Abstract: Percolation has been used as a model for describing a wide range of different phenomena [Saberi Phys. Reports 2015; Eckmann et al, Phys. Reports 2007]. For example, the distribution of sizes of epidemics or neural avalanches has been studied using models of percolation on networks [Faqeeh et al arXiv:1508.05590, 2015]. In particular, using a model of percolation, it was shown [Friedman and Landsberg, Chaos 2013] that the hierarchical modular structure observed in brain networks can contribute to a power-law distribution of avalanche sizes and durations, and critical behavior of neural dynamics. This study shows the potential of percolation models to help understand the origin of the various power-law behaviors, observed in experimental setups [Freidman et al, PRL 2012] and computational simulations [Rubinov, Plos. Comp. Biol. 2011] of neural systems. Here, we use methods employed in percolation theories, popularity dynamics, and critical branching processes to investigate the distribution of avalanche sizes on random networks with arbitrary degree distribution. We show, using theoretical and numerical calculations, that even a simple model of neural dynamics can produce a range of distinct power-law behaviors: For scale-free networks with degree distribution exponent $3<\nu<4$, the avalanche size distribution $P(s)$, at the critical point of the neural dynamics model, has a power-law form with exponent $(2\nu-3)/(\nu-2)$. Interestingly, for such scale-free networks, in the subcritical regime, $P(s)$ is also a power-law with exponent $\nu$. This refutes the previous analysis [Cohen PRE 66, 036113, 2002] that indicated that away from the critical point, $P(s)$ should be a power-law (with exponent 5/2) with exponential cutoff. In networks with $\nu>4$ or in non-scale-free networks, at the critical point, $P(s)$ is a pure power-law with exponent 5/2; nonetheless, even away from the critical point, a power-law with exponent 5/2 which is extended for several orders of magnitude can be observed for $P(s)$.

Abstract: It is known that the noise during gene expression comes about in two ways. The inherent stochasticity of biochemical processes generates "intrinsic" noise. "Extrinsic" noise refers to variation in identically-regulated quantities between different cells. The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in protein concentrations. To study the spatial effects of intrinsic and extrinsic noises on the gene regulation determining the emergence of tumor we have applied a multiscale chemo-mechanical model of cancer development in epithelial tissue proposed recently in [1]. The epithelium is represented by an elastic 2D array of polygonal cells with its own gene regulation dynamics. The model allows for the simulation of evolution of multiple cells interacting via the chemical signaling or mechanically induced strain. The algorithm includes the transformation of normal cells into a cancerous state triggered by a local failure of spatial synchronization of the cellular rhythms. To model the delay-induced stochastic chemical signaling we have used a generalization of the Gillespie algorithm that accounts for delay suggested in [2]. The possibility of the stochastic pattern formation produced by the joint action of time delay and noise was demonstrated in [3]. In this work, we study the effect of the stochastic oscillations responsible for cell-to-cell communications on the emergence of tumor. Both the intrinsic and extrinsic contributions to stochastic pattern formation and circadian rhythm disruption have been explored numerically. [1] Bratsun D.A., Merkuriev D.V., Zakharov A.P., Pismen L.M. Multiscale modeling of tumor growth induced by circadian rhythm disruption in epithelial tissue. J. Biol. Phys. 42, 107-132 (2016). [2] Bratsun, D., Volfson, D., Hasty, J., Tsimring, L.S. Delay-induced stochastic oscillations in gene regulation. PNAS 102, 14593-14598 (2005). [3]Bratsun D.A., Zakharov A.P. Spatial Effects of Delay-Induced Stochastic Oscillations in a Multi-scale Cellular System. Springer Proceedings in Complexity, 93-103 (2016).

Abstract: Tropical forests have long been recognised as one of the largest pools of biodiversity, and tree inventory database from closed canopy forests have recently been used to estimate their species richness. Global patterns of empirical abundance distributions for vascular plant species show that tropical forests vary in their absolute number of species, but display surprising similarities in the distribution of populations across species. In the Amazonia hyper-dominant species are only 1.4% of the total, but they account for half of all trees; on the other spectrum, hyper-rare species make up nearly 70% of the entire pool, but their total population is only 0.12% of all trees. This extreme heterogeneity in abundances across species forms the core of the Fisher’s paradox, an important open question in ecology. Here we introduce an analytical framework which allows one to provide robust and accurate estimates of species richness and abundance distributions in biodiversity-rich ecosystems. We find that previous methods have systematically overestimated the total number of species. Also, our analysis of 15 empirical forest plots highlights that ecosystems at stationarity tend to maximise their relative fluctuation of abundances. This produces a large number of rare species and only a few common species. We argue that a large number of rare species provides a buffer against declines. When biotic factors or environmental conditions change, some of the rare species may be abler than others in maintaining ecosystem’s functions, because different species respond differently to environmental changes. This further underscores the importance of rare species and their link with the insurance effect.

