Phase diagram for the Winfree model of coupled nonlinear oscillators

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Pulse Shape and Voltage-Dependent Synchronization in Spiking Neuron Networks

Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the θ-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses are contradictory, and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse coupling in networks of QIF and θ-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage coupling is not very effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism at the heart of emergent collective behavior, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission in spiking neuron networks.

Complexified synchrony

The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.

Periodic solutions in next generation neural field models

We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.

Exact finite-dimensional description for networks of globally coupled spiking neurons

We consider large networks of globally coupled spiking neurons and derive an exact low-dimensional description of their collective dynamics in the thermodynamic limit. Individual neurons are described by the Ermentrout-Kopell canonical model that can be excitable or tonically spiking and interact with other neurons via pulses. Utilizing the equivalence of the quadratic integrate-and-fire and the theta-neuron formulations, we first derive the dynamical equations in terms of the Kuramoto-Daido order parameters (Fourier modes of the phase distribution) and relate them to two biophysically relevant macroscopic observables, the firing rate and the mean voltage. For neurons driven by Cauchy white noise or for Cauchy-Lorentz distributed input currents, we adapt the results by Cestnik and Pikovsky [Chaos 32, 113126 (2022)1054-150010.1063/5.0106171] and show that for arbitrary initial conditions the collective dynamics reduces to six dimensions. We also prove that in this case the dynamics asymptotically converges to a two-dimensional invariant manifold first discovered by Ott and Antonsen. For identical, noise-free neurons, the dynamics reduces to three dimensions, becoming equivalent to the Watanabe-Strogatz description. We illustrate the exact six-dimensional dynamics outside the invariant manifold by calculating nontrivial basins of different asymptotic regimes in a bistable situation.

Entrainment degree of globally coupled Winfree oscillators under external forcing

We consider globally connected coupled Winfree oscillators under the influence of an external periodic forcing. Such systems exhibit many qualitatively different regimes of collective dynamics. Our aim is to understand this collective dynamics and, in particular, the system's capability of entrainment to the external forcing. To quantify the entrainment of the system, we introduce the entrainment degree, that is, the proportion of oscillators that synchronize to the forcing, as the main focus of this paper. Through a series of numerical simulations, we study the entrainment degree for different inter-oscillator coupling strengths, external forcing strengths, and distributions of natural frequencies of the Winfree oscillators, and we compare the results for the different cases. In the case of identical oscillators, we give a precise description of the parameter regions where oscillators are entrained. Finally, we use a mean-field method, based on the Ott-Antonsen ansatz, to obtain a low-dimensional description of the collective dynamics and to compute an approximation of the entrainment degree. The mean-field results turn out to be strikingly similar to the results obtained through numerical simulations of the full system dynamics.

Role of phase-dependent influence function in the Winfree model of coupled oscillators

We consider a globally coupled Winfree model comprised of a phase-dependent influence function and sensitive function, and unravel the impact of offset and integer parameters, characterizing the shape of the influence function, on the phase diagram of the Winfree model. The decreasing value of the offset parameter decreases the degree of positive phase shift among the oscillators by promoting the negative phase shift, which indeed favors the onset of multistability among the synchronous oscillatory state and asynchronous stable steady states in a large region of the phase diagram. Further, large integer parameters lead to brief pulses of the influence function, which again enhances the effect of the offset parameter. There is an explosive transition to a synchronous oscillatory state from an asynchronous steady state via a Hopf bifurcation. Dynamical transitions and multistability emerge through saddle-node, pitchfork, and homoclinic bifurcations in the phase diagram. We deduce two ordinary differential equations corresponding to the two macroscopic variables from the population of globally coupled Winfree oscillators using the Ott-Antonsen ansatz. We also deduce various bifurcation curves analytically from the reduced low-dimensional macroscopic variables for the exactly solvable case. The analytical curves exactly match the simulation boundaries in the phase diagram.

The Sakaguchi-Kuramoto model in presence of asymmetric interactions that break phase-shift symmetry

The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The Sakaguchi-Kuramoto model is a generalization of the basic model that considers the presence of a phase lag parameter in the interaction, thereby making it asymmetric between oscillator pairs. Here, we consider a further generalization by adding an interaction that breaks the phase-shift symmetry of the model. The highlight of our study is the unveiling of a very rich bifurcation diagram comprising of both oscillatory and non-oscillatory synchronized states as well as an incoherent state: There are regions of two-state as well as an interesting and hitherto unexplored three-state coexistence arising from asymmetric interactions in our model.

