Percolation of spatially constrained Erdős-Rényi networks with degree correlations

A statistical analysis method for probability distributions in Erdös-Rényi random networks with preferential cutting-rewiring operation

The study of specific physiological processes from the perspective of network physiology has gained recent attention. Modeling the global information integration among the separated functionalized modules in structural and functional brain networks is a central problem. In this article, the preferentially cutting-rewiring operation (PCRO) is introduced to approximatively describe the above physiological process, which consists of the cutting procedure and the rewiring procedure with specific preferential constraints. By applying the PCRO on the classical Erdös-Rényi random network (ERRN), three types of isolated nodes are generated, based on which the common leaves (CLs) are formed between the two hubs. This makes the initially homogeneous ERRN experience drastic changes and become heterogeneous. Importantly, a statistical analysis method is proposed to theoretically analyze the statistical properties of an ERRN with a PCRO. Specifically, the probability distributions of these three types of isolated nodes are derived, based on which the probability distribution of the CLs can be obtained easily. Furthermore, the validity and universality of our statistical analysis method have been confirmed in numerical experiments. Our contributions may shed light on a new perspective in the interdisciplinary field of complexity science and biological science and would be of great and general interest to network physiology.

Regional Neurodegeneration in vitro: The Protective Role of Neural Activity

Traumatic brain injury is a devastating public health problem, the eighth leading cause of death across the world. To improve our understanding of how injury at the cellular scale affects neural circuit function, we developed a protocol to precisely injure individual neurons within an in vitro neural network. We used high speed calcium imaging to estimate alterations in neural activity and connectivity that occur followed targeted microtrauma. Our studies show that mechanically injured neurons inactivate following microtrauma and eventually re-integrate into the network. Single neuron re-integration is dependent on its activity prior to injury and initial connections in the network: more active and integrated neurons are more resistant to microtrauma and more likely to re-integrate into the network. Micromechanical injury leads to neuronal death 6 h post-injury in a subset of both injured and uninjured neurons. Interestingly, neural activity and network participation after injury were associated with survival in linear discriminate analysis (77.3% correct prediction, Wilks' Lambda = 0.838). Based on this observation, we modulated neuronal activity to rescue neurons after microtrauma. Inhibition of neuronal activity provided much greater survivability than did activation of neurons (ANOVA, p < 0.01 with post-hoc Tukey HSD, p < 0.01). Rescue of neurons by blocking activity in the post-acute period is partially mediated by mitochondrial energetics, as we observed silencing neurons after micromechanical injury led to a significant reduction in mitochondrial calcium accumulation. Overall, the present study provides deeper insight into the propagation of injury within networks, demonstrating that together the initial activity, network structure, and post-injury activity levels contribute to the progressive changes in a neural circuit after mechanical trauma.

Impact of Physical Obstacles on the Structural and Effective Connectivity of in silico Neuronal Circuits

Scaffolds and patterned substrates are among the most successful strategies to dictate the connectivity between neurons in culture. Here, we used numerical simulations to investigate the capacity of physical obstacles placed on a flat substrate to shape structural connectivity, and in turn collective dynamics and effective connectivity, in biologically-realistic neuronal networks. We considered μ-sized obstacles placed in mm-sized networks. Three main obstacle shapes were explored, namely crosses, circles and triangles of isosceles profile. They occupied either a small area fraction of the substrate or populated it entirely in a periodic manner. From the point of view of structure, all obstacles promoted short length-scale connections, shifted the in- and out-degree distributions toward lower values, and increased the modularity of the networks. The capacity of obstacles to shape distinct structural traits depended on their density and the ratio between axonal length and substrate diameter. For high densities, different features were triggered depending on obstacle shape, with crosses trapping axons in their vicinity and triangles funneling axons along the reverse direction of their tip. From the point of view of dynamics, obstacles reduced the capacity of networks to spontaneously activate, with triangles in turn strongly dictating the direction of activity propagation. Effective connectivity networks, inferred using transfer entropy, exhibited distinct modular traits, indicating that the presence of obstacles facilitated the formation of local effective microcircuits. Our study illustrates the potential of physical constraints to shape structural blueprints and remodel collective activity, and may guide investigations aimed at mimicking organizational traits of biological neuronal circuits.

