Search

In the last three decades it has become clear that cortical regions, interconnected via white-matter fibers, form a modular and hierarchical network. This organization, which has also been seen at the microscopic level in the form of interconnected neural assemblies, is believed to support the coexistence of segregation (specialization) and integration (binding) of information. A fundamental open question is to understand how this complex structure can emerge in the brain. Here, we made a first step to address this question and propose that adaptation to various inputs could be the key driving mechanism for the formation of structural assemblies. To test this idea, we develop a model of quadratic integrate-and-fire spiking neurons, trained to stimuli targetting distinct sub-populations. The model is designed to satisfy several biologically plausible constraints: (i) the network contains excitatory and inhibitory neurons with Hebbian and anti-Hebbian STDP; and (ii) neither the neuronal activity nor the synaptic weights are frozen after the learning phase. Instead, the network continues firing spontaneously while synaptic plasticity remains active. We find that only the combination of the two inhibitory STDP sub-populations allows for the formation of stable modular organization in the network, with each sub-population playing a distinct role. The Hebbian sub-population controls for the firing rate and the anti-Hebbian mediates pattern selectivity. After the learning phase, the network activity settles into an asynchronous irregular resting-state, resembling the behaviour typically observed in-vivo in the cortex. This post-learning activity also displays spontaneous memory recalls, which are fundamental for the long-term consolidation of the learned memory items. The model introduced represents a starting point for the joint investigation of neural dynamics, connectivity and plasticity., Comment: 28 pages, 11 figures

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying real systems across various fields of research. Represented as graphs, different systems can be studied using the same analysis methods, which allows for their comparison. Here, we challenge the wide-spread idea that graph theory is a universal analysis tool, uniformly applicable to any kind of network data. Instead, we show that many classical graph metrics (including degree, clustering coefficient and geodesic distance) arise from a common hidden propagation model: the discrete cascade. From this perspective, graph metrics are no longer regarded as combinatorial measures of the graph, but as spatio-temporal properties of the network dynamics unfolded at different temporal scales. Once graph theory is seen as a model-based (and not a purely data-driven) analysis tool, we can freely or intentionally replace the discrete cascade by other canonical propagation models and define new network metrics. This opens the opportunity to design, explicitly and transparently, dedicated analyses for different types of real networks by choosing a propagation model that matches their individual constraints. In this way, we take stand that network topology cannot always be abstracted independently from network dynamics, but shall be jointly studied. Which is key for the interpretability of the analyses. The model-based perspective here proposed serves to integrate into a common context both the classical graph analysis and the more recent network metrics defined in the literature which were, directly or indirectly, inspired by propagation phenomena on networks., Comment: 23 pages, 6 figures. Perspective article

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short.

Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. The intricate topology of networks can also be investigated by means of the collective dynamics observed in the interactions of self-sustained oscillations (synchronization patterns) or propagation-like processes such as random walks. However, networks are often inferred from real data forming dynamic systems, which are different from those employed to reveal their topological characteristics. This stresses the necessity for a theoretical framework dedicated to the mutual relationship between the structure and dynamics in complex networks, as the two sides of the same coin. Here we propose a rigorous framework based on the network response over time (i.e., Green function) to study interactions between nodes across time. For this purpose we define the \emph{flow} that describes the interplay between the network connectivity and external inputs. This multivariate measure relates to the concepts of graph communicability and the map equation. We illustrate our theory using the multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics, but the formalism can be adapted to other local dynamics for which the Green function is known. We provide applications to classical network examples, such as small-world ring and hierarchical networks. Our theory defines a comprehensive framework that is canonically related to directed and weighted networks, thus paving a new way to revise the standards for network analysis, from the pairwise interactions between nodes to the global properties of networks including community detection., Comment: preprint, 8 figures