Feedback-induced desynchronization and oscillation quenching in a population of globally coupled oscillators

Motivated from a wide range of applications, various methods to control synchronization in coupled oscillators have been proposed. Previous studies have demonstrated that global feedback typically induces three macroscopic behaviors: synchronization, desynchronization, and oscillation quenching. However, analyzing all of these transitions within a single theoretical framework is difficult, and thus the feedback effect is only partially understood in each framework. Herein, we analyze a model of globally coupled phase oscillators exposed to global feedback, which shows all of the typical macroscopic dynamical states. Analytical tractability of the model enables us to obtain detailed phase diagrams where transitions and bistabilities between different macroscopic states are identified. Additionally, we propose strategies to steer the oscillators into targeted states with minimal feedback strength. Our study provides a useful overview of the effect of global feedback and is expected to serve as a benchmark when more sophisticated feedback needs to be designed.

Traveling waves in non-local pulse-coupled networks

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.

Dynamics of Structured Networks of Winfree Oscillators

Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an “upstream” and a “downstream” oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.

Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons

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 are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control.

Realizing spin Hamiltonians in nanoscale active photonic lattices

Spin models arise in the microscopic description of magnetic materials and have been recently used to map certain classes of optimization problems involving large degrees of freedom. In this regard, various optical implementations of such Hamiltonians have been demonstrated to quickly converge to the global minimum in the energy landscape. Yet, so far, an integrated nanophotonic platform capable of emulating complex magnetic materials is still missing. Here, we show that the cooperative interplay among vectorial electromagnetic modes in coupled metallic nanolasers can be utilized to implement certain types of spin Hamiltonians. Depending on the topology/geometry of the arrays, these structures can be governed by a classical XY Hamiltonian that exhibits ferromagnetic and antiferromagnetic couplings, as well as geometrical frustration. Our results pave the way towards a scalable nanophotonic platform to study spin exchange interactions and could address a variety of optimization problems.

The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory

A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott-Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collective synchronization and full desynchronization is observed in all cases.

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

Synchronization of heterogeneous oscillator populations in response to weak and strong coupling

Synchronous behavior of a population of chemical oscillators is analyzed in the presence of both weak and strong coupling. In each case, we derive upper bounds on the critical coupling strength which are valid for arbitrary populations of nonlinear, heterogeneous oscillators. For weak perturbations, infinitesimal phase response curves are used to characterize the response to coupling, and graph theoretical techniques are used to predict synchronization. In the strongly perturbed case, we observe a phase dependent perturbation threshold required to elicit an immediate spike and use this behavior for our analytical predictions. Resulting upper bounds on the critical coupling strength agree well with our experimental observations and numerical simulations. Furthermore, important system parameters which determine synchronization are different in the weak and strong coupling regimes. Our results point to new strategies by which limit cycle oscillators can be studied when the applied perturbations become strong enough to immediately reset the phase.

Kuramoto Model for Excitation-Inhibition-Based Oscillations

The Kuramoto model (KM) is a theoretical paradigm for investigating the emergence of rhythmic activity in large populations of oscillators. A remarkable example of rhythmogenesis is the feedback loop between excitatory (E) and inhibitory (I) cells in large neuronal networks. Yet, although the EI-feedback mechanism plays a central role in the generation of brain oscillations, it remains unexplored whether the KM has enough biological realism to describe it. Here we derive a two-population KM that fully accounts for the onset of EI-based neuronal rhythms and that, as the original KM, is analytically solvable to a large extent. Our results provide a powerful theoretical tool for the analysis of large-scale neuronal oscillations.

The Dynamics of Networks of Identical Theta Neurons

We consider finite and infinite all-to-all coupled networks of identical theta neurons. Two types of synaptic interactions are investigated: instantaneous and delayed (via first-order synaptic processing). Extensive use is made of the Watanabe/Strogatz (WS) ansatz for reducing the dimension of networks of identical sinusoidally-coupled oscillators. As well as the degeneracy associated with the constants of motion of the WS ansatz, we also find continuous families of solutions for instantaneously coupled neurons, resulting from the reversibility of the reduced model and the form of the synaptic input. We also investigate a number of similar related models. We conclude that the dynamics of networks of all-to-all coupled identical neurons can be surprisingly complicated.

