Phenotypic plasticity: A missing element in the theory of vegetation pattern formation.

Regular spatial patterns of vegetation are a common sight in drylands. Their formation is a population-level response to water stress that increases water availability for the few via partial plant mortality. At the individual level, plants can also adapt to water stress by changing their phenotype. Phenotypic plasticity of individual plants and spatial patterning of plant populations have extensively been studied independently, but the likely interplay between the two robust mechanisms has remained unexplored. In this paper, we incorporate phenotypic plasticity into a multi-level theory of vegetation pattern formation and use a fascinating ecological phenomenon, the Namibian "fairy circles," to demonstrate the need for such a theory. We show that phenotypic changes in the root structure of plants, coupled with pattern-forming feedback within soil layers, can resolve two puzzles that the current theory fails to explain: observations of multi-scale patterns and the absence of theoretically predicted large-scale stripe and spot patterns along the rainfall gradient. Importantly, we find that multi-level responses to stress unveil a wide variety of more effective stress-relaxation pathways, compared to single-level responses, implying a previously underestimated resilience of dryland ecosystems.

Evidence for scale-dependent root-antation feedback and its role in halting the spread of a pantropical shrub into an endemic sedge.

Vegetation pattern formation is a widespread phenomenon in resource-limited environments, but the driving mechanisms are largely unconfirmed empirically. Combining results of field studies and mathematical modeling, empirical evidence for a generic pattern-formation mechanism is demonstrated with the clonal shrub Guilandina bonduc L. (hereafter Guilandina) on the Brazilian island of Trindade. The mechanism is associated with water conduction by laterally spread roots and root augmentation as the shoot grows-a crucial element in the positive feedback loop that drives spatial patterning. Assuming precipitation-dependent root-shoot relations, the model accounts for the major vegetation landscapes on Trindade Island, substantiating lateral root augmentation as the driving mechanism of Guilandina patterning. Guilandina expands into surrounding communities dominated by the Trindade endemic, Cyperus atlanticus Hemsl. (hereafter Cyperus). It appears to do so by decreasing the water potential in soils below Cyperus through its dense lateral roots, leaving behind a patchy Guilandina-only landscape. We use this system to highlight a novel form of invasion, likely to apply to many other systems where the invasive species is pattern-forming. Depending on the level of water stress, the invasion can take two distinct forms: (i) a complete invasion at low stress that culminates in a patchy Guilandina-only landscape through a spot-replication process, and (ii) an incomplete invasion at high stress that begins but does not spread, forming isolated Guilandina spots of fixed size, surrounded by bare-soil halos, in an otherwise uniform Cyperus grassland. Thus, drier climates may act selectively on pattern-forming invasive species, imposing incomplete invasion and reducing the negative effects on native species.

Linking spatial self-organization to community assembly and biodiversity.

Temporal shifts to drier climates impose environmental stresses on plant communities that may result in community reassembly and threatened ecosystem services, but also may trigger self-organization in spatial patterns of biota and resources, which act to relax these stresses. The complex relationships between these counteracting processes - community reassembly and spatial self-organization - have hardly been studied. Using a spatio-temporal model of dryland plant communities and a trait-based approach, we study the response of such communities to increasing water-deficit stress. We first show that spatial patterning acts to reverse shifts from fast-growing species to stress-tolerant species, as well as to reverse functional-diversity loss. We then show that spatial self-organization buffers the impact of further stress on community structure. Finally, we identify multistability ranges of uniform and patterned community states and use them to propose forms of non-uniform ecosystem management that integrate the need for provisioning ecosystem services with the need to preserve community structure.

Spiral wave chimera-like transient dynamics in three-dimensional grid of diffusive ecological systems.

