A hierarchy index for networks in the brain reveals a complex entangled organizational structure.

Networks involved in information processing often have their nodes arranged hierarchically, with the majority of connections occurring in adjacent levels. However, despite being an intuitively appealing concept, the hierarchical organization of large networks, such as those in the brain, is difficult to identify, especially in absence of additional information beyond that provided by the connectome. In this paper, we propose a framework to uncover the hierarchical structure of a given network, that identifies the nodes occupying each level as well as the sequential order of the levels. It involves optimizing a metric that we use to quantify the extent of hierarchy present in a network. Applying this measure to various brain networks, ranging from the nervous system of the nematode Caenorhabditis elegans to the human connectome, we unexpectedly find that they exhibit a common network architectural motif intertwining hierarchy and modularity. This suggests that brain networks may have evolved to simultaneously exploit the functional advantages of these two types of organizations, viz., relatively independent modules performing distributed processing in parallel and a hierarchical structure that allows sequential pooling of these multiple processing streams. An intriguing possibility is that this property we report may be common to information processing networks in general.

Morphogen-regulated contact-mediated signaling between cells can drive the transitions underlying body segmentation in vertebrates.

We propose a unified mechanism that reproduces the sequence of dynamical transitions observed during somitogenesis, the process of body segmentation during embryonic development, that is invariant across all vertebrate species. This is achieved by combining inter-cellular interactions mediated via receptor-ligand coupling with global spatial heterogeneity introduced through a morphogen gradient known to occur along the anteroposterior axis. Our model reproduces synchronized oscillations in the gene expression in cells at the anterior of the presomitic mesoderm as it grows by adding new cells at its posterior, followed by travelling waves and subsequent arrest of activity, with the eventual appearance of somite-like patterns. This framework integrates a boundary-organized pattern formation mechanism, which uses positional information provided by a morphogen gradient, with the coupling-mediated self-organized emergence of collective dynamics, to explain the processes that lead to segmentation.

Information integration and collective motility in phototactic cyanobacteria.

Cells in microbial colonies integrate information across multiple spatial and temporal scales while sensing environmental cues. A number of photosynthetic cyanobacteria respond in a directional manner to incident light, resulting in the phototaxis of individual cells. Colonies of such bacteria exhibit large-scale changes in morphology, arising from cell-cell interactions, during phototaxis. These interactions occur through type IV pili-mediated physical contacts between cells, as well as through the secretion of complex polysaccharides ('slime') that facilitates cell motion. Here, we describe a computational model for such collective behaviour in colonies of the cyanobacterium Synechocystis. The model is designed to replicate observations from recent experiments on the emergent response of the colonies to varied light regimes. It predicts the complex colony morphologies that arise as a result. We ask if changes in colony morphology during phototaxis can be used to infer if cells integrate information from multiple light sources simultaneously, or respond to these light sources separately at each instant of time. We find that these two scenarios cannot be distinguished from the shapes of colonies alone. However, we show that tracking the trajectories of individual cyanobacteria provides a way of determining their mode of response. Our model allows us to address the emergent nature of this class of collective bacterial motion, linking individual cell response to the dynamics of colony shape.

Epidemic prevalence information on social networks can mediate emergent collective outcomes in voluntary vaccine schemes.

The effectiveness of a mass vaccination program can engender its own undoing if individuals choose to not get vaccinated believing that they are already protected by herd immunity. This would appear to be the optimal decision for an individual, based on a strategic appraisal of her costs and benefits, even though she would be vulnerable during subsequent outbreaks if the majority of the population argues in this manner. We investigate how voluntary vaccination can nevertheless emerge in a social network of rational agents, who make informed decisions whether to be vaccinated, integrated with a model of epidemic dynamics. The information available to each agent includes the prevalence of the disease in their local network neighborhood and/or globally in the population, as well as the fraction of their neighbors that are protected against the disease. Crucially, the payoffs governing the decision of agents vary with disease prevalence, resulting in the vaccine uptake behavior changing in response to contagion spreading. The collective behavior of the agents responding to local prevalence can lead to a significant reduction in the final epidemic size, particularly for less contagious diseases having low basic reproduction number [Formula: see text]. Near the epidemic threshold ([Formula: see text]) the use of local prevalence information can result in divergent responses in the final vaccine coverage. Our results suggest that heterogeneity in the risk perception resulting from the spatio-temporal evolution of an epidemic differentially affects agents' payoffs, which is a critical determinant of the success of voluntary vaccination schemes.