Synchronization scenarios in the Winfree model of coupled oscillators

Fifty years ago Arthur Winfree proposed a deeply influential mean-field model for the collective synchronization of large populations of phase oscillators. Here we provide a detailed analysis of the model for some special, analytically tractable cases. Adopting the thermodynamic limit, we derive an ordinary differential equation that exactly describes the temporal evolution of the macroscopic variables in the Ott-Antonsen invariant manifold. The low-dimensional model is then thoroughly investigated for a variety of pulse types and sinusoidal phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Spatiotemporal dynamics of the Kuramoto-Sakaguchi model with time-dependent connectivity

We study the dynamics of the paradigmatic Kuramoto-Sakaguchi model of identical coupled phase oscillators with various kinds of time-dependent connectivity using Eulerian discretization. We explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities, namely, the strength of incoherence and the discontinuity measure. In the quasistatic limit of the changing of coupling range, we observe how the system relaxes from one state to another and identify a few interesting collective dynamical states along the way. Under a sinusoidal change of the coupling range, the global order parameter characterizing the degree of synchronization in the system is shown to undergo a hysteresis with the coupling range. We also study the low-dimensional spatiotemporal dynamics of the local order parameter in the continuum limit using the recently developed Ott-Antonsen ansatz and justify some of our numerical results. In particular, we identify an intrinsic time scale of the Kuramoto system and show that the simulations exhibit two distinct kinds of qualitative behavior in two cases when the time scale associated with the switching of the coupling radius is very large compared to the intrinsic time scale and when it is comparable to the intrinsic time scale.

Chimeras in networks with purely local coupling

Chimera states in spatially extended networks of oscillators have some oscillators synchronized while the remainder are asynchronous. These states have primarily been studied in networks with nonlocal coupling, and more recently in networks with global coupling. Here, we present three networks with only local coupling (diffusive, to nearest neighbors) which are numerically found to support chimera states. One of the networks is analyzed using a self-consistency argument in the continuum limit, and this is used to find the boundaries of existence of a chimera state in parameter space.

Dynamics of globally coupled oscillators: Progress and perspectives

In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.

Reentrant transition in coupled noisy oscillators

We report on a synchronization-breaking instability observed in a noisy oscillator unidirectionally coupled to a pacemaker. Using a phase oscillator model, we find that, as the coupling strength is increased, the noisy oscillator lags behind the pacemaker more frequently and the phase slip rate increases, which may not be observed in averaged phase models such as the Kuramoto model. Investigation of the corresponding Fokker-Planck equation enables us to obtain the reentrant transition line between the synchronized state and the phase slip state. We verify our theory using the Brusselator model, suggesting that this reentrant transition can be found in a wide range of limit cycle oscillators.

Zero-lag synchronization despite inhomogeneities in a relay system

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

Driving potential and noise level determine the synchronization state of hydrodynamically coupled oscillators

Motile cilia are highly conserved structures in the evolution of organisms, generating the transport of fluid by periodic beating, through remarkably organized behavior in space and time. It is not known how these spatiotemporal patterns emerge and what sets their properties. Individual cilia are nonequilibrium systems with many degrees of freedom. However, their description can be represented by simpler effective force laws that drive oscillations, and paralleled with nonlinear phase oscillators studied in physics. Here a synthetic model of two phase oscillators, where colloidal particles are driven by optical traps, proves the role of the average force profile in establishing the type and strength of synchronization. We find that highly curved potentials are required for synchronization in the presence of noise. The applicability of this approach to biological data is also illustrated by successfully mapping the behavior of cilia in the alga Chlamydomonas onto the coarse-grained model.

An oscillator network exhibiting a long-lasting response to an external signal

This study proposes an oscillator network to model the long-lasting responses observed in neural circuits. The responses of the proposed network model are represented by the temporal synchronization of the oscillators. The response duration does not depend on the natural frequency of the oscillators, which allows the responses to last much longer than the oscillation period of the oscillators. We can control the response duration by tuning the connection strengths between the oscillators and the external signal that triggers the responses. It is possible to break and restart the responses regardless of the way in which the oscillators are connected.

