Emergent scale-free networks.

Many complex systems-from the Internet to social, biological, and communication networks-are thought to exhibit scale-free structure. However, prevailing explanations require that networks grow over time, an assumption that fails in some real-world settings. Here, we explain how scale-free structure can emerge without growth through network self-organization. Beginning with an arbitrary network, we allow connections to detach from random nodes and then reconnect under a mixture of preferential and random attachment. While the numbers of nodes and edges remain fixed, the degree distribution evolves toward a power-law with an exponent γ = 1 + 1 p that depends only on the proportion p of preferential (rather than random) attachment. Applying our model to several real networks, we infer p directly from data and predict the relationship between network size and degree heterogeneity. Together, these results establish how scale-free structure can arise in networks of constant size and density, with broad implications for the structure and function of complex systems.

Perceptual Resolution of Ambiguity: Can Tuned, Divisive Normalization Account for both Interocular Similarity Grouping and Difference Enhancement.

Our visual system usually provides a unique and functional representation of the external world. At times, however, the visual system has more than one compelling interpretation of the same retinal stimulus; in this case, neural populations compete for perceptual dominance to resolve ambiguity. Spatial and temporal context can guide perceptual experience. Recent evidence shows that ambiguous retinal stimuli are sometimes resolved by enhancing either similarity or differences among multiple percepts. Divisive normalization is a canonical neural computation that enables context-dependent sensory processing by attenuating a neuron's response by other neurons. Experiments here show that divisive normalization can account for perceptual representations of either similarity enhancement (so-called grouping) or difference enhancement, offering a unified framework for opposite perceptual outcomes.

Synchrotron-source micro-x-ray computed tomography for examining butterfly eyes.

Comparative anatomy is an important tool for investigating evolutionary relationships among species, but the lack of scalable imaging tools and stains for rapidly mapping the microscale anatomies of related species poses a major impediment to using comparative anatomy approaches for identifying evolutionary adaptations. We describe a method using synchrotron source micro-x-ray computed tomography (syn-µXCT) combined with machine learning algorithms for high-throughput imaging of Lepidoptera (i.e., butterfly and moth) eyes. Our pipeline allows for imaging at rates of ~15 min/mm3 at 600 nm3 resolution. Image contrast is generated using standard electron microscopy labeling approaches (e.g., osmium tetroxide) that unbiasedly labels all cellular membranes in a species-independent manner thus removing any barrier to imaging any species of interest. To demonstrate the power of the method, we analyzed the 3D morphologies of butterfly crystalline cones, a part of the visual system associated with acuity and sensitivity and found significant variation within six butterfly individuals. Despite this variation, a classic measure of optimization, the ratio of interommatidial angle to resolving power of ommatidia, largely agrees with early work on eye geometry across species. We show that this method can successfully be used to determine compound eye organization and crystalline cone morphology. Our novel pipeline provides for fast, scalable visualization and analysis of eye anatomies that can be applied to any arthropod species, enabling new questions about evolutionary adaptations of compound eyes and beyond.

A dynamic scale-mixture model of motion in natural scenes.

Some of the most important tasks of visual and motor systems involve estimating the motion of objects and tracking them over time. Such systems evolved to meet the behavioral needs of the organism in its natural environment, and may therefore be adapted to the statistics of motion it is likely to encounter. By tracking the movement of individual points in videos of natural scenes, we begin to identify common properties of natural motion across scenes. As expected, objects in natural scenes move in a persistent fashion, with velocity correlations lasting hundreds of milliseconds. More subtly, we find that the observed velocity distributions are heavy-tailed and can be modeled as a Gaussian scale-mixture. Extending this model to the time domain leads to a dynamic scale-mixture model, consisting of a Gaussian process multiplied by a positive scalar quantity with its own independent dynamics. Dynamic scaling of velocity arises naturally as a consequence of changes in object distance from the observer, and may approximate the effects of changes in other parameters governing the motion in a given scene. This modeling and estimation framework has implications for the neurobiology of sensory and motor systems, which need to cope with these fluctuations in scale in order to represent motion efficiently and drive fast and accurate tracking behavior.

