Cluster solutions in networks of weakly coupled oscillators on a 2D square torus

IF 0.4 Q4 MATHEMATICS, APPLIED
J. Culp
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引用次数: 1

Abstract

We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.
二维方形环面上弱耦合振子网络的簇解
我们考虑具有周期边界条件的弱耦合神经振荡器的N×N晶格网络的模型(2D方环面),其中神经元之间的耦合被假设在大小为r的von Neumann邻域内,表示为von Neumann-r邻域。使用相位模型约简技术,我们研究了在我们的网络中,沿水平和垂直方向的相邻振荡器之间存在具有恒定相位差(Ψh,Ψv)的簇解,其中Ψh和Ψv不一定相同。应用Kronecker乘积表示和循环矩阵理论,我们提出了一种新的方法来分析具有恒定相位差(即Ψh,Ψv相等)的簇解的稳定性。我们通过导出具有von Neumann 1-邻域和2邻域耦合的这类簇解的稳定性的精确条件来开始我们的分析,然后我们将我们的结果推广到任意邻域大小r≥1的von Neumann-r邻域耦合。这种发展的稳定性分析方法确实可以扩展到我们网络中的任意耦合。最后,通过考虑Morris Lecar神经元的抑制网络,使用数值模拟来验证N和r的不同值的上述分析结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.40
自引率
0.00%
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0
审稿时长
21 weeks
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