Dynamical maximum entropy approach to flocking

Inverse modeling of time-delayed interactions via the dynamic-entropy formalism

Although instantaneous interactions are unphysical, a large variety of maximum entropy statistical inference methods match the model-inferred and the empirically measured equal-time correlation functions. Focusing on collective motion of active units, this constraint is reasonable when the interaction timescale is much faster than that of the interacting units, as in starling flocks, yet it fails in a number of counterexamples, as in leukocyte coordination (where signaling proteins diffuse among two cells). Here, we relax this assumption and develop a path integral approach to maximum-entropy framework, which includes delay in signaling. Our method is able to infer the strength of couplings and fields, but also the time required by the couplings to completely transfer information among the units. We demonstrate the validity of our approach providing excellent results on synthetic datasets of non-Markovian trajectories generated by the Heisenberg-Kuramoto and Vicsek models equipped with delayed interactions. As a proof of concept, we also apply the method to experiments on dendritic migration, where matching equal-time correlations results in a significant information loss.

Detecting and characterizing phase transitions in active matter using entropy

A major challenge in the study of active matter lies in quantitative characterization of phases and transitions between them. We show how the entropy of a collection of active objects can be used to classify regimes and spatial patterns in their collective behavior. Specifically, we estimate the contributions to the total entropy from correlations between the degrees of freedom of position and orientation. This analysis pin-points the flocking transition in the Vicsek model while clarifying the physical mechanism behind the transition. When applied to experiments on swarming Bacillus subtilis with different cell aspect ratios and overall bacterial area fractions, the entropy analysis reveals a rich phase diagram with transitions between qualitatively different swarm statistics. We discuss physical and biological implications of these findings.

Scale-free correlations and potential criticality in weakly ordered populations of brain cancer cells

Collective behavior spans several orders of magnitude of biological organization, from cell colonies to flocks of birds. We used time-resolved tracking of individual glioblastoma cells to investigate collective motion in an ex vivo model of glioblastoma. At the population level, glioblastoma cells display weakly polarized motion in the (directional) velocities of single cells. Unexpectedly, fluctuations in velocities are correlated over distances many times the size of a cell. Correlation lengths scale linearly with the maximum end-to-end length of the population, indicating that they are scale-free and lack a characteristic decay scale other than the size of the system. Last, a data-driven maximum entropy model captures statistical features of the experimental data with only two free parameters: the effective length scale (nc) and strength (J) of local pairwise interactions between tumor cells. These results show that glioblastoma assemblies exhibit scale-free correlations in the absence of polarization, suggesting that they may be poised near a critical point.

Resolving entropy contributions in nonequilibrium transitions

We derive a functional for the entropy contributed by any microscopic degrees of freedom as arising from their measurable pair correlations. Applicable both in and out of equilibrium, this functional yields the maximum entropy which a system can have given a certain correlation function. When applied to different correlations, the method allows us to identify the degrees of freedom governing a certain physical regime, thus capturing and characterizing dynamic transitions. The formalism applies also to systems whose translational invariance is broken by external forces and whose number of particles may vary. We apply it to experimental results for jammed bidisperse emulsions, capturing the crossover of this nonequilibrium system from crystalline to disordered hyperuniform structures as a function of mixture composition. We discover that the cross-correlations between the positions and sizes of droplets in the emulsion play the central role in the formation of the disordered hyperuniform states. We discuss implications of the approach for entropy estimation out of equilibrium and for characterizing transitions in disordered systems.

Inference, Prediction, & Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes

Inferring models, predicting the future, and estimating the entropy rate of discrete-time, discrete-event processes is well-worn ground. However, a much broader class of discrete-event processes operates in continuous-time. Here, we provide new methods for inferring, predicting, and estimating them. The methods rely on an extension of Bayesian structural inference that takes advantage of neural network's universal approximation power. Based on experiments with complex synthetic data, the methods are competitive with the state-of-the-art for prediction and entropy-rate estimation.