Adaptive Control Using an Oscillator Network with Capacitive Couplers

We focus on the oscillatory aspects of neural components and their roles in neural system functionality such as adaptability. This study provides an implementation of adaptive control by using a network system consisting of oscillators and capacitive couplers. The functionality of oscillators alone is fairly limited, but capacitive couplers change interaction between oscillators, enabling logical operations, absolute function, gain variable amplifiers, and adaptive control. We provide basic functional networks of oscillators and capacitive couplers and demonstrate adaptive control implementation based on these functions.

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.

Renormalization of oscillator lattices with disorder

A real-space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dvarphi_{k}/dt=omega_{k}+g summation operator_{l}f_{lk}(varphi_{l},varphi_{k}) . The transformation acts on ensembles of such lattices. Critical properties corresponding to a second-order phase transition toward macroscopic synchronization are deduced. The analysis is potentially exact but relies in part on unproven assumptions. Numerically, second-order phase transitions with the predicted properties are observed as g increases in two structurally different two-dimensional oscillator models. One model has smooth coupling f_{lk}(varphi_{l},varphi_{k})=phi(varphi_{l}-varphi_{k}) , where phi(x) is nonodd. The other model is pulse coupled, with f_{lk}(varphi_{l},varphi_{k})=delta(varphi_{l})phi(varphi_{k}) . Lower bounds for the critical dimensions for different types of coupling are obtained. For nonodd coupling, macroscopic synchronization cannot be ruled out for any dimension D> or =1 , whereas in the case of odd coupling, the well-known result that it can be ruled out for D<3 is regained.

Input-Driven Oscillations in Networks with Excitatory and Inhibitory Neurons with Dynamic Synapses

Previous work has shown that networks of neurons with two coupled layers of excitatory and inhibitory neurons can reveal oscillatory activity. For example, Börgers and Kopell (2003) have shown that oscillations occur when the excitatory neurons receive a sufficiently large input. A constant drive to the excitatory neurons is sufficient for oscillatory activity. Other studies (Doiron, Chacron, Maler, Longtin, & Bastian, 2003; Doiron, Lindner, Longtin, Maler, & Bastian, 2004) have shown that networks of neurons with two coupled layers of excitatory and inhibitory neurons reveal oscillatory activity only if the excitatory neurons receive correlated input, regardless of the amount of excitatory input. In this study, we show that these apparently contradictory results can be explained by the behavior of a single model operating in different regimes of parameter space. Moreover, we show that adding dynamic synapses in the inhibitory feedback loop provides a robust network behavior over a broad range of stimulus intensities, contrary to that of previous models. A remarkable property of the introduction of dynamic synapses is that the activity of the network reveals synchronized oscillatory components in the case of correlated input, but also reflects the temporal behavior of the input signal to the excitatory neurons. This allows the network to encode both the temporal characteristics of the input and the presence of spatial correlations in the input simultaneously.

Phase diagram of a generalized Winfree model

We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling k be a function of the Kuramoto order parameter r through an exponent z such that z=1 corresponds to the standard Winfree model, z<1 strengthens the coupling at low r (low amount of synchronization), and at z>1 , the coupling is weakened for low r . Using both analytical and numerical approaches, we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing z<1 . We also find that the original Winfree model is a rather special case; indeed, the partial locked behavior disappears for z>1 . At fixed k and varying gamma , the stability boundary of the locked phase corresponds to a transition that is continuous for z<1 and first order for z>1 . This change in the nature of the transition is in accordance with a previous study of a similarly modified Kuramoto model.

Singular unlocking transition in the Winfree model of coupled oscillators

The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the unlocking transition, in which the frequency-locked state disappears abruptly as the spread of natural frequencies exceeds a critical width. Viewed as a function of the coupling strength, this critical width defines a bifurcation curve in parameter space. For the special case where the frequency distribution is uniform, earlier work had uncovered a puzzling singularity in this bifurcation curve. Here we seek to understand what causes the singularity. Using the Poincaré-Lindstedt method of perturbation theory, we analyze the locked state and its associated unlocking transition, first for an arbitrary distribution of natural frequencies, and then for discrete systems of N oscillators. We confirm that the bifurcation curve becomes singular for a continuum uniform distribution, yet find that it remains well behaved for any finite N , suggesting that the continuum limit is responsible for the singularity.