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 N~100 neurons. As N 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. This principle becomes tractable if we restrict attention to correlations among pairs of neurons that link together into a tree; we can find the best tree efficiently, and the underlying statistical physics models are exactly solved. We use this approach to analyze experiments on N~1500 neurons in the mouse hippocampus, and show that the resulting model captures the distribution of synchronous activity in the network.

The double-drift illusion biases the marmoset oculomotor system.

The double-drift illusion has two unique characteristics: The error between the perceived and physical position of the stimulus grows over time, and saccades to the moving target land much closer to the physical than the perceived location. These results suggest that the perceptual and saccade targeting systems integrate visual information over different time scales. Functional imaging studies in humans have revealed several potential cortical areas of interest, including the prefrontal cortex. However, we currently lack an animal model to study the neural mechanisms of location perception that underlie the double-drift illusion. To fill this gap, we trained two marmoset monkeys to fixate and then saccade to the double-drift stimulus. In line with human observers for radial double-drift trajectories with fast internal motion, we find that saccade endpoints show a significant bias that is, nevertheless, smaller than the bias seen in human perceptual reports. This bias is modulated by changes in the external and internal speeds of the stimulus. These results demonstrate that the saccade targeting system of the marmoset monkey is influenced by the double-drift illusion.

Stimulus invariant aspects of the retinal code drive discriminability of natural scenes.

Everything that the brain sees must first be encoded by the retina, which maintains a reliable representation of the visual world in many different, complex natural scenes while also adapting to stimulus changes. Decomposing the population code into independent and cell-cell interactions reveals how broad scene structure is encoded in the adapted retinal output. By recording from the same retina while presenting many different natural movies, we see that the population structure, characterized by strong interactions, is consistent across both natural and synthetic stimuli. We show that these interactions contribute to encoding scene identity. We also demonstrate that this structure likely arises in part from shared bipolar cell input as well as from gap junctions between retinal ganglion cells and amacrine cells.

Predictive saccades and decision making in the beetle-predating saffron robber fly.

Internal predictions about the sensory consequences of self-motion, encoded by corollary discharge, are ubiquitous in the animal kingdom, including for fruit flies, dragonflies, and humans. In contrast, predicting the future location of an independently moving external target requires an internal model. With the use of internal models for predictive gaze control, vertebrate predatory species compensate for their sluggish visual systems and long sensorimotor latencies. This ability is crucial for the timely and accurate decisions that underpin a successful attack. Here, we directly demonstrate that the robber fly Laphria saffrana, a specialized beetle predator, also uses predictive gaze control when head tracking potential prey. Laphria uses this predictive ability to perform the difficult categorization and perceptual decision task of differentiating a beetle from other flying insects with a low spatial resolution retina. Specifically, we show that (1) this predictive behavior is part of a saccade-and-fixate strategy, (2) the relative target angular position and velocity, acquired during fixation, inform the subsequent predictive saccade, and (3) the predictive saccade provides Laphria with additional fixation time to sample the frequency of the prey's specular wing reflections. We also demonstrate that Laphria uses such wing reflections as a proxy for the wingbeat frequency of the potential prey and that consecutively flashing LEDs to produce apparent motion elicits attacks when the LED flicker frequency matches that of the beetle's wingbeat cycle.

Gaussian Information Bottleneck and the Non-Perturbative Renormalization Group.

The renormalization group (RG) is a class of theoretical techniques used to explain the collective physics of interacting, many-body systems. It has been suggested that the RG formalism may be useful in finding and interpreting emergent low-dimensional structure in complex systems outside of the traditional physics context, such as in biology or computer science. In such contexts, one common dimensionality-reduction framework already in use is information bottleneck (IB), in which the goal is to compress an "input" signal X while maximizing its mutual information with some stochastic "relevance" variable Y. IB has been applied in the vertebrate and invertebrate processing systems to characterize optimal encoding of the future motion of the external world. Other recent work has shown that the RG scheme for the dimer model could be "discovered" by a neural network attempting to solve an IB-like problem. This manuscript explores whether IB and any existing formulation of RG are formally equivalent. A class of soft-cutoff non-perturbative RG techniques are defined by families of non-deterministic coarsening maps, and hence can be formally mapped onto IB, and vice versa. For concreteness, this discussion is limited entirely to Gaussian statistics (GIB), for which IB has exact, closed-form solutions. Under this constraint, GIB has a semigroup structure, in which successive transformations remain IB-optimal. Further, the RG cutoff scheme associated with GIB can be identified. Our results suggest that IB can be used to impose a notion of "large scale" structure, such as biological function, on an RG procedure.