Decomposing the Local Arrow of Time in Interacting Systems

We show that the evidence for a local arrow of time, which is equivalent to the entropy production in thermodynamic systems, can be decomposed. In a system with many degrees of freedom, there is a term that arises from the irreversible dynamics of the individual variables, and then a series of non-negative terms contributed by correlations among pairs, triplets, and higher-order combinations of variables. We illustrate this decomposition on simple models of noisy logical computations, and then apply it to the analysis of patterns of neural activity in the retina as it responds to complex dynamic visual scenes. We find that neural activity breaks detailed balance even when the visual inputs do not, and that this irreversibility arises primarily from interactions between pairs of neurons.

Emergence of local irreversibility in complex interacting systems

Living systems are fundamentally irreversible, breaking detailed balance and establishing an arrow of time. But how does the evident arrow of time for a whole system arise from the interactions among its multiple elements? We show that the local evidence for the arrow of time, which is the entropy production for thermodynamic systems, can be decomposed. First, it can be split into two components: an independent term reflecting the dynamics of individual elements and an interaction term driven by the dependencies among elements. Adapting tools from nonequilibrium physics, we further decompose the interaction term into contributions from pairs of elements, triplets, and higher-order terms. We illustrate our methods on models of cellular sensing and logical computations, as well as on patterns of neural activity in the retina as it responds to visual inputs. We find that neural activity can define the arrow of time even when the visual inputs do not, and that the dominant contribution to this breaking of detailed balance comes from interactions among pairs of neurons.

On the dimensionality of behavior

How do we characterize animal behavior? Psychophysics started with human behavior in the laboratory, and focused on simple contexts, such as the decision among just a few alternative actions in response to sensory inputs. In contrast, ethology focused on animal behavior in the natural environment, emphasizing that evolution selects potentially complex behaviors that are useful in specific contexts. New experimental methods now make it possible to monitor animal and human behaviors in vastly greater detail. This “physics of behavior” holds the promise of combining the psychophysicist’s quantitative approach with the ethologist’s appreciation of natural context. One question surrounding this growing body of data concerns the dimensionality of behavior. Here I try to give this concept a precise definition.

There is a growing effort in the “physics of behavior” that aims at complete quantitative characterization of animal movements under more complex, naturalistic conditions. One reaction to the resulting explosion of high-dimensional data is the search for low-dimensional structure. Here I try to define more clearly what we mean by the dimensionality of behavior, where observable behavior may consist of either continuous trajectories or sequences of discrete states. This discussion also serves to isolate situations in which the dimensionality of behavior is effectively infinite.

TOLOMEO, a Novel Machine Learning Algorithm to Measure Information and Order in Correlated Networks and Predict Their State

We present ToloMEo (TOpoLogical netwOrk Maximum Entropy Optimization), a program implemented in C and Python that exploits a maximum entropy algorithm to evaluate network topological information. ToloMEo can study any system defined on a connected network where nodes can assume N discrete values by approximating the system probability distribution with a Pottz Hamiltonian on a graph. The software computes entropy through a thermodynamic integration from the mean-field solution to the final distribution. The nature of the algorithm guarantees that the evaluated entropy is variational (i.e., it always provides an upper bound to the exact entropy). The program also performs machine learning, inferring the system’s behavior providing the probability of unknown states of the network. These features make our method very general and applicable to a broad class of problems. Here, we focus on three different cases of study: (i) an agent-based model of a minimal ecosystem defined on a square lattice, where we show how topological entropy captures a crossover between hunting behaviors; (ii) an example of image processing, where starting from discretized pictures of cell populations we extract information about the ordering and interactions between cell types and reconstruct the most likely positions of cells when data are missing; and (iii) an application to recurrent neural networks, in which we measure the information stored in different realizations of the Hopfield model, extending our method to describe dynamical out-of-equilibrium processes.

