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Abstract
In real-world complex systems, heterogeneous components often interact in complex connection patterns and could be schematized by a formalism of multilayer network. In this work, the synchronization characteristics of multilayer network composed of semiconductor lasers (SLs) are investigated systematically. It is demonstrated that the interplay between different layers plays an important role on the synchronization patterns. We elucidate that the performance of cluster synchronization could be facilitated effectively with the introduction of disjoint layer symmetry into network topology. Intertwined stability of clusters from different layers could be decoupled into independent, and the parameter spaces for stable synchronization are extended significantly. The robustness of our proposed regulation scheme on operation parameters is numerically evaluated. Furthermore, the generality of presented theoretical results is validated in networks with more complex topology and multiple layers.
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1. Introduction
Complex networks have been widely adopted in the field of sociology, biology, physics and neuroscience [1]. Synchronization is a very important nonlinear phenomenon in complex networks, which exists in natural and engineered systems playing an essential role in those proper functioning [2,3]. Although global synchronization (GS) has been a well-studied theory [4,5], populations in real-world network may exhibit more complex synchronization phenomenons, such as cluster synchronization (CS) [6–8]. In cluster synchronization, members within same cluster will synchronize to uniform dynamics, and different clusters evolve on diverse trajectories [9]. Because of its advantages including low-cost and easy integration characteristics, synchronization of semiconductor lasers (SLs) has received considerable attention for its widespread applications in chaotic radar, secure communication and neuromorphic photonics [10–21]. In recent years, the synchronization characteristics of SLs in mutually coupled systems have been studied extensively. I. Kanter et. al investigate the isochronous synchronization in mutually coupled SLs [22], Jiang et. al discussed the leader-laggard chaos synchronization in mutually coupled external-cavity SLs [23]. Then they demonstrated that the synchronization of mutually coupled SLs could be stabilized with the introduction of common injection [24]. Furthermore, the researches begin to extend the mutually coupled system to small motifs. Chaotic synchronization and bubbling effect in star-type network are discussed systematically [25,26]. Global synchronization and cluster synchronization is investigated in ring and fully-connected SL networks [27,28]. However, the investigation of SLs network with complex topology still need be considered from practical application perspective. Recently, as network theory has became an excellent foundation for modeling different kind of complex systems, the investigation of synchronization patterns in SLs network has been extended from motifs to networks with complex topology with the introduction of network theory. SLs in network could be partitioned into equitable clusters based on the inherent symmetry of network topology, and the stability of cluster synchronization is decoupled into independent groups associated with these symmetries [29,30]. R. Roy et. al present the experimental research on the synchronization of complex SLs networks with arbitrary topology with optoelectronic feedback loop [31]. Different control strategies are proposed to control the stability of cluster synchronization, such as intervention of network connections [32,33], pinning control [34], and nodes’ dynamics [35].
Moreover, recent developments have enabled the accessibility to further details in complex networks, and it is shown that real-world complex systems are not single isolated networks, heterogeneous components often interact in complex connection patterns, and mathematical formalism to schematize these complex scenarios is the multilayer network [36,37]. Networks are coupled and interconnected, disrupting the working state of a node in one layer of the network can cause the entire system to malfunction, multilayer complex networks are necessary to solve the more realistic and difficult problems, coupling that facilitates the synchronization of nodes in single-layer networks may work in reverse in multilayer networks [38,39]. While, most of the investigations on complex SLs networks are still based on a one-dimensional plane, multilayer complex networks have rarely been applied to the study of synchronization in complex networks composed of SLs.
Therefore, in this work, we bridge the gap by constructing feasible model for multilayer SLs network and explore the synchronization characteristics systematically in this scenario. The relationship between network topology and synchronization patterns is revealed. Moreover, we numerically evaluate the influence of intra-coupling strength inside each layers and inter-coupling strength among different layers on the stability of cluster synchronization, and it is demonstrated that there is intertwined synchronization among the clusters in different layers. Even though the dynamical evolution of SLs in these clusters are incoherent, they will always realize synchrony and desynchrony state at the same time. The crucial role of interconnectivity pattern between different layers on the stability of cluster synchronization is the interplay between different layers and we elucidate that the stability of cluster synchronization in multilayer SLs network could be facilitated effectively with the introduction of disjoint layer symmetry in network topology. At last, we extend our method for cluster synchronization regulation to a SLs network with three-layers.