In the present article, we demonstrate the emergence and existence of the spiral wave chimera-like transient pattern in coupled ecological systems, composed of prey-predator patches, where the patches are connected in a three-dimensional medium through local diffusion. We explore the transition scenarios among several collective dynamical behaviors together with transient spiral wave chimera-like states and investigate the long time behavior of these states. The transition from the transient spiral chimera-like pattern to the long time synchronized or desynchronized pattern appears through the deformation of the incoherent region of the spiral core. We discuss the transient dynamics under the influence of the species diffusion at different time instants. By calculating the instantaneous strength of incoherence of the populations, we estimate the duration of the transient dynamics characterized by the persistence of the chimera-like spatial coexistence of coherent and incoherent patterns over the spatial domain. We generalize our observations on the transient dynamics in a three-dimensional grid of diffusive ecological systems by considering two different prey-predator systems.

Additional repulsion reduces the dynamical resilience in the damaged networks.

In this paper, we investigate the dynamical robustness of diffusively coupled oscillatory networks under the influence of an additional repulsive link. Such a dynamical resilience property is realized through the aging process of the damaged network of active and inactive oscillators. The aging process is one type of phase transition, mainly appearing at a critical threshold of a fraction of the inactive oscillator node where the mean oscillation amplitude of the entire network suddenly vanishes. These critical fractions of the failure nodes in the network are broadly used as a measure of network resilience. Here, we analytically derived the critical fraction of the aging process in the dynamical network. We find that the addition of the repulsive link enhances the critical threshold of the aging transition of diffusively coupled oscillators, which indicated that the dynamical robustness of the coupled network decreases with the presence of the repulsive interaction. Furthermore, we investigate the dynamical robustness of the network against the number of deteriorating repulsive links. We observed that a certain percentage of the repulsive link is enabled to produce the aging process in the entire network. Finally, the effect of symmetry-breaking coupling and the targeted inactivation process on the dynamical robustness property of damaged networks were investigated. The analytically obtained results are verified numerically in the network of coupled Stuart-Landau oscillators. These findings may help us to better understand the role of the coupling mechanism on the phase transition in the damaged network.

Invariance and stability conditions of interlayer synchronization manifold.

We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

Spike chimera states and firing regularities in neuronal hypernetworks.

A complex spatiotemporal pattern with coexisting coherent and incoherent domains in a network of identically coupled oscillators is known as a chimera state. Here, we report the emergence and existence of a novel type of nonstationary chimera pattern in a network of identically coupled Hindmarsh-Rose neuronal oscillators in the presence of synaptic couplings. The development of brain function is mainly dependent on the interneuronal communications via bidirectional electrical gap junctions and unidirectional chemical synapses. In our study, we first consider a network of nonlocally coupled neurons where the interactions occur through chemical synapses. We uncover a new type of spatiotemporal pattern, which we call "spike chimera" induced by the desynchronized spikes of the coupled neurons with the coherent quiescent state. Thereafter, imperfect traveling chimera states emerge in a neuronal hypernetwork (which is characterized by the simultaneous presence of electrical and chemical synapses). Using suitable characterizations, such as local order parameter, strength of incoherence, and velocity profile, the existence of several dynamical states together with chimera states is identified in a wide range of parameter space. We also investigate the robustness of these nonstationary chimera states together with incoherent, coherent, and resting states with respect to initial conditions by using the basin stability measurement. Finally, we extend our study for the effect of firing regularity in the observed states. Interestingly, we find that the coherent motion of the neuronal network promotes the entire system to regular firing.

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 Erdos Rényi random networks of Stuart-Landau oscillators.

Chimera patterns in three-dimensional locally coupled systems.