Learning the distribution of single-cell chromosome conformations in bacteria reveals emergent order across genomic scales

The order and variability of bacterial chromosome organization, contained within the distribution of chromosome conformations, are unclear. Here, we develop a fully data-driven maximum entropy approach to extract single-cell 3D chromosome conformations from Hi-C experiments on the model organism Caulobacter crescentus. The predictive power of our model is validated by independent experiments. We find that on large genomic scales, organizational features are predominantly present along the long cell axis: chromosomal loci exhibit striking long-ranged two-point axial correlations, indicating emergent order. This organization is associated with large genomic clusters we term Super Domains (SuDs), whose existence we support with super-resolution microscopy. On smaller genomic scales, our model reveals chromosome extensions that correlate with transcriptional and loop extrusion activity. Finally, we quantify the information contained in chromosome organization that may guide cellular processes. Our approach can be extended to other species, providing a general strategy to resolve variability in single-cell chromosomal organization.

The Maximum Caliber Variational Principle for Nonequilibria

Ever since Clausius in 1865 and Boltzmann in 1877, the concepts of entropy and of its maximization have been the foundations for predicting how material equilibria derive from microscopic properties. But, despite much work, there has been no equally satisfactory general variational principle for nonequilibrium situations. However, in 1980, a new avenue was opened by E.T. Jaynes and by Shore and Johnson. We review here maximum caliber, which is a maximum-entropy-like principle that can infer distributions of flows over pathways, given dynamical constraints. This approach is providing new insights, particularly into few-particle complex systems, such as gene circuits, protein conformational reaction coordinates, network traffic, bird flocking, cell motility, and neuronal firing.

Efficiency of one-dimensional active transport conditioned on motility

By conditioning a stochastic process on the value of an observable, one obtains a new stochastic process with different properties. We apply this idea in the context of active matter and condition interacting self-propelled particles on their individual motility. Using the effective process formalism from dynamical large deviations theory, we derive the interactions that actuate the imposed mobility against jamming interactions in two toy models-the totally asymmetric exclusion process and run-and-tumble particles, in the case of two or three particles. We provide a framework which takes into account the energy-consumption required for self-propulsion and address the question of how energy-efficient the emergent interactions are. Upon conditioning, run-and-tumble particles develop an alignment interaction and achieve a higher gain in efficiency than TASEP particles. A point of diminishing returns in efficiency is reached beyond a certain level of conditioning. With recourse to a general formula for the change in energy efficiency upon conditioning, we conclude that the most significant gains occur when there are large fluctuations in mobility to exploit. From a detailed comparison of the emergent effective interaction in a two- versus a three-body scenario, we discover evidence of a screening effect which suggests that conditioning can produce topological rather than metric interactions.

A Comparison of the Maximum Entropy Principle Across Biological Spatial Scales

Despite their differences, biological systems at different spatial scales tend to exhibit common organizational patterns. Unfortunately, these commonalities are often hard to grasp due to the highly specialized nature of modern science and the parcelled terminology employed by various scientific sub-disciplines. To explore these common organizational features, this paper provides a comparative study of diverse applications of the maximum entropy principle, which has found many uses at different biological spatial scales ranging from amino acids up to societies. By presenting these studies under a common approach and language, this paper aims to establish a unified view over these seemingly highly heterogeneous scenarios.

Screening by changes in stereotypical behavior during cell motility

Stereotyped behaviors are series of postures that show very little variability between repeats. They have been used to classify the dynamics of individuals, groups and species without reference to the lower-level mechanisms that drive them. Stereotypes are easily identified in animals due to strong constraints on the number, shape, and relative positions of anatomical features, such as limbs, that may be used as landmarks for posture identification. In contrast, the identification of stereotypes in single cells poses a significant challenge as the cell lacks these landmark features, and finding constraints on cell shape is a non-trivial task. Here, we use the maximum caliber variational method to build a minimal model of cell behavior during migration. Without reference to biochemical details, we are able to make behavioral predictions over timescales of minutes using only changes in cell shape over timescales of seconds. We use drug treatment and genetics to demonstrate that maximum caliber descriptors can discriminate between healthy and aberrant migration, thereby showing potential applications for maximum caliber methods in automated disease screening, for example in the identification of behaviors associated with cancer metastasis.