Impact of targeted attack on the spontaneous activity in spatial and biologically-inspired neuronal networks

We study the structural and dynamical consequences of damage in spatial neuronal networks. Inspired by real in vitro networks, we construct directed networks embedded in a two-dimensional space and follow biological rules for designing the wiring of the system. As a result, synthetic cultures display strong metric correlations similar to those observed in real experiments. In its turn, neuronal dynamics is incorporated through the Izhikevich model adopting the parameters derived from observation in real cultures. We consider two scenarios for damage, targeted attacks on those neurons with the highest out-degree and random failures. By analyzing the evolution of both the giant connected component and the dynamical patterns of the neurons as nodes are removed, we observe that network activity halts for a removal of 50% of the nodes in targeted attacks, much lower than the 70% node removal required in the case of random failures. Notably, the decrease of neuronal activity is not gradual. Both damage scenarios portray "boosts" of activity just before full silencing that are not present in equivalent random (Erdös-Rényi) graphs. These boosts correspond to small, spatially compact subnetworks that are able to maintain high levels of activity. Since these subnetworks are absent in the equivalent random graphs, we hypothesize that metric correlations facilitate the existence of local circuits sufficiently integrated to maintain activity, shaping an intrinsic mechanism for resilience.

The angular nature of road networks

Road networks are characterised by several structural and geometrical properties. The topological structure determines partially the hierarchical arrangement of roads, but since these are networks that are spatially constrained, geometrical properties play a fundamental role in determining the network’s behaviour, characterising the influence of each of the street segments on the system. In this work, we apply percolation theory to the UK’s road network using the relative angle between street segments as the occupation probability. The appearance of the spanning cluster is marked by a phase transition, indicating that the system behaves in a critical way. Computing Shannon’s entropy of the cluster sizes, different stages of the percolation process can be discerned, and these indicate that roads integrate to the giant cluster in a hierarchical manner. This is used to construct a hierarchical index that serves to classify roads in terms of their importance. The obtained classification is in very good correspondence with the official designations of roads. This methodology hence provides a framework to consistently extract the main skeleton of an urban system and to further classify each road in terms of its hierarchical importance within the system.

Dominance of Metric Correlations in Two-Dimensional Neuronal Cultures Described through a Random Field Ising Model

We introduce a novel random field Ising model, grounded on experimental observations, to assess the importance of metric correlations in cortical circuits in vitro. Metric correlations arise from both the finite axonal length and the heterogeneity in the spatial arrangement of neurons. The experiments consider the response of neuronal cultures to an external electric stimulation for a gradually weaker connectivity strength between neurons, and in cultures with different spatial configurations. The model can be analytically solved in the metric-free, mean-field scenario. The presence of metric correlations precipitates a strong deviation from the mean field. Null models of the same networks that preserve the distribution of connections recover the mean field. Our results show that metric-inherited correlations in spatial networks dominate the connectivity blueprint, mask the actual distribution of connections, and may emerge as the asset that shapes network dynamics.

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

In this article, we investigate explosive bond percolation (EBP) with the product rule, formally known as the Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice. One of the key features of the EBP transition is the delay, compared to the corresponding random bond percolation (RBP), in the onset of the spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, although ultimately proved wrong, that explosive percolation (EP) exhibits a first-order transition. In the case of EP, much effort has been devoted to resolving the issue of its order of transition and almost no effort has been devoted to finding the critical point, critical exponents, etc., to classify it into universality classes. This is in sharp contrast to the situation for classical random percolation. We do not even know all the exponents of EP for a regular planar lattice or for an Erdös-Renyi network. We first find the critical point p_{c} numerically and then obtain all the critical exponents, β, γ, and ν, as well as the Fisher exponent τ and the fractal dimension d_{f} of the spanning cluster. We also compare our results for EBP with those for RBP and find that all the exponents of EBP obey the same scaling relations as do those for RBP. Our findings suggest that EBP is not special except for the fact that the exponent β is unusually small compared to that for RBP.