Abstract: Rapidly mutating pathogens may be able to persist in the population and reach an endemic equilibrium by escaping acquired immunity of hosts. For such diseases, multiple biological, environmental and population level mechanisms determine epidemic dynamics, including pathogen’s epidemiological traits, seasonality, interaction with other circulating strains and spatial fragmentation of hosts and their mixing. We focus on the two latter factors and study the impact of the heterogeneities characterizing population distribution and mobility network on the equilibrium dynamics of the infection both with one strain and with multiple competing strains. We consider a susceptible-infected-recovered-susceptible model on a metapopulation system where individuals are distributed in subpopulations connected with a network of mobility flows. We simulate disease spreading by means of a mechanistic stochastic model and we systematically explore different levels of spatial disaggregation, probability of traveling among subpopulations and mobility network topology, reconstructing the phase space of pathogen persistence and the dynamics out of the equilibrium. Results depict a rich dynamical behaviour. The increase in the average duration of immunity reduces the chance of persistence until extinction is certain above a threshold value. Such critical parameter, however, is crucially affected by the traveling probability, being larger for intermediate levels of mobility coupling. The dynamical regimes observed are very diversified and present oscillations and metastable states. Topological heterogeneities leave their signature on the spatial dynamics, where subpopulation connectivity affects recurrence of epidemic waves, spreading velocity and chance to be infected. The present work uncovers the crucial role of hosts’ space structure on the ecological dynamics of rapidly mutating pathogens, opening the path for further studies on disease ecology in presence of a complex and heterogeneous environment.

Abstract: The role of correlations in the communication process in the functional brain networks is still highly debated in neuroscience. In this study, we apply a simple SIS epidemic spreading model on the human connectome to analyze the structural topological properties that drive these correlations of activity. We first verify results from previous discrete-time studies with our continuous-time simulations. Then, we introduce a small time delay and analyze the so-called delayed correlations of one brain region to the others. We find that just above the critical threshold direct structural connections induce higher cross-correlations between two brain regions and that the larger the distance between two nodes in the structural network, the lower is their delayed correlation. We prove analytically that the delayed auto-correlation is decreasing for small time lags and show with simulations that it even seems to be exponentially decaying for very small time lags. Hubs seem to have a lower auto-correlation than other nodes, but their delayed correlation with direct neighbors seems to be much higher than with other nodes. Previous studies found that the direction of activity spreading in the human connectome seems to be mostly from the back to the front. Using the delayed correlations and the measure of transfer entropy we can confirm this dominant back-to-front pattern with our SIS model. We show that the "rich club" structure of densely connected hubs seems to be responsible for this observed spreading pattern.

Abstract: It has been argued that complex behavior in many biological systems, including but not limited to human and animal brains, is to a great extent a consequence of high interconnectedness among the individual elements, such as neurons in brains. As a very crude approximation, brain can be viewed as an associative memory that is implemented as a large network of heavily interconnected neurons. Hopfield Networks are a popular model of associative memory. From a dynamical systems perspective, it has been posited that the complexity of possible behaviors of a Hopfield network is largely due to the aforementioned high level of interconnectedness. We show, however, that many aspects of provably complex – and, in particular, unpredictable within realistic computational resources – behavior can also be obtained in very sparsely connected Hopfield networks and related classes of Boolean Network Automata. In fact, it turns out that the most fundamental problems about the memory capacity of a Hopfield network are computationally intractable, even for restricted types of networks that are uniformly sparse, with only a handful neighbors per node. This is significant not only from a theoretical computer science standpoint, but also from connectionist science and neuroscience perspectives: animal brains viewed as networks of neurons are relatively sparse and have local structure. One implication of our work is that some of the most fundamental aspects of biological (and other) networks’ dynamics do not require high density, in order to exhibit provably complex, computationally intractable to predict behavior.