Solitary states in multiplex neural networks: onset and vulnerability
Leonhard Schuelen, David A. Janzen, Everton S. Medeiros, Anna Zakharova
SSolitary states in multiplex neural networks: onset andvulnerability
Leonhard Sch¨ulen a , David A. Janzen a , Everton S. Medeiros a , AnnaZakharova a, ∗ a Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623Berlin, Germany
Abstract
We investigate solitary states in a two-layer multiplex network of FitzHugh-Nagumo neurons in the oscillatory regime. We demonstrate how solitary statescan be induced in a multiplex network consisting of two non-identical layers.More specifically, we show that these patterns can be introduced via weak mul-tiplexing into a network that is fully synchronized in isolation. We show thatthis result is robust under variations of the inter-layer coupling strength andlargely independent of the choice of initial conditions. Moreover, we study thevulnerability of solitary states with respect to changes in the inter-layer topol-ogy. In more detail, we remove links that connect two solitary nodes of eachlayer and evaluate the resulting pattern. We find a highly non-trivial depen-dence of the survivability of the solitary states on topological (position in thenetwork) and dynamical (phase of the oscillation) characteristics.
Keywords: solitary states, multiplex networks, FitzHugh-Nagumo model,synchronization, phase sensitivity
1. Introduction
Synchronization phenomena in networks of coupled oscillators are of greatimportance in many fields of research ranging from physics and chemistry tobiology, neuroscience, physiology, ecology, socio-economic systems, computerscience and engineering[1–4]. In neural systems, synchronization can play a sig-nificant and constructive role in learning and in the context of cognition [5], butis also linked to pathological states such as Parkinson’s disease [6] or epilepsy[7, 8]. It is, therefore, particularly important to understand the mechanismsof synchronization of such systems. Moreover, it crucial to investigate tran-sitions from synchronized states to desynchronized regimes and vise versa, aswell as complex partial synchronization patterns [9, 10] occurring during these ∗ Corresponding author
Email address: [email protected] (Anna Zakharova )
Preprint submitted to Elsevier September 25, 2020 a r X i v : . [ n li n . AO ] S e p ransitions. Whereas chimera states [11], which represent a peculiar type ofpartial synchronization pattern defined by a spatial coexistence of synchronousand asynchronous behavior in networks of identical oscillators [12], have beenstudied extensively in the context of neuronal networks [13–17], other complexpartial synchronization patterns and, in particular, solitary states are still poorlyunderstood.Solitary states emerge in networks of coupled dynamical units, such as maps[18, 19] and oscillators [20–23] and consist of a large cluster of synchronizedoscillators and very few nodes (compared to the network size) that are split offfrom the synchronized cluster and distributed randomly along the network. Theterm “solitary” stems from the Latin ”solitarius” meaning ”alone” or ”lonely”.The solitary elements of the network are ”alone” in the sense that the vastmajority of the nodes they are coupled to show a uniform behavior differentfrom the dynamics of the node itself. We refer to such solutions as ”solitarystates” whereas we name the oscillators that are split off from the synchronizedcluster ”solitary nodes”. In other words, ”solitary state” always refers to thestate of the whole network, while ”solitary node” denotes oscillators that do notbelong to the synchronized group.Multilayer networks have recently gained attention of researchers from vari-ous fields since they offer a better representation of the topology and dynamicsof real-world systems [24, 25]. Moreover, they open up new possibilities of con-trol allowing to regulate nonlinear systems by means of the interplay betweendynamics and multiplexing. On of the advantages of the multiplexing control isthe possibility of inducing the desired state in one of the layers without manip-ulating its parameters by solely adjusting the parameters of the other layer. Anumber of challenging problems occurring in the context of multilayer networksare related to the control of synchronization and complex spatio-temporal pat-terns. For chimera states, for example, the control through multiplexing [26, 27]and through the interplay of time delay and multiplexing [28, 29] has been in-vestigated. Recently, the so-called weak multiplexing control has been reportedand applied to chimera states [15] and coherence resonance [30]. The distinctivefeature of this control scheme is the possibility of achieving the desired statein a certain layer in the presence of weak coupling between the layers (i.e., thecoupling between the layers is much smaller than that inside the layers). Therearises a question whether weak multiplexing control can also be applied to soli-tary states. Previously, solitary states have been investigated in one-layer neuralnetworks [22] including those with time-delayed connections [31].In this study, we investigate the onset and possible vulnerabilities of solitarystates occurring in a two-layer multiplex network of coupled FitzHugh-Nagumo(FHN) oscillators with non-identical layers. We select the internal control pa-rameters in a way that only one of them is allowed to exhibit solitary stateswhen in isolation, whereas in the other one only the fully synchronized solutionis stable in isolation. By letting the two layers interact via a diffusive couplingscheme, we establish the onset of solitary states in the two-layer system forboth controlled and random initial conditions. Next, we analyze the robustnessof such patterns by investigating their existence for different sets of intra- and2nter-layer control parameters. As the solitary states occurring in the two-layerconfiguration rely on the diffusive interaction between the individual layers, theremoval of connections between them (inter-layer links) might pose a threat tosolitary states. Indeed, we show that solitary states are vulnerable with respectto the removal of inter-layer links. In more detail, the following two factorsplay important role: (i) the spatial location of the nodes for which the inter-layer links are removed; (ii) the dynamical phases of the FHN oscillators at thislocation.
