at different distances and angles from the environmental boundary by encoding boundary information in the environment; these cells can be used to complement path integration information [24; 25; 26]. In this paper, the boundary cells are modeled by the mutual excitation and inhibition of direction-aware and distance-aware information of the mobile robot, and the construction process is shown below. Step 1: The mobile robot explores in an unknown environment (its exploration schematic diagram is shown in Figure 2). The surrounding shaded part is the wall, the circular runner is the mobile robot, and the single black arrow is its movement direction. Figure 1: Block diagram of the overall system structure. When the mobile robot moves away from the wall, the boundary cells go through two processes, the activity growth zone and the attenuation zone. When they then reach the boundary cell inhibition zone their activity attenuates to zero. The rules for updating the activity state of boundary cell neurons are as follows:  
\[h(t+1)=\left\{ \begin{array}{ll}h(t)+\tau\frac{b-d_{r}(t)}{b},&\quad\text{if }S(t)=S_{exgr}\\ 0,&\quad\text{if }S(t)=S_{inh}\\ \left[h(t)-\tau\frac{b-d_{r}(t)}{b}\right]_{+},&\quad\text{if }S(t)=S_{excar} \end{array} \right. \tag{5}\]  
where \(h(t)\) is the current time boundary cell activity value, \(h(t+1)\) is the next time boundary cell activity value, and \(\tau\) is the activity factor (which is used to regulate the rate of change in boundary cell neuron activity and the value of 0.8 is taken for this factor in this paper). \(\left[\bullet\right]_{+}\) indicates that the output value is non-negative. Step 3: When the mobile robot runs into the boundary cell excitation zone, the boundary cell distance excitation value \(b_{border}(t)\) is updated with the distance-aware information between the current position of the mobile robot and the environmental boundary. \[b_{border}(t)=\exp\left(-\frac{h(t)(R(t)-d_{r}(t))^{2}}{2\sigma_{rad}^{2} \left(d_{r}(t)\right)}\right) \tag{6}\]  
where \(\sigma_{rad}(\bullet)\) is the boundary cell distance sensitivity function and the relationship between the sensitivity of the boundary cell to the environmental boundary. Moreover, the vertical distance from the mobile robot to the environmental boundary is expressed as per the following:  
\[\sigma_{rad}\left(d_{r}(t)\right)=\delta_{0}*\left(d_{r}(t)/\beta+1\right) \tag{7}\]  
where \(\delta_{0}\) is the boundary cell sensitivity enhancement constant, which is adapted to the complexity of the environmental boundary and which is set to 1.2 in this paper. \(\beta\) is the boundary cell distance perception correction factor, which avoids the boundary cell response value being too large due to the small distance information that is detected in the small environmental scenarios and which is taken as 0.8 in this paper. Figure 3: Diagram of the six scenarios of mobile robots in the environment. Subsequently, the boundary cell angular excitation value \(r_{border}(t)\) is updated based on the angular-aware information between the mobile robot and environmental boundary:  
\[r_{border}(t)=\exp\left(-\frac{h(t)(\theta(t)-\alpha(t))^{2}}{2\sigma_{ang}^{2}}\right) \tag{8}\]  
where \(\sigma_{ang}\) is the adjustment parameter for the angular excitation value of the boundary cell, which is used to regulate the effect of angle on the excitation value of the border cell, and is taken as 0.5 in this paper.