The coexistence of coherent and incoherent domains, namely the appearance of chimera states, has been studied extensively in many contexts of science and technology since the past decade, though the previous studies are mostly built on the framework of one-dimensional and two-dimensional interaction topologies. Recently, the emergence of such fascinating phenomena has been studied in a three-dimensional (3D) grid formation while considering only the nonlocal interaction. Here we study the emergence and existence of chimera patterns in a three-dimensional network of coupled Stuart-Landau limit-cycle oscillators and Hindmarsh-Rose neuronal oscillators with local (nearest-neighbor) interaction topology. The emergence of different types of spatiotemporal chimera patterns is investigated by taking two distinct nonlinear interaction functions. We provide appropriate analytical explanations in the 3D grid of the network formation and the corresponding numerical justifications are given. We extend our analysis on the basis of the Ott-Antonsen reduction approach in the case of Stuart-Landau oscillators containing infinite numbers of oscillators. Particularly, in the Hindmarsh-Rose neuronal network the existence of nonstationary chimera states is characterized by an instantaneous strength of incoherence and an instantaneous local order parameter. Besides, the condition for achieving exact neuronal synchrony is obtained analytically through a linear stability analysis. The different types of collective dynamics together with chimera states are mapped over a wide range of various parameter spaces.

Chimera states in neuronal networks: A review.

Neuronal networks, similar to many other complex systems, self-organize into fascinating emergent states that are not only visually compelling, but also vital for the proper functioning of the brain. Synchronous spatiotemporal patterns, for example, play an important role in neuronal communication and plasticity, and in various cognitive processes. Recent research has shown that the coexistence of coherent and incoherent states, known as chimera states or simply chimeras, is particularly important and characteristic for neuronal systems. Chimeras have also been linked to the Parkinson's disease, epileptic seizures, and even to schizophrenia. The emergence of this unique collective behavior is due to diverse factors that characterize neuronal dynamics and the functioning of the brain in general, including neural bumps and unihemispheric slow-wave sleep in some aquatic mammals. Since their discovery, chimera states have attracted ample attention of researchers that work at the interface of physics and life sciences. We here review contemporary research dedicated to chimeras in neuronal networks, focusing on the relevance of different synaptic connections, and on the effects of different network structures and coupling setups. We also cover the emergence of different types of chimera states, we highlight their relevance in other related physical and biological systems, and we outline promising research directions for the future, including possibilities for experimental verification.

Emergence of synchronization and regularity in firing patterns in time-varying neural hypernetworks.

We study synchronization of dynamical systems coupled in time-varying network architectures, composed of two or more network topologies, corresponding to different interaction schemes. As a representative example of this class of time-varying hypernetworks, we consider coupled Hindmarsh-Rose neurons, involving two distinct types of networks, mimicking interactions that occur through the electrical gap junctions and the chemical synapses. Specifically, we consider the connections corresponding to the electrical gap junctions to form a small-world network, while the chemical synaptic interactions form a unidirectional random network. Further, all the connections in the hypernetwork are allowed to change in time, modeling a more realistic neurobiological scenario. We model this time variation by rewiring the links stochastically with a characteristic rewiring frequency f. We find that the coupling strength necessary to achieve complete neuronal synchrony is lower when the links are switched rapidly. Further, the average time required to reach the synchronized state decreases as synaptic coupling strength and/or rewiring frequency increases. To quantify the local stability of complete synchronous state we use the Master Stability Function approach, and for global stability we employ the concept of basin stability. The analytically derived necessary condition for synchrony is in excellent agreement with numerical results. Further we investigate the resilience of the synchronous states with respect to increasing network size, and we find that synchrony can be maintained up to larger network sizes by increasing either synaptic strength or rewiring frequency. Last, we find that time-varying links not only promote complete synchronization, but also have the capacity to change the local dynamics of each single neuron. Specifically, in a window of rewiring frequency and synaptic coupling strength, we observe that the spiking behavior becomes more regular.

Chimera states in two-dimensional networks of locally coupled oscillators.

