Synchronization of coupled limit cycles

A unified approach for analyzing synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented with a discussion of numerical simulations of a compartmental model of a neuron.Comment: Journal of Nonlinear Science, accepte

Synchronization Probability in Large Random Networks

In a generalized framework, where multi-state and inter-state linkages are allowed, we derive a sufficient condition for the stability of synchronization in a network of chaotic attractors. This condition explicitly relates the network structure and the local and coupling dynamics to synchronization stability. For large Erd\"{o}s-R\'{e}nyi networks, the obtained condition is translated into a lower bound on the probability of stability of synchrony. Our results show that the probability of stability quickly increases as the randomness crosses a threshold which for large networks is inversely proportional to the network size

The geometry of spontaneous spiking in neuronal networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic \beta-cells, which in isolation are known to exhibit irregular spiking. At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity or small total effective resistance are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models

Limitations of perturbative techniques in the analysis of rhythms and oscillations

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience

Sur un système de deux oscillateurs FitzHugh-Nagumo couplés

Ce mémoire consiste en l’étude du comportement dynamique de deux oscillateurs FitzHugh-Nagumo identiques couplés. Les paramètres considérés sont l’intensité du courant injecté et la force du couplage. Juqu’à cinq solutions stationnaires, dont on analyse la stabilité asymptotique, peuvent co-exister selon les valeurs de ces paramètres. Une analyse de bifurcation, effectuée grâce à des méthodes tant analytiques que numériques, a permis de détecter différents types de bifurcations (point de selle, Hopf, doublement de période, hétéroclinique) émergeant surtout de la variation du paramètre de couplage. Une attention particulière est portée aux conséquences de la symétrie présente dans le système.We study the dynamical behaviour of a pair of identical, coupled FitzHugh-Nagumo oscillators. We determine the parameter values leading to the existence of up to five equilibrium solutions, and analyze the asymptotic stability of each one. A combination of analytical and numerical techniques is used to analyze the numerous bifurcations (saddle-node, Hopf, period-doubling, heteroclinic) occurring as parameters, most notably the coupling strength, are varied, attention being paid to the rôle played by symmetries in the system