Resumption of dynamism in damaged networks of coupled oscillators

Patterns of neuronal synchrony in higher-order networks

Synchrony in neuronal networks is crucial for cognitive functions, motor coordination, and various neurological disorders. While traditional research has focused on pairwise interactions between neurons, recent studies highlight the importance of higher-order interactions involving multiple neurons. Both types of interactions lead to complex synchronous spatiotemporal patterns, including the fascinating phenomenon of chimera states, where synchronized and desynchronized neuronal activity coexist. These patterns are thought to resemble pathological states such as schizophrenia and Parkinson's disease, and their emergence is influenced by neuronal dynamics as well as by synaptic connections and network structure. This review integrates the current understanding of how pairwise and higher-order interactions contribute to different synchrony patterns in neuronal networks, providing a comprehensive overview of their role in shaping network dynamics. We explore a broad range of connectivity mechanisms that drive diverse neuronal synchrony patterns, from pairwise long-range temporal interactions and time-delayed coupling to adaptive communication and higher-order, time-varying connections. We cover key neuronal models, including the Hindmarsh-Rose model, the stochastic Hodgkin-Huxley model, the Sherman model, and the photosensitive FitzHugh-Nagumo model. By investigating the emergence and stability of various synchronous states, this review highlights their significance in neurological systems and indicates directions for future research in this rapidly evolving field.

Aging in a weighted ensemble of excitable and self-oscillatory neurons: The role of pairwise and higher-order interactions

We investigate the aging transition in networks of excitable and self-oscillatory units as the fraction of inherently excitable units increases. Two network topologies are considered: a scale-free network with weighted pairwise interactions and a two-dimensional simplicial complex with weighted scale-free pairwise and triadic interactions. Without triadic interactions, the aging transition from collective oscillations to oscillation death (inhomogeneous stationary states) can occur either suddenly or through an intermediate state of partial oscillation. However, when triadic interactions are present, the network becomes less resilient, and the transition occurs without partial oscillation at any coupling strength. Furthermore, we observe the presence of inhomogeneous steady states within the complete oscillation death regime, regardless of the network interaction models.

Time delays shape the eco-evolutionary dynamics of cooperation

We study the intricate interplay between ecological and evolutionary processes through the lens of the prisoner's dilemma game. But while previous studies on cooperation amongst selfish individuals often assume instantaneous interactions, we take into consideration delays to investigate how these might affect the causes underlying prosocial behavior. Through analytical calculations and numerical simulations, we demonstrate that delays can lead to oscillations, and by incorporating also the ecological variable of altruistic free space and the evolutionary strategy of punishment, we explore how these factors impact population and community dynamics. Depending on the parameter values and the initial fraction of each strategy, the studied eco-evolutionary model can mimic a cyclic dominance system and even exhibit chaotic behavior, thereby highlighting the importance of complex dynamics for the effective management and conservation of ecological communities. Our research thus contributes to the broader understanding of group decision-making and the emergence of moral behavior in multidimensional social systems.

Additional complex conjugate feedback-induced explosive death and multistabilities

Many natural and man-made systems require suitable feedback to function properly. In this study, we aim to investigate the impact of additional complex conjugate feedback on globally coupled Stuart-Landau oscillators. We find that this additional feedback results in the onset of symmetry breaking clusters and out-of-phase clusters. Interestingly, we also find the existence of explosive amplitude death along with disparate multistable states. We characterize the first-order transition to explosive death through the amplitude order parameter and show that the transition from oscillatory to death state indeed shows a hysteresis nature. Further, we map the global dynamical transitions in the parametric spaces. In addition, to understand the existence of multistabilities and their transitions, we analyze the bifurcation scenarios of the reduced model and also explore their basin stability. Our study will shed light on the emergent dynamics in the presence of additional feedback.

Dynamical robustness of complex networks subject to long-range connectivity

In spite of a few attempts in understanding the dynamical robustness of complex networks, this extremely important subject of research is still in its dawn as compared to the other dynamical processes on networks. We hereby consider the concept of long-range interactions among the dynamical units of complex networks and demonstrate for the first time that such a characteristic can have noteworthy impacts on the dynamical robustness of networked systems, regardless of the underlying network topology. We present a comprehensive analysis of this phenomenon on top of diverse network architectures. Such dynamical damages being able to substantially affect the network performance, determining mechanisms that boost the robustness of networks becomes a fundamental question. In this work, we put forward a prescription based upon self-feedback that can efficiently resurrect global rhythmicity of complex networks composed of active and inactive dynamical units, and thus can enhance the network robustness. We have been able to delineate the whole proposition analytically while dealing with all d -path adjacency matrices, having an excellent agreement with the numerical results. For the numerical computations, we examine scale-free networks, Watts–Strogatz small-world model and also Erdös–Rényi random network, along with Landau–Stuart oscillators for casting the local dynamics.