Maximum entropy principle analysis in network systems with short-time recordings

In many realistic systems, maximum entropy principle (MEP) analysis provides an effective characterization of the probability distribution of network states. However, to implement the MEP analysis, a sufficiently long-time data recording in general is often required, e.g., hours of spiking recordings of neurons in neuronal networks. The issue of whether the MEP analysis can be successfully applied to network systems with data from short-time recordings has yet to be fully addressed. In this work, we investigate relationships underlying the probability distributions, moments, and effective interactions in the MEP analysis and then show that, with short-time recordings of network dynamics, the MEP analysis can be applied to reconstructing probability distributions of network states that is much more accurate than the one directly measured from the short-time recording. Using spike trains obtained from both Hodgkin-Huxley neuronal networks and electrophysiological experiments, we verify our results and demonstrate that MEP analysis provides a tool to investigate the neuronal population coding properties for short-time recordings.

Balancing selfishness and norm conformity can explain human behavior in large-scale prisoner's dilemma games and can poise human groups near criticality

Cooperation is central to the success of human societies as it is crucial for overcoming some of the most pressing social challenges of our time; still, how human cooperation is achieved and may persist is a main puzzle in the social and biological sciences. Recently, scholars have recognized the importance of social norms as solutions to major local and large-scale collective action problems, from the management of water resources to the reduction of smoking in public places to the change in fertility practices. Yet a well-founded model of the effect of social norms on human cooperation is still lacking. Using statistical-physics techniques and integrating findings from cognitive and behavioral sciences, we present an analytically tractable model in which individuals base their decisions to cooperate both on the economic rewards they obtain and on the degree to which their action complies with social norms. Results from this parsimonious model are in agreement with observations in recent large-scale experiments with humans. We also find the phase diagram of the model and show that the experimental human group is poised near a critical point, a regime where recent work suggests living systems respond to changing external conditions in an efficient and coordinated manner.

Detecting intermittent switching leadership in coupled dynamical systems

Leader-follower relationships are commonly hypothesized as a fundamental mechanism underlying collective behaviour in many biological and physical systems. Understanding the emergence of such behaviour is relevant in science and engineering to control the dynamics of complex systems toward a desired state. In prior works, due in part to the limitations of existing methods for dissecting intermittent causal relationships, leadership is assumed to be consistent in time and space. This assumption has been contradicted by recent progress in the study of animal behaviour. In this work, we leverage information theory and time series analysis to propose a novel and simple method for dissecting changes in causal influence. Our approach computes the cumulative influence function of a given individual on the rest of the group in consecutive time intervals and identify change in the monotonicity of the function as a change in its leadership status. We demonstrate the effectiveness of our approach to dissect potential changes in leadership on self-propelled particles where the emergence of leader-follower relationship can be controlled and on tandem flights of birds recorded in their natural environment. Our method is expected to provide a novel methodological tool to further our understanding of collective behaviour.

An introduction to the maximum entropy approach and its application to inference problems in biology

A cornerstone of statistical inference, the maximum entropy framework is being increasingly applied to construct descriptive and predictive models of biological systems, especially complex biological networks, from large experimental data sets. Both its broad applicability and the success it obtained in different contexts hinge upon its conceptual simplicity and mathematical soundness. Here we try to concisely review the basic elements of the maximum entropy principle, starting from the notion of 'entropy', and describe its usefulness for the analysis of biological systems. As examples, we focus specifically on the problem of reconstructing gene interaction networks from expression data and on recent work attempting to expand our system-level understanding of bacterial metabolism. Finally, we highlight some extensions and potential limitations of the maximum entropy approach, and point to more recent developments that are likely to play a key role in the upcoming challenges of extracting structures and information from increasingly rich, high-throughput biological data.