Intrinsic brain activity is characterized by highly structured co-activations between different regions, whose origin is still under debate. In this paper, we address the question whether it is possible to unveil how the underlying anatomical connectivity shape the brain's spontaneous correlation structure. We start from the assumption that in order for two nodes to exhibit large covariation, they must be exposed to similar input patterns from the entire network. We then acknowledge that information rarely spreads only along an unique route, but rather travels along all possible paths. In real networks the strength of local perturbations tends to decay as they propagate away from the sources, leading to a progressive attenuation of the original information content and, thus, of their influence. We use these notions to derive a novel analytical measure, $\mathcal{T}$ , which quantifies the similarity of the whole-network input patterns arriving at any two nodes only due to the underlying topology, in what is a generalization of the matching index. We show that this measure of topological similarity can indeed be used to predict the contribution of network topology to the expected correlation structure, thus unveiling the mechanism behind the tight but elusive relationship between structure and function in complex networks. Finally, we use this measure to investigate brain connectivity, showing that information about the topology defined by the complex fabric of brain axonal pathways specifies to a large extent the time-average functional connectivity observed at rest.

The large-scale structural ingredients of the brain and neural connectomes have been identified in recent years. These are, similar to the features found in many other real networks: the arrangement of brain regions into modules and the presence of highly connected regions (hubs) forming rich-clubs. Here, we examine how modules and hubs shape the collective dynamics on networks and we find that both ingredients lead to the emergence of complex dynamics. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which either modules or hubs are destroyed, we find that functional complexity always decreases in the perturbed networks. A comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, lies in a dynamical state that reflects the largest complexity its anatomical connectome can host. Last, we generalise the topology of neural connectomes into a new hierarchical network model that successfully combines modular organisation with rich-club forming hubs. This is achieved by centralising the cross-modular connections through a preferential attachment rule. Our network model hosts more complex dynamics than other hierarchical models widely used as benchmarks.

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider the neural network of the \textit{C.elegans} soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter $\rho$, we also employ other measures of coherence, such as the chimera-like $\chi$ and metastability $\lambda$ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the \textit{C.elegans} neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks., Comment: 10 pages, 3 figures

Recent studies have pointed out the importance of transient synchronization between widely distributed neural assemblies to understand conscious perception. These neural assemblies form intricate networks of neurons and synapses whose detailed map for mammals is still unknown and far from our experimental capabilities. Only in a few cases, for example the C. elegans, we know the complete mapping of the neuronal tissue or its mesoscopic level of description provided by cortical areas. Here we study the process of transient and global synchronization using a simple model of phase-coupled oscillators assigned to cortical areas in the cerebral cat cortex. Our results highlight the impact of the topological connectivity in the developing of synchronization, revealing a transition in the synchronization organization that goes from a modular decentralized coherence to a centralized synchronized regime controlled by a few cortical areas forming a Rich-Club connectivity pattern., Comment: 24 pages, 8 figures. Final version published in PLoS One

Modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and function of complex systems. Generally speaking, modules are islands of highly connected nodes separated by a relatively small number of links. Every module can have contributions of links from any node in the network. The challenge is to disentangle these contributions to understand how the modular structure is built. The main problem is that the analysis of a certain partition into modules involves, in principle, as many data as number of modules times number of nodes. To confront this challenge, here we first define the contribution matrix, the mathematical object containing all the information about the partition of interest, and after, we use a Truncated Singular Value Decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allow us to scrutinize the skeleton of the modular structure, revealing the structure of individual modules and their interrelations., Comment: 21 pages, 10 figures

Although most of the real networks contain a mixture of directed and bidirectional (reciprocal) connections, the reciprocity $r$ has received little attention as a subject of theoretical understanding. We study the expected reciprocity of networks with an arbitrary degree sequence and a broad class of degree correlations by means of statistical ensemble approach. We demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to $r$ is revealed. Numerical experiments using novel network randomization methods show very good agreement to our analytical estimations., Comment: 8 pages, 3 figures, added a new table and a new figure, accepted for publication in Phys.Rev.E