Structure and function in artificial, zebrafish and human neural networks.

Network science provides a set of tools for the characterization of the structure and functional behavior of complex systems. Yet a major problem is to quantify how the structural domain is related to the dynamical one. In other words, how the diversity of dynamical states of a system can be predicted from the static network structure? Or the reverse problem: starting from a set of signals derived from experimental recordings, how can one discover the network connections or the causal relations behind the observed dynamics? Despite the advances achieved over the last two decades, many challenges remain concerning the study of the structure-dynamics interplay of complex systems. In neuroscience, progress is typically constrained by the low spatio-temporal resolution of experiments and by the lack of a universal inferring framework for empirical systems. To address these issues, applications of network science and artificial intelligence to neural data have been rapidly growing. In this article, we review important recent applications of methods from those fields to the study of the interplay between structure and functional dynamics of human and zebrafish brain. We cover the selection of topological features for the characterization of brain networks, inference of functional connections, dynamical modeling, and close with applications to both the human and zebrafish brain. This review is intended to neuroscientists who want to become acquainted with techniques from network science, as well as to researchers from the latter field who are interested in exploring novel application scenarios in neuroscience.

Publisher's Note: Entropy of weighted recurrence plots [Phys. Rev. E 90, 042919 (2014)].

This corrects the article DOI: 10.1103/PhysRevE.90.042919.

Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings.

We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling wave and π-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.

Outgrowth of the axillary bud in rose is controlled by sugar metabolism and signalling.

Shoot branching is a pivotal process during plant growth and development, and is antagonistically orchestrated by auxin and sugars. In contrast to extensive investigations on hormonal regulatory networks, our current knowledge on the role of sugar signalling pathways in bud outgrowth is scarce. Based on a comprehensive stepwise strategy, we investigated the role of glycolysis/the tricarboxylic acid (TCA) cycle and the oxidative pentose phosphate pathway (OPPP) in the control of bud outgrowth. We demonstrated that these pathways are necessary for bud outgrowth promotion upon plant decapitation and in response to sugar availability. They are also targets of the antagonistic crosstalk between auxin and sugar availability. The two pathways act synergistically to down-regulate the expression of BRC1, a conserved inhibitor of shoot branching. Using Rosa calluses stably transformed with GFP-fused promoter sequences of RhBRC1 (pRhBRC1), glycolysis/TCA cycle and the OPPP were found to repress the transcriptional activity of pRhBRC1 cooperatively. Glycolysis/TCA cycle- and OPPP-dependent regulations involve the -1973/-1611 bp and -1206/-709 bp regions of pRhBRC1, respectively. Our findings indicate that glycolysis/TCA cycle and the OPPP are integrative parts of shoot branching control and can link endogenous factors to the developmental programme of bud outgrowth, likely through two distinct mechanisms.

Comparison of Different Spike Train Synchrony Measures Regarding Their Robustness to Erroneous Data From Bicuculline-Induced Epileptiform Activity.

As synchronized activity is associated with basic brain functions and pathological states, spike train synchrony has become an important measure to analyze experimental neuronal data. Many measures of spike train synchrony have been proposed, but there is no gold standard allowing for comparison of results from different experiments. This work aims to provide guidance on which synchrony measure is best suited to quantify the effect of epileptiform-inducing substances (e.g., bicuculline, BIC) in in vitro neuronal spike train data. Spike train data from recordings are likely to suffer from erroneous spike detection, such as missed spikes (false negative) or noise (false positive). Therefore, different timescale-dependent (cross-correlation, mutual information, spike time tiling coefficient) and timescale-independent (Spike-contrast, phase synchronization (PS), A-SPIKE-synchronization, A-ISI-distance, ARI-SPIKE-distance) synchrony measures were compared in terms of their robustness to erroneous spike trains. For this purpose, erroneous spike trains were generated by randomly adding (false positive) or deleting (false negative) spikes (in silico manipulated data) from experimental data. In addition, experimental data were analyzed using different spike detection threshold factors in order to confirm the robustness of the synchrony measures. All experimental data were recorded from cortical neuronal networks on microelectrode array chips, which show epileptiform activity induced by the substance BIC. As a result of the in silico manipulated data, Spike-contrast was the only measure that was robust to false-negative as well as false-positive spikes. Analyzing the experimental data set revealed that all measures were able to capture the effect of BIC in a statistically significant way, with Spike-contrast showing the highest statistical significance even at low spike detection thresholds. In summary, we suggest using Spike-contrast to complement established synchrony measures because it is timescale independent and robust to erroneous spike trains.

