We report parabolic bursting in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducitng device is used for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. In this SNIC regime, the bursting appears in a broad paramater space of the ensemble of mixed junctions. For a coupling value above a threshold, the network splits into two synchronized clusters when a reductionism approach is applied to reproduce the bursting behavior of the large network. The excitable junctions effectively induces a slow dynamics in the network to generate bursting. This bursting is a generic property of a globally coupled network with a mixed population of dynamical nodes where each node posseses the SNIC property.
We study the collective dynamics in a network of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the network, whereas the feedback from the resources to the network is comprised of components acting homogeneously or inhomogeneously on individual units of the network. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the network and the resources can give rise to collective activity bursting in the network. To explain the…
We show that a cumulative action of noise and delayed feedback on an excitable theta-neuron leads to rather coherent stochastic bursting. An idealized point process, valid if the characteristic time scales in the problem are well-separated, is used to describe statistical properties such as the power spectrum and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate and the probability to induce a spike during the delay action, can be calculated from the solutions of a stationary and a forced Fokker-Planck equation.
We study the robustness of self-sustained oscillatory activity in a globally coupled ensemble of excitable and oscillatory units. The critical balance to achieve collective self-sustained oscillations is analytically established. We also report a universal scaling function for the ensemble's mean frequency. Our results extend the framework of the `Aging Transition' [Phys. Rev. Lett. 93, 104101 (2004)] including a broad class of dynamical systems potentially relevant in biology.
We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling…
We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influe…
We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which …
We explore the influence of a block of excitable units on the existence and behavior of chimera states in a nonlocally coupled ring-network of FitzHugh-Nagumo elements. The FitzHugh-Nagumo system, a paradigmatic model in many fields from neuroscience to chemical pattern formation and nonlinear electronics, exhibits oscillatory or excitable behavior depending on the values of its parameters. Until now, chimera states have been studied in networks of coupled oscillatory FitzHugh-Nagumo elements. In the present work, we find that introducing a block of excitable units into the network may lead to…
Small-world networks describe many important practical systems among which neural networks consisting of excitable nodes are the most typical ones. In this paper we study self-sustained oscillations of target waves in excitable small-world networks. A novel dominant phase-advanced driving (DPAD) method, which is generally applicable for analyzing all oscillatory complex networks consisting of nonoscillatory nodes, is proposed to reveal the self-organized structures supporting this type of oscillations. The DPAD method explicitly explores the oscillation sources and wave propagation paths of th…
We consider the long-time dynamics of a general class of nonlinear Fokker-Planck equations, describing the large population behavior of mean-field interacting units. The main motivation of this work concerns the case where the individual dynamics is excitable, i.e. when each isolated dynamics rests in a stable state, whereas a sufficiently strong perturbation induces a large excursion in the phase space. We address the question of the emergence of oscillatory behaviors induced by noise and interaction in such systems. We tackle this problem by considering this model as a slow-fast system (the …
In this paper, we focus on the emergence of diverse neuronal oscillations arising in a mixed population of neurons with different excitability properties. These properties produce mixed mode oscillations (MMOs) characterized by the combination of large amplitudes and alternate subthreshold or small amplitude oscillations. Considering the biophysically plausible, Izhikevich neuron model, we demonstrate that various MMOs, including MMBOs (mixed mode bursting oscillations) and synchronized tonic spiking appear in a randomly connected network of neurons, where a fraction of them is in a quiescent …
In this paper, we study pairs of oscillators that are indirectly coupled via active (excitable) cells. We introduce a scalar phase model for coupled oscillators and excitable cells. We first show that one excitable and one oscillatory cell will exhibit phase locking at a variety of $m:n$ patterns. We next introduce a second oscillatory cell and show that the only attractor is synchrony between the oscillators. We will also study the robustness to heterogeneity when excitable cell fires or is quiescent. We next examine the dynamics when the oscillators are coupled via two excitable cells. In th…
We study synchronization in delay-coupled neural networks of heterogeneous nodes. It is well known that heterogeneities in the nodes hinder synchronization when becoming too large. We show that an adaptive tuning of the overall coupling strength can be used to counteract the effect of the heterogeneity. Our adaptive controller is demonstrated on ring networks of FitzHugh-Nagumo systems which are paradigmatic for excitable dynamics but can also -- depending on the system parameters -- exhibit self-sustained periodic firing. We show that the adaptively tuned time-delayed coupling enables synchro…
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 β-cells, which in isolation are known to exhibit irregular spiking. At the heart of this intrigui…
Recently, self-sustained oscillations in complex networks consisting of nonoscillatory nodes (network oscillators) have attracted great interest in diverse natural and social fields. Due to complexity of network behaviors, little is known so far about the basic structures and fundamental rules underlying the oscillations, not to mention the principles of how to control it. In this article we propose a common design principle for oscillations; predict novel and universal Branched Circle (BC) structures of oscillatory networks based on this principle; and suggest an operable Complexity Reduction…
The spread of an infectious disease is known to change people's behavior, which in turn affects the spread of disease. Adaptive network models that account for both epidemic and behavior change have found oscillations, but in an extremely narrow region of the parameter space, which contrasts with intuition and available data. In this paper we propose a simple SIS epidemic model on an adaptive network with time-delayed rewiring, and show that oscillatory solutions are now present in a wide region of the parameter space. Altering the transmission or rewiring rates reveals the presence of an ende…
We use a generic model for type-I excitability (known as the SNIPER or SNIC model) to describe the local dynamics of nodes within a network in the presence of non-zero coupling delays. Utilising the method of the Master Stability Function, we investigate the stability of the zero-lag synchronised dynamics of the network nodes and its dependence on the two coupling parameters, namely the coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model for type-II excitability), there are parameter ranges where the stability of synchronisation depends on the coupling strength and d…
Slow fluctuations shape cell signaling near a bifurcation in embryonic stem cells
Instituto de Investigación en Biomedicina de Buenos Aires (IBioBA) – CONICET（布宜诺斯艾利斯生物医学研究所）；Partner Institute of the Max Planck Society（马普学会合作所）；Research and Technology Institute, AREA Science Park（AREA科学园研究与技术研究所）；Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology（马克斯·普朗克分子生理学生态细胞生物学系）；Leiden Academic Centre for Drug Research, Leiden University（莱顿大学莱顿药物研究学术中心）
A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual sync…
We investigate the complex spatiotemporal dynamics of an ecological network with species dispersal mediated via a mean-field coupling. The local dynamics of the network are governed by the Truscott--Brindley model, which is an important ecological model showing excitability. Our results focus on the interplay of excitability and dispersal by always considering that the individual nodes are in their (excitable) steady states. In contrast to the previous studies, we not only observe the dispersal induced generation of oscillation but we also report two distinct mechanisms of cessation of oscilla…
We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but ar…
We consider Hodgkin-Huxley-type model that is a stiff ODE system with two fast and one slow variables. For the parameter ranges under consideration the original version of the model has unstable fixed point and the oscillating attractor that demonstrates bifurcation from bursting to spiking dynamics. Also a modified version is considered where the bistability occurs such that an area in the parameter space appears where the fixed point becomes stable and coexists with the bursting attractor. For these two systems we create artificial neural networks that are able to reproduce their dynamics. T…
