Using multilayer network analysis to explore the temporal dynamics of collective behavior

Social organisms often show collective behaviors such as group foraging or movement.Collective behaviors can emerge from interactions between group members and may depend on the behavior of key individuals.When social interactions change over time,collective behaviors may change because these behaviors emerge from interactions among individuals.Despite the importance of,and growing interest in,the temporal dynamics of social interactions,it is not clear how to quantify changes in interactions over time or measure their stability.Furthermore,the temporal scale at which we should observe changes in social networks to detect biologically meaningful changes is not always apparent.Here we use multilayer network analysis to quantify temporal dynamics of social networks of the social spider Stegodyphus dumicola and determine how these dynamics relate to individual and group behaviors.We found that social interactions changed over time at a constant rate.Variation in both network structure and the identity of a keystone individual was not related to the mean or variance of the collective prey attack speed.Individuals that maintained a large and stable number of connections,despite changes in network structure,were the boldest individuals in the group.Therefore,social interactions and boldness are linked across time,but group collective behavior is not influenced by the stability of the social network.Our work demonstrates that dynamic social networks can be modeled in a multilayer framework.This approach may reveal biologically important temporal changes to social structure in other systems.     

Multi-cellular logistics of collective cell migration

During development, the formation of biological networks (such as organs and neuronal networks) is controlled by multicellular transportation phenomena based on cell migration. In multi-cellular systems, cellular locomotion is restricted by physical interactions with other cells in a crowded space, similar to passengers pushing others out of their way on a packed train. The motion of individual cells is intrinsically stochastic and may be viewed as a type of random walk. However, this walk takes place in a noisy environment because the cell interacts with its randomly moving neighbors. Despite this randomness and complexity, development is highly orchestrated and precisely regulated, following genetic (and even epigenetic) blueprints. Although individual cell migration has long been studied, the manner in which stochasticity affects multi-cellular transportation within the precisely controlled process of development remains largely unknown. To explore the general principles underlying multicellular migration, we focus on the migration of neural crest cells, which migrate collectively and form streams. We introduce a mechanical model of multi-cellular migration. Simulations based on the model show that the migration mode depends on the relative strengths of the noise from migratory and non-migratory cells. Strong noise from migratory cells and weak noise from surrounding cells causes "collective migration," whereas strong noise from non-migratory cells causes "dispersive migration." Moreover, our theoretical analyses reveal that migratory cells attract each other over long distances, even without direct mechanical contacts. This effective interaction depends on the stochasticity of the migratory and non-migratory cells. On the basis of these findings, we propose that stochastic behavior at the single-cell level works effectively and precisely to achieve collective migration in multi-cellular systems.     

Network Events on Multiple Space and Time Scales in Cultured Neural Networks and in a Stochastic Rate Model

Cortical networks, in-vitro as well as in-vivo, can spontaneously generate a variety of collective dynamical events such as network spikes, UP and DOWN states, global oscillations, and avalanches. Though each of them has been variously recognized in previous works as expression of the excitability of the cortical tissue and the associated nonlinear dynamics, a unified picture of the determinant factors (dynamical and architectural) is desirable and not yet available. Progress has also been partially hindered by the use of a variety of statistical measures to define the network events of interest. We propose here a common probabilistic definition of network events that, applied to the firing activity of cultured neural networks, highlights the co-occurrence of network spikes, power-law distributed avalanches, and exponentially distributed ‘quasi-orbits’, which offer a third type of collective behavior. A rate model, including synaptic excitation and inhibition with no imposed topology, synaptic short-term depression, and finite-size noise, accounts for all these different, coexisting phenomena. We find that their emergence is largely regulated by the proximity to an oscillatory instability of the dynamics, where the non-linear excitable behavior leads to a self-amplification of activity fluctuations over a wide range of scales in space and time. In this sense, the cultured network dynamics is compatible with an excitation-inhibition balance corresponding to a slightly sub-critical regime. Finally, we propose and test a method to infer the characteristic time of the fatigue process, from the observed time course of the network’s firing rate. Unlike the model, possessing a single fatigue mechanism, the cultured network appears to show multiple time scales, signalling the possible coexistence of different fatigue mechanisms.     