Inverse statistical physics of protein sequences: a key issues review

In the course of evolution, proteins undergo important changes in their amino acid sequences, while their three-dimensional folded structure and their biological function remain remarkably conserved. Thanks to modern sequencing techniques, sequence data accumulate at unprecedented pace. This provides large sets of so-called homologous, i.e. evolutionarily related protein sequences, to which methods of inverse statistical physics can be applied. Using sequence data as the basis for the inference of Boltzmann distributions from samples of microscopic configurations or observables, it is possible to extract information about evolutionary constraints and thus protein function and structure. Here we give an overview over some biologically important questions, and how statistical-mechanics inspired modeling approaches can help to answer them. Finally, we discuss some open questions, which we expect to be addressed over the next years.

Three-dimensional trajectories and network analyses of group behaviour within chimney swift flocks during approaches to the roost

Chimney swifts (Chaetura pelagica) are highly manoeuvrable birds notable for roosting overnight in chimneys, in groups of hundreds or thousands of birds, before and during their autumn migration. At dusk, birds gather in large numbers from surrounding areas near a roost site. The whole flock then employs an orderly, but dynamic, circling approach pattern before rapidly entering a small aperture en masse. We recorded the three-dimensional trajectories of ≈1 800 individual birds during a 30 min period encompassing flock formation, circling, and landing, and used these trajectories to test several hypotheses relating to flock or group behaviour. Specifically, we investigated whether the swifts use local interaction rules based on topological distance (e.g. the n nearest neighbours, regardless of their distance) rather than physical distance (e.g. neighbours within x m, regardless of number) to guide interactions, whether the chimney entry zone is more or less cooperative than the surrounding flock, and whether the characteristic subgroup size is constant or varies with flock density. We found that the swift flock is structured around local rules based on physical distance, that subgroup size increases with density, and that there exist regions of the flock that are less cooperative than others, in particular the chimney entry zone.

Perspectives on theory at the interface of physics and biology

Theoretical physics is the search for simple and universal mathematical descriptions of the natural world. In contrast, much of modern biology is an exploration of the complexity and diversity of life. For many, this contrast is prima facie evidence that theory, in the sense that physicists use the word, is impossible in a biological context. For others, this contrast serves to highlight a grand challenge. I am an optimist, and believe (along with many colleagues) that the time is ripe for the emergence of a more unified theoretical physics of biological systems, building on successes in thinking about particular phenomena. In this essay I try to explain the reasons for my optimism, through a combination of historical and modern examples.

Body size affects the strength of social interactions and spatial organization of a schooling fish (Pseudomugil signifer)

While a rich variety of self-propelled particle models propose to explain the collective motion of fish and other animals, rigorous statistical comparison between models and data remains a challenge. Plausible models should be flexible enough to capture changes in the collective behaviour of animal groups at their different developmental stages and group sizes. Here, we analyse the statistical properties of schooling fish (Pseudomugil signifer) through a combination of experiments and simulations. We make novel use of a Boltzmann inversion method, usually applied in molecular dynamics, to identify the effective potential of the mean force of fish interactions. Specifically, we show that larger fish have a larger repulsion zone, but stronger attraction, resulting in greater alignment in their collective motion. We model the collective dynamics of schools using a self-propelled particle model, modified to include varying particle speed and a local repulsion rule. We demonstrate that the statistical properties of the fish schools are reproduced by our model, thereby capturing a number of features of the behaviour and development of schooling fish.

Local equilibrium in bird flocks

The correlated motion of flocks is an instance of global order emerging from local interactions. An essential difference with analogous ferromagnetic systems is that flocks are active: animals move relative to each other, dynamically rearranging their interaction network. The effect of this off-equilibrium element is well studied theoretically, but its impact on actual biological groups deserves more experimental attention. Here, we introduce a novel dynamical inference technique, based on the principle of maximum entropy, which accodomates network rearrangements and overcomes the problem of slow experimental sampling rates. We use this method to infer the strength and range of alignment forces from data of starling flocks. We find that local bird alignment happens on a much faster timescale than neighbour rearrangement. Accordingly, equilibrium inference, which assumes a fixed interaction network, gives results consistent with dynamical inference. We conclude that bird orientations are in a state of local quasi-equilibrium over the interaction length scale, providing firm ground for the applicability of statistical physics in certain active systems.