Oscillation suppression and chimera states in time-varying networks

Complex network theory has offered a powerful platform for the study of several natural dynamic scenarios, based on the synergy between the interaction topology and the dynamics of its constituents. With research in network theory being developed so fast, it has become extremely necessary to move from simple network topologies to more sophisticated and realistic descriptions of the connectivity patterns. In this context, there is a significant amount of recent works that have emerged with enormous evidence establishing the time-varying nature of the connections among the constituents in a large number of physical, biological, and social systems. The recent review article by Ghosh et al. [Phys. Rep. 949, 1-63 (2022)] demonstrates the significance of the analysis of collective dynamics arising in temporal networks. Specifically, the authors put forward a detailed excerpt of results on the origin and stability of synchronization in time-varying networked systems. However, among the complex collective dynamical behaviors, the study of the phenomenon of oscillation suppression and that of other diverse aspects of synchronization are also considered to be central to our perception of the dynamical processes over networks. Through this review, we discuss the principal findings from the research studies dedicated to the exploration of the two collective states, namely, oscillation suppression and chimera on top of time-varying networks of both static and mobile nodes. We delineate how temporality in interactions can suppress oscillation and induce chimeric patterns in networked dynamical systems, from effective analytical approaches to computational aspects, which is described while addressing these two phenomena. We further sketch promising directions for future research on these emerging collective behaviors in time-varying networks.

Complex evolutionary dynamics due to punishment and free space in ecological multigames

The concurrence of ecological and evolutionary processes often arises as an integral part of various biological and social systems. We here study eco-evolutionary dynamics by adopting two paradigmatic metaphors of social dilemmas with contrasting outcomes. We use the Prisoner’s Dilemma and Snowdrift games as the backbone of the proposed mathematical model. Since cooperation is a costly proposition in the face of the Darwinian theory of evolution, we go beyond the traditional framework by introducing punishment as an additional strategy. Punishers bare an additional cost from their own resources to try and discourage or prohibit free-riding from selfish defectors. Our model also incorporates the ecological signature of free space, which has an altruistic-like impact because it allows others to replicate and potentially thrive. We show that the consideration of these factors has broad implications for better understanding the emergent complex evolutionary dynamics. In particular, we report the simultaneous presence of different subpopulations through the spontaneous emergence of cyclic dominance, and we determine various stationary points using traditional game-theoretic concepts and stability analysis.

Persistence in multilayer ecological network consisting of harvested patches

Complex network theory yields a powerful approach to solve the difficulties arising in a major section of ecological systems, prey-predator interaction being one among them. A large variety of ecological systems have been successfully investigated employing the theory of complex networks, and one of the most significant advancements in this theory is the emerging field of multilayer networks. The field of multilayer networks provides a natural framework to accommodate multiple layers of complexities emerging in ecosystems. In this article, we consider prey-predator patches communicating among themselves while being connected by distinct small-world dispersal topologies in two layers of the network. We scrutinize the robustness of the multilayer ecological network sustaining gradually over harvested patches. We thoroughly report the consequences of introducing asymmetries in both interlayer and intralayer dispersal strengths as well as the network topologies on the global persistence of species in the network. Besides numerical simulation, we analytically derive the critical point up to which the network can sustain species in the network. Apart from the results on a purely multiplex framework, we validate our claims for multilayer formalism in which the patches of the layers are different. Interestingly, we observe that due to the interaction between the two layers, species are recovered in the layer that we assume to be extinct initially. Moreover, we find similar results while considering two completely different prey-predator systems, which eventually attests that the outcomes are not model specific.

Enhancement of dynamical robustness in a mean-field coupled network through self-feedback delay

The network of self-sustained oscillators plays an important role in exploring complex phenomena in many areas of science and technology. The aging of an oscillator is referred to as turning non-oscillatory due to some local perturbations that might have adverse effects in macroscopic dynamical activities of a network. In this article, we propose an efficient technique to enhance the dynamical activities for a network of coupled oscillators experiencing aging transition. In particular, we present a control mechanism based on delayed negative self-feedback, which can effectively enhance dynamical robustness in a mean-field coupled network of active and inactive oscillators. Even for a small value of delay, robustness gets enhanced to a significant level. In our proposed scheme, the enhancing effect is more pronounced for strong coupling. To our surprise even if all the oscillators perturbed to equilibrium mode were delayed negative self-feedback is able to restore oscillatory activities in the network for strong coupling strength. We demonstrate that our proposed mechanism is independent of coupling topology. For a globally coupled network, we provide numerical and analytical treatment to verify our claim. To show that our scheme is independent of network topology, we also provide numerical results for the local mean-field coupled complex network. Also, for global coupling to establish the generality of our scheme, we validate our results for both Stuart-Landau limit cycle oscillators and chaotic Rössler oscillators.