Spacing ratio characterization of the spectra of directed random networks.

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdos-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Sugar availability suppresses the auxin-induced strigolactone pathway to promote bud outgrowth.

Apical dominance occurs when the growing shoot tip inhibits the outgrowth of axillary buds. Apically-derived auxin in the nodal stem indirectly inhibits bud outgrowth via cytokinins and strigolactones. Recently, sugar deprivation was found to contribute to this phenomenon. Using rose and pea, we investigated whether sugar availability interacts with auxin in bud outgrowth control, and the role of cytokinins and strigolactones, in vitro and in planta. We show that sucrose antagonises auxin's effect on bud outgrowth, in a dose-dependent and coupled manner. Sucrose also suppresses strigolactone inhibition of outgrowth and the rms3 strigolactone-perception mutant is less affected by reducing sucrose supply. However, sucrose does not interfere with the regulation of cytokinin levels by auxin and stimulates outgrowth even with optimal cytokinin supply. These observations were assembled into a computational model in which sucrose represses bud response to strigolactones, largely independently of cytokinin levels. It quantitatively captures our observed dose-dependent sucrose-hormones effects on bud outgrowth and allows us to express outgrowth response to various combinations of auxin and sucrose levels as a simple quantitative law. This study places sugars in the bud outgrowth regulatory network and paves the way for a better understanding of branching plasticity in response to environmental and genotypic factors.

Onset of synchronization of Kuramoto oscillators in scale-free networks.

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 2<γ≤3, in line with what has been observed for other dynamical processes in such networks. For γ>3, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

Traveling phase waves in asymmetric networks of noisy chaotic attractors.

We explore identical Rössler systems organized into two equally sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. With this asymmetric coupling, we analyze patterns in the phase dynamics that coexist with chaotic amplitudes. We specifically investigate traveling phase waves where the oscillators settle on a new rhythm different from their own. We show that these waves are possible even without coherence in the phase angles. It is further demonstrated that the emergence of these incoherent traveling waves depends on the type of coupling, not on the individual dynamics of the Rössler systems. Together with the study of noise effects, our results suggest a promising new avenue toward the interplay of chaotic, noisy, coherent, and incoherent collective dynamics.

Cytokinins Are Initial Targets of Light in the Control of Bud Outgrowth.

Bud outgrowth is controlled by environmental and endogenous factors. Through the use of the photosynthesis inhibitor norflurazon and of masking experiments, evidence is given here that light acts mainly as a morphogenic signal in the triggering of bud outgrowth and that initial steps in the light signaling pathway involve cytokinins (CKs). Indeed, in rose (Rosa hybrida), inhibition of bud outgrowth by darkness is suppressed solely by the application of CKs. In contrast, application of sugars has a limited effect. Exposure of plants to white light (WL) induces a rapid (after 3-6 h of WL exposure) up-regulation of CK synthesis (RhIPT3 and RhIPT5), of CK activation (RhLOG8), and of CK putative transporter RhPUP5 genes and to the repression of the CK degradation RhCKX1 gene in the node. This leads to the accumulation of CKs in the node within 6 h and in the bud at 24 h and to the triggering of bud outgrowth. Molecular analysis of genes involved in major mechanisms of bud outgrowth (strigolactone signaling [RwMAX2], metabolism and transport of auxin [RhPIN1, RhYUC1, and RhTAR1], regulation of sugar sink strength [RhVI, RhSUSY, RhSUC2, and RhSWEET10], and cell division and expansion [RhEXP and RhPCNA]) reveal that, when supplied in darkness, CKs up-regulate their expression as rapidly and as intensely as WL Additionally, up-regulation of CKs by WL promotes xylem flux toward the bud, as evidenced by Methylene Blue accumulation in the bud after CK treatment in the dark. Altogether, these results suggest that CKs are initial components of the light signaling pathway that controls the initiation of bud outgrowth.

