Search

Many systems of physical and biological interest are characterized by assemblies of phase oscillators whose interaction is mediated by a diffusing chemical. The coupling effect results from the fact that the local concentration of the mediating chemical affects both its production and absorption by each oscillator. Since the chemical diffuses through the medium in which the oscillators are embedded, the coupling among oscillators is non-local; it considers all the oscillators depending on their relative spatial distances. We considered a mathematical model for this coupling, when the diffusion time is arbitrary with respect to the characteristic oscillator periods, yielding a system of coupled nonlinear integro-differential equations which can be solved using Green functions for appropriate boundary conditions. In this paper, we show numerical solutions of these equations for three finite domains: a linear one-dimensional interval, a rectangular, and a circular region, with absorbing boundary conditions. From the numerical solutions, we investigate phase and frequency synchronization of the oscillators, with respect to changes in the coupling parameters for the three considered geometries. [ABSTRACT FROM AUTHOR]

Adaptive interactions are an important property of many real-word network systems. A feature of such networks is the change in their connectivity depending on the current states of the interacting elements. In this work, we study the question of how the heterogeneous character of adaptive couplings influences the emergence of new scenarios in the collective behavior of networks. Within the framework of a two-population network of coupled phase oscillators, we analyze the role of various factors of heterogeneous interaction, such as the rules of coupling adaptation and the rate of their change in the formation of various types of coherent behavior of the network. We show that various schemes of heterogeneous adaptation lead to the formation of transient phase clusters of various types. [ABSTRACT FROM AUTHOR]

We explore the model of a population of nonlocally coupled identical phase oscillators on a ring (Abrams and Strogatz 2004 Phys. Rev. Lett. 93 174102) and describe traveling patterns. In the continuous in space formulation, we find families of traveling wave solutions for left-right symmetric and asymmetric couplings. Only the simplest of these waves are stable, which is confirmed by numerical simulations for a finite population. We demonstrate that for asymmetric coupling, a weakly turbulent traveling chimera regime is established, both from an initial standing chimera or an unstable traveling wave profile. The weakly turbulent chimera is a macroscopically chaotic state, with a well-defined synchronous domain and partial coherence in the disordered domain. We characterize it through the correlation function and the Lyapunov spectrum.

The purpose of this study is to acquaint the reader with one of the effective approaches to describing processes in adaptive networks, built in the framework of the well-known Kuramoto model. Methods. The solution to this problem is based on the analysis of the results of works devoted to the study of the dynamics of oscillatory networks with adaptive couplings. Main classes of models of dynamical couplings used in the description of adaptive networks are considered, and the dynamical and structural effects caused by the presence of the corresponding law of coupling adaptation are analyzed. Results. Principles of constructing models of adaptive networks based on the phase description developed by Kuramoto are presented. Materials presented in the review show that the Kuramoto system with dynamic couplings demonstrates a wide range of fundamentally new phenomena and modes. Considered networks include well-known models of dynamical couplings that implement various laws of adaptation of inter-element interactions depending on the states of the elements, in particular, on their relative phase difference. For each network model, a class of possible solutions is established, and general properties of collective dynamics are identified, due to the presence of adaptability of couplings. One of the features of such networks is the multistability of behavior, determined by the possibility of the formation in the network of many different cluster states, including chimera ones. It was found that the implemented coupling adaptation mechanism affects not only the configuration of clusters formed in the network, but also the nature of phase distributions within them. The processes of cluster formation are accompanied by a restructuring of the interaction topology, leading to the formation of hierarchical and modular structures. Conclusion. In conclusion, we briefly summarize the results presented in the review.