Connecting the dots in ethology: applying network theory to understand neural and animal collectives

A major goal shared by neuroscience and collective behavior is to understand how dynamic interactions between individual elements give rise to behaviors in populations of neurons and animals, respectively. This goal has recently become within reach, thanks to techniques providing access to the connectivity and activity of neuronal ensembles as well as to behaviors among animal collectives. The next challenge using these datasets is to unravel network mechanisms generating population behaviors. This is aided by network theory, a field that studies structure–function relationships in interconnected systems. Here we review studies that have taken a network view on modern datasets to provide unique insights into individual and collective animal behaviors. Specifically, we focus on how analyzing signal propagation, controllability, symmetry, and geometry of networks can tame the complexity of collective system dynamics. These studies illustrate the potential of network theory to accelerate our understanding of behavior across ethological scales.     

Bursting Reverberation as a Multiscale Neuronal Network Process Driven by Synaptic Depression-Facilitation

Neuronal networks can generate complex patterns of activity that depend on membrane properties of individual neurons as well as on functional synapses. To decipher the impact of synaptic properties and connectivity on neuronal network behavior, we investigate the responses of neuronal ensembles from small (5–30 cells in a restricted sphere) and large (acute hippocampal slice) networks to single electrical stimulation: in both cases, a single stimulus generated a synchronous long-lasting bursting activity. While an initial spike triggered a reverberating network activity that lasted 2–5 seconds for small networks, we found here that it lasted only up to 300 milliseconds in slices. To explain this phenomena present at different scales, we generalize the depression-facilitation model and extracted the network time constants. The model predicts that the reverberation time has a bell shaped relation with the synaptic density, revealing that the bursting time cannot exceed a maximum value. Furthermore, before reaching its maximum, the reverberation time increases sub-linearly with the synaptic density of the network. We conclude that synaptic dynamics and connectivity shape the mean burst duration, a property present at various scales of the networks. Thus bursting reverberation is a property of sufficiently connected neural networks, and can be generated by collective depression and facilitation of underlying functional synapses.     

Resilience and Controllability of Dynamic Collective Behaviors

The network paradigm is used to gain insight into the structural root causes of the resilience of consensus in dynamic collective behaviors, and to analyze the controllability of the swarm dynamics. Here we devise the dynamic signaling network which is the information transfer channel underpinning the swarm dynamics of the directed interagent connectivity based on a topological neighborhood of interactions. The study of the connectedness of the swarm signaling network reveals the profound relationship between group size and number of interacting neighbors, which is found to be in good agreement with field observations on flock of starlings [Ballerini et al. (2008) Proc. Natl. Acad. Sci. USA, 105: 1232]. Using a dynamical model, we generate dynamic collective behaviors enabling us to uncover that the swarm signaling network is a homogeneous clustered small-world network, thus facilitating emergent outcomes if connectedness is maintained. Resilience of the emergent consensus is tested by introducing exogenous environmental noise, which ultimately stresses how deeply intertwined are the swarm dynamics in the physical and network spaces. The availability of the signaling network allows us to analytically establish for the first time the number of driver agents necessary to fully control the swarm dynamics.     

Motility Enhancement through Surface Modification Is Sufficient for Cyanobacterial Community Organization during Phototaxis

The emergent behaviors of communities of genotypically identical cells cannot be easily predicted from the behaviors of individual cells. In many cases, it is thought that direct cell-cell communication plays a critical role in the transition from individual to community behaviors. In the unicellular photosynthetic cyanobacterium Synechocystis sp. PCC 6803, individual cells exhibit light-directed motility (“phototaxis”) over surfaces, resulting in the emergence of dynamic spatial organization of multicellular communities. To probe this striking community behavior, we carried out time-lapse video microscopy coupled with quantitative analysis of single-cell dynamics under varying light conditions. These analyses suggest that cells secrete an extracellular substance that modifies the physical properties of the substrate, leading to enhanced motility and the ability for groups of cells to passively guide one another. We developed a biophysical model that demonstrates that this form of indirect, surface-based communication is sufficient to create distinct motile groups whose shape, velocity, and dynamics qualitatively match our experimental observations, even in the absence of direct cellular interactions or changes in single-cell behavior. Our computational analysis of the predicted community behavior, across a matrix of cellular concentrations and light biases, demonstrates that spatial patterning follows robust scaling laws and provides a useful resource for the generation of testable hypotheses regarding phototactic behavior. In addition, we predict that degradation of the surface modification may account for the secondary patterns occasionally observed after the initial formation of a community structure. Taken together, our modeling and experiments provide a framework to show that the emergent spatial organization of phototactic communities requires modification of the substrate, and this form of surface-based communication could provide insight into the behavior of a wide array of biological communities.     