2. Solitary states in disconnected layers
We start our investigations on the occurrence of solitary states in two-layernetworks by considering an isolated layer of FitzHugh-Nagumo (FHN) oscillatorsnon-locally coupled in a ring topology. For such networks, the occurrence ofsolitary states has been recently discussed, see Ref. [22]. Therefore, we usethis reference as a corner stone for our investigation. The system of equationsdescribing the network of FHN oscillators is given by [22]: ε du i dt = u i − u i − v i + σ R i + R (cid:88) j = i − R [ b uu ( u j − u i ) + b uv ( v j − v i )] ,dv i dt = u i + a i + σ R i + R (cid:88) j = i − R [ b vu ( u j − u i ) + b vv ( v j − v i )] , (1)where u i and v i are, respectively, the activator and inhibitor variables of eachFHN oscillator i , with i = 1 , . . . , N . The parameter N is the total number ofoscillators in the network. The strength of the coupling is given by σ . Thecoupling range R indicates the number of nearest neighbors in each direction onthe ring. The quantity R can be normalized by the total number of oscillatorsin the network, allowing us to introduce a quasi-continuous parameter calledcoupling radius r = R/N . For an individual FHN oscillator, the value of thevariable a i defines the excitability threshold and determines whether the systemis in the excitable ( | a i | > | a i | <
1) regime. Since we studynetworks of identical oscillators, we set a i = a = 0 . i .Finally, parameter ε characterizes time-scale separation between activator u and inhibitor v and is fixed to ε = 0 .
05 throughout the paper.The coupling function in Eq. (1) contains not only direct, but also crossinputs between activator ( u ) and inhibitor ( v ) variables. This aspect is modeledby a rotational coupling matrix as discussed in [32]: B = (cid:18) b uu b uv b vu b vv (cid:19) = (cid:18) cos φ sin φ − sin φ cos φ (cid:19) , (2)where φ ∈ [ − π ; π ). Originally developed in the context of chimera states, it hasbeen shown that solitary states may arise for a variety of values of φ , dependingon the coupling strength of the network [22]. Here we fix the parameter φ =3 igure 1: Examples for solitary states in a single layer network of nonlocally coupled FHNunits with one (first row), five (second row), nine (third row) and thirteen (fourth row) solitarynodes. The left panels show the space-time plots, the middle ones illustrate snapshots of theactivator variable u i and the right ones depict the mean-phase velocity profiles. Parametersare: N = 300, σ = 0 . φ = π/ − . r = 0 . a = 0 . ε = 0 . π/ − .
2, as this guarantees the occurrence of solitary states for a rather largeinterval of the coupling strength.The spatio-temporal patterns occurring in the network described by Eq. (1)also depend on the choice of the system’s initial conditions (ICs). Specifically,for prescribed values of σ = 0 . φ = π/ − .
2, the number of solitarynodes in the asymptotic solutions varies with the system ICs. Therefore, wefirst identify the ICs (basins of attraction) corresponding to the patterns withdifferent number of solitary nodes and then let the system ICs evolve into thedesired pattern. This procedure is implemented by randomly selecting 1, 5,9 and 13 oscillators on the ring and assigning to them the initial conditions u sol ( t = 0) = 1 . v sol (0) = 0 . u sync (0) = − . v sync (0) = − . σ on the onset and termi-nation of solitary states for the four cases shown in Fig. 1, we study the numberof solitary nodes as function of σ in the interval [0 . , . σ = 0 . σ = 0 .