The level of firing rate heterogeneity in a population of cortical neurons has consequences for how stimuli are processed. Recent studies have shown that the right amount of firing rate heterogeneity (not too much or too little) is a signature of efficient coding, thus quantifying the relative amount of firing rate heterogeneity is important. In a feedforward network of stochastic neural oscillators, we study the firing rate heterogeneity stemming from two sources: intrinsic (different individual cells) and network (different effects from presynaptic inputs). We find that the relationship between these two forms of heterogeneity can lead to significant changes in firing rate heterogeneity. We consider several networks, including noisy excitatory synaptic inputs, and noisy inputs with both excitatory and inhibitory inputs. To mathematically explain these results, we apply a phase reduction and derive asymptotic approximations of the firing rate statistics assuming weak noise and coupling. Our analytic calculations reveals how the interaction between intrinsic and network heterogeneity results in different firing rate distributions. Our work shows the importance of the phase-resetting curve (and various transformations of it revealed by our analytic calculations) in controlling firing rate statistics.

There is evidence in mammals that recovering from jetlag after westward travel is faster than after eastward travel. To understand why, mathematical models have been used, along with theories of entrainment rooted in experimental evidence. The most complete understanding relies on detailed mathematical modeling, so it is helpful to develop an intuition about why there is an east-west asymmetry. One such intuition is that humans have long periods and therefore recover better when they can delay. Although this is part of the reason, it does not explain why short-period mice also recover from westward travel faster. Our goal is to provide a simple intuition consistent with detailed mathematical theories, but which does not require mathematical expertise to follow. Here, we present the intuition that westward travel is easier to recover from because of a simple principle: delays are self-enhancing. [ABSTRACT FROM AUTHOR]

We investigated interactions within chimera states in a phase oscillator network with two coupled subpopulations. To quantify interactions within and between these subpopulations, we estimated the corresponding (delayed) mutual information that—in general—quantifies the capacity or the maximum rate at which information can be transferred to recover a sender's information at the receiver with a vanishingly low error probability. After verifying their equivalence with estimates based on the continuous phase data, we determined the mutual information using the time points at which the individual phases passed through their respective Poincaré sections. This stroboscopic view on the dynamics may resemble, e.g., neural spike times, that are common observables in the study of neuronal information transfer. This discretization also increased processing speed significantly, rendering it particularly suitable for a fine-grained analysis of the effects of experimental and model parameters. In our model, the delayed mutual information within each subpopulation peaked at zero delay, whereas between the subpopulations it was always maximal at non-zero delay, irrespective of parameter choices. We observed that the delayed mutual information of the desynchronized subpopulation preceded the synchronized subpopulation. Put differently, the oscillators of the desynchronized subpopulation were “driving” the ones in the synchronized subpopulation. These findings were also observed when estimating mutual information of the full phase trajectories. We can thus conclude that the delayed mutual information of discrete time points allows for inferring a functional directed flow of information between subpopulations of coupled phase oscillators.

Coupled populations of identical phase oscillators with higher-order interactions can give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. For these heteroclinic cycles to be observable, they have to have some stability properties. In this paper, we complement the existence results for heteroclinic cycles given in a companion paper by proving stability results for heteroclinic cycles for coupled oscillator populations consisting of two oscillators each. Moreover, we show that for systems with four coupled phase oscillator populations, there are distinct heteroclinic cycles that form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators. [ABSTRACT FROM AUTHOR]

Many real-world systems can be modeled as networks of interacting oscillatory units. Collective dynamics that are of functional relevance for the oscillator network, such as switching between metastable states, arise through the interplay of network structure and interaction. Here, we give results for small networks on the existence of heteroclinic cycles between dynamically invariant sets on which the oscillators show localized frequency synchrony. Trajectories near these heteroclinic cycles will exhibit sequential switching of localized frequency synchrony: a population oscillators in the network will oscillate faster (or slower) than others and which population has this property sequentially changes over time. Since we give explicit conditions on the system parameters for such dynamics to arise, our results give insights into how network structure and interactions (which include higher-order interactions between oscillators) facilitate heteroclinic switching between localized frequency synchrony. [ABSTRACT FROM AUTHOR]

We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott–Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles. [ABSTRACT FROM AUTHOR]

Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks, and neuronal networks. We investigate a paradigmatic system of adaptively coupled phase oscillators inspired by neuronal networks with synaptic plasticity. One important behavior of such systems reveals splitting of the network into clusters of oscillators with the same frequencies, where different clusters correspond to different frequencies. Starting from one-cluster solutions we provide existence criteria for multicluster solutions and present their explicit form. The phases of the oscillators within one cluster can be organized in different patterns: antipodal, double antipodal, and splay type. Interestingly, multiclusters are shown to exist where different clusters exhibit different patterns. For instance, an antipodal cluster can coexist with a splay cluster. We also provide stability conditions for one- and multicluster solutions. These conditions, in particular, reveal a high level of multistability. [ABSTRACT FROM AUTHOR]

We describe an oscillatory neural network designed as a system of generalized phase oscillators with a central element. It is shown that a winner-take-all principle can be realized in this system in terms of the competition of peripheral oscillators for the synchronization with a central oscillator. Several examples illustrate how this network can be used for the simulation of various cognitive functions: consecutive selection of objects in the image, visual search, and multiple object tracking. [ABSTRACT FROM AUTHOR]

Highlights • Observables distribution asymmetry can cause false connectivity estimations. • To avoid this error the measured phase should be transformed to an invariant phase. • We extended Dynamic Causal Modelling for phase coupling with such a transformation. • The new extension of Dynamic Causal Modelling can characterize m:n phase coupling. • The proposed new method is able to resolve spurious coupling between brain regions. Abstract Background The interaction of different brain regions is supported by transient synchronization between neural oscillations at different frequencies. Different measures based on synchronization theory are used to assess the strength of the interactions from experimental data. One method of estimating the effective connectivity between brain regions, within the framework of the theory of weakly coupled phase oscillators, was implemented in Dynamic Causal Modelling (DCM) for phase coupling (Penny et al., 2009). However, the results of such an approach strongly depend on the observables used to reconstruct the equations (Kralemann et al., 2008). In particular, an asymmetric distribution of the observables could result in a false estimation of the effective connectivity between the network nodes. New method In this work we built a new modelling part into DCM for phase coupling, and extended it with a distortion function that accommodates departures from purely sinusoidal oscillations. Results By analysing numerically generated data sets with an asymmetric phase distribution, we demonstrated that the extended DCM for phase coupling with the additional modelling component, correctly estimates the coupling functions. Comparison with existing methods The new method allows for different intrinsic frequencies among coupled neuronal populations and provides results that do not depend on the distribution of the observables. Conclusions The proposed method can be used to analyse effective connectivity between brain regions within and between different frequency bands, to characterize m:n phase coupling, and to unravel underlying mechanisms of the transient synchronization. [ABSTRACT FROM AUTHOR]

Large-scale neurophysiological networks are often reconstructed from band-pass filtered time series derived from magnetoencephalography (MEG) data. Common practice is to reconstruct these networks separately for different frequency bands and to treat them independently. Recent evidence suggests that this separation may be inadequate, as there can be significant coupling between frequency bands (interlayer connectivity). A multilayer network approach offers a solution to analyze frequency-specific networks in one framework. We propose to use a recently developed network reconstruction method in conjunction with phase oscillator models to estimate interlayer connectivity that optimally fits the empirical data. This approach determines interlayer connectivity based on observed frequency-specific time series of the phase and a connectome derived from diffusion weighted imaging. The performance of this interlayer reconstruction method was evaluated in-silico . Our reconstruction of the underlying interlayer connectivity agreed to very high degree with the ground truth. Subsequently, we applied our method to empirical resting-state MEG data obtained from healthy subjects and reconstructed two-layered networks consisting of either alpha-to-beta or theta-to-gamma band connectivity. Our analysis revealed that interlayer connectivity is dominated by a multiplex structure, i.e. by one-to-one interactions for both alpha-to-beta band and theta-to-gamma band networks. For theta–gamma band networks, we also found a plenitude of interlayer connections between distant nodes, though weaker connectivity relative to the one-to-one connections. Our work is an stepping stone towards the identification of interdependencies across frequency-specific networks. Our results lay the ground for the use of the promising multilayer framework in this field with more-informed and justified interlayer connections.