2. Theory and model
We consider a multilayer network composed of heterogeneous SLs that coupled in multiple patterns as shown in Fig.1 (a), each layer is formed of SLs with homogeneous internal parameters, but different layers with heterogeneous parameters, and SLs are coupled through intra-links inside layers and inter-links among different layers. A set of adjacency matrix is introduced to describe the complex network topology mathematically. Adjacency matrix $A^{pp}$ ($p=\eta, \beta$) describes the intra-coupling inside layers. $A_{mn}^{pp}=1$ if there is an oriented optical injection from $\mathrm {SL}_n^p$ to $\mathrm {SL}_m^p$, and in other cases $A_{mn}^{pp}$ equals to 0. Inter-layer coupling between different layers can also be schematized by adjacency matrix $A^{pq}$, ($p,q=\eta,\beta$ and $p\neq q$ ). Similarly, $A_{mn}^{pq}=1$ when there is an injection from $\mathrm {SL}_n^q$ to $\mathrm {SL}_m^p$. As all the couplings in network are assumed to be undirected, $A^{pq}=A^{qp^T}$ and $A^{pp}=A^{pp^T}$. Therefore, the adjacency matrices of multilayer SLs network in Fig. 1 (a) are equal to Eq. (1). The symmetry of network topology could be interpreted as a permutation matrix $P$ that re-orders the SLs in a way that keep the dynamical equations invariant (that is, $PA^{pp}=A^{pp}P$). Since the internal parameters of SLs in different layers are different, symmetric permutations will be constrained to move SLs within each layers, i.e. $P^{\eta }$ and $P^{\beta }$. Then, in order to preserve flow invariance, the compatibility between symmetries of different layers need to be guaranteed either, i.e. $P^{\eta }A^{\eta \beta }=A^{\eta \beta }P^{\beta }$ and $P^{\beta }A^{\beta \eta }=A^{\beta \eta }P^{\eta }$. After all the inherent symmetries of network topology are identified, SLs in multilayer network of Fig. 1 (a) will be divided into synchronizable clusters by the disjoint set of SLs that map to each other in the symmetric permutation, namely, $C_{1}^{\eta }=\{1\}$, $C_{2}^{\eta }=\{2,3\}$ in layer $\eta$, and $C_{1}^{\beta }=\{1\}$, $C_{2}^{\beta }=\{2,3\}$ in layer $\beta$.
Fig. 1. Structural diagram of multilayer SLs network composed of layer $\eta$ and $\beta$. Nodes marked with same color belong to the same cluster. (a) multilayer networks with intertwined cluster, solid lines represent intra-coupling within each layer, and dashed lines indicate inter-coupling between different layers,(b) multilayer network with the introduction of disjoint layer symmetry into network topology, the red dashed lines represent the modification of network structure compared to (a).
The general dynamical equations that describe multi-layer network are [39]
In order to investigate the synchronization characteristics of multilayer network composed of delay-coupled SLs, the dynamics of isolated node $F$ is represented by the Lang-Kobayashi equations and the output function $H$ is intensity of optical field emitting from SLs. Then the dynamical equations for multilayer SLs network in Fig. 1 (a) will be written as [29,39,40] :
Table 1. Value of parameters used in simulations
To evaluate the stability of cluster synchronization in multilayer SLs network quantitatively, the cross correlation function (CCF) is introduced and defined as Eq. (5) by calculating the correlation coefficient between the dynamics of SLs within identical clusters.