A Current Perspective on Wound Healing and Tumour-Induced Angiogenesis.

Angiogenesis, or capillary growth from pre-existing vasculature, is an essential component of several physiological processes, both vital and pathological. These include dermal wound healing and tumour growth that together pose some of the most significant challenges to healthcare systems worldwide. Over the last few decades, mathematical modelling has proven to be a valuable tool for unravelling the complex network of interactions that underlie such processes. Moreover, theoretical frameworks that describe some of the mechanical and chemical aspects of angiogenesis inherent in wound healing and tumour growth have revealed intriguing similarities between the two processes. In this review, we highlight some of the significant contributions made by mathematical models of tumour-induced and wound healing angiogenesis and illustrate how advances in each field have been made using insights from the other. We also detail some open problems that could be addressed through a combination of theoretical and experimental approaches.

Precision of morphogen-driven tissue patterning during development is enhanced through contact-mediated cellular interactions.

Cells in developing embryos reliably differentiate to attain location-specific fates, despite fluctuations in morphogen concentrations that provide positional information and in molecular processes that interpret it. We show that local contact-mediated cell-cell interactions utilize inherent asymmetry in the response of patterning genes to the global morphogen signal yielding a bimodal response. This results in robust developmental outcomes with a consistent identity for the dominant gene at each cell, substantially reducing the uncertainty in the location of boundaries between distinct fates.

Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes.

The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor-ß, platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly composed of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process, such as re-epithelialization. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds; and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations and provide an alternative pathway for gaining insight into this complex biological process.

Spontaneous generation of persistent activity in diffusively coupled cellular assemblies.

The spontaneous generation of electrical activity underpins a number of essential physiological processes, and is observed even in tissues where specialized pacemaker cells have not been identified. The emergence of periodic oscillations in diffusively coupled assemblies of excitable and electrically passive cells (which are individually incapable of sustaining autonomous activity) has been suggested as a possible mechanism underlying such phenomena. In this paper we investigate the dynamics of such assemblies in more detail by considering simple motifs of coupled electrically active and passive cells. The resulting behavior encompasses a wide range of dynamical phenomena, including chaos. However, embedding such assemblies in a lattice yields spatiotemporal patterns that either correspond to a quiescent state or to partial or globally synchronized oscillations. The resulting reduction in dynamical complexity suggests an emergent simplicity in the collective dynamics of such large, spatially extended systems. Furthermore, we show that such patterns can be reproduced by a reduced model comprising only excitatory and oscillatory elements. Our results suggest a generalization of the mechanism by which periodic activity can emerge in a heterogeneous system comprising nonoscillatory elements by coupling them diffusively, provided their steady states in isolation are sufficiently dissimilar.

Mesoscopic architecture enhances communication across the macaque connectome revealing structure-function correspondence in the brain.

Analyzing the brain in terms of organizational structures at intermediate scales provides an approach to unravel the complexity arising from interactions between its large number of components. Focusing on a wiring diagram that spans the cortex, basal ganglia, and thalamus of the macaque brain, we identify robust modules in the network that provide a mesoscopic-level description of its topological architecture. Surprisingly, we find that the modular architecture facilitates rapid communication across the network, instead of localizing activity as is typically expected in networks having community organization. By considering processes of diffusive spreading and coordination, we demonstrate that the specific pattern of inter- and intramodular connectivity in the network allows propagation to be even faster than equivalent randomized networks with or without modular structure. This pattern of connectivity is seen at different scales and is conserved across principal cortical divisions, as well as subcortical structures. Furthermore, we find that the physical proximity between areas is insufficient to explain the modular organization, as similar mesoscopic structures can be obtained even after factoring out the effect of distance constraints on the connectivity. By supplementing the topological information about the macaque connectome with physical locations, volumes, and functions of the constituent areas and analyzing this augmented dataset, we reveal a counterintuitive role played by the modular architecture of the brain in promoting global coordination of its activity. It suggests a possible explanation for the ubiquity of modularity in brain networks.

Contact-mediated signaling enables disorder-driven transitions in cellular assemblies.