Learning low-dimensional generalizable natural features from retina using a U-net.

Much of sensory neuroscience focuses on presenting stimuli that are chosen by the experimenter because they are parametric and easy to sample and are thought to be behaviorally relevant to the organism. However, it is not generally known what these relevant features are in complex, natural scenes. This work focuses on using the retinal encoding of natural movies to determine the presumably behaviorally-relevant features that the brain represents. It is prohibitive to parameterize a natural movie and its respective retinal encoding fully. We use time within a natural movie as a proxy for the whole suite of features evolving across the scene. We then use a task-agnostic deep architecture, an encoder-decoder, to model the retinal encoding process and characterize its representation of "time in the natural scene" in a compressed latent space. In our end-to-end training, an encoder learns a compressed latent representation from a large population of salamander retinal ganglion cells responding to natural movies, while a decoder samples from this compressed latent space to generate the appropriate future movie frame. By comparing latent representations of retinal activity from three movies, we find that the retina has a generalizable encoding for time in the natural scene: the precise, low-dimensional representation of time learned from one movie can be used to represent time in a different movie, with up to 17 ms resolution. We then show that static textures and velocity features of a natural movie are synergistic. The retina simultaneously encodes both to establishes a generalizable, low-dimensional representation of time in the natural scene.

Inferring couplings in networks across order-disorder phase transitions.

Statistical inference is central to many scientific endeavors, yet how it works remains unresolved. Answering this requires a quantitative understanding of the intrinsic interplay between statistical models, inference methods, and the structure in the data. To this end, we characterize the efficacy of direct coupling analysis (DCA) - a highly successful method for analyzing amino acid sequence data-in inferring pairwise interactions from samples of ferromagnetic Ising models on random graphs. Our approach allows for physically motivated exploration of qualitatively distinct data regimes separated by phase transitions. We show that inference quality depends strongly on the nature of data-generating distributions: optimal accuracy occurs at an intermediate temperature where the detrimental effects from macroscopic order and thermal noise are minimal. Importantly our results indicate that DCA does not always outperform its local-statistics-based predecessors; while DCA excels at low temperatures, it becomes inferior to simple correlation thresholding at virtually all temperatures when data are limited. Our findings offer insights into the regime in which DCA operates so successfully, and more broadly, how inference interacts with the structure in the data.

Remembering the Past to See the Future.

In addition to the role that our visual system plays in determining what we are seeing right now, visual computations contribute in important ways to predicting what we will see next. While the role of memory in creating future predictions is often overlooked, efficient predictive computation requires the use of information about the past to estimate future events. In this article, we introduce a framework for understanding the relationship between memory and visual prediction and review the two classes of mechanisms that the visual system relies on to create future predictions. We also discuss the principles that define the mapping from predictive computations to predictive mechanisms and how downstream brain areas interpret the predictive signals computed by the visual system.

Spatially displaced excitation contributes to the encoding of interrupted motion by a retinal direction-selective circuit.

Spatially distributed excitation and inhibition collectively shape a visual neuron's receptive field (RF) properties. In the direction-selective circuit of the mammalian retina, the role of strong null-direction inhibition of On-Off direction-selective ganglion cells (On-Off DSGCs) on their direction selectivity is well-studied. However, how excitatory inputs influence the On-Off DSGC's visual response is underexplored. Here, we report that On-Off DSGCs have a spatially displaced glutamatergic receptive field along their horizontal preferred-null motion axes. This displaced receptive field contributes to DSGC null-direction spiking during interrupted motion trajectories. Theoretical analyses indicate that population responses during interrupted motion may help populations of On-Off DSGCs signal the spatial location of moving objects in complex, naturalistic visual environments. Our study highlights that the direction-selective circuit exploits separate sets of mechanisms under different stimulus conditions, and these mechanisms may help encode multiple visual features.