Chimera state is defined as a mixed type of collective state in which synchronized and desynchronized subpopulations of a network of coupled oscillators coexist and the appearance of such anomalous behavior has strong connection to diverse neuronal developments. Most of the previous studies on chimera states are not extensively done in two-dimensional ensembles of coupled oscillators by taking neuronal systems with nonlinear coupling function into account while such ensembles of oscillators are more realistic from a neurobiological point of view. In this paper, we report the emergence and existence of chimera states by considering locally coupled two-dimensional networks of identical oscillators where each node is interacting through nonlinear coupling function. This is in contrast with the existence of chimera states in two-dimensional nonlocally coupled oscillators with rectangular kernel in the coupling function. We find that the presence of nonlinearity in the coupling function plays a key role to produce chimera states in two-dimensional locally coupled oscillators. We analytically verify explicitly in the case of a network of coupled Stuart-Landau oscillators in two dimensions that the obtained results using Ott-Antonsen approach and our analytical finding very well matches with the numerical results. Next, we consider another type of important nonlinear coupling function which exists in neuronal systems, namely chemical synaptic function, through which the nearest-neighbor (locally coupled) neurons interact with each other. It is shown that such synaptic interacting function promotes the emergence of chimera states in two-dimensional lattices of locally coupled neuronal oscillators. In numerical simulations, we consider two paradigmatic neuronal oscillators, namely Hindmarsh-Rose neuron model and Rulkov map for each node which exhibit bursting dynamics. By associating various spatiotemporal behaviors and snapshots at particular times, we study the chimera states in detail over a large range of coupling parameter. The existence of chimera states is confirmed by instantaneous angular frequency, order parameter and strength of incoherence.

Coexisting synchronous and asynchronous states in locally coupled array of oscillators by partial self-feedback control.

We report the emergence of coexisting synchronous and asynchronous subpopulations of oscillators in one dimensional arrays of identical oscillators by applying a self-feedback control. When a self-feedback is applied to a subpopulation of the array, similar to chimera states, it splits into two/more sub-subpopulations coexisting in coherent and incoherent states for a range of self-feedback strength. By tuning the coupling between the nearest neighbors and the amount of self-feedback in the perturbed subpopulation, the size of the coherent and the incoherent sub-subpopulations in the array can be controlled, although the exact size of them is unpredictable. We present numerical evidence using the Landau-Stuart system and the Kuramoto-Sakaguchi phase model.

Basin stability for chimera states.

Chimera states, namely complex spatiotemporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics, are investigated in a network of coupled identical oscillators. These intriguing spatiotemporal patterns were first reported in nonlocally coupled phase oscillators, and it was shown that such mixed type behavior occurs only for specific initial conditions in nonlocally and globally coupled networks. The influence of initial conditions on chimera states has remained a fundamental problem since their discovery. In this report, we investigate the robustness of chimera states together with incoherent and coherent states in dependence on the initial conditions. For this, we use the basin stability method which is related to the volume of the basin of attraction, and we consider nonlocally and globally coupled time-delayed Mackey-Glass oscillators as example. Previously, it was shown that the existence of chimera states can be characterized by mean phase velocity and a statistical measure, such as the strength of incoherence, by using well prepared initial conditions. Here we show further how the coexistence of different dynamical states can be identified and quantified by means of the basin stability measure over a wide range of the parameter space.

Basin stability measure of different steady states in coupled oscillators.

In this report, we investigate the stabilization of saddle fixed points in coupled oscillators where individual oscillators exhibit the saddle fixed points. The coupled oscillators may have two structurally different types of suppressed states, namely amplitude death and oscillation death. The stabilization of saddle equilibrium point refers to the amplitude death state where oscillations are ceased and all the oscillators converge to the single stable steady state via inverse pitchfork bifurcation. Due to multistability features of oscillation death states, linear stability theory fails to analyze the stability of such states analytically, so we quantify all the states by basin stability measurement which is an universal nonlocal nonlinear concept and it interplays with the volume of basins of attractions. We also observe multi-clustered oscillation death states in a random network and measure them using basin stability framework. To explore such phenomena we choose a network of coupled Duffing-Holmes and Lorenz oscillators which are interacting through mean-field coupling. We investigate how basin stability for different steady states depends on mean-field density and coupling strength. We also analytically derive stability conditions for different steady states and confirm by rigorous bifurcation analysis.