Fig. 2. Isochronous cluster synchronization in multilayer SLs network. Dynamical evolution of SLs in cluster $C_2^{\eta }$ and $C_2^{\beta }$ a(1)-a(2), the corresponding cross correlation function b(1)-b(2). For all cases considered, $\sigma _{\eta \eta }=\sigma _{\beta \beta }=3\text {ns}^{-1}$, $\sigma _{\eta \beta }=1\text {ns}^{-1}$
3. Results
Here, the characteristics of cluster synchronization in multilayer network composed of SLs is investigated systematically. The stability of isochronous cluster synchronization is evaluated by calculating the values of CCF between intensity time series $I_m^{p}=|E_m^{p}(t)^2|$ of SLs within identical clusters with random initial conditions. $C_{\text {max}}^p$ denotes the maximized absolute value of CCF over a range of $\Delta t$, and the threshold value for stable cluster synchronization is $C_{\text {max}}^p>0.98$. Throughout these results, the intensity of spontaneous-emission noise is set equal to zero. To discuss how the network parameter and the inter-layer connection structure impact the cluster synchronization properties, firstly, we consider the effects of intra-layer and inter-layer coupling strengths on cluster synchronization stability and patterns. Fig. 3 plots the correlation coefficient for the non-trivial clusters $C_2^{\eta }$ and $C_2^{\beta }$ shown in Fig. 1 (a) as function of $\sigma _{\eta \eta }$ and $\sigma _{\beta \beta }$, holding inter-coupling strength $\sigma _{\eta \beta }$ constant. In Fig. 3, it is noticeable that the parameter spaces for stable cluster synchronization of layer $\eta$ is coincident with that of layer $\beta$. The chaotic dynamics of SLs in cluster $C_2^{\eta }$ and $C_2^{\beta }$ will always synchronize together or not at all, i.e. the stability of these two clusters are intertwined [41].
The intertwined synchronization properties can also be inspected by the topology of network. Each SL in cluster $C_2^{\eta }$ receives optical injection from exactly one SL in cluster $C_2^{\beta }$, and vice versa. If the dynamics of cluster $C_2^{\eta }$ is not synchronized, then SLs in cluster $C_2^{\beta }$ will receive totally different optical injection, and they will not achieve synchrony either. By the time the network becomes more sophisticated, the intertwined property could also be discerned by symmetric permutation $P^\eta$ and $P^\beta$ of multilayer network which can be calculated from adjacency matrix using discrete algebra routines [39,41] . In all the symmetric permutations, nodes in intertwined clusters and layers need to be swapped simultaneously to preserve the adjacency of network topology. Furthermore, it can also be found in Fig. 3 that stable cluster synchronization could merely be realized with weak inter-coupling strength. And the clusters will lose synchrony completely with the increment of inter-coupling strength. Since in intertwined synchronization patterns, disturbances from different layers will transfer and affect each other, perturbations from the other layer will increase with inter-coupling strength, leading to the difficulty in stabilizing clusters. We thus conclude that connections among different layers may result in a variety of synchronization patterns, and the interplay between different layers plays an essential role on the stability of cluster synchronization.
Fig. 3. The stability of cluster $C_2^{\eta }$ and $C_2^{\beta }$ as function of coupling strength. a(1)-a(2) $C_{\text {max}}$ as function of intra-coupling strength $\sigma _{\eta \eta }$ with $\sigma _{\beta \beta }=3\text {ns}^{-1}$, b(1)-b(2) $C_{\text {max}}$ as function of intra-coupling strength $\sigma _{\beta \beta }$ with $\sigma _{\eta \eta }=3\text {ns}^{-1}$.
Therefore, in order to improve the synchronization performance of multilayer SLs network, connections between different layers is modified to decouple the stability of intertwined cluster. In multilayer network with intertwined cluster, symmetric permutations of different layers are dependent. Nodes need to be swapped simultaneously to preserve the adjacency of network topology. For example, in Fig.1 (a), a symmetric permutation between nodes 2 and 3 in layer $\eta$ must be accompanied with a permutation between nodes 2 and 3 in layer $\beta$. However, the interplay between different layers could be regulated to isolate the layer symmetry into disjoint sets. In disjoint layer symmetry, symmetric permutations in one layer are independent of swaps in all the other layers. As shown in Fig.1 (b), additional interlayer coupling links labeled as red dotted line are added between SLs in intertwined clusters, and symmetric permutations in layer $\eta$ become completely independent of permutations in layer $\beta$. Symmetry-breaking bifurcation in one layer will not alter the formation of cluster in other layers. Moreover, with the introduction of disjoint layer symmetry, SLs in cluster $C_2^{\eta }$ ($C_2^{\beta }$) receive the same input sum from the SLs of cluster $C_2^{\beta }$ ($C_2^{\eta }$) no matter they are synchronized or not. When the clusters in one layer lose stability, clusters in other layers could remain synchronized, stability of cluster synchronization in different layers become independent. As a result, the synchronization stability of intertwined cluster is decoupled successfully, perturbations from other layers could be isolated and the performance of cluster synchronization is improved effectively. Figs. 4 a(1)-a(2) depict the synchronization performance of $C_2^{\eta }$ and $C_2^{\beta }$ as function of inter-coupling strength $\sigma _{\eta \beta }$ with network topology in Fig. 1 (a) and Fig. 1 (b), respectively. As can be seen from the comparison between maximum CCF, the operation range for stable cluster synchronization of our regulated scenario in Fig. 1 (b) is much wider than that of the intertwined synchronization in Fig. 1 (a), indicating the improvement of cluster synchronization stability by our method. Since the introduction of disjoint layer symmetry could decouple the intertwined clusters in different layers into independent synchronizable groups, synchronization stability of different groups become independent to each other. Figs. 4 b(1)-b(2), c(1)-c(2) present the effects of intra-coupling strength in each layer on the values of maximum CCF for cluster $C_2^{\eta }$ and $C_2^{\beta }$ with different inter-coupling strength $\sigma _{\eta \beta }$ for network in Fig. 1 (b). The parameters for calculation in Figs. 4 b(1) to c(2) are $p_f=1.5$, $\alpha ^\eta =5$, $\varepsilon ^\eta =2.5\times 10^{-23}\text {m}^{3}$ and $\alpha ^\beta =4$, $\varepsilon ^\beta =3.5\times 10^{-23}\text {m}^{3}$, to allow the network characteristics to be clearly observed. It is clearly shown that cluster $C_2^{\beta }$ in layer $\beta$ could remain synchronized while cluster $C_2^{\eta }$ in layer $\eta$ lose stability, validating the independence between the stability of cluster synchronization in two layers. Moreover, different from the situation for network in Fig. 1 (a) that stable cluster synchronization is constrained to weak inter-coupling strength $\sigma _{\eta \beta }$, strong $\sigma _{\eta \beta }$ will facilitate the cluster synchronization in multilayer network with disjoint layer symmetry, conversely. In Fig. 1 (b), SLs receive common injections from different layers, which drive the dynamical evolution of SLs within cluster to isochronous synchronization and overcome the symmetry-breaking effects stemming from different initial conditions.
Fig. 4. Regulation of cluster synchronization with the introduction of disjoint layer symmetry. a(1)-a(2) Maximum CCF $C_{\text {max}}$ as function of inter-coupling strength $\sigma _{\eta \beta }$ with different network topology. $C_{\text {max}}$ as function of intra-coupling strength $\sigma _{\eta \eta }$ b(1)-b(2) and $\sigma _{\beta \beta }$ c(1)-c(2) with network topology in Fig. 1(b). The remaining unchanged constant coupling strength in each plot are $3\text {ns}^{-1}$.
The outperformance of network topology with disjoint layer symmetry on regulation of cluster synchronization in multilayer SLs network is further emphasized in Fig. 5. The landscape of stable cluster synchronization for different layers in parameter spaces of coupling strength is displayed, when the coupling strength is not a variable parameter, its constant value is $3\text {ns}^{-1}$. As shown in Figs. 5 a(1)-a(3), b(1)-b(3), coupling between different layers will inhibit cluster synchronization in multilayer networks (Fig. 1 (a)) with intertwined property between layers. Stability region (cream) lies only around weak inter-coupling strength. Moreover, as strong intra-coupling strength in other layers could contribute to inter-layer interaction either, performance of cluster synchronization will decline with the increment of intra-coupling strength in different layers either. However, there are widespread region of stable synchronization in multilayer network with disjoint layer symmetry [see Figs. 5 c(1)-c(3), d(1)-d(3)]. Cluster synchronization is regulated and improved effectively with the modulation of network topology (Fig. 1 (b)). Synchronization stability of different layers is decoupled into independent groups, parameter spaces for stable cluster synchronization of each layers are untangled.
Fig. 5. Landscape for synchronous state of clusters in different layers and network topology. Evolution maps of maximum CCF $C_{max}$ of cluster $C_2^{\eta }$ a(1)-a(3), cluster $C_2^{\beta }$ b(1)-b(3) in parameter spaces of $\sigma _{\eta \eta } \times \sigma _{\beta \beta }$, $\sigma _{\eta \eta } \times \sigma _{\eta \beta }$ and $\sigma _{\beta \beta } \times \sigma _{\eta \beta }$ for the case in Fig. 1 (a). Parameter landscape of $C_{max}$ for network with modified topology in Fig. 1 (b). c(1)-c(3) illustrates the results for cluster $C_2^{\eta }$ and d(1)-d(3) illustrates the results for cluster $C_2^{\beta }$.