Optimal prediction with resource constraints using the information bottleneck.

Responding to stimuli requires that organisms encode information about the external world. Not all parts of the input are important for behavior, and resource limitations demand that signals be compressed. Prediction of the future input is widely beneficial in many biological systems. We compute the trade-offs between representing the past faithfully and predicting the future using the information bottleneck approach, for input dynamics with different levels of complexity. For motion prediction, we show that, depending on the parameters in the input dynamics, velocity or position information is more useful for accurate prediction. We show which motion representations are easiest to re-use for accurate prediction in other motion contexts, and identify and quantify those with the highest transferability. For non-Markovian dynamics, we explore the role of long-term memory in shaping the internal representation. Lastly, we show that prediction in evolutionary population dynamics is linked to clustering allele frequencies into non-overlapping memories.

Supervised learning through physical changes in a mechanical system.

Mechanical metamaterials are usually designed to show desired responses to prescribed forces. In some applications, the desired force-response relationship is hard to specify exactly, but examples of forces and desired responses are easily available. Here, we propose a framework for supervised learning in thin, creased sheets that learn the desired force-response behavior by physically experiencing training examples and then, crucially, respond correctly (generalize) to previously unseen test forces. During training, we fold the sheet using training forces, prompting local crease stiffnesses to change in proportion to their experienced strain. We find that this learning process reshapes nonlinearities inherent in folding a sheet so as to show the correct response for previously unseen test forces. We show the relationship between training error, test error, and sheet size (model complexity) in learning sheets and compare them to counterparts in machine-learning algorithms. Our framework shows how the rugged energy landscape of disordered mechanical materials can be sculpted to show desired force-response behaviors by a local physical learning process.

Nonlinear mixed selectivity supports reliable neural computation.

Neuronal activity in the brain is variable, yet both perception and behavior are generally reliable. How does the brain achieve this? Here, we show that the conjunctive coding of multiple stimulus features, commonly known as nonlinear mixed selectivity, may be used by the brain to support reliable information transmission using unreliable neurons. Nonlinearly mixed feature representations have been observed throughout primary sensory, decision-making, and motor brain areas. In these areas, different features are almost always nonlinearly mixed to some degree, rather than represented separately or with only additive (linear) mixing, which we refer to as pure selectivity. Mixed selectivity has been previously shown to support flexible linear decoding for complex behavioral tasks. Here, we show that it has another important benefit: in many cases, it makes orders of magnitude fewer decoding errors than pure selectivity even when both forms of selectivity use the same number of spikes. This benefit holds for sensory, motor, and more abstract, cognitive representations. Further, we show experimental evidence that mixed selectivity exists in the brain even when it does not enable behaviorally useful linear decoding. This suggests that nonlinear mixed selectivity may be a general coding scheme exploited by the brain for reliable and efficient neural computation.

Aristaless Controls Butterfly Wing Color Variation Used in Mimicry and Mate Choice.

Neotropical Heliconius butterflies display a diversity of warningly colored wing patterns, which serve roles in both Müllerian mimicry and mate choice behavior. Wing pattern diversity in Heliconius is controlled by a small number of unlinked, Mendelian "switch" loci [1]. One of these, termed the K locus, switches between yellow and white color patterns, important mimicry signals as well as mating cues [2-4]. Furthermore, mate preference behavior is tightly linked to this locus [4]. K controls the distribution of white versus yellow scales on the wing, with a dominant white allele and a recessive yellow allele. Here, we combine fine-scale genetic mapping, genome-wide association studies, gene expression analyses, population and comparative genomics, and genome editing with CRISPR/Cas9 to characterize the molecular basis of the K locus in Heliconius and to infer its evolutionary history. We show that white versus yellow color variation in Heliconius cydno is due to alternate haplotypes at a putative cis-regulatory element (CRE) downstream of a tandem duplication of the homeodomain transcription factor aristaless. Aristaless1 (al1) and aristaless2 (al2) are differentially regulated between white and yellow wings throughout development with elevated expression of al1 in developing white wings, suggesting a role in repressing pigmentation. Consistent with this, knockout of al1 causes white wings to become yellow. The evolution of wing color in this group has been marked by retention of the ancestral yellow color in many lineages, a single origin of white coloration in H. cydno, and subsequent introgression of white color from H. cydno into H. melpomene.