Maintenance of Motility Bias during Cyanobacterial Phototaxis

Signal transduction in bacteria is complex, ranging across scales from molecular signal detectors and effectors to cellular and community responses to stimuli. The unicellular, photosynthetic cyanobacterium Synechocystis sp. PCC6803 transduces a light stimulus into directional movement known as phototaxis. This response occurs via a biased random walk toward or away from a directional light source, which is sensed by intracellular photoreceptors and mediated by Type IV pili. It is unknown how quickly cells can respond to changes in the presence or directionality of light, or how photoreceptors affect single-cell motility behavior. In this study, we use time-lapse microscopy coupled with quantitative single-cell tracking to investigate the timescale of the cellular response to various light conditions and to characterize the contribution of the photoreceptor TaxD1 (PixJ1) to phototaxis. We first demonstrate that a community of cells exhibits both spatial and population heterogeneity in its phototactic response. We then show that individual cells respond within minutes to changes in light conditions, and that movement directionality is conferred only by the current light directionality, rather than by a long-term memory of previous conditions. Our measurements indicate that motility bias likely results from the polarization of pilus activity, yielding variable levels of movement in different directions. Experiments with a photoreceptor (taxD1) mutant suggest a supplementary role of TaxD1 in enhancing movement directionality, in addition to its previously identified role in promoting positive phototaxis. Motivated by the behavior of the taxD1 mutant, we demonstrate using a reaction-diffusion model that diffusion anisotropy is sufficient to produce the observed changes in the pattern of collective motility. Taken together, our results establish that single-cell tracking can be used to determine the factors that affect motility bias, which can then be coupled with biophysical simulations to connect changes in motility behaviors at the cellular scale with group dynamics.     

Nonparametric simulation of signal transduction networks with semi-synchronized update

Simulating signal transduction in cellular signaling networks provides predictions of network dynamics by quantifying the changes in concentration and activity-level of the individual proteins. Since numerical values of kinetic parameters might be difficult to obtain, it is imperative to develop non-parametric approaches that combine the connectivity of a network with the response of individual proteins to signals which travel through the network. The activity levels of signaling proteins computed through existing non-parametric modeling tools do not show significant correlations with the observed values in experimental results. In this work we developed a non-parametric computational framework to describe the profile of the evolving process and the time course of the proportion of active form of molecules in the signal transduction networks. The model is also capable of incorporating perturbations. The model was validated on four signaling networks showing that it can effectively uncover the activity levels and trends of response during signal transduction process.     

Nutrient complexity triggers transitions between solitary and colonial growth in bacterial populations

Microbial populations often experience fluctuations in nutrient complexity in their natural environment such as between high molecular weight polysaccharides and simple monosaccharides. However, it is unclear if cells can adopt growth behaviors that allow individuals to optimally respond to differences in nutrient complexity. Here, we directly control nutrient complexity and use quantitative single-cell analysis to study the growth dynamics of individuals within populations of the aquatic bacterium Caulobacter crescentus. We show that cells form clonal microcolonies when growing on the polysaccharide xylan, which is abundant in nature and degraded using extracellular cell-linked enzymes; and disperse to solitary growth modes when the corresponding monosaccharide xylose becomes available or nutrients are exhausted. We find that the cellular density required to achieve maximal growth rates is four-fold higher on xylan than on xylose, indicating that aggregating is advantageous on polysaccharides. When collectives on xylan are transitioned to xylose, cells start dispersing, indicating that colony formation is no longer beneficial and solitary behaviors might serve to reduce intercellular competition. Our study demonstrates that cells can dynamically tune their behaviors when nutrient complexity fluctuates, elucidates the quantitative advantages of distinct growth behaviors for individual cells and indicates why collective growth modes are prevalent in microbial populations.Subject terms: Microbial ecology, Water microbiology, Biofilms     

From single-cell genetic architecture to cell population dynamics: quantitatively decomposing the effects of different population heterogeneity sources for a genetic network with positive feedback architecture