From phenotype to genotype in complex brain networks

Generative models are a popular instrument for illuminating the relationships between the hidden variables driving the growth of a complex network and its final topological characteristics, a process known as the "genotype to phenotype problem". However, the definition of a complete methodology encompassing all stages of the analysis, and in particular the validation of the final model, is still an open problem. We here discuss a framework that allows to quantitatively optimise and validate each step of the model creation process. It is based on the execution of a classification task, and on estimating the additional precision provided by the modelled genotype. This encompasses the three main steps of the model creation, namely the selection of topological features, the optimisation of the parameters of the generative model, and the validation of the obtained results. We provide a minimum requirement for a generative model to be useful, prescribing the function mapping genotype to phenotype to be non-monotonic; and we further show how a previously published model does not fulfil such condition, casting doubts on its fitness for the study of neurological disorders. The generality of such framework guarantees its applicability beyond neuroscience, like the emergence of social or technological networks.

Bootstrap percolation on spatial networks

Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of long-range links' lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameter-dependent critical value for the power-law exponent, above which there is a double phase transition, mixed of a second-order phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a second-order phase transition. We further find a parameter-independent critical value around -1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value -1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the self-organization of spatial structure of online social networks, in terms of the effective function for information spreading.

Activity and High-Order Effective Connectivity Alterations in Sanfilippo C Patient-Specific Neuronal Networks

Induced pluripotent stem cell (iPSC) technology has been successfully used to recapitulate phenotypic traits of several human diseases in vitro. Patient-specific iPSC-based disease models are also expected to reveal early functional phenotypes, although this remains to be proved. Here, we generated iPSC lines from two patients with Sanfilippo type C syndrome, a lysosomal storage disorder with inheritable progressive neurodegeneration. Mature neurons obtained from patient-specific iPSC lines recapitulated the main known phenotypes of the disease, not present in genetically corrected patient-specific iPSC-derived cultures. Moreover, neuronal networks organized in vitro from mature patient-derived neurons showed early defects in neuronal activity, network-wide degradation, and altered effective connectivity. Our findings establish the importance of iPSC-based technology to identify early functional phenotypes, which can in turn shed light on the pathological mechanisms occurring in Sanfilippo syndrome. This technology also has the potential to provide valuable readouts to screen compounds, which can prevent the onset of neurodegeneration.

•

Fibroblasts from two Sanfilippo C patients were reprogrammed to obtain iPSCs

•

iPSCs were successfully differentiated to neural cells that mimic the disease

•

Networks of patients’ neurons show altered activity and connectivity

•

Early functional phenotypes are prevented in gene-corrected patients’ neurons

Emergence of assortative mixing between clusters of cultured neurons

The analysis of the activity of neuronal cultures is considered to be a good proxy of the functional connectivity of in vivo neuronal tissues. Thus, the functional complex network inferred from activity patterns is a promising way to unravel the interplay between structure and functionality of neuronal systems. Here, we monitor the spontaneous self-sustained dynamics in neuronal cultures formed by interconnected aggregates of neurons (clusters). Dynamics is characterized by the fast activation of groups of clusters in sequences termed bursts. The analysis of the time delays between clusters' activations within the bursts allows the reconstruction of the directed functional connectivity of the network. We propose a method to statistically infer this connectivity and analyze the resulting properties of the associated complex networks. Surprisingly enough, in contrast to what has been reported for many biological networks, the clustered neuronal cultures present assortative mixing connectivity values, meaning that there is a preference for clusters to link to other clusters that share similar functional connectivity, as well as a rich-club core, which shapes a 'connectivity backbone' in the network. These results point out that the grouping of neurons and the assortative connectivity between clusters are intrinsic survival mechanisms of the culture.

Transfer entropy reconstruction and labeling of neuronal connections from simulated calcium imaging

Neuronal dynamics are fundamentally constrained by the underlying structural network architecture, yet much of the details of this synaptic connectivity are still unknown even in neuronal cultures in vitro. Here we extend a previous approach based on information theory, the Generalized Transfer Entropy, to the reconstruction of connectivity of simulated neuronal networks of both excitatory and inhibitory neurons. We show that, due to the model-free nature of the developed measure, both kinds of connections can be reliably inferred if the average firing rate between synchronous burst events exceeds a small minimum frequency. Furthermore, we suggest, based on systematic simulations, that even lower spontaneous inter-burst rates could be raised to meet the requirements of our reconstruction algorithm by applying a weak spatially homogeneous stimulation to the entire network. By combining multiple recordings of the same in silico network before and after pharmacologically blocking inhibitory synaptic transmission, we show then how it becomes possible to infer with high confidence the excitatory or inhibitory nature of each individual neuron.