Synchronization of networks of oscillators with distributed delay coupling

Complex dynamics in a fractional order nephron pressure and flow regulation model

Cardiovascular diseases can be attributed to irregular blood pressure, which may be caused by malfunctioning kidneys that regulate blood pressure. Research has identified complex oscillations in the mechanisms used by the kidney to regulate blood pressure. This study uses established physiological knowledge and earlier autoregulation models to derive a fractional order nephron autoregulation model. The dynamical behaviour of the model is analyzed using bifurcation plots, revealing periodic oscillations, chaotic regions, and multistability. A lattice array of the model is used to study collective behaviour and demonstrates the presence of chimeras in the network. A ring network of the fractional order model is also considered, and a diffusion coupling strength is adopted. A basin of synchronization is derived, considering coupling strength, fractional order or number of neighbours as parameters, and measuring the strength of incoherence. Overall, the study provides valuable insights into the complex dynamics of the nephron autoregulation model and its potential implications for cardiovascular diseases.

Blinking coupling enhances network synchronization

This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.

Phase response approaches to neural activity models with distributed delay

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

Dynamics of coupled Kuramoto oscillators with distributed delays

This paper studies the effects of two different types of distributed-delay coupling in the system of two mutually coupled Kuramoto oscillators: one where the delay distribution is considered inside the coupling function and the other where the distribution enters outside the coupling function. In both cases, the existence and stability of phase-locked solutions is analyzed for uniform and gamma distribution kernels. The results show that while having the distribution inside the coupling function only changes parameter regions where phase-locked solutions exist, when the distribution is taken outside the coupling function, it affects both the existence, as well as stability properties of in- and anti-phase states. For both distribution types, various branches of phase-locked solutions are computed, and regions of their stability are identified for uniform, weak, and strong gamma distributions.

The Impact of Small Time Delays on the Onset of Oscillations and Synchrony in Brain Networks

The human brain constitutes one of the most advanced networks produced by nature, consisting of billions of neurons communicating with each other. However, this communication is not in real-time, with different communication or time-delays occurring between neurons in different brain areas. Here, we investigate the impacts of these delays by modeling large interacting neural circuits as neural-field systems which model the bulk activity of populations of neurons. By using a Master Stability Function analysis combined with numerical simulations, we find that delays (1) may actually stabilize brain dynamics by temporarily preventing the onset to oscillatory and pathologically synchronized dynamics and (2) may enhance or diminish synchronization depending on the underlying eigenvalue spectrum of the connectivity matrix. Real eigenvalues with large magnitudes result in increased synchronizability while complex eigenvalues with large magnitudes and positive real parts yield a decrease in synchronizability in the delay vs. instantaneously coupled case. This result applies to networks with fixed, constant delays, and was robust to networks with heterogeneous delays. In the case of real brain networks, where the eigenvalues are predominantly real, owing to the nearly symmetric nature of these weight matrices, biologically plausible, small delays, are likely to increase synchronization, rather than decreasing it.

Collective behaviors of mean-field coupled Stuart-Landau limit-cycle oscillators under additional repulsive links

We study the collective behaviors of a large population of Stuart-Landau limit-cycle oscillators that coupled diffusively and equally with all of the others via the conjugate of the mean field, where the underlying interaction is shown to break the rotational symmetry of the coupled system. In the model, an ensemble of Stuart-Landau oscillators are in fact diffusively coupled via the mean field in the real parts, whereas additional repulsive links are present in the imaginary parts. All the oscillators are linked via the similar state variables, which distinctly differs from the conjugate coupling through dissimilar variables in the previous studies. We show that depending on the strength of coupling and the distribution of natural frequencies, the coupled system exhibits three qualitatively different types of collective stationary behaviors: amplitude death (AD), oscillation death (OD), and incoherent state. Our goal is to analytically characterize the onset of the above three typical macrostates by performing the rigorous linear stability analyses of the corresponding mean-field coupled system. We prove that AD is able to occur in the coupled system with identical frequencies, where the stable coupling interval of AD is found to be independent on the system's size N. When the natural frequencies are distributed according to a general density function, we obtain the analytic equations that govern the exact stability boundaries of AD, OD, and the incoherence for a coupled system in the thermodynamic limit N→∞. All the theoretical predictions are well confirmed via numerical simulations of the coupled system with a specific Lorentzian frequency distribution.

Desynchronization Transitions in Adaptive Networks

Adaptive networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability approach for a large class of adaptive networks. This approach allows for reducing the synchronization problem for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. We show how the interplay between adaptivity and network structure gives rise to the formation of stability islands. Moreover, we report a desynchronization transition and the emergence of complex partial synchronization patterns induced by an increasing overall coupling strength. We illustrate our findings using adaptive networks of coupled phase oscillators and FitzHugh-Nagumo neurons with synaptic plasticity.

Dynamics of unidirectionally-coupled ring neural network with discrete and distributed delays

In this paper, we consider a ring neural network with one-way distributed-delay coupling between the neurons and a discrete delayed self-feedback. In the general case of the distribution kernels, we are able to find a subset of the amplitude death regions depending on even (odd) number of neurons in the network. Furthermore, in order to show the full region of the amplitude death, we use particular delay distributions, including Dirac delta function and gamma distribution. Stability conditions for the trivial steady state are found in parameter spaces consisting of the synaptic weight of the self-feedback and the coupling strength between the neurons, as well as the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses stability. We also perform numerical simulations of the fully nonlinear system to confirm theoretical findings.

Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh-Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Propagation delays determine neuronal activity and synaptic connectivity patterns emerging in plastic neuronal networks

In plastic neuronal networks, the synaptic strengths are adapted to the neuronal activity. Specifically, spike-timing-dependent plasticity (STDP) is a fundamental mechanism that modifies the synaptic strengths based on the relative timing of pre- and postsynaptic spikes, taking into account the spikes' temporal order. In many studies, propagation delays were neglected to avoid additional dynamic complexity or computational costs. So far, networks equipped with a classic STDP rule typically rule out bidirectional couplings (i.e., either loops or uncoupled states) and are, hence, not able to reproduce fundamental experimental findings. In this review paper, we consider additional features, e.g., extensions of the classic STDP rule or additional aspects like noise, in order to overcome the contradictions between theory and experiment. In addition, we review in detail recent studies showing that a classic STDP rule combined with realistic propagation patterns is able to capture relevant experimental findings. In two coupled oscillatory neurons with propagation delays, bidirectional synapses can be preserved and potentiated. This result also holds for large networks of type-II phase oscillators. In addition, not only the mean of the initial distribution of synaptic weights, but also its standard deviation crucially determines the emergent structural connectivity, i.e., the mean final synaptic weight, the number of two-neuron loops, and the symmetry of the final connectivity pattern. The latter is affected by the firing rates, where more symmetric synaptic configurations emerge at higher firing rates. Finally, we discuss these findings in the context of the computational neuroscience-based development of desynchronizing brain stimulation techniques.

Enhancing noise-induced switching times in systems with distributed delays

The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution leads to an increase in the switching times even for longer values of mean time delays for both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution. Analytical predictions are compared to the direct numerical simulations and show excellent agreement between theory and numerical experiment.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Spatial splay states and splay chimera states in coupled map lattices

We study the existence and stability of splay states in the coupled sine circle map lattice system using analytic and numerical techniques. The splay states are observed for very low values of the nonlinearity parameter, i.e., for maps which deviate very slightly from the shift map case. We also observe that depending on the parameters of the system the splay state bifurcates to a mixed or chimera splay state consisting of a mixture of splay and synchronized states, together with kinks in the phases of some of the maps and then to a stable globally synchronized state. We show that these pure states and the mixed states are all temporally chaotic for our systems, and we explore the stability of these states to perturbations. Our studies may provide pointers to the behavior of systems in diverse application contexts such as Josephson junction arrays and chemical oscillations.

Master stability islands for amplitude death in networks of delay-coupled oscillators

This paper presents a master stability function (MSF) approach for analyzing the stability of amplitude death (AD) in networks of delay-coupled oscillators. Unlike the familiar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay-coupled networks we show that such MSFs generally require two additional inputs: the time delay and the coupling strength. To utilize the MSF for determining the stability of AD of general networks for a chosen nonlinear system (node dynamics) and coupling function, we introduce the concept of master stability islands (MSIs), which are two-dimensional stability islands of the delay-coupling parameter space together with a third dimension ("altitude") encoding for eigenvalues that result in stable AD. We numerically compute the MSFs and visualize the corresponding MSIs for several common chaotic systems including the Rössler, the Lorenz, and Chen's system and find that it is generally possible to achieve AD and that a nonzero time delay is necessary for the stabilization of the AD states.

Time-Delayed Models of Gene Regulatory Networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.