We show that, when cells communicate by contact-mediated interactions, heterogeneity in cell shapes and sizes leads to qualitatively distinct collective behavior in the tissue. For intercellular coupling that implements lateral inhibition, such disorder-driven transitions can substantially alter the asymptotic pattern of differentiated cells by modulating their fate choice through changes in the neighborhood geometry. In addition, when contact-induced signals influence inherent cellular oscillations, disorder leads to the emergence of functionally relevant partially-ordered dynamical states.

Mathematical Modeling Can Advance Wound Healing Research.

Significance: For over 30 years, there has been sustained interest in the development of mathematical models for investigating the complex mechanisms underlying each stage of the wound healing process. Despite the immense associated challenges, such models have helped usher in a paradigm shift in wound healing research. Recent Advances: In this article, we review contributions in the field that span epidermal, dermal, and corneal wound healing, and treatments of nonhealing wounds. The recent influence of mathematical models on biological experiments is detailed, with a focus on wound healing assays and fibroblast-populated collagen lattices. Critical Issues: We provide an overview of the field of mathematical modeling of wound healing, highlighting key advances made in recent decades, and discuss how such models have contributed to the development of improved treatment strategies and/or an enhanced understanding of the tightly regulated steps that comprise the healing process. Future Directions: We detail some of the open problems in the field that could be addressed through a combination of theoretical and/or experimental approaches. To move the field forward, we need to have a common language between scientists to facilitate cross-collaboration, which we hope this review can support by highlighting progress to date.

Contact-mediated cellular communication supplements positional information to regulate spatial patterning during development.

Development in multicellular organisms is marked by a high degree of spatial organization of the cells attaining distinct fates in the embryo. Recent experiments showing that suppression of intercellular interactions can alter the spatial patterns arising during development suggest that cell fates cannot be determined by the exclusive regulation of differential gene expression by morphogen gradients (the conventional view encapsulated in the French flag model). Using a mathematical model that describes the receptor-ligand interaction between cells in close physical proximity, we show that such intercellular signaling can regulate the process of selective gene expression within each cell, allowing information from the cellular neighborhood to influence the process by which the thresholds of morphogen concentration that dictate cell fates adaptively emerge. This results in local modulations of the positional cues provided by the global field set up by the morphogen, allowing interaction-mediated self-organized pattern formation to complement boundary-organized mechanisms in the context of development.

Disorder in cellular packing can alter proliferation dynamics to regulate growth.

The mechanisms by which an organ regulates its growth are not yet fully understood, especially when the cells are closely packed as in epithelial tissues. We explain growth arrest as a collective dynamical transition in coupled oscillators on disordered lattices. As the cellular morphologies become homogeneous over the course of development, the signals induced by cell-cell contact increase beyond a critical value that triggers coordinated cessation of the cell-cycle oscillators driving cell division. Thus, control of cell proliferation is causally related to the geometry of cellular packing.

Phototaxis in Cyanobacteria: From Mutants to Models of Collective Behavior.

Cyanobacteria rely on photosynthesis, and thus have evolved complex responses to light. These include phototaxis, the ability of cells to sense light direction and move towards or away from it. Analysis of mutants has demonstrated that phototaxis requires the coordination of multiple photoreceptors and signal transduction networks. The output of these networks is relayed to type IV pili (T4P) that attach to and exert forces on surfaces or other neighboring cells to drive "twitching" or "gliding" motility. This, along with the extrusion of polysaccharides or "slime" by cells, facilitates the emergence of group behavior. We evaluate recent models that describe the emergence of collective colony-scale behavior from the responses of individual, interacting cells. We highlight the advantages of "active matter" approaches in the study of bacterial communities, discussing key differences between emergent behavior in cyanobacterial phototaxis and similar behavior in chemotaxis or quorum sensing.

Milling and meandering: Flocking dynamics of stochastically interacting agents with a field of view.

We introduce a stochastic agent-based model for the flocking dynamics of self-propelled particles that exhibit nonlinear velocity-alignment interactions with neighbors within their field of view. The stochasticity in the dynamics is spatially heterogeneous and arises implicitly from the nature of the interparticle interactions. We observe long-time spatial cohesion in the emergent flocking dynamics, despite the absence of attractive forces that explicitly depend on the relative positions of particles. The wide array of flocking patterns exhibited by this model are characterized by identifying spatially distinct clusters and computing their corresponding angular momenta.

Lateral inhibition provides a unifying framework for spatiotemporal pattern formation in media comprising relaxation oscillators.

