# Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network
## 2 Model Description  
Proposed initially as a simplification of neural cell activity, the FHN oscillator is a paradigmatic model of excitable systems and is wildly used to describe the dynamic behavior of neurons. Here, a neural network described as N non-locally coupled FHN oscillators is analyzed [12]:  
\[\epsilon\frac{du_{k}}{dt}=u_{k}-\frac{u_{k}^{3}}{3}-v_{k}+\frac{\sigma}{2R} \sum_{j=k-R}^{j=k+R}\bigl{[}b_{uu}(u_{j}-u_{k})+b_{uv}(v_{j}-v_{k})\bigr{]} \tag{1}\]  
\[\frac{dv_{k}}{dt}=u_{k}+a+\frac{\sigma}{2R}\sum_{j=k-R}^{j=k+R}\bigl{[}b_{uu}( u_{j}-u_{k})+b_{vv}(v_{j}-v_{k})\bigr{]} \tag{2}\]  
where \(u_{k}\) and \(v_{k}\) are the activator and inhibitor variables, respectively, \(\varepsilon>0\) is a timescale parameter, \(\sigma\) is the coupling strength among neurons, \(a\) is the threshold parameter, and \(B\) is the rotational coupling matrix:\[B= \begin{pmatrix}b_{uu}&b_{uv}\\ b_{vu}&b_{vv}\end{pmatrix} = \begin{pmatrix}\cos\phi&\sin\phi\\ -\sin\phi&\cos\phi\end{pmatrix}  \tag{3}\]  
which is determined only by the coupling phase \(\phi\). Here, all the neurons make up a ring. That is to say, all indices are modulo \(N\), where \(N\) is the total number of neurons. As it can be seen, each element is coupled to \(R\) neighbors on either side. \(r=R/N\) is called coupling radius. The value of each parameter mentioned above has its own effect on the performance of the system. \(\varepsilon\) and \(a\) determine the behavior of a single neuron. \(\varepsilon\) determines the slow and fast dynamics associated with \(u_{k}\) and \(v_{k}\), respectively. \(a\) determines whether a neuron exhibits oscillatory (\(|a|<1\)) or excitable (\(|a|>1\)) behavior [10; 24]. \(\phi\), \(r\), and \(\sigma\) determine the collective behavior of the network. In general, chimera states are expected for a pronounced off-diagonal coupling (\(\phi\approx\pi/2\)), but not for a diagonal one (\(\phi\approx 0\) or \(\phi\approx\pi\)). When changing the value of \(r\) and \(\sigma\), the number of incoherent domains will change accordingly [12]. To compare with literature, \(a,\phi,r,\varepsilon,\sigma\) are chosen to be the same as in Ref. [12]. Namely, \(a=0.5\), \(\phi=\pi/2-0.1\), \(r=0.35\), \(\varepsilon=0.05\), \(\sigma=0.1\). The attractive behaviors of this system are chimera states, which are characterized by the coexistence of coherent domains (periodic in-phase oscillations) and incoherent domains (chaotic behavior in time and space). And the coherent domains can represent phase synchronization in a neural network. However, the spatial distribution of coherent domains is sensitive to the initial condition. This means that there are some mechanisms in neural networks that can control the position of the coherent structures, and synaptic pruning may be one of them. In order to conduct synaptic pruning that occurs in the neural network, the coupling among neurons is modified. We choose a few contiguous neurons and change their coupling so that the neurons in this region are only affected by each other, but the other neurons in the system are still affected by them. Additionally, the synaptic connections among these neurons are enhanced to keep them synchronizing with each other. In the case of 300 oscillators, a pruned region of 5 oscillators is used. In order to make these 5 oscillators synchronize, the coupling strength between every two nodes in the region is set to 0.2, while the strength between nodes outside this region is 0.000476. The single direction coupling strength between nodes inside and outside this region is also 0.000476. In fact, after the coupling strength of the pruned area exceeds a critical value, these oscillators will synchronize, and the increase of the strength will make no difference anymore. The difference in coupling strength between neurons is realistic, considering the diversification of synaptic strength [25].