State dependence of stimulus-induced variability tuning in macaque MT.

Behavioral states marked by varying levels of arousal and attention modulate some properties of cortical responses (e.g. average firing rates or pairwise correlations), yet it is not fully understood what drives these response changes and how they might affect downstream stimulus decoding. Here we show that changes in state modulate the tuning of response variance-to-mean ratios (Fano factors) in a fashion that is neither predicted by a Poisson spiking model nor changes in the mean firing rate, with a substantial effect on stimulus discriminability. We recorded motion-sensitive neurons in middle temporal cortex (MT) in two states: alert fixation and light, opioid anesthesia. Anesthesia tended to lower average spike counts, without decreasing trial-to-trial variability compared to the alert state. Under anesthesia, within-trial fluctuations in excitability were correlated over longer time scales compared to the alert state, creating supra-Poisson Fano factors. In contrast, alert-state MT neurons have higher mean firing rates and largely sub-Poisson variability that is stimulus-dependent and cannot be explained by firing rate differences alone. The absence of such stimulus-induced variability tuning in the anesthetized state suggests different sources of variability between states. A simple model explains state-dependent shifts in the distribution of observed Fano factors via a suppression in the variance of gain fluctuations in the alert state. A population model with stimulus-induced variability tuning and behaviorally constrained information-limiting correlations explores the potential enhancement in stimulus discriminability by the cortical population in the alert state.

Learning to make external sensory stimulus predictions using internal correlations in populations of neurons.

To compensate for sensory processing delays, the visual system must make predictions to ensure timely and appropriate behaviors. Recent work has found predictive information about the stimulus in neural populations early in vision processing, starting in the retina. However, to utilize this information, cells downstream must be able to read out the predictive information from the spiking activity of retinal ganglion cells. Here we investigate whether a downstream cell could learn efficient encoding of predictive information in its inputs from the correlations in the inputs themselves, in the absence of other instructive signals. We simulate learning driven by spiking activity recorded in salamander retina. We model a downstream cell as a binary neuron receiving a small group of weighted inputs and quantify the predictive information between activity in the binary neuron and future input. Input weights change according to spike timing-dependent learning rules during a training period. We characterize the readouts learned under spike timing-dependent synaptic update rules, finding that although the fixed points of learning dynamics are not associated with absolute optimal readouts they convey nearly all of the information conveyed by the optimal readout. Moreover, we find that learned perceptrons transmit position and velocity information of a moving-bar stimulus nearly as efficiently as optimal perceptrons. We conclude that predictive information is, in principle, readable from the perspective of downstream neurons in the absence of other inputs. This suggests an important role for feedforward prediction in sensory encoding.

Perceptual interaction of local motion signals.

Motion signals are a rich source of information used in many everyday tasks, such as segregation of objects from background and navigation. Motion analysis by biological systems is generally considered to consist of two stages: extraction of local motion signals followed by spatial integration. Studies using synthetic stimuli show that there are many kinds and subtypes of local motion signals. When presented in isolation, these stimuli elicit behavioral and neurophysiological responses in a wide range of species, from insects to mammals. However, these mathematically-distinct varieties of local motion signals typically co-exist in natural scenes. This study focuses on interactions between two kinds of local motion signals: Fourier and glider. Fourier signals are typically associated with translation, while glider signals occur when an object approaches or recedes. Here, using a novel class of synthetic stimuli, we ask how distinct kinds of local motion signals interact and whether context influences sensitivity to Fourier motion. We report that local motion signals of different types interact at the perceptual level, and that this interaction can include subthreshold summation and, in some subjects, subtle context-dependent changes in sensitivity. We discuss the implications of these observations, and the factors that may underlie them.