Phenotypic cell-to-cell variability or cell population heterogeneity originates from two fundamentally different sources: unequal partitioning of cellular material at cell division and stochastic fluctuations associated with intracellular reactions. We developed a mathematical and computational framework that can quantitatively isolate both heterogeneity sources and applied it to a genetic network with positive feedback architecture. The framework consists of three vastly different mathematical formulations: a), a continuum model, which completely neglects population heterogeneity; b), a deterministic cell population balance model, which accounts for population heterogeneity originating only from unequal partitioning at cell division; and c), a fully stochastic model accommodating both sources of population heterogeneity. The framework enables the quantitative decomposition of the effects of the different population heterogeneity sources on system behavior. Our results indicate the importance of cell population heterogeneity in accurately predicting even average population properties. Moreover, we find that unequal partitioning at cell division and sharp division rates shrink the region of the parameter space where the population exhibits bistable behavior, a characteristic feature of networks with positive feedback architecture. In addition, intrinsic noise at the single-cell level due to slow operator fluctuations and small numbers of molecules further contributes toward the shrinkage of the bistability regime at the cell population level. Finally, the effect of intrinsic noise at the cell population level was found to be markedly different than at the single-cell level, emphasizing the importance of simulating entire cell populations and not just individual cells to understand the complex interplay between single-cell genetic architecture and behavior at the cell population level.     

AgentCell: a digital single-cell assay for bacterial chemotaxis

MOTIVATION: In recent years, single-cell biology has focused on the relationship between the stochastic nature of molecular interactions and variability of cellular behavior. To describe this relationship, it is necessary to develop new computational approaches at the single-cell level. RESULTS: We have developed AgentCell, a model using agent-based technology to study the relationship between stochastic intracellular processes and behavior of individual cells. As a test-bed for our approach we use bacterial chemotaxis, one of the best characterized biological systems. In this model, each bacterium is an agent equipped with its own chemotaxis network, motors and flagella. Swimming cells are free to move in a 3D environment. Digital chemotaxis assays reproduce experimental data obtained from both single cells and bacterial populations.     

Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.     

The role of single-cell analyses in understanding cell lineage commitment

The study of cell lineage commitment is critical for improving our understanding of tissue development and regeneration, and for realizing stem cell-based therapies and engineered tissue replacements. Recently, the discovery of an unanticipated degree of variability in fundamental biological processes, including divergent responses of genetically identical cells to various stimuli, has provided mechanistic insight into cellular decision making and the collective behavior of cell populations. Therefore, the study of lineage commitment with single-cell resolution could provide greater knowledge of cellular differentiation mechanisms and the influence of noise on cellular processes. This will require the adoption of new technologies for single-cell analysis as traditional methods typically measure average values of bulk population behavior. This review discusses the recent developments in methods for analyzing the behavior of individual cells, and how these approaches are leading to a deeper understanding and better control of cellular decision making.     

Using Synchronous Boolean Networks to Model Several Phenomena of Collective Behavior

In this paper, we propose an approach for modeling and analysis of a number of phenomena of collective behavior. By collectives we mean multi-agent systems that transition from one state to another at discrete moments of time. The behavior of a member of a collective (agent) is called conforming if the opinion of this agent at current time moment conforms to the opinion of some other agents at the previous time moment. We presume that at each moment of time every agent makes a decision by choosing from the set (where 1-decision corresponds to action and 0-decision corresponds to inaction). In our approach we model collective behavior with synchronous Boolean networks. We presume that in a network there can be agents that act at every moment of time. Such agents are called instigators. Also there can be agents that never act. Such agents are called loyalists. Agents that are neither instigators nor loyalists are called simple agents. We study two combinatorial problems. The first problem is to find a disposition of instigators that in several time moments transforms a network from a state where the majority of simple agents are inactive to a state with the majority of active agents. The second problem is to find a disposition of loyalists that returns the network to a state with the majority of inactive agents. Similar problems are studied for networks in which simple agents demonstrate the contrary to conforming behavior that we call anticonforming. We obtained several theoretical results regarding the behavior of collectives of agents with conforming or anticonforming behavior. In computational experiments we solved the described problems for randomly generated networks with several hundred vertices. We reduced corresponding combinatorial problems to the Boolean satisfiability problem (SAT) and used modern SAT solvers to solve the instances obtained.