Neurons in the inhibitory network of the striatum display cell assembly firing patterns which recent results suggest may consist of spatially compact neural clusters. Previous computational modeling of striatal neural networks has indicated that non-monotonic, distance-dependent coupling may promote spatially localized cluster firing. Here, we identify conditions for the existence and stability of cluster firing solutions in which clusters consist of spatially adjacent neurons in inhibitory neural networks. We consider simple non-monotonic, distance-dependent connectivity schemes in weakly coupled 1-D networks where cells make stronger connections with their $k$th nearest neighbors on each side and weaker connections with closer neighbors. Using the phase model reduction of the network system, we prove the existence of cluster solutions where neurons that are spatially close together are also synchronized in the same cluster, and find stability conditions for these solutions. Our analysis predicts the long-term behavior for networks of neurons, and we confirm our results by numerical simulations of biophysical neuron network models. Our results demonstrate that an inhibitory network with non-monotonic, distance-dependent connectivity can exhibit cluster solutions where adjacent cells fire together.

纹状体抑制网络中的神经元显示细胞组装放电模式，最近的结果表明可能由空间紧凑的神经簇组成。先前纹状体神经网络的计算模型表明，非单调、距离相关的耦合可能会促进空间局部集群发射。在这里，我们确定了集群激发解决方案的存在和稳定性的条件，其中集群由抑制神经网络中的空间相邻神经元组成。我们在弱耦合的一维网络中考虑简单的非单调、依赖于距离的连接方案，其中单元与它们的连接更强。$克$每边的最近邻居和与更近邻居的较弱连接。使用网络系统的相位模型约简，我们证明了在空间上靠近的神经元在同一簇中也同步的簇解的存在，并找到这些解的稳定性条件。我们的分析预测了神经元网络的长期行为，并且我们通过生物物理神经元网络模型的数值模拟证实了我们的结果。我们的结果表明，具有非单调、距离相关连接的抑制网络可以展示相邻单元格一起发射的集群解决方案。