Time-varying multiplex network: Intralayer and interlayer synchronization.

A large class of engineered and natural systems, ranging from transportation networks to neuronal networks, are best represented by multiplex network architectures, namely a network composed of two or more different layers where the mutual interaction in each layer may differ from other layers. Here we consider a multiplex network where the intralayer coupling interactions are switched stochastically with a characteristic frequency. We explore the intralayer and interlayer synchronization of such a time-varying multiplex network. We find that the analytically derived necessary condition for intralayer and interlayer synchronization, obtained by the master stability function approach, is in excellent agreement with our numerical results. Interestingly, we clearly find that the higher frequency of switching links in the layers enhances both intralayer and interlayer synchrony, yielding larger windows of synchronization. Further, we quantify the resilience of synchronous states against random perturbations, using a global stability measure based on the concept of basin stability, and this reveals that intralayer coupling strength is most crucial for determining both intralayer and interlayer synchrony. Lastly, we investigate the robustness of interlayer synchronization against a progressive demultiplexing of the multiplex structure, and we find that for rapid switching of intralayer links, the interlayer synchronization persists even when a large number of interlayer nodes are disconnected.

Excitation and suppression of chimera states by multiplexing.

We study excitation and suppression of chimera states in an ensemble of nonlocally coupled oscillators arranged in a framework of multiplex network. We consider the homogeneous network (all identical oscillators) with different parametric cases and interlayer heterogeneity by introducing parameter mismatch between the layers. We show the feasibility to suppress chimera states in the multiplex network via moderate interlayer interaction between a layer exhibiting chimera state and other layers which are in a coherent or incoherent state. On the contrary, for larger interlayer coupling, we observe the emergence of identical chimera states in both layers which we call an interlayer chimera state. We map the spatiotemporal behavior in a wide range of parameters, varying interlayer coupling strength and phase lag in two and three multiplexing layers. We also prove the emergence of interlayer chimera states in a multiplex network via evaluation of a continuous model. Furthermore, we consider the two-layered network of Hindmarsh-Rose neurons and reveal that in such a system multiplex interaction between layers is capable of exciting not only the synchronous interlayer chimera state but also nonidentical chimera patterns.

Imperfect traveling chimera states induced by local synaptic gradient coupling.

In this paper, we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through local, synaptic gradient coupling. We discover a new chimera pattern, namely the imperfect traveling chimera state, where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for one-way local coupling, which is in contrast to the earlier belief that only nonlocal, global, or nearest-neighbor local coupling can give rise to chimera state; this find further relaxes the essential connectivity requirement of getting a chimera state. We choose a network of identical bursting Hindmarsh-Rose neuronal oscillators, and we show that depending upon the relative strength of the synaptic and gradient coupling, several chimera patterns emerge. We map all the spatiotemporal behaviors in parameter space and identify the transitions among several chimera patterns, an in-phase synchronized state, and a global amplitude death state.

Chimera states in purely local delay-coupled oscillators.

We study the existence of chimera states in a network of locally coupled chaotic and limit-cycle oscillators. The necessary condition for chimera state in purely local coupled oscillators is discussed. At first, we numerically observe the existence of chimera or multichimera states in the locally coupled Hindmarsh-Rose neuron model. We find that delay time in the nonlinear local coupling reduces the domain of the coherent island in the parameter space of the synaptic coupling strength and time delay, and thus the coherent region can be completely eliminated once the time delay exceeds a certain threshold. We then consider another form of nonlinearity in the local coupling, and the existence of chimera states is observed in the time-delayed Mackey-Glass system and in a Van der Pol oscillator. We also discuss the effect of time delay in local coupling for the existence of chimera states in Mackey-Glass systems. The nonlinearity present in the coupling function plays a key role in the emergence of chimera or multichimera states. A phase diagram for the chimera state is identified over a wide parameter space.