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem. [ABSTRACT FROM AUTHOR]

Highlights • Adapted a real pacemaker model that has calcium handling and perturbations to the funny current to model other cell types. • Analyzed spatio-temporal dynamics of tissue and did not find central wave generation despite biophysical realism. • Found peripheral wave generation that is consistent with experiments that removed the atrium. • Reduced model captured frequency, stability, and transient times of traveling waves; transients showed center-to-peripheral. • Thus, central wave generation in these realistic models must occur transiently or require atrium and/or fibrous tissue. Abstract The sinoatrial-node (SAN) is a complex heterogeneous tissue that generates a stable rhythm in healthy hearts, yet a general mechanistic explanation for when and how this tissue remains stable is lacking. Although computational and theoretical analyses could elucidate these phenomena, such methods have rarely been used in realistic (large-dimensional) gap-junction coupled heterogeneous pacemaker tissue models. In this study, we adapt a recent model of pacemaker cells (Severi et al., 2012), incorporating biophysical representations of ion channel and intracellular calcium dynamics, to capture physiological features of a heterogeneous population of pacemaker cells, in particular "center" and "peripheral" cells with distinct intrinsic frequencies and action potential morphology. Large-scale simulations of the SAN tissue, represented by a heterogeneous tissue structure of pacemaker cells, exhibit a rich repertoire of behaviors, including complete synchrony, traveling waves of activity originating from periphery to center, and transient traveling waves originating from the center. We use phase reduction methods that do not require fully simulating the large-scale model to capture these observations. Moreover, the phase reduced models accurately predict key properties of the tissue electrical dynamics, including wave frequencies when synchronization occurs, and wave propagation direction in a variety of tissue models. With the reduced phase models, we analyze the relationship between cell distributions and coupling strengths and the resulting transient dynamics. Further, the reduced phase model predicts parameter regimes of irregular electrical dynamics. Thus, we demonstrate that phase reduced oscillator models applied to realistic pacemaker tissue is a useful tool for investigating the spatial-temporal dynamics of cardiac pacemaker activity. [ABSTRACT FROM AUTHOR]

Systems of N identical globally coupled phase oscillators can demonstrate a multitude of complex behaviors. Such systems can have chaotic dynamics for N > 4 when a coupling function is biharmonic. The case N = 4 does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies have shown that some of chaotic attractors in this system are organized by heteroclinic networks. In the present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria. [ABSTRACT FROM AUTHOR]

We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i.e., when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative dynamics in the phase space. We relate this phenomenon to the time-reversibility of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger rings with skew-symmetric coupling, some suffcient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings. [ABSTRACT FROM AUTHOR]

The chimera state, exhibiting a hybrid state of coexisting coherent and incoherent behaviors, has become a fast growing field in the past decade. In this paper, we investigate bipartite networks of nonlocally coupled phase oscillators. We report different types of chimera states such as in-phase chimera states, antiphase chimera states, and syn-desyn chimera states. Phase diagrams on different parameter planes are presented. [ABSTRACT FROM AUTHOR]

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here, we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector-for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling. [ABSTRACT FROM AUTHOR]

We investigate the transition to phase and frequency synchronization in a one-dimensional chain of phase oscillator “cells” where the coupling is mediated by the local concentration of a chemical which can diffuse in the inter-oscillator medium and it is both secreted and absorbed by the oscillator “cells”, influencing their dynamical behavior. This coupling has the advantage of having a tunable parameter which makes it possible to pass continuously from a global (all-to-all) to a local (nearest-neighbor) coupling form. We have verified that synchronous behavior depends on the coupling strength and coupling length. [ABSTRACT FROM AUTHOR]

