Deriving a genetic regulatory network from an optimization principle.

Many biological systems operate near the physical limits to their performance, suggesting that aspects of their behavior and underlying mechanisms could be derived from optimization principles. However, such principles have often been applied only in simplified models. Here, we explore a detailed mechanistic model of the gap gene network in the embryo, optimizing its 50+ parameters to maximize the information that gene expression levels provide about nuclear positions.

Finding the last bits of positional information.

In a developing embryo, information about the position of cells is encoded in the concentrations of morphogen molecules. In the fruit fly, the local concentrations of just a handful of proteins encoded by the gap genes are sufficient to specify position with a precision comparable to the spacing between cells along the anterior-posterior axis. This matches the precision of downstream events such as the striped patterns of expression in the pair-rule genes, but is not quite sufficient to define unique identities for individual cells.

Scale invariance in early embryonic development.

The expression of a few key genes determines the body plan of the fruit fly. We show that the spatial expression patterns for several of these genes scale precisely with embryo size. Discrete positional markers such as the peaks in striped patterns or the boundaries of expression domains have positions along the embryo's major axis proportional to embryo length, accurate to within 1%.

Long Timescales, Individual Differences, and Scale Invariance in Animal Behavior.

The explosion of data on animal behavior in more natural contexts highlights the fact that these behaviors exhibit correlations across many timescales. However, there are major challenges in analyzing these data: records of behavior in single animals have fewer independent samples than one might expect. In pooling data from multiple animals, individual differences can mimic long-ranged temporal correlations; conversely, long-ranged correlations can lead to an overestimate of individual differences.

Scale invariance in early embryonic development.

The body plan of the fruit fly is determined by the expression of just a handful of genes. We show that the spatial patterns of expression for several of these genes scale precisely with the size of the embryo. Concretely, discrete positional markers such as the peaks in striped patterns have absolute positions along the anterior-posterior axis that are proportional to embryo length, with better than 1% accuracy.

Exactly solvable statistical physics models for large neuronal populations.

Maximum entropy methods provide a principled path connecting measurements of neural activity directly to statistical physics models, and this approach has been successful for populations of neurons. As increases in new experiments, we enter an undersampled regime where we have to choose which observables should be constrained in the maximum entropy construction. The best choice is the one that provides the greatest reduction in entropy, defining a "minimax entropy" principle.

Long time scales, individual differences, and scale invariance in animal behavior.

The explosion of data on animal behavior in more natural contexts highlights the fact that these behaviors exhibit correlations across many time scales. But there are major challenges in analyzing these data: records of behavior in single animals have fewer independent samples than one might expect; in pooling data from multiple animals, individual differences can mimic long-ranged temporal correlations; conversely long-ranged correlations can lead to an over-estimate of individual differences. We suggest an analysis scheme that addresses these problems directly, apply this approach to data on the spontaneous behavior of walking flies, and find evidence for scale invariant correlations over nearly three decades in time, from seconds to one hour.

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.

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.

On the dimensionality of behavior.

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.

Trading bits in the readout from a genetic network.

In the regulation of gene expression, information of relevance to the organism is represented by the concentrations of transcription factor molecules. To extract this information the cell must effectively "measure" these concentrations, but there are physical limits to the precision of these measurements. We use the gap gene network in the early fly embryo as an example of the tradeoff between the precision of concentration measurements and the transmission of relevant information.

Optimal Local Estimates of Visual Motion in a Natural Environment.

Many organisms use visual signals to estimate motion, and these estimates typically are biased. Here, we ask whether these biases may reflect physical rather than biological limitations. Using a camera-gyroscope system, we sample the joint distribution of images and rotational motions in a natural environment, and from this distribution we construct the optimal estimator of velocity based on local image intensities.

Coarse Graining, Fixed Points, and Scaling in a Large Population of Neurons.

We develop a phenomenological coarse-graining procedure for activity in a large network of neurons, and apply this to recordings from a population of 1000+ cells in the hippocampus. Distributions of coarse-grained variables seem to approach a fixed non-Gaussian form, and we see evidence of scaling in both static and dynamic quantities. These results suggest that the collective behavior of the network is described by a nontrivial fixed point.