001 up to σ = 0 .
36. Similarly, we perform the same procedure in the downward direc-tion by starting again at σ = 0 . σ = 0 .
25. We observe that solitary states in general occur for a limited intervalsof σ . Also, depending on the number of solitary nodes present in the spatio-temporal pattern, the respective interval of σ is different. Namely, the plateausizes for the curves in the case of 13 (blue), 9 (red), 5 (green) and 1 (black)solitary nodes indicate that the higher the number of solitary nodes in a spatio-temporal pattern, the smaller the respective region of occurrence. These resultssuggest that the spatio-temporal patterns with lower number of solitary nodesare structurally more persistent than the patterns with large solitary cluster.
3. Onset of solitary states in a two-layer network
We now analyze the onset of solitary states in a two-layer multiplex network.The two-layer multiplex architecture is a subclass of multilayer networks, wherethe only inter-layer connections are between the replica nodes of every layer.Both of the layers consist, therefore, of the same number of elements. The5 igure 2: a) Exemplary phase portrait of a solitary state showing the trajectory followed bythe synchronized cluster (black) and the one followed by the solitary node (red) with σ = 0 . σ for 1 (black), 5 (green), 9(red) and 13 (blue) initial solitary nodes corresponding to the ones depicted in Fig. 1. Otherparameters are: N = 300, φ = π/ − . r = 0 . a = 0 . ε = 0 . system of equations describing the two-layer multiplex network is given by: ε du i dt = u i − u i − v i + σ R i + R (cid:88) j = i − R [ b uu ( u j − u i ) + b uv ( v j − v i )]+ σ ( u i − u i ) ,dv i dt = u i + a + σ R i + R (cid:88) j = i − R [ b vu ( u j − u i ) + b vv ( v j − v i )] ,ε du i dt = u i − u i − v i + σ R i + R (cid:88) j = i − R [ b uu ( u j − u i ) + b uv ( v j − v i )]+ σ ( u i − u i ) ,dv i dt = u i + a + σ R i + R (cid:88) j = i − R [ b vu ( u j − u i ) + b vv ( v j − v i )] , (3)where u i ( u i ) and v i ( v i ) are the activator and inhibitor variables of theFHN oscillators in the first (second) layer, respectively. Both layers have thesame number of elements N , thus i = 1 , . . . , N . The control parameters a = 0 . ε = 0 .
05 are the same as in Section 2. The coupling radii of both layers areidentical and also kept from the previous section, i.e., r = r = r = 0 .
35. The6ntra-layer dynamics is made distinct by mismatching the respective intra-layercoupling strength specified by σ and σ . In contrast to the intra-layer couplingscheme described in Section 2, the inter-layer coupling function is of a simplediffusive type controlled by the inter-layer coupling strength σ .In the following, we investigate the onset of solitary states in the two-layernetwork (Eq. (3)) with non-identical layers. First, we fix the intra-layer couplingstrength of the second layer to σ = 0 .
4. In accordance to Fig. 2(b), regardlessof the layer ICs, this value guarantees that this layer is fully synchronized whenconsidered in isolation. Next, in order to observe the generation of solitarystates in the coupled layers, we set the coupling strength of the first layer tolevels supporting the occurrence of such states. We start with σ = 0 . σ = 0 .
05. Then, we proceed to define the ICs of bothlayers in the same fashion as described in Section 2, i.e., we prescribe fourspatio-temporal patterns with 1, 5, 9 and 13 solitary nodes. With this, we letthe system evolve in time. After discarding a transient phase ( τ = 4000 arb.time units), in Fig. 3, we show snapshots of the obtained pattern for the fourdifferent cases a) 1 b) 5 c) 9 and d) 13 initial solitary nodes. We observe the onsetof solitary states in both non-identical layers. The solitary nodes are locatedat the same positions in both layers for all our simulations. Interestingly, theresulting pattern is different when compared to the first layer in isolation. Moststrikingly, the case with 13 solitary nodes in the isolated first layer collapses tojust a state with just one single solitary node in both layers (Fig. 3(d)). Further,the pattern with 5 solitary nodes in the single network develops into the statewith 8 in the multiplex system as shown in Fig. 3(b) and the pattern with 9solitary nodes in the isolated layer results in a pattern with 7 in Fig. 3(c).In the next step, we analyze the robustness of the spatio-temporal patternscontaining solitary states to variations in the intra-layer coupling strength ofthe first layer, i.e., the layer inducing such patterns. As already mentioned, wefix the intra-layer coupling strength of the second layer to σ = 0 . σ = 0 .