In many applications of biological interest spatially localized phase oscillators present a non-local coupling characterized by the diffusion of a mediating chemical. The latter is both produced and absorbed by the phase oscillators according to their dynamical behavior. We investigate numerically phase and frequency synchronization in a model for chemical coupling of uniform phase oscillators with natural frequencies randomly distributed according to some statistical distribution. We study the effect of removal of oscillators, caused by lesions, on the synchronization properties of the network. The partial or complete suppression of synchronization is also investigated through the influence of a time-delayed feedback external signal. • Chemical coupling among phase oscillators is mediated by the a substance which diffuses through the medium in which the oscillators are embedded. • The oscillators both release and absorb this chemical depending on their dynamical behavior. • The resulting interaction is nonlocal and affects collective properties like phase and frequency synchronization. • We investigate the effects of removing oscillators, and of an external time-delayed feedback control signal, designed to suppress synchronization. [ABSTRACT FROM AUTHOR]

Although it is very important to scrutinize recurrent structures of neural networks for elucidating brain functions, conventional methods often have difficulty in characterizing global loops within a network systematically. Here we applied the Hodge–Kodaira decomposition, a topological method, to an evolving neural network model in order to characterize its loop structure. By controlling a learning rule parametrically, we found that a model with an STDP-rule, which tends to form paths coincident with causal firing orders, had the most loops. Furthermore, by counting the number of global loops in the network, we detected the inhomogeneity inside the chaotic region, which is usually considered intractable. [ABSTRACT FROM AUTHOR]

We investigated the organization of a recurrent network under ongoing synaptic plasticity using a model of neural oscillators coupled by dynamic synapses. In this model, the coupling weights changed dynamically, depending on the timing between the oscillators. We determined the phase coupling function of the oscillator model, Γ ( ϕ ) , using conductance-based neuron models. Furthermore, we examined the effects of the Fourier zero mode of Γ ( ϕ ) , which has a critical role in the case of spike-time-dependent plasticity-organized recurrent networks. Heterogeneous layered clusters with different frequencies emerged from homogeneous populations as the Fourier zero mode increased. Our findings may provide new insights into the self-assembly mechanisms of neural networks related to synaptic plasticity. [ABSTRACT FROM AUTHOR]

Cockroaches are remarkably stable runners, exhibiting rapid recovery from external perturbations. To uncover the mechanisms behind this important behavioral trait, we recorded leg kinematics of freely running animals in both undisturbed and perturbed trials. Functional coupling underlying inter-leg coordination was monitored before and during localized perturbations, which were applied to single legs via magnetic impulses. The resulting transient effects on all legs and the recovery times to normal pre-perturbation kinematics were studied. We estimated coupling architecture and strength by fitting experimental data to a six-leg-unit phase oscillator model. Using maximum-likelihood techniques, we found that a network with nearest-neighbor inter-leg coupling best fitted the data and that, although coupling strengths vary among preparations, the overall inputs entering each leg are approximately balanced and consistent. Simulations of models with different coupling strengths encountering perturbations suggest that the coupling schemes estimated from our experiments allow animals relatively fast and uniform recoveries from perturbations. [ABSTRACT FROM AUTHOR]

Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations. [ABSTRACT FROM AUTHOR]

Background: Recent experimental results suggest that impairment of auditory information processing in the thalamo-cortical loop is crucially related to schizophrenia. Large differences between schizophrenia patients and healthy controls were found in the cortical EEG signals. Methods: We derive a phenomenological mathematical model, based on coupled phase oscillators with continuously distributed frequencies to describe the neural activity of the thalamo-cortical loop. We examine the influence of the bidirectional coupling strengths between the thalamic and the cortical area with regard to the phase-locking effects observed in the experiments. We extend this approach to a model consisting of a thalamic area coupled to two cortical areas, each comprising a set of nonidentical phase oscillators. In the investigations of our model, we applied the Ott-Antonsen theory and the Pikovsky-Rosenblum reduction methods to the original system. Results: The results derived from our mathematical model satisfactorily reproduce the experimental data obtained by EEG measurements. Furthermore, they show that modifying the coupling strength from the thalamic region to a cortical region affects the duration of phase synchronization, while a change in the feedback to the thalamus affects the strength of synchronization in the cortex. In addition, our model provides an explanation in terms of nonlinear dynamics as to why brain waves desynchronize after a given phase reset. Conclusion: Our model can explain functional differences seen between EEG records of healthy subjects and schizophrenia patients on a system theoretic basis. Because of this and its predictive character, the model may be considered to pave the way towards an early and reliable clinical detection of schizophrenia that is dependent on the interconnections between the thalamic and cortical regions. In particular, the model parameter that describes the strength of this connection can be used for a diagnostic classification of schizophrenia patients. [ABSTRACT FROM AUTHOR]

