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.     

From intracellular signaling to population oscillations: bridging size- and time-scales in collective behavior

Collective behavior in cellular populations is coordinated by biochemical signaling networks within individual cells. Connecting the dynamics of these intracellular networks to the population phenomena they control poses a considerable challenge because of network complexity and our limited knowledge of kinetic parameters. However, from physical systems, we know that behavioral changes in the individual constituents of a collectively behaving system occur in a limited number of well-defined classes, and these can be described using simple models. Here, we apply such an approach to the emergence of collective oscillations in cellular populations of the social amoeba Dictyostelium discoideum. Through direct tests of our model with quantitative in vivo measurements of single-cell and population signaling dynamics, we show how a simple model can effectively describe a complex molecular signaling network at multiple size and temporal scales. The model predicts novel noise-driven single-cell and population-level signaling phenomena that we then experimentally observe. Our results suggest that like physical systems, collective behavior in biology may be universal and described using simple mathematical models.     

A Dynamic Analysis of IRS-PKR Signaling in Liver Cells: A Discrete Modeling Approach

A major challenge in systems biology is to develop a detailed dynamic understanding of the functions and behaviors in a particular cellular system, which depends on the elements and their inter-relationships in a specific network. Computational modeling plays an integral part in the study of network dynamics and uncovering the underlying mechanisms. Here we proposed a systematic approach that incorporates discrete dynamic modeling and experimental data to reconstruct a phenotype-specific network of cell signaling. A dynamic analysis of the insulin signaling system in liver cells provides a proof-of-concept application of the proposed methodology. Our group recently identified that double-stranded RNA-dependent protein kinase (PKR) plays an important role in the insulin signaling network. The dynamic behavior of the insulin signaling network is tuned by a variety of feedback pathways, many of which have the potential to cross talk with PKR. Given the complexity of insulin signaling, it is inefficient to experimentally test all possible interactions in the network to determine which pathways are functioning in our cell system. Our discrete dynamic model provides an in silico model framework that integrates potential interactions and assesses the contributions of the various interactions on the dynamic behavior of the signaling network. Simulations with the model generated testable hypothesis on the response of the network upon perturbation, which were experimentally evaluated to identify the pathways that function in our particular liver cell system. The modeling in combination with the experimental results enhanced our understanding of the insulin signaling dynamics and aided in generating a context-specific signaling network.     

Quantifying and tracing information cascades in swarms

We propose a novel, information-theoretic, characterisation of cascades within the spatiotemporal dynamics of swarms, explicitly measuring the extent of collective communications. This is complemented by dynamic tracing of collective memory, as another element of distributed computation, which represents capacity for swarm coherence. The approach deals with both global and local information dynamics, ultimately discovering diverse ways in which an individual's spatial position is related to its information processing role. It also allows us to contrast cascades that propagate conflicting information with waves of coordinated motion. Most importantly, our simulation experiments provide the first direct information-theoretic evidence (verified in a simulation setting) for the long-held conjecture that the information cascades occur in waves rippling through the swarm. Our experiments also exemplify how features of swarm dynamics, such as cascades' wavefronts, can be filtered and predicted. We observed that maximal information transfer tends to follow the stage with maximal collective memory, and principles like this may be generalised in wider biological and social contexts.     

Minimal Network Topologies for Signal Processing during Collective Cell Chemotaxis

Cell-cell communication plays an important role in collective cell migration. However, it remains unclear how cells in a group cooperatively process external signals to determine the group’s direction of motion. Although the topology of signaling pathways is vitally important in single-cell chemotaxis, the signaling topology for collective chemotaxis has not been systematically studied. Here, we combine mathematical analysis and simulations to find minimal network topologies for multicellular signal processing in collective chemotaxis. We focus on border cell cluster chemotaxis in the Drosophila egg chamber, in which responses to several experimental perturbations of the signaling network are known. Our minimal signaling network includes only four elements: a chemoattractant, the protein Rac (indicating cell activation), cell protrusion, and a hypothesized global factor responsible for cell-cell interaction. Experimental data on cell protrusion statistics allows us to systematically narrow the number of possible topologies from more than 40,000,000 to only six minimal topologies with six interactions between the four elements. This analysis does not require a specific functional form of the interactions, and only qualitative features are needed; it is thus robust to many modeling choices. Simulations of a stochastic biochemical model of border cell chemotaxis show that the qualitative selection procedure accurately determines which topologies are consistent with the experiment. We fit our model for all six proposed topologies; each produces results that are consistent with all experimentally available data. Finally, we suggest experiments to further discriminate possible pathway topologies.     