05. Hence, in Fig. 4(a) we show thenumber of solitary nodes depending on σ . We restrict ourselves to the casesof 1 (black curve), 5(green) and 9 (red) initial solitary nodes. As in Section2, we start our investigation with σ = 0 .
3, vary strength by ∆ σ = 0 .
001 inthe ”upward” direction to σ = 0 .
34 and in the ”downward” direction until σ = 0 .
25. When compared to the results for this layer in isolation (shownin Fig. 2(b)), we find in Fig. 4(a) that the lower boundary of the interval ofoccurrence of solitary states is slightly shifted to the left. The solitary statesarise at lower values of σ . For instance, the threshold for the onset of a singlesolitary node is at σ = 0 .
276 for an isolated layer, while it is at σ = 0 . σ = 0 . σ .In our approach, the states occurring in the two-layer network are induced7 igure 3: Snapshots of the activator u i in the first layer (black circles) and second layer (pinkdiamonds) of solitary states in a two-layer network with the same initial conditions as usedin Fig. 1 in both layers for a) 1, b) 5, c) 9 and d) 13 initial solitary nodes. Parameters are: N = N = 300, σ = 0 . σ = 0 . σ = 0 . φ = π/ − . r = 0 . a = 0 . ε = 0 . igure 4: a) Number of solitary nodes depending on the intra-layer coupling strength of thefirst layer σ for 1 (black), 5 (green), 9 (red) initial solitary nodes as shown in Fig. 3. b) Theinter-layer synchronization error E depending on the intra-layer coupling strength of thefirst layer σ for 1 (black), 5 (green), 9 (red) initial solitary nodes as shown in Fig. 3. Otherparameters are: N = 300, σ = 0 . σ = 0 . φ = π/ − . r = 0 . a = 0 . ε = 0 . by the existence of such states in one of the layers in isolation. The question ofhow high the level of synchronization between the two heterogeneous layers is,arises therefore naturally, especially given the fact that the solitary nodes arelocated at the same positions in both layers. In order to address this issue, weestimate the inter-layer synchronization error given by: E = lim T →∞ N T (cid:90) T N (cid:88) i =1 (cid:107) x i ( t ) − x i ( t ) (cid:107) dt, (4)where x i = ( u i , v i ), with i = 1 , . . . , N , is the vector containing the activatorand inhibitor variables of the FHN oscillators in each layer. The operation (cid:107) . (cid:107) denotes the euclidean norm. The superscript indices in Eq. (4) identify layers.The parameter T indicates the time interval considered in the averaging process.Hence, the definition of E takes into account differences in the state variablesof corresponding oscillators in each layer, lower values of E indicate highersynchronization level between the layers. In Fig. 4(b), we obtain E for theinterval of intra-layer coupling strength σ at which solitary states occur. Weobserve that, despite solitary nodes occurring in the same spatial location ofboth layers, the synchronization among them is imperfect. This fact is expectedonce the layers are non-identical, namely σ (cid:54) = σ . In accordance, E decreasesas the coupling strength σ increases towards the value of σ , i.e., the differencebetween the layers is reduced. On the other hand, the solitary states abruptlycease to exist prior to the identical case σ = σ .Up to this point we concentrated our analysis of solitary states in a two-layernetwork considering restricted sets of initial conditions for both layers. We now9eneralize this approach by taking the initial conditions randomly in the statespace of the two-layer system. Hence, in Fig. 5(a) we show a snapshot ofa spatio-temporal pattern containing 11 solitary nodes obtained from randomICs. We observe that, similarly to the case with specially prepared ICs, thesolitary nodes are also at the same position in both layers. This pattern isobtained for σ = 0 .