Differential excitatory and inhibitory interactions, specifically lateral inhibition, between the constituent elements of complex systems underlie a wide range of spatiotemporal patterns in nature. Here, we show that when systems of relaxation oscillators, whose dynamics involve widely separate timescales, are coupled primarily through diffusion of the inactivation component, they exhibit strikingly similar patterns regardless of specific details of the model kinetics and spatial topology. This universality stems from the fact that all observed patterns can be viewed as either specific manifestations of, or arising through interactions between, two fundamental classes of collective dynamics, viz., a state comprising clusters of synchronized oscillators, and a time-invariant spatially inhomogeneous state resulting from oscillator death. Our work provides an unifying framework for understanding the emergent global behavior of various chemical, biological, and ecological systems spanning several time and length scales.

Emergent memory in cell signaling: Persistent adaptive dynamics in cascades can arise from the diversity of relaxation time-scales.

The mitogen-activated protein kinase (MAPK) signaling cascade, an evolutionarily conserved motif present in all eukaryotic cells, is involved in coordinating crucial cellular functions. While the asymptotic dynamical behavior of the pathway stimulated by a time-invariant signal is relatively well-understood, we show using a computational model that it exhibits a rich repertoire of transient adaptive responses to changes in stimuli. When the signal is switched on, the response is characterized by long-lived modulations in frequency as well as amplitude. On withdrawing the stimulus, the activity decays over long timescales, exhibiting reverberations characterized by repeated spiking in the activated MAPK concentration. The long-term persistence of such post-stimulus activity suggests that the cascade retains memory of the signal for a significant duration following its removal. The molecular mechanism underlying the reverberatory activity is related to the existence of distinct relaxation rates for the different cascade components. This results in the imbalance of fluxes between different layers of the cascade, with the reuse of activated kinases as enzymes when they are released from sequestration in complexes. The persistent adaptive response, indicative of a cellular "short-term" memory, suggests that this ubiquitous signaling pathway plays an even more central role in information processing by eukaryotic cells.

Bifurcations of flame filaments in chaotically mixed combustion reactions.

We study the behaviour of steady-state solutions of a two-component flame filament system subject to chaotic mixing. This system exhibits a saddle-node bifurcation at a critical Damköhler number. We analyze the system through a one-dimensional phenomenological lamellar model. We present a nonperturbative technique, which allows us to describe the behaviour of the reduced lamellar model near the saddle node bifurcation. The influence of the Lewis number on the solution behaviour is investigated. We present a simple empirical formula for the wave speed valid for large Damköhler and large Lewis numbers. This formula allows us to describe the solution far away from the bifurcation. Numerical simulations show good agreement with the results.

Emergence of coupling-induced oscillations and broken symmetries in heterogeneously driven nonlinear reaction networks.

Many natural systems including the brain comprise coupled elements that are stimulated non-uniformly. In this paper we show that heterogeneously driven networks of excitatory-inhibitory units exhibit a diverse range of collective phenomena, including the appearance of spontaneous oscillations upon coupling quiescent elements. On varying the coupling strength a previously unreported transition is seen wherein the symmetries of the synchronization patterns in the stimulated and unstimulated groups undergo mutual exchange. The system also exhibits coexisting chaotic and non-chaotic attractors - a result that may be of interest in connection to earlier reports of varying degrees of chaoticity in the brain.

Phototaxis as a Collective Phenomenon in Cyanobacterial Colonies.

Cyanobacteria are a diverse group of photosynthetic bacteria that exhibit phototaxis, or motion in response to light. Cyanobacteria such as Synechocystis sp. secrete a mixture of complex polysaccharides that facilitate cell motion, while their type 4 pili allow them to physically attach to each other. Even though cells can respond individually to light, colonies are observed to move collectively towards the light source in dense finger-like projections. We present an agent-based model for cyanobacterial phototaxis that accounts for slime deposition as well as for direct physical links between bacteria, mediated through their type 4 pili. We reproduce the experimentally observed aggregation of cells at the colony boundary as a precursor to finger formation. Our model also describes the changes in colony morphology that occur when the location of the light source is abruptly changed. We find that the overall motion of cells toward light remains relatively unimpaired even if a fraction of them do not sense light, allowing heterogeneous populations to continue to mount a robust collective response to stimuli. Our work suggests that in addition to bio-chemical signalling via diffusible molecules in the context of bacterial quorum-sensing, short-ranged physical interactions may also contribute to collective effects in bacterial motility.