Stability of synchronization manifolds and its nonlinear behaviour in memristive coupled discrete neuron model

In this study, we investigate the impact of first and second-order coupling strengths on the stability of a synchronization manifold in a Discrete FitzHugh-Nagumo (DFHN) neuron model with memristor coupling. Master Stability Function (MSF) is used to estimate the stability of the synchronized manifold. The MSF of the DFHN model exhibits two zero crossings as we vary the coupling strengths, which is categorized as class Gamma 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{2}$$\end{document}. Interestingly, both zero-crossing points demonstrate a power-law relationship with respect to both the first-order coupling strength and flux coefficient, as well as the second-order coupling strength and flux coefficient. In contrast, the zero crossings follow a linear relationship between first-order and second-order coupling strength. These linear and nonlinear relationships enable us to forecast the zero-crossing point and, consequently, determine the coupling strengths at which the stability of the synchronization manifold changes for any given set of parameters. We further explore the regime of the stable synchronization manifold within a defined parameter space. Lower values of both first and second-order coupling strengths have minimal impact on the transition between stable and unstable synchronization regimes. Conversely, higher coupling strengths lead to a shrinking regime of the stable synchronization manifold. This reduction follows an exponential relationship with the coupling strengths. This study is helpful in brain-inspired computing systems by understanding synchronization stability in neuron models with memristor coupling. It helps to create more efficient neural networks for tasks like pattern recognition and data processing.