Recording Multicellular Behavior in Myxococcus xanthus Biofilms using Time-lapse Microcinematography

A swarm of the δ-proteobacterium Myxococcus xanthus contains millions of cells that act as a collective, coordinating movement through a series of signals to create complex, dynamic patterns as a response to environmental cues. These patterns are self-organizing and emergent; they cannot be predicted by observing the behavior of the individual cells. Using a time-lapse microcinematography tracking assay, we identified a distinct emergent pattern in M. xanthus called chemotaxis, defined as the directed movement of a swarm up a nutrient gradient toward its source ¹.In order to efficiently characterize chemotaxis via time-lapse microcinematography, we developed a highly modifiable plate complex (Figure 1) and constructed a cluster of 8 microscopes (Figure 2), each capable of capturing time-lapse videos. The assay is rigorous enough to allow consistent replication of quantifiable data, and the resulting videos allow us to observe and track subtle changes in swarm behavior. Once captured, the videos are transferred to an analysis/storage computer with enough memory to process and store thousands of videos. The flexibility of this setup has proven useful to several members of the M. xanthus community.Download video file.^{(133M, mp4)}     

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.     

Oscillations and synchrony in a cortical neural network

In this paper, the oscillations and synchronization status of two different network connectivity patterns based on Izhikevich model are studied. One of the connectivity patterns is a randomly connected neuronal network, the other one is a small-world neuronal network. This Izhikevich model is a simple model which can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Detailed investigations reveal that by varying some key parameters, such as the connection weights of neurons, the external current injection, the noise of intensity and the neuron number, this neuronal network will exhibit various collective behaviors in randomly coupled neuronal network. In addition, we show that by changing the number of nearest neighbor and connection probability in small-world topology can also affect the collective dynamics of neuronal activity. These results may be instructive in understanding the collective dynamics of mammalian cortex.     

Cell-cell signaling interactions coordinate multiple cell behaviors that drive morphogenesis of the lateral line

The zebrafish sensory lateral line system has emerged as a powerful model for the mechanistic study of collective cell migration and morphogenesis. Recent work has uncovered the details of a signaling network involving the Wnt/β-catenin, Fgf and Delta-Notch pathways that patterns the migrating lateral line primordium into distinct regions. Cells within these regions exhibit different fundamental behaviors that together orchestrate normal lateral line morphogenesis. In this review, we summarize the signaling network that patterns the migrating lateral line primordium and describe how this patterning coordinates crucial morphogenic cell behaviors.Key words: lateral line, Wnt signaling, Fgf signaling, collective migration, morphogenesis     

Information flow principles for plasticity in foraging robot swarms

An important characteristic of a robot swarm that must operate in the real world is the ability to cope with changeable environments by exhibiting behavioural plasticity at the collective level. For example, a swarm of foraging robots should be able to repeatedly reorganise in order to exploit resource deposits that appear intermittently in different locations throughout their environment. In this paper, we report on simulation experiments with homogeneous foraging robot teams and show that analysing swarm behaviour in terms of information flow can help us to identify whether a particular behavioural strategy is likely to exhibit useful swarm plasticity in response to dynamic environments. While it is beneficial to maximise the rate at which robots share information when they make collective decisions in a static environment, plastic swarm behaviour in changeable environments requires regulated information transfer in order to achieve a balance between the exploitation of existing information and exploration leading to acquisition of new information. We give examples of how information flow analysis can help designers to decide on robot control strategies with relevance to a number of applications explored in the swarm robotics literature.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