275 and σ = 0 . x -axis being the intra-layer coupling strength of the first layer σ and the y -axis being the inter-layer coupling strength σ . Hence, in Figs. 5(c) and5(d), we show the occurrence of solitary states obtained from random initialconditions in both the first c) and the second d) layer. On the x -axis the intra-layer coupling strength of the first layer σ is varied in the interval [0 . , . y -axis the inter-layer coupling strength σ is varied in the interval[0 . , . σ axis. This result is in agreement with the comparison between thesingle network (isolated layer) shown in Fig. 2(b) and the two-layer networkwith σ = 0 .
05 shown in Fig. 4(a). Another interesting conclusion obtainedfrom Figs. 5(c) and Fig. 5(d) is that solitary states are observed for any valueof σ . Only for very small values of σ , the interaction between the layers isnot strong enough to induce solitary states in the second layer. Furthermore,in the lower left region of these figures, one can see fuzzy behavior typical formultistability. For this parameter region, it is very difficult to ascertain if thesystem converges to a solitary state or completely synchronizes. Finally, in Fig.5(b), we analyze the quality of the inter-layer synchronization for the studiedcombinations of σ and σ . The color code represents the error E . We observethat for the parameters corresponding to solitary states, the synchronizationerror between the layers is higher when compared to the cases at which thelayers are internally synchronized. For low values of σ , the error E increasesas the pulling between the layers is simply not large enough to ensure theirmutual synchronization. Another interesting aspect captured from Fig. 5(b) isthat a higher number of solitary nodes implies higher values of E .
4. Phase-sensitivity to inter-layer links removal
As we demonstrated in the previous sections, the onset of solitary statesin the considered two-layer network relies on characteristics of the first layerallowing to induce the solitary states in the second one. One can expect thatthe systematic removal of inter-layers links would release the second layer and,consequently, the solitary states would eventually remain confined in the firstlayer. However, despite such a straightforward mechanism, further analysis of10 igure 5: a) Snapshot of the activator u i of the first layer (black circles) and the second layer(pink diamonds) for 11 solitary nodes obtained from random initial conditions with σ = 0 . σ = 0 .
05. b) Inter-layer synchronization error for solitary states obtained from randominitial conditions and various values of σ and σ . Regions of existence of solitary statesobtained from random initial conditions for various values of σ and σ in layer 1 (c)) andlayer 2 (d)). Other parameters are: N = N = 300, σ = 0 . φ = π/ − . r = 0 . a = 0 . ε = 0 . i ) of the nodes containing solitary nodes are identical in both layers.Thus, there are inter-layer links connecting oscillators that are in the synchro-nized clusters and inter-layer links connecting the solitary nodes of the layers.We refer to the former as ”synchronized links” and to the latter as ”solitarylinks”. Importantly, we restrict the removal procedure only to solitary links,since those have the strongest impact on the system’s spatio-temporal pattern.First, we consider the two-layer network containing only one solitary nodeas shown in Fig. 3(a). For this case, we have only one choice for the removalof an inter-layer link. The spatial location plays no role here, as all stateswith a single solitary node are identical upon index renaming. Therefore weentirely focus on the influence of the oscillation phase in the solitary node.After discarding an initial transient phase of the trajectory ( τ = 1010 arb.units), we initiate the removal of the solitary link along the system trajectoryfor times t r equally spaced in time by ∆ t = 0 .
01. After each removal, weevolve the system for another time interval of τ to check the survivability ofsolitary states. Hence, in Fig. 6, we investigate the phase sensitivity for twosituations in which the two-layer network exhibits a single solitary node, namelyfor σ = 0 .
28 in the left-hand column of this figure and for σ = 0 .
30 in theright-hand column. Specifically, in Figs. 6(a) and 6(b), we show the number ofsurviving solitary nodes in each layer ( y -axis) as function of the removal time t r during the period of oscillation ( x -axis). In blue, we illustrate the number ofsolitary nodes surviving in the first layer, while in orange is the same quantityfor the second layer. As can easily be seen, the survivability of solitary nodes isindeed sensitive to the oscillation phase at which the solitary link is removed.For σ = 0 .
28 in Fig. 6(a), we observe intervals of removal times with differentsizes corresponding to phases at which the solitary state is extinct in both layers,e.g. t r ∼ [1011 . , . t r ∼ [1013 , σ = 0 . σ = 0 .
28 and in Fig. 6(d) for σ = 0 .