Abstract: We present a perceptron model with processing units consisting of coupled phase oscillators. The processing units are able to compute the input signals through a high order synapse mechanism. We show how a network of these elements can be used in analogy to the classical multilayer feedforward neural network. The main characteristics of the classical multilayer perceptron model are conserved, as for example, the backpropagation algorithm for learning. This model of coupled phase oscillators can be seen as a generic study in order to use different kind of oscillators for computational tasks. [Copyright &y& Elsevier]

Abstract: The problem of growing complexity of the dynamics of the coupled phase oscillators as the number of oscillators in the chain increases is considered. The organization of the parameter space (parameter of the frequency detuning between the second and the first oscillators versus parameter of dissipative coupling) is discussed. The regions of complete synchronization, quasi-periodic regimes of different dimensions and chaos are identified. We discuss transformation of the domains of different dynamics as the number of oscillators grows. We use the method of charts of Lyapunov exponents and modification of the method of the chart of dynamical regimes to visualize two-frequency regimes of different type. Limits of applicability of the quasi-harmonic approximation and the features of the dynamics of the original system which are not described by the approximate phase equations are discussed for the case of three coupled oscillators. [Copyright &y& Elsevier]

Abstract: We design an artificial central pattern generator and we exemplify its integration using different prototypic virtual robot models. The artificial central pattern generator is modeled through a network of phase oscillators. Virtual robots are controlled by this central pattern generator using an interface that interpolates stored angular states of their target poses. Finally, we consider control signals coming from the robot''s environment to the central pattern generator and to the interface between the central pattern generator and the robot. We show that the control signals can change the dynamics of the system in order to modify the robot''s movement patterns. We study three cases: changing the frequency of the pattern sequences, switching between gaits of locomotion and the reorientation of a robotic entity. [Copyright &y& Elsevier]

Abstract: For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all stationary two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator’s phase to perturbations. For large systems with a PRC, which is zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together with its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony. [Copyright &y& Elsevier]

Abstract: We consider moving fronts and stationary “bumps” in networks of non-locally coupled phase oscillators. Fronts connect regions of high local synchrony with regions of complete asynchrony, while bumps consist of spatially-localised regions of partially-synchronous oscillators surrounded by complete asynchrony. Using the Ott–Antonsen ansatz we derive non-local differential equations which describe the network dynamics in the continuum limit. Front and bump solutions of these equations are studied by either “freezing” them in a travelling coordinate frame or analysing them as homoclinic or heteroclinic orbits. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable. [Copyright &y& Elsevier]

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and travelling wave solutions in the parameter space characterizing the nonlocality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of phase oscillators. [ABSTRACT FROM AUTHOR]

Abstract: Nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena. For a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative, we provide analytic formulas for the stationary statistical quantities of the random period. This is an important quantity which we term ‘response’ (i.e., the spike times, instantaneous frequency in neuroscience, the cycle time in chemical reactions, etc.) that is often analytically intractable in noisy oscillator systems. The analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time-varying Fokker–Planck equation. The steady-state and dynamic responses are also analyzed with deterministic forcing. A second order analytic formula is derived for the steady-state response, whereas the dynamic response with time-varying forcing is more complicated. We focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation. By utilizing various techniques in the weak noise regime, this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing. Comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory. [Copyright &y& Elsevier]