Co-evolutionary Dynamics of Collective Action with Signaling for a Quorum

Collective signaling for a quorum is found in a wide range of organisms that face collective action problems whose successful solution requires the participation of some quorum of the individuals present. These range from humans, to social insects, to bacteria. The mechanisms involved, the quorum required, and the size of the group may vary. Here we address the general question of the evolution of collective signaling at a high level of abstraction. We investigate the evolutionary dynamics of a population engaging in a signaling N-person game theoretic model. Parameter settings allow for loners and cheaters, and for costly or costless signals. We find a rich dynamics, showing how natural selection, operating on a population of individuals endowed with the simplest strategies, is able to evolve a costly signaling system that allows individuals to respond appropriately to different states of Nature. Signaling robustly promotes cooperative collective action, in particular when coordinated action is most needed and difficult to achieve. Two different signaling systems may emerge depending on Nature’s most prevalent states.     

Design of ant-inspired stochastic control policies for collective transport by robotic swarms

In this paper, we present an approach to designing decentralized robot control policies that mimic certain microscopic and macroscopic behaviors of ants performing collective transport tasks. In prior work, we used a stochastic hybrid system model to characterize the observed team dynamics of ant group retrieval of a rigid load. We have also used macroscopic population dynamic models to design enzyme-inspired stochastic control policies that allocate a robotic swarm around multiple boundaries in a way that is robust to environmental variations. Here, we build on this prior work to synthesize stochastic robot attachment–detachment policies for tasks in which a robotic swarm must achieve non-uniform spatial distributions around multiple loads and transport them at a constant velocity. Three methods are presented for designing robot control policies that replicate the steady-state distributions, transient dynamics, and fluxes between states that we have observed in ant populations during group retrieval. The equilibrium population matching method can be used to achieve a desired transport team composition as quickly as possible; the transient matching method can control the transient population dynamics of the team while driving it to the desired composition; and the rate matching method regulates the rates at which robots join and leave a load during transport. We validate our model predictions in an agent-based simulation, verify that each controller design method produces successful transport of a load at a regulated velocity, and compare the advantages and disadvantages of each method.     

Dynamics and Energetics: A Consensus Analysis of the Impact of Calcium on EF-CaM Protein Complex

We have studied the relationship between dynamical correlations and energetic contributions in an attempt to model the transmission of information inside protein-protein complexes. The complex formed between the edema factor (EF) of Bacillus anthracis and calmodulin (CaM) was taken as an example, as the formation and stability of the complex depend on the calcium complexation level. The effect of calcium through EF-CaM residue network has been investigated with various approaches: 1), the elastic network model; 2), the local feature analysis; 3), the generalized correlations; and 4), the energetic dependency maps (EDMs), on 15-ns molecular dynamics simulations of the complex loaded with 0, 2, or 4 Ca²⁺ ions. The elastic network model correctly describes the basic architecture of the complex but is poorly sensitive to the level of calcium compared to the other methods. The local feature analysis allows us to characterize the local dynamics of the complex and the propagation of the calcium signal through CaM. The analyses of global dynamics and energetics—through generalized correlations and EDMs—provide a comprehensive picture of EF-CaM architecture and can be unified by using the concept of residue network connectedness. A medium connectedness, defined as the ability of each residue to communicate with all remaining parts of the complex, is observed for the 2Ca²⁺ level, which was experimentally identified as the most stable form of EF-CaM. The hierarchy of relative stabilities given by the EDMs sheds a new light on the EF-CaM interaction mechanism described experimentally and supports an organization of the complex architecture centered around nucleation points.     

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain’s division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains—but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.     

Formation,collective motion,and merging of macroscopic bacterial aggregates

Chemotactic bacteria form emergent spatial patterns of variable cell density within cultures that are initially spatially uniform. These patterns are the result of chemical gradients that are created from the directed movement and metabolic activity of billions of cells. A recent study on pattern formation in wild bacterial isolates has revealed unique collective behaviors of the bacteria Enterobacter cloacae. As in other bacterial species, Enterobacter cloacae form macroscopic aggregates. Once formed, these bacterial clusters can migrate several millimeters, sometimes resulting in the merging of two or more clusters. To better understand these phenomena, we examine the formation and dynamics of thousands of bacterial clusters that form within a 22 cm square culture dish filled with soft agar over two days. At the macroscale, the aggregates display spatial order at short length scales, and the migration of cell clusters is superdiffusive, with a merging acceleration that is correlated with aggregate size. At the microscale, aggregates are composed of immotile cells surrounded by low density regions of motile cells. The collective movement of the aggregates is the result of an asymmetric flux of bacteria at the boundary. An agent-based model is developed to examine how these phenomena are the result of both chemotactic movement and a change in motility at high cell density. These results identify and characterize a new mechanism for collective bacterial motility driven by a transient, density-dependent change in motility.     

