Paper Info
Title
Stability in a Hebbian Network of Kuramoto Oscillators with Second-Order Couplings for Binary Pattern Retrieve
Abstract
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 E. 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], Holzel and Krischer considered the model with epsilon = 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 epsilon > 0. For any other binary pattern we prove that there exists a critical value of s 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.
Year
DOI
Venue
2020
10.1137/19M1269397
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
Keywords
DocType
Volume
Kuramoto oscillators,binary pattern retrieve,Hebbian rule,stability,second-order Fourier term
Journal
19
Issue
ISSN
Citations 
2
1536-0040
0
PageRank 
References 
Authors
0.34
0
3
Name
Order
Citations
PageRank
Xiaoxue Zhao101.01
Zhuchun Li2165.00
Xiaoping Xue318617.00