To demonstrate the generality of our proposed scheme, we investigate the synchronization property in a multilayer SLs network with more complex topology and more SLs as shown in Figs. 6 a(1)-a(2). Based on the inherent symmetry of network topology, the SLs can be partitioned into 9 different clusters: $C_1^{\eta }=\{1\}$, $C_2^{\eta }=\{2, 4\}$, $C_3^{\eta }=\{3, 7\}$, $C_4^{\eta }=\{5\}$, $C_5^{\eta }=\{6\}$ in layer $\eta$, $C_1^{\beta }=\{1, 7\}$, $C_2^{\beta }=\{2, 8\}$, $C_3^{\beta }=\{3, 5\}$, $C_4^{\beta }=\{4, 6\}$ in layer $\beta$. Figures 6 b(1)-b(2) plot the stability of cluster synchronization as function of inter-coupling strength between deifferent layers. It can be found that there is a intertwined relationship between the stability of cluster $C_2^{\eta }=\{2, 4\}$ and $C_3^{\beta }=\{3, 5\}$ as well as $C_3^{\eta }=\{3, 7\}$ and $C_4^{\beta }=\{4, 6\}$. They will always achieve and lose synchrony at the same time. Then with the introduction of disjoint symmetry into network topology (Fig. 6 a(2)), the parameter spaces for stable cluster synchronization are extended significantly as shown in Figs. 6 c(1)-c(2). Moreover, Figs. 6 d(1)-d(2) present the influence of inter-coupling strength with different values of $\sigma _{\eta \eta }$ and $\sigma _{\beta \beta }$. It can be clearly seen that the stability of synchronization for intertwined cluster is decoupled successfully, parameter spaces for stable synchronization of different clusters become independent.
Fig. 6. Schematic structure of more complicated two-layer SLs network a(1)-a(2) and the influence of coupling strength and inter-layer connection patterns on stability of synchronization. b(1)-c(2) $C_{\text {max}}$ as function of inter-coupling strength $\sigma _{\eta \beta }$ with $\sigma _{\eta \eta }=\sigma _{\beta \beta }=3\text {ns}^{-1}$, d(1)-d(2) with $\sigma _{\eta \eta }=\sigma _{\beta \beta }=6\text {ns}^{-1}$.
As previous calculations in this work are conducted in a two layers network with identical intra-layer interactions, a common question is whether our proposed scheme is robust to multilayer networks with more complex topology. To validate the generality, we perform the regulation of cluster synchronization in a three layers SLs network with different coupling topology within each layer as shown in Fig. 7 a(1). The internal parameters linewidth enhancement factor $\alpha ^p$ and gain saturation coefficient $\varepsilon ^p$ of SLs in different layers are set to be heterogeneous, $\alpha ^\eta =3$, $\varepsilon ^\eta =3.5\times 10^{-23}\text {m}^{3}$, $\alpha ^\beta =4$, $\varepsilon ^\beta =4.5\times 10^{-23}\text {m}^{3}$ and $\alpha ^\gamma =3.5$, $\varepsilon ^\gamma =3\times 10^{-23}\text {m}^{3}$, and current factor $p_f=1.5$. Based on the symmetry of multilayer network topology, SLs in this multilayer network are divided into 13 clusters as: $C_1^{\eta }=\{1\}$, $C_2^{\eta }=\{2, 4\}$, $C_3^{\eta }=\{3, 7\}$, $C_4^{\eta }=\{5\}$, $C_5^{\eta }=\{6\}$ in layer $\eta$, $C_1^{\beta }=\{1, 7\}$, $C_2^{\beta }=\{2, 8\}$, $C_3^{\beta }=\{3, 5\}$, $C_4^{\beta }=\{4, 6\}$ in layer $\beta$ and $C_1^{\gamma }=\{1, 4\}$, $C_2^{\gamma }=\{2, 3\}$, $C_3^{\gamma }=\{3, 6\}$, $C_4^{\gamma }=\{5, 7\}$ in layer $\gamma$. Figures 7 b(1)-b(2) focus on the behavior of intertwined synchronization pattern for networks in Fig. 7 a(1) and illustrate the values of maximum CCF as function inter layer coupling strength. It can be found that synchronization of SLs within cluster $C_3^{\eta }$, $C_4^{\beta }$ and $C_4^{\gamma }$ lost simultaneously with $\sigma _{\eta \beta }>4\text {ns}^{-1}$, $\sigma _{\beta \gamma }>5\text {ns}^{-1}$. However, these clusters will be decoupled into independent synchronizable groups with the introduction of independent layer symmetry with the topology displayed in Fig. 7 a(2). Regulation of network topology enables a significant range of parameter spaces for stable cluster synchronization, and stability region of clusters in different layers is independent to each other. Figures 7 b(1)-b(2) and7 c(1)-c(2) possess the same