Synchronization is an emergent phenomenon in coupled dynamical networks. The Master Stability Function (MSF) is a highly elegant and powerful tool for characterizing the stability of synchronization states. However, a significant challenge lies in determining the MSF for complex dynamical networks driven by nonlinear interaction mechanisms. These mechanisms introduce additional complexity through the intricate connectivity of interacting elements within the network and the intrinsic dynamics, which are governed by nonlinear processes with diverse parameters and higher dimensionality of systems. Over the past 25 years, extensive research has focused on determining the MSF for pairwise coupled identical systems with diffusive coupling. Our literature survey highlights two significant advancements in recent years: the consideration of multilayer networks instead of single-layer networks and the extension of MSF analysis to incorporate higher-order interactions alongside pairwise interactions. In this review article, we revisit the analysis of the MSF for diffusively pairwise coupled dynamical systems and extend this framework to more general coupling schemes. Furthermore, we systematically derive the MSF for multilayer dynamical networks and single-layer coupled systems by incorporating higher-order interactions alongside pairwise interactions. The primary focus of our review is on the analytical derivation and numerical computation of the MSF for complex dynamical networks. Finally, we demonstrate the application of the MSF in data science, emphasizing its relevance and potential in this rapidly evolving field.

We investigate a primal-dual (PD) method for the saddle point problem (SPP) that uses a linear approximation of the primal function instead of the standard proximal step, resulting in a linearized PD (LPD) method. For convex-strongly concave SPP, we observe that the LPD method has a suboptimal dependence on the Lipschitz constant of the primal function. To fix this issue, we combine features of Accelerated Gradient Descent with the LPD method resulting in a single-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD method achieves the optimal gradient complexity when the SPP has a semi-linear coupling function. We also present an inexact ALPD method for SPPs with a general nonlinear coupling function that maintains the optimal gradient evaluations of the primal parts and significantly improves the gradient evaluations of the coupling term compared to the ALPD method. We verify our findings with numerical experiments.

We consider a class of stochastic dynamical networks whose governing dynamics can be modeled using a coupling function. It is shown that the dynamics of such networks can generate geometrically ergodic trajectories under some reasonable assumptions. We show that a general class of coupling functions can be learned using only one sample trajectory from the network. This is practically plausible as in numerous applications it is desired to run an experiment only once but for a longer period of time, rather than repeating the same experiment multiple times from different initial conditions. Building upon ideas from the concentration inequalities for geometrically ergodic Markov chains, we formulate several results about the convergence of the empirical estimator to the true coupling function. Our theoretical findings are supported by extensive simulation results.

In a complex system, the interactions between individual agents often lead to emergent collective behavior like spontaneous synchronization, swarming, and pattern formation. The topology of the network of interactions can have a dramatic influence over those dynamics. In many studies, researchers start with a specific model for both the intrinsic dynamics of each agent and the interaction network, and attempt to learn about the dynamics that can be observed in the model. Here we consider the inverse problem: given the dynamics of a system, can one learn about the underlying network? We investigate arbitrary networks of coupled phase-oscillators whose dynamics are characterized by synchronization. We demonstrate that, given sufficient observational data on the transient evolution of each oscillator, one can use machine learning methods to reconstruct the interaction network and simultaneously identify the parameters of a model for the intrinsic dynamics of the oscillators and their coupling. Keywords: nonlinear dynamics, phase oscillators, Kuramoto oscillators, network reconstruction, network topology, machine learning, computational methods 1. Introduction Nature and society brim with systems of coupled oscillators, including pacemaker cells in the heart, insulin-secreting cells in the pancreas, neural networks in the brain, fireflies that synchronize their flashing, chemical reactions, Josephson junctions, power grids, metronomes, and applause in human crowds, to name merely a few [1-9]. The dynamics of coupled oscillators in complex networks have been studied extensively.

We have developed and tested an analog/digital VLSI system that models the coordination of biological segmental oscillators underlying axial locomotion in animals such as leeches and lampreys. In its current form the system consists of a chain of twelve pattern generating circuits that are capable of arbitrary contralateral inhibitory synaptic coupling. Each pattern generating circuit is implemented with two independent silicon Morris-Lecar neurons with a total of 32 programmable (floating-gate based) inhibitory synapses, and an asynchronous address-event interconnection element that provides synaptic connectivity and implements axonal delay. We describe and analyze the data from a set of experiments exploring the system behavior in terms of synaptic coupling.

We have developed and tested an analog/digital VLSI system that models the coordination of biological segmental oscillators underlying axial locomotion in animals such as leeches and lampreys. In its current form the system consists of a chain of twelve pattern generating circuits that are capable of arbitrary contralateral inhibitory synaptic coupling. Each pattern generating circuit is implemented with two independent silicon Morris-Lecar neurons with a total of 32 programmable (floating-gate based) inhibitory synapses, and an asynchronous address-event interconnection element that provides synaptic connectivity and implements axonal delay. We describe and analyze the data from a set of experiments exploring the system behavior in terms of synaptic coupling.

We have developed and tested an analog/digital VLSI system that models thecoordination of biological segmental oscillators underlying axial locomotion in animals such as leeches and lampreys. In its current form the system consists of a chain of twelve pattern generating circuits that are capable of arbitrary contralateral inhibitory synaptic coupling. Each pattern generating circuit is implemented with two independent silicon Morris-Lecar neurons with a total of 32 programmable (floating-gate based) inhibitory synapses, and an asynchronous address-event interconnection elementthat provides synaptic connectivity and implements axonal delay. We describe and analyze the data from a set of experiments exploringthe system behavior in terms of synaptic coupling.