The idea that synchronized oscillations are important in cognitive tasks is receiving significant attention. In this view, single neurons are no longer elementary computational units. Rather, coherent oscillating groups of neurons are seen as nodes of networks performing cognitive tasks. From this assumption, we develop a model of stimulus-pattern learning and recognition. The three most salient features of our model are: 1) a new definition of synchronization; 2) demonstrated robustness in the presence of noise; and 3) pattern learning. [ABSTRACT FROM AUTHOR]

Abstract: We study a variety of mixed synchronous/incoherent (“chimera”) states in several heterogeneous networks of coupled phase oscillators. For each network, the recently-discovered Ott–Antonsen ansatz is used to reduce the number of variables in the partial differential equation (PDE) governing the evolution of the probability density function by one, resulting in a time-evolution PDE for a variable with as many spatial dimensions as the network. Bifurcation analysis is performed on the steady states of these PDEs. The results emphasise the commonality of the dynamics of the different networks, and provide stability information that was previously inferred. [Copyright &y& Elsevier]

We develop and study two neural network models of perceptual alternations. Both models have a star-like architecture of connections with a central element connected to a set of peripheral elements. A particular perception is simulated in terms of partial synchronization between the central element and some sub-group of peripheral elements. The first model is constructed from phase oscillators and the mechanism of perceptual alternations is based on chaotic intermittency under fixed parameter values. Similar to experimental evidence, the distribution of times between perceptual alternations is represented by the gamma distribution. The second model is built of spiking neurons of the Hodgkin–Huxley type. The mechanism of perceptual alternations is based on plasticity of inhibitory synapses which increases the inhibition from the central unit to the neural assembly representing the current percept. As a result another perception is formed. Simulations show that the second model is in good agreement with behavioural data on switching times between percepts of ambiguous figures and with experimental results on binocular rivalry of two and four percepts. [ABSTRACT FROM AUTHOR]

Abstract: We study the synchronization of coupled phase oscillators in random complex networks. The topology of the networks is assumed to be vary over time. Here we mainly study the onset of global phase synchronization when the topology switches rapidly over time. We find that the results are, to some extent, different from those in deterministic situations. In particular, the synchronizability of coupled oscillators can be enhanced in ER networks and scale-free networks under fast switching, while in stochastic small-world networks such enhancement is not significant. [Copyright &y& Elsevier]

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain. Newtonian body-limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning, and learning. We outline relevant background material from biomechanics and neurobiology, explain key properties of the hybrid dynamical systems that underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble "peg-leg walker" to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. This final integration in a tractable and illuminating model is an outstanding challenge. [ABSTRACT FROM AUTHOR]

Abstract: Inspired by the discovery of possible roles of synchronization of oscillations in the brain, networks of coupled phase oscillators have been proposed before as models of associative memory based on the concept of temporal coding of information. Here we show, however, that error-free retrieval states of such networks turn out to be typically unstable regardless of the network size, in contrast to the classical Hopfield model. We propose a remedy for this undesirable property, and provide a systematic study of the improved model. In particular, we show that the error-free capacity of the network is at least patterns per neuron, where is the number of oscillators (neurons) and the strength of the second-order mode in the coupling function. [Copyright &y& Elsevier]

A model consisting of two identical unidirectionally coupled 1D arrays of phase oscillators is studied. The master array is in the spatio-temporal chaos regime and is periodically driven. The time series of the distance between the arrays is the main object of our study. As the unidirectional coupling between the arrays increases, a transition from a Markovian regime to a noise regime in the time series is observed. Typically, this time series has a PDF whose core depends strongly on the stroboscopic section and typically displays tails with power-law dependence. The kind of intermittency of this time series is not characteristic of on–off intermittency, which typically arises in models of chaotic synchronization. [Copyright &y& Elsevier]