Maximum caliber inference and the stochastic Ising model

We investigate the maximum caliber variational principle as an inference algorithm used to predict dynamical properties of complex nonequilibrium, stationary, statistical systems in the presence of incomplete information. Specifically, we maximize the path entropy over discrete time step trajectories subject to normalization, stationarity, and detailed balance constraints together with a path-dependent dynamical information constraint reflecting a given average global behavior of the complex system. A general expression for the transition probability values associated with the stationary random Markov processes describing the nonequilibrium stationary system is computed. By virtue of our analysis, we uncover that a convenient choice of the dynamical information constraint together with a perturbative asymptotic expansion with respect to its corresponding Lagrange multiplier of the general expression for the transition probability leads to a formal overlap with the well-known Glauber hyperbolic tangent rule for the transition probability for the stochastic Ising model in the limit of very high temperatures of the heat reservoir.

Fluctuating fitness shapes the clone-size distribution of immune repertoires

The adaptive immune system relies on the diversity of receptors expressed on the surface of B- and T cells to protect the organism from a vast amount of pathogenic threats. The proliferation and degradation dynamics of different cell types (B cells, T cells, naive, memory) is governed by a variety of antigenic and environmental signals, yet the observed clone sizes follow a universal power-law distribution. Guided by this reproducibility we propose effective models of somatic evolution where cell fate depends on an effective fitness. This fitness is determined by growth factors acting either on clones of cells with the same receptor responding to specific antigens, or directly on single cells with no regard for clones. We identify fluctuations in the fitness acting specifically on clones as the essential ingredient leading to the observed distributions. Combining our models with experiments, we characterize the scale of fluctuations in antigenic environments and we provide tools to identify the relevant growth signals in different tissues and organisms. Our results generalize to any evolving population in a fluctuating environment.

Emergent behavioural phenotypes of swarming models revealed by mimicking a frustrated anti-ferromagnet

Self-propelled particle (SPP) models are often compared with animal swarms. However, the collective animal behaviour observed in experiments often leaves considerable unconstrained freedom in the structure of a proposed model. Essentially, multiple models can describe the observed behaviour of animal swarms in simple environments. To tackle this degeneracy, we study swarms of SPPs in non-trivial environments as a new approach to distinguish between candidate models. We restrict swarms of SPPs to circular (periodic) channels where they polarize in one of two directions (like spins) and permit information to pass through windows between neighbouring channels. Co-alignment between particles then couples the channels (anti-ferromagnetically) so that they tend to counter-rotate. We study channels arranged to mimic a geometrically frustrated anti-ferromagnet and show how the effects of this frustration allow us to better distinguish between SPP models. Similar experiments could therefore improve our understanding of collective motion in animals. Finally, we discuss how the spin analogy can be exploited to construct universal logic gates, and therefore swarming systems that can function as Turing machines.

Inferring Transition Rates of Networks from Populations in Continuous-Time Markov Processes

We are interested inferring rate processes on networks. In particular, given a network's topology, the stationary populations on its nodes, and a few global dynamical observables, can we infer all the transition rates between nodes? We draw inferences using the principle of maximum caliber (maximum path entropy). We have previously derived results for discrete-time Markov processes. Here, we treat continuous-time processes, such as dynamics among metastable states of proteins. The present work leads to a particularly important analytical result: namely, that when the network is constrained only by a mean jump rate, the rate matrix is given by a square-root dependence of the rate, kab ∝ (πb/πa)(1/2), on πa and πb, the stationary-state populations at nodes a and b. This leads to a fast way to estimate all of the microscopic rates in the system. As an illustration, we show that the method accurately predicts the nonequilibrium transition rates in an in silico gene expression network and transition probabilities among the metastable states of a small peptide at equilibrium. We note also that the method makes sensible predictions for so-called extra-thermodynamic relationships, such as those of Bronsted, Hammond, and others.