Searching for collective behavior in a small brain.

In large neuronal networks, it is believed that functions emerge through the collective behavior of many interconnected neurons. Recently, the development of experimental techniques that allow simultaneous recording of calcium concentration from a large fraction of all neurons in Caenorhabditis elegans-a nematode with 302 neurons-creates the opportunity to ask whether such emergence is universal, reaching down to even the smallest brains. Here, we measure the activity of 50+ neurons in C.

Optimal Decoding of Cellular Identities in a Genetic Network.

In developing organisms, spatially prescribed cell identities are thought to be determined by the expression levels of multiple genes. Quantitative tests of this idea, however, require a theoretical framework capable of exposing the rules and precision of cell specification over developmental time. We use the gap gene network in the early fly embryo as an example to show how expression levels of the four gap genes can be jointly decoded into an optimal specification of position with 1% accuracy.

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.

Collective Behavior of Place and Non-place Neurons in the Hippocampal Network.

Discussions of the hippocampus often focus on place cells, but many neurons are not place cells in any given environment. Here we describe the collective activity in such mixed populations, treating place and non-place cells on the same footing. We start with optical imaging experiments on CA1 in mice as they run along a virtual linear track and use maximum entropy methods to approximate the distribution of patterns of activity in the population, matching the correlations between pairs of cells but otherwise assuming as little structure as possible.

PCA meets RG.

A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis (PCA) focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group (RG).

Complexity of generic biochemical circuits: topology versus strength of interactions.

The historical focus on network topology as a determinant of biological function is still largely maintained today, illustrated by the rise of structure-only approaches to network analysis. However, biochemical circuits and genetic regulatory networks are defined both by their topology and by a multitude of continuously adjustable parameters, such as the strength of interactions between nodes, also recognized as important. Here we present a class of simple perceptron-based Boolean models within which comparing the relative importance of topology versus interaction strengths becomes a quantitatively well-posed problem.

Predictability and hierarchy in Drosophila behavior.

Even the simplest of animals exhibit behavioral sequences with complex temporal dynamics. Prominent among the proposed organizing principles for these dynamics has been the idea of a hierarchy, wherein the movements an animal makes can be understood as a set of nested subclusters. Although this type of organization holds potential advantages in terms of motion control and neural circuitry, measurements demonstrating this for an animal's entire behavioral repertoire have been limited in scope and temporal complexity.

Entropic effects in a nonequilibrium system: Flocks of birds.

When European starlings come together to form a flock, the distribution of their individual velocities narrows around the mean velocity of the flock. We argue that, in a broad class of models for the joint distribution of positions and velocities, this narrowing generates an entropic effect that opposes the cohesion of the flock. The strength of this effect depends strongly on the nature of the interactions among birds: If birds are coupled to a fixed number of neighbors, the entropic forces are weak, while if they couple to all other birds within a fixed distance, the entropic effects are sufficient to tear a flock apart.

Extending the dynamic range of transcription factor action by translational regulation.

A crucial step in the regulation of gene expression is binding of transcription factor (TF) proteins to regulatory sites along the DNA. But transcription factors act at nanomolar concentrations, and noise due to random arrival of these molecules at their binding sites can severely limit the precision of regulation. Recent work on the optimization of information flow through regulatory networks indicates that the lower end of the dynamic range of concentrations is simply inaccessible, overwhelmed by the impact of this noise.

The Neural Code for Motor Control in the Cerebellum and Oculomotor Brainstem.

A single extra spike makes a difference. Here, the size of the eye velocity in the initiation of smooth eye movements in the right panel depends on whether a cerebellar Purkinje cell discharges 3 (red), 4 (green), 5 (blue), or 6 (black) spikes in the 40-ms window indicated by the gray shading in the rasters on the left. Spike trains are rich in information that can be extracted to guide behaviors at millisecond time resolution or across longer time intervals.

Thermodynamics and signatures of criticality in a network of neurons.

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics.