Complex Network Geometry and Frustrated Synchronization

Synchronization of Kuromoto Oscillators on Simplicial Complexes: Hysteresis, Cluster Formation and Partial Synchronization

The analysis of the synchronization of oscillator systems based on simplicial complexes presents some interesting features. The transition to synchronization can be abrupt or smooth depending on the substrate, the frequency distribution of the oscillators and the initial distribution of the phase angles. Both partial and complete synchronization can be seen as quantified by the order parameter. The addition of interactions of a higher order than the usual pairwise ones can modify these features further, especially when the interactions tend to have the opposite signs. Cluster synchronization is seen on sparse lattices and depends on the spectral dimension and whether the networks are mixed, sparse or compact. Topological effects and the geometry of shared faces are important and affect the synchronization patterns. We identify and analyze factors, such as frustration, that lead to these effects. We note that these features can be observed in realistic systems such as nanomaterials and the brain connectome.

Frustrated Synchronization of the Kuramoto Model on Complex Networks

We present a synchronization transition study of the locally coupled Kuramoto model on extremely large graphs. We compare regular 405 and 1004 lattice results with those of 12,0002 lattice substrates with power-law decaying long links (ll). The latter heterogeneous network exhibits ds>4 spectral dimensions. We show strong corrections to scaling and mean-field type of criticality at d=5, with logarithmic corrections at d=4 Euclidean dimensions. Contrarily, the ll model exhibits a non-mean-field smeared transition, with oscillating corrections at similarly high spectral dimensions. This suggests that the network heterogeneity is relevant, causing frustrated synchronization akin to Griffiths effects.

Unraveling the mesoscale organization induced by network-driven processes

Complex systems are characterized by emergent patterns created by the nontrivial interplay between dynamical processes and the networks of interactions on which these processes unfold. Topological or dynamical descriptors alone are not enough to fully embrace this interplay in all its complexity, and many times one has to resort to dynamics-specific approaches that limit a comprehension of general principles. To address this challenge, we employ a metric-that we name Jacobian distance-which captures the spatiotemporal spreading of perturbations, enabling us to uncover the latent geometry inherent in network-driven processes. We compute the Jacobian distance for a broad set of nonlinear dynamical models on synthetic and real-world networks of high interest for applications from biological to ecological and social contexts. We show, analytically and computationally, that the process-driven latent geometry of a complex network is sensitive to both the specific features of the dynamics and the topological properties of the network. This translates into potential mismatches between the functional and the topological mesoscale organization, which we explain by means of the spectrum of the Jacobian matrix. Finally, we demonstrate that the Jacobian distance offers a clear advantage with respect to traditional methods when studying human brain networks. In particular, we show that it outperforms classical network communication models in explaining functional communities from structural data, therefore highlighting its potential in linking structure and function in the brain.

Chimera-like states in neural networks and power systems

Partial, frustrated synchronization, and chimera-like states are expected to occur in Kuramoto-like models if the spectral dimension of the underlying graph is low: ds<4. We provide numerical evidence that this really happens in the case of the high-voltage power grid of Europe (ds<2), a large human connectome (KKI113) and in the case of the largest, exactly known brain network corresponding to the fruit-fly (FF) connectome (ds<4), even though their graph dimensions are much higher, i.e., dgEU≃2.6(1) and dgFF≃5.4(1), dgKKI113≃3.4(1). We provide local synchronization results of the first- and second-order (Shinomoto) Kuramoto models by numerical solutions on the FF and the European power-grid graphs, respectively, and show the emergence of chimera-like patterns on the graph community level as well as by the local order parameters.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.

Synchronization Transition of the Second-Order Kuramoto Model on Lattices

The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Herein, we investigate this transition on large 2D and 3D lattices and provide numerical evidence of hybrid phase transitions, whereby the oscillator phases θi exhibit a crossover, while the frequency is spread over a real phase transition in 3D. Thus, a lower critical dimension dlO=2 is expected for the frequencies and dlR=4 for phases such as that in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as ∼t-d/2 in the case of an aligned initial state of the phases in agreement with the linear approximation. In 3D, however, in the case of the initially random distribution of θi, we find a faster decay, characterized by ∼t-1.8(1) as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.

Hypergraph geometry reflects higher-order dynamics in protein interaction networks

Protein interactions form a complex dynamic molecular system that shapes cell phenotype and function; in this regard, network analysis is a powerful tool for studying the dynamics of cellular processes. Current models of protein interaction networks are limited in that the standard graph model can only represent pairwise relationships. Higher-order interactions are well-characterized in biology, including protein complex formation and feedback or feedforward loops. These higher-order relationships are better represented by a hypergraph as a generalized network model. Here, we present an approach to analyzing dynamic gene expression data using a hypergraph model and quantify network heterogeneity via Forman-Ricci curvature. We observe, on a global level, increased network curvature in pluripotent stem cells and cancer cells. Further, we use local curvature to conduct pathway analysis in a melanoma dataset, finding increased curvature in several oncogenic pathways and decreased curvature in tumor suppressor pathways. We compare this approach to a graph-based model and a differential gene expression approach.

Explosive transitions to synchronization in networks of frequency dipoles

We reveal that an introduction of frequency-weighted inter-layer coupling term in networks of frequency dipoles can induce explosive synchronization transitions. The reason for explosive synchronization is that the oscillators with synchronization superiority are moderately suppressed. The findings show that a super-linear correlation induces explosive synchronization in networks of frequency dipoles, while a linear or sub-linear correlation excites chimera-like states. Clearly, the synchronization transition mode of networks of frequency dipoles is controlled by the power-law exponent. In addition, by means of the mean-field approximation, we obtain the critical values of the coupling strength within and between layers in two limit cases. The results of theoretical analysis are in good agreement with those of numerical simulation. Compared with the previous models, the model proposed in this paper retains the topological structure of network and the intrinsic properties of oscillators, so it is easy to realize pinning control.

Synchronization dynamics on power grids in Europe and the United States

Dynamical simulation of the cascade failures on the Europe and United States (U.S.) high-voltage power grids has been done via solving the second-order Kuramoto equation. We show that synchronization transition happens by increasing the global coupling parameter K with metasatble states depending on the initial conditions so that hysteresis loops occur. We provide analytic results for the time dependence of frequency spread in the large-K approximation and by comparing it with numerics of d=2,3 lattices, we find agreement in the case of ordered initial conditions. However, different power-law (PL) tails occur, when the fluctuations are strong. After thermalizing the systems we allow a single line cut failure and follow the subsequent overloads with respect to threshold values T. The PDFs p(N_{f}) of the cascade failures exhibit PL tails near the synchronization transition point K_{c}. Near K_{c} the exponents of the PLs for the U.S. power grid vary with T as 1.4≤τ≤2.1, in agreement with the empirical blackout statistics, while on the Europe power grid we find somewhat steeper PLs characterized by 1.4≤τ≤2.4. Below K_{c}, we find signatures of T-dependent PLs, caused by frustrated synchronization, reminiscent of Griffiths effects. Here we also observe stability growth following the blackout cascades, similar to intentional islanding, but for K>K_{c} this does not happen. For T<T_{c}, bumps appear in the PDFs with large mean values, known as "dragon king" blackout events. We also analyze the delaying or stabilizing effects of instantaneous feedback or increased dissipation and show how local synchronization behaves on geographic maps.

Mirroring of synchronization in a bi-layer master-slave configuration of Kuramoto oscillators

The phenomenon of mirroring of synchronization is investigated in dynamically dissimilar, unidirectionally coupled, bi-layer master-slave configuration of globally coupled Kuramoto oscillators. The dynamics of the master layer depends solely on the distribution of the natural frequencies of its oscillators. On the other hand, the slave layer dynamics depends not only on the distribution of the natural frequencies of its oscillators but also on the unidirectional coupling with the master layer. The standard Kuramoto order parameter is used to study synchronization in the individual layers and of the bi-layer network. A transition to a completely mirroring state is observed in the dynamics of the slave layer, as the mirroring coefficient in the unidirectional coupling is increased. We derive analytically and verify numerically the conditions for the slave layer to fully mimic the synchronization properties of the master layer. It is further shown that while the master and slave layers are individually synchronized, the bi-layer network exhibits a state of frustrated synchronization.

Epidemic models characterize seizure propagation and the effects of epilepsy surgery in individualized brain networks based on MEG and invasive EEG recordings

Epilepsy surgery is the treatment of choice for drug-resistant epilepsy patients. However, seizure-freedom is currently achieved in only 2/3 of the patients after surgery. In this study we have developed an individualized computational model based on MEG brain networks to explore seizure propagation and the efficacy of different virtual resections. Eventually, the goal is to obtain individualized models to optimize resection strategy and outcome. We have modelled seizure propagation as an epidemic process using the susceptible-infected (SI) model on individual brain networks derived from presurgical MEG. We included 10 patients who had received epilepsy surgery and for whom the surgery outcome at least one year after surgery was known. The model parameters were tuned in in order to reproduce the patient-specific seizure propagation patterns as recorded with invasive EEG. We defined a personalized search algorithm that combined structural and dynamical information to find resections that maximally decreased seizure propagation for a given resection size. The optimal resection for each patient was defined as the smallest resection leading to at least a 90% reduction in seizure propagation. The individualized model reproduced the basic aspects of seizure propagation for 9 out of 10 patients when using the resection area as the origin of epidemic spreading, and for 10 out of 10 patients with an alternative definition of the seed region. We found that, for 7 patients, the optimal resection was smaller than the resection area, and for 4 patients we also found that a resection smaller than the resection area could lead to a 100% decrease in propagation. Moreover, for two cases these alternative resections included nodes outside the resection area. Epidemic spreading models fitted with patient specific data can capture the fundamental aspects of clinically observed seizure propagation, and can be used to test virtual resections in silico. Combined with optimization algorithms, smaller or alternative resection strategies, that are individually targeted for each patient, can be determined with the ultimate goal to improve surgery outcome. MEG-based networks can provide a good approximation of structural connectivity for computational models of seizure propagation, and facilitate their clinical use.

Local topological moves determine global diffusion properties of hyperbolic higher-order networks

From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, still represents a challenge and is the object of intense research. Here, we introduce two nonequilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the intertwined appearance of small-world behavior, δ-hyperbolicity, and community structure. We show that different topological moves, determining the nonequilibrium growth of the higher-order hyperbolic network models, induce tuneable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, then the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.

Higher-order percolation processes on multiplex hypergraphs

Higher-order interactions are increasingly recognized as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraphs as well as simplicial complexes capture the higher-order interactions of complex systems and allow us to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We observe that in presence of the structural cutoff the ensemble of multiplex hypergraphs can be mapped to an ensemble of multiplex bipartite networks. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks, and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.

Higher-order temporal network effects through triplet evolution

We study the evolution of networks through 'triplets'-three-node graphlets. We develop a method to compute a transition matrix to describe the evolution of triplets in temporal networks. To identify the importance of higher-order interactions in the evolution of networks, we compare both artificial and real-world data to a model based on pairwise interactions only. The significant differences between the computed matrix and the calculated matrix from the fitted parameters demonstrate that non-pairwise interactions exist for various real-world systems in space and time, such as our data sets. Furthermore, this also reveals that different patterns of higher-order interaction are involved in different real-world situations. To test our approach, we then use these transition matrices as the basis of a link prediction algorithm. We investigate our algorithm's performance on four temporal networks, comparing our approach against ten other link prediction methods. Our results show that higher-order interactions in both space and time play a crucial role in the evolution of networks as we find our method, along with two other methods based on non-local interactions, give the best overall performance. The results also confirm the concept that the higher-order interaction patterns, i.e., triplet dynamics, can help us understand and predict the evolution of different real-world systems.

