SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1124-1159, January 2020.
We study the stability of an oscillatory associative memory network consisting of $N$ coupled Kuramoto oscillators with applications in binary pattern retrieve. In this model, the coupling function consists of a Hebbian term and a second-order Fourier term with nonnegative strength $\varepsilon$. In [Phys. D, 197 (2004), pp. 134--148] Nishikawa, Hoppensteadt, and Lai studied the stability using the approach of linearization; the criteria for stability/instability is given by the spectrum of linearization which is a matrix of order $N$. In recent literature [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 188--201], Hölzel and Krischer considered the model with $\varepsilon=0$ and introduced the orthogonality of binary patterns so that the eigenvalues of linearization can be calculated. In this paper, we will present conditions for stability/instability based on the gradient formulation. First, we use the potential estimate to derive a criteria for stability/instability by the spectrum of a matrix of order $N-1$. This potential estimate also gives convergence rate under some conditions. Second, we focus on the special case with mutually orthogonal memorized patterns. We find a sufficient and necessary condition for a binary pattern to be stable for any $\varepsilon > 0$. For any other binary pattern we prove that there exists a critical value of $\varepsilon$ below which it is unstable. A lower bound for this critical strength is provided. A significant advantage of the results in this case is that the conditions for stability/instability are easy to verify and the lower bound of critical strength is easy to compute. Third, when the memorized patterns are not mutually orthogonal, we suggest a framework to transform it into the case of orthogonal memorized patterns. Simulations are presented to illustrate our results.

中文翻译：

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具有二阶模式检索的二阶耦合的仓本振荡器Hebbian网络的稳定性

SIAM应用动力系统杂志，第19卷，第2期，第1124-1159页，2020年1月。
我们研究了由$ N $耦合的Kuramoto振荡器组成的振荡联想存储网络的稳定性，以及在二进制模式检索中的应用。在此模型中，耦合函数由具有非负强度$ \ varepsilon $的Hebbian项和二阶Fourier项组成。在[Phys。D，197（2004），134--148页] Nishikawa，Hoppensteadt和Lai使用线性化方法研究了稳定性；稳定性/不稳定性的标准由线性化频谱给出，线性化频谱是$ N $阶矩阵。在最近的文献中[SIAM J.Appl。达因 Syst。，14（2015），pp。188--201]，Hölzel和Krischer考虑了$ \ varepsilon = 0 $的模型，并引入了二进制模式的正交性，以便可以计算线性化的特征值。在本文中，我们将根据梯度公式介绍稳定性/不稳定性的条件。首先，我们使用潜在的估计值，通过阶次为N $ -1 $的矩阵的频谱来得出稳定性/不稳定性的标准。在某些情况下，此潜在估计值还可以得出收敛速度。其次，我们关注具有相互正交的记忆模式的特殊情况。我们发现对于任何大于0的$ \ varepsilon，二元模式稳定的充分必要条件。对于任何其他二进制模式，我们证明存在$ \ varepsilon $的临界值，低于该临界值是不稳定的。提供了该临界强度的下限。在这种情况下，结果的显着优势是，易于验证稳定性/不稳定性的条件，并且易于计算临界强度的下限。第三，当存储的图案不相互正交时，我们建议使用一个框架将其转换为正交存储的图案。进行仿真以说明我们的结果。