Tweaking synchronization by connectivity modifications.

Natural and man-made networks often possess locally treelike substructures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach, we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behavior. We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum, dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes. We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either discrete or continuous. Here our main result is that a certain number of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization.

New Insights into Phloem Unloading and Expression of Sucrose Transporters in Vegetative Sinks of the Parasitic Plant Phelipanche ramosa L. (Pomel).

The plant-parasitic plant interaction is a interesting model to study sink-source relationship and phloem unloading. The parasitic plants, such as the achlorophyllous plant Phelipanche ramosa, connect to the host phloem through the haustorium and act as supernumerary sinks for the host-derived photoassimilates, primarily sucrose. The application of the fluorescent symplastic tracer, carboxyfluorescein (CF) derived from carboxyfluorescein diacetate (CFDA), to the leaves of the host plant (Brassica napus) showed direct phloem connections at the host-parasite interface. These experiments also evidenced the dominant apoplastic pathway for phloem unloading in major vegetative sinks of the parasite, including tubercles and shoots, except the adventitious root apices. The CF experiments showed also the symplastic isolation of the phloem tissues from the sink tissues in tubercle and shoot of the parasite, then suggesting the pivotal role of sucrose transporters in sucrose unloading in P. ramosa sinks. Three cDNAs encoding sucrose transporters (PrSUT) were isolated from the parasitic plant. PrSUT1 transcripts accumulated at the same level in the tubercle throughout the parasite growth while a significant increase in transcript accumulation occurred after emergence in the flowering shoot, notably in the growing apical part. The in situ hybridization experiments revealed the PrSUT1 transcript accumulation in the mature phloem cells of both subterranean and flowering shoots, as well as in shoot terminal sinks corresponding to apical meristem, scale leaf primordia and immature vasculature. The transient expression experiments in Arabidopsis protoplasts showed that PrSUT1 was localized at the plasma membrane, suggesting its role in phloem functioning and sucrose uptake by the sink cells in P. ramosa. Conversely, the PrSUT2 transcript accumulation was constantly low in tubercles and shoots but PrSUT3 transcripts accumulated markedly in the subterranean and flowering shoots, in concordance with the PrSUT3 mRNA accumulation in multiple sink areas including apical meristem, scale-leaf primordia, immature vasculature and even storage parenchyma. However, the PrSUT3 transcripts did not accumulate in the mature phloem cells. The transient expression experiments in Arabidopsis protoplasts suggested a tonoplast localization of PrSUT3, for which nevertheless the involvement in intracellular sucrose transport needs clarification.

Thermodynamic characterization of networks using graph polynomials.

In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.

Collective dynamics in two populations of noisy oscillators with asymmetric interactions.

We study two intertwined globally coupled networks of noisy Kuramoto phase oscillators that have the same natural frequency but differ in their perception of the mean field and their contribution to it. Such a give-and-take mechanism is given by asymmetric in- and out-coupling strengths which can be both positive and negative. We uncover in this minimal network of networks intriguing patterns of discordance, where the ensemble splits into two clusters separated by a constant phase lag. If it differs from π, then traveling wave solutions emerge. We observe a second route to traveling waves via traditional one-cluster states. Bistability is found between the various collective states. Analytical results and bifurcation diagrams are derived with a reduced system.

Effects of assortative mixing in the second-order Kuramoto model.

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping. Our results show that it is possible to control collective behavior of damped Kuramoto oscillators by tuning the network structure or by adjusting the dissipation related to the phases' movement.