3, we show the phase-spaceprojection u × v in the direction of the solitary node in the first layer. Inthese figures, the color code stands for the number of surviving solitary nodes,namely one (red) and zero (blue). The most important feature to be highlightedis the location of the higher phase-sensitivity in the left lower corner of the limitcycle. We attribute this sensitivity pattern to trajectory disturbances generatedby an unstable equilibrium dwelling nearby.12 igure 6: Two cases of vulnerability of a single solitary node for a two-layer multiplex networkwith a), c) intra-layer coupling strength of the first layer σ = 0 .
28 and b), d) σ = 0 .
3; a) andb) show the number of surviving solitary nodes for the first (blue circles) and the second layer(red triangle) for different removal times t r . c) and d) show the dependency of the survival inlayer 1 on the position in the phase space where either one (blue) or no (red) node survives assolitary. Other parameters are: N = N = N = 300, σ = 0 . σ = 0 . a = 0 . ε = 0 . r = r = 0 .
35 , φ = π/ − . σ = 0 . σ = 0 .
05. For this configuration, the dependence of solitary states on thespatial location of solitary links can also be demonstrated. Hence, in order toinvestigate this dependency, we apply the same removal methodology as in theprevious case and we restrict ourselves to the removal of a single solitary linkat every network realization. With this, in Figs. 7(a) and 7(b), we show space-time plots at which the effect of removing different solitary links is visualizedfor different removal times in the first and second layer, respectively. In thesefigures, the x -axis depicts the indices of the 11 solitary nodes, while the y -axisshows the removal time t r of the respective solitary link. Color coded is thenumber of surviving solitary nodes. Note that each time instance and eachbar in these plots corresponds to a different network realization. Hence, theseplots do not show any kind of time-series, but encapsulate the information formany different simulations. The most important feature of the results shownin Figs. 7(a) and 7(b) is that the survivability of solitary states depends ona topological (node index) and and a dynamical (phase) characteristics. Forinstance, comparing the nodes 53 and 227 of both layers, we can clearly see thatthe former shows a regular well-behaved phase-sensitivity, while in the latterthe phase-sensitivity is very high. On one hand, for node 53 the phase at whichthe link is removed does not matter all that much, the number of survivingsolitary nodes is almost always the same, except for phases that are close tothe unstable equilibrium. This result is demonstrated in Figs. 7(c) and 7(d)for the first and second layer, respectively. On the other hand, for node 227,the picture is drastically different. Here, we observe an extremely high phase-sensitivity, as depicted in Figs. 7(e) and 7(f). A large variety of phases alongwith its oscillatory pattern affect the survivability of solitary states.We attribute the asymmetries observed in the vulnerability of solitary nodesessentially to two different mechanisms: First, the non-uniform distributionof solitary nodes across the spatial extension of our layers. Due to the non-locality of the intra-layer coupling scheme, the removal of a given solitary linkindirectly affects a neighborhood of the corresponding node which, in turn,correlates with a different number of solitary nodes. Second, the occurrenceof unstable chaotic sets in the systems high-dimensional phase-space results intransients for which the duration depends non-trivially on the spatial direction ofperturbations (index of the removed solitary link) and the phase of the respectiveoscillations at which such perturbations are applied. The occurrence of chaotictransients with different lengths has been previously found to produce intricatephase-sensitivities in synchronized solutions [33, 34].
5. Conclusion
In summary, we report the onset of solitary states for a two-layer networkcomposed of FitzHugh-Nagumo oscillators. For different parameter sets, wefound that the number of solitary nodes contained in the solution patterns14 igure 7: Sensitivity of solitary states depending on the position and the time of removal ofthe solitary link. a) and b) Color coded number of surviving solitary nodes depending onthe position ( x -axis) and the time ( y -axis) of the removal of a solitary link. Sensitivity ofthe number of surviving solitary nodes depending on the phase of the node 53 (c) and d))and node 227 (e) and f)). After the removal of a link, a transient of τ = 3000 time units ischosen before evaluating the number of surviving solitary nodes. Other parameters: N = 300, σ = 0 . a = 0 . ε = 0 . r = r = 0 .
35 , φ = π/ − . Acknowledgments
We thank Eckehard Sch¨oll, Yuri Maistrenko, Cristina Masoller and RicoBerner for fruitful discussions. This work was supported by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) - Projektnum-mer - 163436311 - SFB 910.
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