Coupling effects on energy transduction in coupled polymer chains with perturbation of noise

Noise-assistant transduction was investigated in coupled polymer chains where one subsystem was exposed to environment noise. It was found that coupling could transfer oscillation from one subsystem disturbed by noise to the other not disturbed by noise and play a role of a noise filtering for the other. Then, a sort of coupling-induced synchronization was investigated as a function of noise intensity and coupling strength. In particular, we calculated the minimum coupling strength to reach synchronization and pointed out that noise dominated at small coupling strength, otherwise, coupling dominated.

Coarse graining the dynamics of coupled oscillators

We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.

Phase resetting and coupling of noisy neural oscillators

A number of experimental groups have recently computed Phase Response Curves (PRCs) for neurons. There is a great deal of noise in the data. We apply methods from stochastic nonlinear dynamics to coupled noisy phase-resetting maps and obtain the invariant density of phase distributions. By exploiting the special structure of PRCs, we obtain some approximations for the invariant distributions. Comparisons to Monte-Carlo simulations are made. We show how phase-dependence of the noise can move the peak of the invariant density away from the peak expected from the analysis of the deterministic system and thus lead to noise-induced bifurcations.

Wave patterns in frequency-entrained oscillator lattices

We study and classify firing waves in two-dimensional oscillator lattices. To do so, we simulate a pulse-coupled oscillator model aimed to resemble a group of pacemaker cells in the heart. The oscillators are assigned random natural frequencies, and we focus on frequency entrained states. Depending on the initial condition, three types of wave landscapes are seen asymptotically. A concentric landscape contains concentric waves with one or more foci. Spiral landscapes contain one or more spiral waves. A mixed landscape contains both concentric and spiral waves. Mixed landscapes are only seen for moderate coupling strengths g, since for higher g, spiral waves have higher frequency than concentric waves, so that they cannot mix in frequency entrained states. If the initial condition is random, the probability to get a concentric landscape increases with increasing coupling strength g, but decreases with increasing lattice size. The g dependence of the probability enables hysteresis, where the system jumps between the two landscape types as g is continuously changed. For moderate g, spiral tips rotate around a suppressed oscillator that never fires. We call such an oscillator an oscillator defect. A spiral may also rotate around a point defect situated between the oscillators. In that case all oscillators fire at the entrained frequency. For larger g, a spiral tip either moves around a row of suppressed oscillators, a row defect, or around an open curve situated between the oscillators, which may be called a line defect. The length of a row or line defect increases with g. Our results may help understand sinus node reentry, where the natural pacemaker of the heart suddenly shifts to a higher frequency. Some of the observed phenomena seem generic, based on simulations of other models.

Modular synchronization in complex networks

We study the synchronization transition (ST) of a modified Kuramoto model on two different types of modular complex networks. It is found that the ST depends on the type of intermodular connections. For the network with decentralized (centralized) intermodular connections, the ST occurs at finite coupling constant (behaves abnormally). Such distinct features are found in the yeast protein interaction network and the Internet, respectively. Moreover, by applying the finite-size scaling analysis to an artificial network with decentralized intermodular connections, we obtain the exponent associated with the order parameter of the ST to be beta approximately 1 different from beta(MF) approximately 1/2 obtained from the scale-free network with the same degree distribution but the absence of modular structure, corresponding to the mean field value.

Phase chaos in coupled oscillators

A complex high-dimensional chaotic behavior, phase chaos, is found in the finite-dimensional Kuramoto model of coupled phase oscillators. This type of chaos is characterized by half of the spectrum of Lyapunov exponents being positive and the Lyapunov dimension equaling almost the total system dimension. Intriguingly, the strongest phase chaos occurs for intermediate-size ensembles. Phase chaos is a common property of networks of oscillators of very different natures, such as phase oscillators, limit-cycle oscillators, and chaotic oscillators, e.g., Rössler systems.