Renormalization group theory of percolation on pseudofractal simplicial and cell complexes

Simplicial complexes are gaining increasing scientific attention as they are generalized network structures that can represent the many-body interactions existing in complex systems ranging from the brain to high-order social networks. Simplicial complexes are formed by simplicies, such as nodes, links, triangles, and so on. Cell complexes further extend these generalized network structures as they are formed by regular polytopes, such as squares, pentagons, etc. Pseudofractal simplicial and cell complexes are a major example of generalized network structures and they can be obtained by gluing two-dimensional m-polygons (m=3 triangles, m=4 squares, m=5 pentagons, etc.) along their links according to a simple iterative rule. Here we investigate the interplay between the topology of pseudofractal simplicial and cell complexes and their dynamics by characterizing the critical properties of link percolation defined on these structures. By using the renormalization group we show that the pseudofractal simplicial and cell complexes have a continuous percolation threshold at p_{c}=0. When the pseudofractal structure is formed by polygons of the same size m, the transition is characterized by an exponential suppression of the order parameter P_{∞} that depends on the number of sides m of the polygons forming the pseudofractal cell complex, i.e., P_{∞}∝pexp(-α/p^{m-2}). Here these results are also generalized to random pseudofractal cell complexes formed by polygons of different number of sides m.

Power-Law Distributions of Dynamic Cascade Failures in Power-Grid Models

Power-law distributed cascade failures are well known in power-grid systems. Understanding this phenomena has been done by various DC threshold models, self-tuned at their critical point. Here, we attempt to describe it using an AC threshold model, with a second-order Kuramoto type equation of motion of the power-flow. We have focused on the exploration of network heterogeneity effects, starting from homogeneous two-dimensional (2D) square lattices to the US power-grid, possessing identical nodes and links, to a realistic electric power-grid obtained from the Hungarian electrical database. The last one exhibits node dependent parameters, topologically marginally on the verge of robust networks. We show that too weak quenched heterogeneity, coming solely from the probabilistic self-frequencies of nodes (2D square lattice), is not sufficient for finding power-law distributed cascades. On the other hand, too strong heterogeneity destroys the synchronization of the system. We found agreement with the empirically observed power-law failure size distributions on the US grid, as well as on the Hungarian networks near the synchronization transition point. We have also investigated the consequence of replacing the usual Gaussian self-frequencies to exponential distributed ones, describing renewable energy sources. We found a drop in the steady state synchronization averages, but the cascade size distribution, both for the US and Hungarian systems, remained insensitive and have kept the universal tails, being characterized by the exponent τ≃1.8. We have also investigated the effect of an instantaneous feedback mechanism in case of the Hungarian power-grid.

Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes

The higher-order interactions of complex systems, such as the brain, are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τ_t ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τ_t ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τ_t ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in nonamenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with interlayer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.

Renormalization group for link percolation on planar hyperbolic manifolds

Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. Boettcher et al. [Nat. Comm. 3, 787 (2012)2041-172310.1038/ncomms1774] have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing m-polygons to single edges. We show that when the size m of the polygons is drawn from a distribution q_{m} with asymptotic power-law scaling q_{m}≃Cm^{-γ} for m≫1, different universality classes can be observed depending on the value of the power-law exponent γ. Interestingly, the percolation transition is hybrid for γ∈(3,4) and becomes continuous for γ∈(2,3].

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.

Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.

Heterogeneity effects in power grid network models

We have compared the phase synchronization transition of the second-order Kuramoto model on two-dimensional (2D) lattices and on large, synthetic power grid networks, generated from real data. The latter are weighted, hierarchical modular networks. Due to the inertia the synchronization transitions are of first-order type, characterized by fast relaxation and hysteresis by varying the global coupling parameter K. Finite-size scaling analysis shows that there is no real phase transition in the thermodynamic limit, unlike in the mean-field model. The order parameter and its fluctuations depend on the network size without any real singular behavior. In case of power grids the phase synchronization breaks down at lower global couplings, than in case of 2D lattices of the same sizes, but the hysteresis is much narrower or negligible due to the low connectivity of the graphs. The temporal behavior of desynchronization avalanches after a sudden quench to low K values has been followed and duration distributions with power-law tails have been detected. This suggests rare region effects, caused by frozen disorder, resulting in heavy-tailed distributions, even without a self-organization mechanism as a consequence of a catastrophic drop event in the couplings.