Swarming and Aggregation in the Parasitic Diplomonad Flagellate Spironucleus vortens

Pathogenicity, evolutionary history, and unusual cell organization of diplomonads are well known, particularly for Giardia and Spironucleus; however, behavior of these aerotolerant anaerobes is largely unknown. Addressing this deficit, we studied behavior of the piscine diplomonad Spironucleus vortens (ATCC 50386) in in vitro culture. Spironucleus vortens trophozoites from Angelfish, Pterophyllum scalare, were maintained axenically in modified liver digest, yeast extract, and iron (LYI) medium, at 22 °C in the dark, and subcultured weekly. Cultures were monitored every 1–2 d, by removing an aliquot, and loading cells into a hemocytometer chamber, or onto a regular microscope slide. We observed three distinct swimming behaviors: (i) spontaneous formation of swarms, reaching 200 μm in diameter, persisting for up to several min in situ, (ii) directional movement of the swarm, via collective motility, and (iii) independent swimming of trophozoites to form a band (aggregation), presumably at the location of optimal environmental conditions. These behaviors have not previously been reported in Spironucleus. The observation that flagellate motility can change, from individual self‐propulsion to complex collective swarming motility, prompts us to advocate S. vortens as a new model for study of group behavioral dynamics, complementing emerging studies of collective swimming in flagellated bacteria.     

Nodal signaling regulates endodermal cell motility and actin dynamics via Rac1 and Prex1

Embryo morphogenesis is driven by dynamic cell behaviors, including migration, that are coordinated with fate specification and differentiation, but how such coordination is achieved remains poorly understood. During zebrafish gastrulation, endodermal cells sequentially exhibit first random, nonpersistent migration followed by oriented, persistent migration and finally collective migration. Using a novel transgenic line that labels the endodermal actin cytoskeleton, we found that these stage-dependent changes in migratory behavior correlated with changes in actin dynamics. The dynamic actin and random motility exhibited during early gastrulation were dependent on both Nodal and Rac1 signaling. We further identified the Rac-specific guanine nucleotide exchange factor Prex1 as a Nodal target and showed that it mediated Nodal-dependent random motility. Reducing Rac1 activity in endodermal cells caused them to bypass the random migration phase and aberrantly contribute to mesodermal tissues. Together, our results reveal a novel role for Nodal signaling in regulating actin dynamics and migration behavior, which are crucial for endodermal morphogenesis and cell fate decisions.     

Modeling the spread of invasive species using dynamic network models

Spread dynamics of biological invasions are influenced by both the availability and spatial arrangement of suitable habitat. As such, invasive spread can be considered to occur across a network of nodes, representing patches of suitable habitat, with linkages representing the potential for movement between habitat patches. While static network models can provide valuable insight into the potential framework of nodes and linkages across which spread could occur, they offer little information on the actual spatiotemporal dynamics of range expansion processes. Here, we explore the development and application of dynamic network models (DNMs) to model the spread of invasive species. DNMs accommodate temporal dynamics in the utilization of nodes and the connections between them and can flexibly perform simulations at the spatial scales of observational data. As case studies, we develop DNMs to simulate the spread of a generalist forest pathogen and the hemlock woolly adelgid (Adelges tsugae Annand). We highlight the utility of DNMs for identifying habitat patches that contribute most to spread across the landscape and for visualizing emergent spread dynamics. While currently underutilized in ecology as compared to static network models, DNMs are potentially applicable to numerous research and management questions relevant to biological invasions and the more general phenomena of range expansion.