Abnormal route to aging transition in a network of coupled oscillators

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

Aging transition in the absence of inactive oscillators

The role of counter-rotating oscillators in an ensemble of coexisting co- and counter-rotating oscillators is examined by increasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratio of the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillators to a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored. The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strength, and, subsequently, aging transition was observed at a critical value of the feedback strength surprisingly without any counter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variables, and it was observed that the symmetry breaking coupling can facilitate the onset of aging transition even in the absence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter space of aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportion of the counter-rotating oscillators in both symmetry preserving and symmetry breaking couplings. Further, the transition from oscillatory to aging occurs via a Hopf bifurcation, while the transition from aging to oscillation death state emerges via the pitchfork bifurcation. Analytical expressions for the critical ratio of the counter-rotating oscillators are deduced to find the stable boundaries of the aging transition.

Revival and death of oscillation under mean-field coupling: Interplay of intrinsic and extrinsic filtering

Mean-field diffusive coupling was known to induce the phenomenon of quenching of oscillations even in identical systems, where the standard diffusive coupling (without mean-field) fails to do so [Phys. Rev. E 89, 052912 (2014)PLEEE81539-375510.1103/PhysRevE.89.052912]. In particular, the mean-field diffusive coupling facilitates the transition from amplitude to oscillation death states and the onset of a nontrivial amplitude death state via a subcritical pitchfork bifurcation. In this paper, we show that an adaptive coupling using a low-pass filter in both the intrinsic and extrinsic variables in the coupling is capable of inducing the counterintuitive phenomenon of reviving of oscillations from the death states induced by the mean-field coupling. In particular, even a weak filtering of the extrinsic (intrinsic) variable in the mean-field coupling facilitates the onset of revival (quenching) of oscillations, whereas a strong filtering of the extrinsic (intrinsic) variable results in quenching (revival) of oscillations. Our results reveal that the degree of filtering plays a predominant role in determining the effect of filtering in the extrinsic or intrinsic variables, thereby engineering the dynamics as desired. We also extend the analysis to networks of mean-field coupled limit-cycle and chaotic oscillators along with the low-pass filters to illustrate the generic nature of our results. Finally, we demonstrate the observed dynamical transition experimentally to elucidate the robustness of our results despite the presence of inherent parameter fluctuations and noise.

Low pass filtering mechanism enhancing dynamical robustness in coupled oscillatory networks

A network that consists of a set of active and inactive nodes is called a damaged network and this type of network shows an aging effect (degradation of dynamical activity). This dynamical deterioration affects the normal functioning of the network and also its performance. Therefore, it is necessary to design a proper mechanism to avoid undesired dynamical activity like degradation. In this work, an efficient mechanism, called the low pass filtering technique, is proposed to enhance the dynamical activity of damaged networks of coupled oscillators. Using this mechanism, the dynamic behavior of the damaged network of coupled active and inactive dynamical units is improved and the network survivability is ensured. Because a minor deviation of the controlling parameter is sufficient to restore the oscillatory behavior when the entire network undergoes an aging transition. Even when the whole network degrades due to the deterioration of each node, the larger values of the interaction strength and the controlling parameter play a key role in favor of the revival of dynamic activity in the entire network. Our proposed mechanism is very simple and effective to recover the dynamic features of a damaged network. The effectiveness of this technique has been testified in globally coupled and Erdős Rényi random networks of Stuart-Landau oscillators.

Emergent dynamics in delayed attractive-repulsively coupled networks

We investigate different emergent dynamics, namely, oscillation quenching and revival of oscillation, in a global network of identical oscillators coupled with diffusive (positive) delay coupling as it is perturbed by symmetry breaking localized repulsive delayed interaction. Starting from the oscillatory state (OS), we systematically identify three types of transition phenomena in the parameter space: (1) The system may reach inhomogeneous steady states from the homogeneous steady state sometimes called as the transition from amplitude death (AD) to oscillation death (OD) state, i.e., OS-AD-OD scenario, (2) Revival of oscillation (OS) from the AD state (OS-AD-OS), and (3) Emergence of the OD state from the oscillatory state (OS) without passing through AD, i.e., OS-OD. The dynamics of each node in the network is assumed to be governed either by the identical limit cycle Stuart-Landau system or by the chaotic Rössler system. Based on clustering behavior observed in the oscillatory network, we derive a reduced low-dimensional model of the large network. Using the reduced model, we investigate the effect of time delay on these transitions and demarcate OS, AD, and OD regimes in the parameter space. We also explore and characterize the bifurcation transitions present in both systems. The generic behavior of the low dimensional model and full network is found to match satisfactorily.