# Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network
## 4 Mechanism Explanation  
To explain the phenomena observed, the simplest coupled-FHN system, namely the system with only two oscillators, is considered. Figure 6 shows the dynamic behavior of an FHN oscillator coupled with another free-oscillating FHN oscillator. \(\varepsilon=0.05\), \(a=0.5\), \(\phi=\pi/2-0.1\), the strength of the unidirectional coupling \(\sigma=0.1\). The phases of the oscillators are described as \(\theta=\arctan(v,u)\). The initial values of \(\theta\) are both in the interval \((-\pi,\pi]\), and they increase monotonically. Two points on the trajectory of a free FHN oscillator with a time difference of 0.30 are used as the initial conditions of the two oscillators. The initial phase difference of the oscillators is \(\Delta\theta=2.055\) in Figure 6a, and \(\Delta\theta=2.182\) in Figure 6b.  
As shown in Figure 6a, when a coupled oscillator is leading the free oscillator, it will be accelerated. The phase difference between them will keep increasing and finally be \(2\pi\), which means the coupled oscillator catches up with the free one again.  
However, the case is different if the coupled oscillator is falling behind, as shown in Figure 6b. The coupled oscillator will still be accelerated, but the phase difference will keep decreasing and finally be 0. The different effects on positive and negative phase difference is a kind of symmetry breaking, which is one of the reasons that caused chimera states.  
From the cases of two coupled oscillators above, it can be known that when one FHN oscillator is coupled with another, there is a phase difference between them, and if the coupling strength is chosen properly, the coupling may always accelerate this oscillator and increase its phase velocity. This explains why the frequencies of all the neurons in the network are higher than their natural frequencies. This is a combined effect of the nonlinearity of a single oscillator and the coupling between oscillators, and this may explain why chimera states only occur when coupling strength is in a specific range.  
In the ring-shaped neural network, the initial phases of all the neurons are random. Some neurons' phases can be leading the average of their neighbors, so their phase velocities will be accelerated by their neighbors, and when their phases are almost \(2\pi\) ahead of their neighbors, they can be regarded as falling behind, while some other neurons become leading and accelerated. The neurons in the pruned region synchronize in a short time, and their phase velocities become fixed. However, for the other neurons, there will always be some leading ones, so the process of mutual acceleration will never end, and the neurons cannot all synchronize. In this process, synchronized neurons gradually form a coherent domain. As these neurons hardly have phase differences with their neighbors, they will hardly be accelerated and have lower phase velocities. On the other hand, desynchronized neurons always have phase differences with their neighbors, therefore largely accelerated and have higher phase velocities. These neurons finally form an incoherent domain. As the phase velocities of neurons in the pruned region are much lower than all the other neurons, there will always be phase differences between these neurons and the neurons located near this region. Therefore, the neurons near the pruned region will be accelerated the most and  
Figure 6: The effect of unidirectional coupling between two FHN oscillators. (**a**) The coupled oscillator leads the free oscillator, (**b**) the coupled oscillator falls behind the free oscillator.  
finally get the highest phase velocities. That is to say, the center of the incoherent structure will drift to this region.