# Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network
## 3 Effect of Pruning on Neural Network  
The effect of synaptic pruning on a neural network with 300 neurons is shown in Figure 2 (non-dimensional time \(t=1000\)). The range 148-152 is the pruned region. It can be seen in Figure 2a that the pruned neurons are synchronized, and they attract the incoherent domain and repel the coherent domain. In Figure 2b, it is shown that the phase velocities of neurons in the coherent domain are lower than those in the incoherent domain, and the phase velocities of pruned neurons are much lower than all the others. The neurons that are closest to the pruned region have the highest phase velocities. Figure 2c shows the local order parameter of the nodes. The parameter is defined as:  
\[Z_{k}=\big{|}\frac{1}{2\delta}\sum_{|j-k|\leq\delta}e^{i\Theta_{j}}\big{|},k=1,...,N \tag{6}\]  
where \(\Theta_{j}=\arctan(v_{j}/u_{j})\) denotes the geometric phase of the j-th node. A window size \(\delta=5\) is used. A large value of \(Z_{k}\) shows that the neuron \(k\) is synchronized with its near neighbors, indicating the neuron is in the coherent domain. A small value of \(Z_{k}\) means that the neuron \(k\) is not synchronized with its near neighbors, indicating the neuron is in the incoherent domain. As the local order parameter's window size is larger than the pruned region, the synchronization of the pruned region can not be observed in Figure 2c, though it can be observed in Figure 2a. For a system without pruning, the coherent domain will be generated at a random place in the system [19,26]. In Figure 2c, similarly, the initial position of the coherent domain is not determined. However, the coherent domain is repelled to the opposite position of the pruned region in 500 non-dimensional time. Though this domain drifts back and forth in a very limited region, it is kept away from the pruned region. This phenomenon is counter-intuitive, because one may expect the short synchronized region will be part of the coherent structure, but it turns out to repel the coherent structure. The explanation for such a counter-intuitive phenomenon will be given in Section 4. The possibility of controlling the positions of chimera states by modifying only one oscillator has been revealed by [PERSON] [22]. Therefore, it is obliged to study if pruning one neuron in an FHN system is sufficient to determine its performance or how the number of pruned neurons affects the system's performance. To make this comparison, four scenarios with different numbers of pruned neurons are simulated (\(N_{p}=1,2,3,5\)), each for 50 cases. Figure 3 shows the average local order parameter in different scenarios. When only 1 neuron is pruned, as shown in Figure 3a,b, the coherent domain is repelled by the pruned neurons. However, the maximum value of the average local order parameter is much lower than 1, and it is hard to tell the boundary between the coherent domain and the incoherent domain. This indicates that the coherent structure's position is not the same for all the cases, and the effect on domains' positions is not precise. Figure 4 shows an example that the pruned neuron failed to repel the coherent domain in 1000 non-dimensional time. When 2 or 3 neurons are pruned, as shown in Figure 3b,c, the values of \(Z_{k}\) on two sides of the figure get higher. The boundaries between different regions are  
Figure 1: Matrix of coupling strength. Dark blue: not coupled; light blue: default coupling strength; yellow: highly coupled. clearer. This indicates that the coherent structure's positions are similar for most of all the 50 cases. When 5 neurons are pruned, as shown in Figure 3d, the values of \(Z_{k}\) on two sides of the figure are very close to 1, and the boundaries between different regions are very clear. This indicates that the coherent structure's positions are the same for all 50 cases.