Sucrose is an early modulator of the key hormonal mechanisms controlling bud outgrowth in Rosa hybrida.

Sugar has only recently been identified as a key player in triggering bud outgrowth, while hormonal control of bud outgrowth is already well established. To get a better understanding of sugar control, the present study investigated how sugar availability modulates the hormonal network during bud outgrowth in Rosa hybrida. Other plant models, for which mutants are available, were used when necessary. Buds were grown in vitro to manipulate available sugars. The temporal patterns of the hormonal regulatory network were assessed in parallel with bud outgrowth dynamics. Sucrose determined bud entrance into sustained growth in a concentration-dependent manner. Sustained growth was accompanied by sustained auxin production in buds, and sustained auxin export in a DR5::GUS-expressing pea line. Several events occurred ahead of sucrose-stimulated bud outgrowth. Sucrose upregulated early auxin synthesis genes (RhTAR1, RhYUC1) and the auxin efflux carrier gene RhPIN1, and promoted PIN1 abundance at the plasma membrane in a pPIN1::PIN1-GFP-expressing tomato line. Sucrose downregulated both RwMAX2, involved in the strigolactone-transduction pathway, and RhBRC1, a repressor of branching, at an early stage. The presence of sucrose also increased stem cytokinin content, but sucrose-promoted bud outgrowth was not related to that pathway. In these processes, several non-metabolizable sucrose analogues induced sustained bud outgrowth in R. hybrida, Pisum sativum, and Arabidopsis thaliana, suggesting that sucrose was involved in a signalling pathway. In conclusion, we identified potential hormonal candidates for bud outgrowth control by sugar. They are central to future investigations aimed at disentangling the processes that underlie regulation of bud outgrowth by sugar.

Universality in the spectral and eigenfunction properties of random networks.

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdos-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

Entropy of weighted recurrence plots.

The Shannon entropy of a time series is a standard measure to assess the complexity of a dynamical process and can be used to quantify transitions between different dynamical regimes. An alternative way of quantifying complexity is based on state recurrences, such as those available in recurrence quantification analysis. Although varying definitions for recurrence-based entropies have been suggested so far, for some cases they reveal inconsistent results. Here we suggest a method based on weighted recurrence plots and show that the associated Shannon entropy is positively correlated with the largest Lyapunov exponent. We demonstrate the potential on a prototypical example as well as on experimental data of a chemical experiment.

Interplay of sugar, light and gibberellins in expression of Rosa hybrida vacuolar invertase 1 regulation.

Our previous findings showed that the expression of the Rosa hybrida vacuolar invertase 1 gene (RhVI1) was tightly correlated with the ability of buds to grow out and was under sugar, gibberellin and light control. Here, we aimed to provide an insight into the mechanistic basis of this regulation. In situ hybridization showed that RhVI1 expression was localized in epidermal cells of young leaves of bursting buds. We then isolated a 895 bp fragment of the promoter of RhVI1. In silico analysis identified putative cis-elements involved in the response to sugars, light and gibberellins on its proximal part (595 bp). To carry out functional analysis of the RhVI1 promoter in a homologous system, we developed a direct method for stable transformation of rose cells. 5' deletions of the proximal promoter fused to the uidA reporter gene were inserted into the rose cell genome to study the cell's response to exogenous and endogenous stimuli. Deletion analysis revealed that the 468 bp promoter fragment is sufficient to trigger reporter gene activity in response to light, sugars and gibberellins. This region confers sucrose- and fructose-, but not glucose-, responsive activation in the dark. Inversely, the -595 to -468 bp region that carries the sugar-repressive element (SRE) is required to down-regulate the RhVI1 promoter in response to sucrose and fructose in the dark. We also demonstrate that sugar/light and gibberellin/light act synergistically to up-regulate ß-glucuronidase (GUS) activity sharply under the control of the 595 bp pRhVI1 region. These results reveal that the 127 bp promoter fragment located between -595 and -468 bp is critical for light and sugar and light and gibberellins to act synergistically.