Stationary properties of maximum-entropy random walks

Maximum-entropy (ME) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic systems, how state space topology and path-dependent constraints affect ME-inferred state probabilities remains unknown. To that end, we derive the transition probabilities and the stationary distribution of a maximum path entropy Markov process subject to state- and path-dependent constraints. A main finding is that the stationary distribution over states differs significantly from the Boltzmann distribution and reflects a competition between path multiplicity and imposed constraints. We illustrate our results with particle diffusion on a two-dimensional landscape. Connections with the path integral approach to diffusion are discussed.

Short-range interactions versus long-range correlations in bird flocks

Bird flocks are a paradigmatic example of collective motion. One of the prominent traits of flocking is the presence of long range velocity correlations between individuals, which allow them to influence each other over the large scales, keeping a high level of group coordination. A crucial question is to understand what is the mutual interaction between birds generating such nontrivial correlations. Here we use the maximum entropy (ME) approach to infer from experimental data of natural flocks the effective interactions between individuals. Compared to previous studies, we make a significant step forward as we retrieve the full functional dependence of the interaction on distance, and find that it decays exponentially over a range of a few individuals. The fact that ME gives a short-range interaction even though its experimental input is the long-range correlation function, shows that the method is able to discriminate the relevant information encoded in such correlations and single out a minimal number of effective parameters. Finally, we show how the method can be used to capture the degree of anisotropy of mutual interactions.

Dynamical criticality in the collective activity of a population of retinal neurons

Recent experimental results based on multielectrode and imaging techniques have reinvigorated the idea that large neural networks operate near a critical point, between order and disorder. However, evidence for criticality has relied on the definition of arbitrary order parameters, or on models that do not address the dynamical nature of network activity. Here we introduce a novel approach to assess criticality that overcomes these limitations, while encompassing and generalizing previous criteria. We find a simple model to describe the global activity of large populations of ganglion cells in the rat retina, and show that their statistics are poised near a critical point. Taking into account the temporal dynamics of the activity greatly enhances the evidence for criticality, revealing it where previous methods would not. The approach is general and could be used in other biological networks.

A general pairwise interaction model provides an accurate description of in vivo transcription factor binding sites

The identification of transcription factor binding sites (TFBSs) on genomic DNA is of crucial importance for understanding and predicting regulatory elements in gene networks. TFBS motifs are commonly described by Position Weight Matrices (PWMs), in which each DNA base pair contributes independently to the transcription factor (TF) binding. However, this description ignores correlations between nucleotides at different positions, and is generally inaccurate: analysing fly and mouse in vivo ChIPseq data, we show that in most cases the PWM model fails to reproduce the observed statistics of TFBSs. To overcome this issue, we introduce the pairwise interaction model (PIM), a generalization of the PWM model. The model is based on the principle of maximum entropy and explicitly describes pairwise correlations between nucleotides at different positions, while being otherwise as unconstrained as possible. It is mathematically equivalent to considering a TF-DNA binding energy that depends additively on each nucleotide identity at all positions in the TFBS, like the PWM model, but also additively on pairs of nucleotides. We find that the PIM significantly improves over the PWM model, and even provides an optimal description of TFBS statistics within statistical noise. The PIM generalizes previous approaches to interdependent positions: it accounts for co-variation of two or more base pairs, and predicts secondary motifs, while outperforming multiple-motif models consisting of mixtures of PWMs. We analyse the structure of pairwise interactions between nucleotides, and find that they are sparse and dominantly located between consecutive base pairs in the flanking region of TFBS. Nonetheless, interactions between pairs of non-consecutive nucleotides are found to play a significant role in the obtained accurate description of TFBS statistics. The PIM is computationally tractable, and provides a general framework that should be useful for describing and predicting TFBSs beyond PWMs.