parameter conditions, and it is clear that when disjoint layer symmetry is introduced to the topology of the network, the scope of parameters for stable cluster synchronization is significantly expanded. In order to characterize more details about the synchronization properties of network in Fig. 7 a(2), we set the value of $\sigma _{\eta \eta }=\sigma _{\beta \beta }=\sigma _{\gamma \gamma }=6\text {ns}^{-1}$. Then it can be clearly seen from Figs. 7 d(1)-d(2) that clusters from different layers are synchronized in entirely different parameter spaces. This is due to the fact that, compared to Fig. 7 a(1), disjoint layer symmetry is introduced in Fig. 7 a(2), which invalidates the intertwined interactions among $C_3^{\eta }$, $C_4^{\beta }$, and $C_4^{\gamma }$, making the stability of clusters independent from each other. At the same time, the synchronization stability of clusters in each layer is enhanced. Therefore, we have a strong argument for the conclusions presented prior to the article.
Fig. 7. Structral diagram of tri-layer SLs Network a(1)-a(2) and the effect of coupling strength and inter-layer connection patterns on synchronization stability. The results of a(1) are displayed in b(1)-b(2), and a(2) results are displayed in c(1)-d(2). b(1) $C_{\text {max}}$ as function of inter-coupling strength $\sigma _{\eta \beta }$ with $\sigma _{\eta \eta }=\sigma _{\beta \beta }=\sigma _{\gamma \gamma }=\sigma _{\beta \gamma }=3\text {ns}^{-1}$. b(2) $C_{\text {max}}$ as function of inter-coupling strength $\sigma _{\beta \gamma }$ with $\sigma _{\eta \eta }=\sigma _{\beta \beta }=\sigma _{\gamma \gamma }=\sigma _{\eta \beta }=3\text {ns}^{-1}$.
4. Discussion
In this paper, synchronization characteristics of cluster synchronization in multilayer networks composed of SLs is investigated systematically. It is demonstrated that the interplay between different layers plays an essential role on the cluster synchronization patterns and stability within each layers. And we elucidate that the intertwined synchronization pattern in multilayer network constrains stable cluster synchronization to weak inter-coupling strength. Disjoint layer symmetry of network topology is introduced to regulate the synchronization stability, intertwined clusters in different layers will be decoupled into independent synchronizable groups. Stability of cluster synchronization in different layers is improved effectively with our proposed scheme.
Moreover, as the complex neuronal network of human brain with multiple regions could be modelled by multilayer network. And in the field of neuromorphic computing, it has been demonstrated that the network topology and synchronization patterns play an important role on the performance of reservoir computing [42]. Compared to electronic schems, photonic reservoir computing (RC) based on semiconductor lasers can process information much faster. As our results investigate the relationship between network topology and synchronization properties in multilayer SLs network which will help to guide the design of reservoir network topology and then make a contribution to the optimization of computing performance for photonic RC. Furthermore, as the neuromorphic computing could also be adopted to improve the performance of chaotic optical communication [43], our results may also suggest a new insight in the research of optical communication. While, it still needs to be noted that experimental demonstration is not presented in this work and we will try to overcome this limitation in the further work.
Funding
National Key Research and Development Program of China (2021YFB2801900); National Natural Science Foundation of China (U22A2089, 62104203, 62375228); Science Fund for Distinguished Young Scholars of Sichuan Province (2023NSFSC1969); Sichuan Province Science and Technology Support Program (2022YFG0026, 2022ZYD0119); Fundamental Research Funds for the Central Universities (2682022CX024, 2682023CG003, 2682023ZTPY015).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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