Mechanism of desynchronization in the finite-dimensional Kuramoto model

We study how a decrease of the coupling strength causes a desynchronization in the Kuramoto model of N globally coupled phase oscillators. We show that, if the natural frequencies are distributed uniformly or close to that, the synchronized state can robustly split into any number of phase clusters with different average frequencies, even culminating in complete desynchronization. In the simplest case of N=3 phase oscillators, the course of the splitting is controlled by a Cherry flow. The general N-dimensional desynchronization mechanism is numerically illustrated for N=5.

Phase shifts of synchronized oscillators and the systolic-diastolic blood pressure relation

We study the phase-synchronization properties of systolic and diastolic arterial pressure in healthy subjects. We find that delays in the oscillatory components of the time series depend on the frequency bands that are considered, in particular we find a change of sign in the phase shift going from the very low frequency band to the high frequency band. This behavior should reflect a collective behavior of a system of nonlinear interacting elementary oscillators. We prove that some models describing such systems, e.g., the Winfree and the Kuramoto models, offer a clue to this phenomenon. For these theoretical models there is a linear relationship between phase shifts and the difference of natural frequencies of oscillators and a change of sign in the phase shift naturally emerges.

Internal stochastic resonance in two coupled liquid membrane oscillators

Internal stochastic resonance (ISR) is investigated in two coupled liquid membrane oscillators when only one oscillator is subjected to environmental noise in the absence of an external signal. Comparing the responses of both subsystems, it is found that enhancement or suppression of ISR for each oscillator depends on the coupling strength, that synchronization of the two oscillators can occur only at strong coupling strength, and that ISR without tuning can also occur under certain conditions in the above-mentioned models.

A Notch feeling of somite segmentation and beyond

Vertebrate segmentation is manifested during embryonic development as serially repeated units termed somites that give rise to vertebrae, ribs, skeletal muscle and dermis. Many theoretical models including the "clock and wavefront" model have been proposed. There is compelling genetic evidence showing that Notch-Delta signaling is indispensable for somitogenesis. Notch receptor and its target genes, Hairy/E(spl) homologues, are known to be crucial for the ticking of the segmentation clock. Through the work done in mouse, chick, Xenopus and zebrafish, an oscillator operated by cyclical transcriptional activation and delayed negative feedback regulation is emerging as the fundamental mechanism underlying the segmentation clock. Ubiquitin-dependent protein degradation and probably other posttranslational regulations are also required. Fgf8 and Wnt3a gradients are important in positioning somite boundaries and, probably, in coordinating tail growth and segmentation. The circadian clock is another biochemical oscillator, which, similar to the segmentation clock, is operated with a negative transcription-regulated feedback mechanism. While the circadian clock uses a more complicated network of pathways to achieve homeostasis, it appears that the segmentation clock exploits the Notch pathway to achieve both signal generation and synchronization. We also discuss mathematical modeling and future directions in the end.

Noise-controlled oscillations and their bifurcations in coupled phase oscillators

We derive in Gaussian approximation dynamical equations for the first two cumulants of the mean field fluctuations in a system of globally coupled stochastic phase oscillators. In these equations the intensity of noise serves as an explicit control parameter. Its variation generates transitions between three dynamical regimes: (i) stationary, (ii) rotatory and (iii) locally oscillatory (breathing). The latter regime has previously not been reported in studies of globally coupled noisy phase oscillators. Our detailed bifurcation analysis is supported by numerical simulations of an ensemble of coupled stochastic phase oscillators. Similar regimes are also found in simulations of globally coupled stochastic FitzHugh-Nagumo elements.

Synchronization law for a van der Pol array

We explore the transition to in-phase synchronization in globally coupled oscillator arrays, and compare results for van der Pol arrays with Josephson junction arrays. Our approach yields in each case an analytically tractable iterative map; the resulting stability formulas are simple because the expansion procedure identifies natural parameter groups. A third example, an array of Duffing-van der Pol oscillators, is found to be of the same fundamental type as the van der Pol arrays, but the Josephson arrays are fundamentally different owing to the absence of self-resonant interactions.

Go and chemoresistance of malignant cells

In general terms the paper discusses the relevance of mitotically quiescent, malignant cells for increasing resistance to chemotherapy. Detailed are some routes to Go for malignancies either dependent or autonomous for their mitotic potential.