Learning probabilistic neural representations with randomly connected circuits.

The brain represents and reasons probabilistically about complex stimuli and motor actions using a noisy, spike-based neural code. A key building block for such neural computations, as well as the basis for supervised and unsupervised learning, is the ability to estimate the surprise or likelihood of incoming high-dimensional neural activity patterns. Despite progress in statistical modeling of neural responses and deep learning, current approaches either do not scale to large neural populations or cannot be implemented using biologically realistic mechanisms. Inspired by the sparse and random connectivity of real neuronal circuits, we present a model for neural codes that accurately estimates the likelihood of individual spiking patterns and has a straightforward, scalable, efficient, learnable, and realistic neural implementation. This model's performance on simultaneously recorded spiking activity of >100 neurons in the monkey visual and prefrontal cortices is comparable with or better than that of state-of-the-art models. Importantly, the model can be learned using a small number of samples and using a local learning rule that utilizes noise intrinsic to neural circuits. Slower, structural changes in random connectivity, consistent with rewiring and pruning processes, further improve the efficiency and sparseness of the resulting neural representations. Our results merge insights from neuroanatomy, machine learning, and theoretical neuroscience to suggest random sparse connectivity as a key design principle for neuronal computation.

Discrete modes of social information processing predict individual behavior of fish in a group.

Individual computations and social interactions underlying collective behavior in groups of animals are of great ethological, behavioral, and theoretical interest. While complex individual behaviors have successfully been parsed into small dictionaries of stereotyped behavioral modes, studies of collective behavior largely ignored these findings; instead, their focus was on inferring single, mode-independent social interaction rules that reproduced macroscopic and often qualitative features of group behavior. Here, we bring these two approaches together to predict individual swimming patterns of adult zebrafish in a group. We show that fish alternate between an "active" mode, in which they are sensitive to the swimming patterns of conspecifics, and a "passive" mode, where they ignore them. Using a model that accounts for these two modes explicitly, we predict behaviors of individual fish with high accuracy, outperforming previous approaches that assumed a single continuous computation by individuals and simple metric or topological weighing of neighbors' behavior. At the group level, switching between active and passive modes is uncorrelated among fish, but correlated directional swimming behavior still emerges. Our quantitative approach for studying complex, multimodal individual behavior jointly with emergent group behavior is readily extensible to additional behavioral modes and their neural correlates as well as to other species.

Information socialtaxis and efficient collective behavior emerging in groups of information-seeking agents.

Individual behavior, in biology, economics, and computer science, is often described in terms of balancing exploration and exploitation. Foraging has been a canonical setting for studying reward seeking and information gathering, from bacteria to humans, mostly focusing on individual behavior. Inspired by the gradient-climbing nature of chemotaxis, the infotaxis algorithm showed that locally maximizing the expected information gain leads to efficient and ethological individual foraging. In nature, as well as in theoretical settings, conspecifics can be a valuable source of information about the environment. Whereas the nature and role of interactions between animals have been studied extensively, the design principles of information processing in such groups are mostly unknown. We present an algorithm for group foraging, which we term "socialtaxis," that unifies infotaxis and social interactions, where each individual in the group simultaneously maximizes its own sensory information and a social information term. Surprisingly, we show that when individuals aim to increase their information diversity, efficient collective behavior emerges in groups of opportunistic agents, which is comparable to the optimal group behavior. Importantly, we show the high efficiency of biologically plausible socialtaxis settings, where agents share little or no information and rely on simple computations to infer information from the behavior of their conspecifics. Moreover, socialtaxis does not require parameter tuning and is highly robust to sensory and behavioral noise. We use socialtaxis to predict distinct optimal couplings in groups of selfish vs. altruistic agents, reflecting how it can be naturally extended to study social dynamics and collective computation in general settings.

High-order feature-based mixture models of classification learning predict individual learning curves and enable personalized teaching.

Pattern classification learning tasks are commonly used to explore learning strategies in human subjects. The universal and individual traits of learning such tasks reflect our cognitive abilities and have been of interest both psychophysically and clinically. From a computational perspective, these tasks are hard, because the number of patterns and rules one could consider even in simple cases is exponentially large. Thus, when we learn to classify we must use simplifying assumptions and generalize. Studies of human behavior in probabilistic learning tasks have focused on rules in which pattern cues are independent, and also described individual behavior in terms of simple, single-cue, feature-based models. Here, we conducted psychophysical experiments in which people learned to classify binary sequences according to deterministic rules of different complexity, including high-order, multicue-dependent rules. We show that human performance on such tasks is very diverse, but that a class of reinforcement learning-like models that use a mixture of features captures individual learning behavior surprisingly well. These models reflect the important role of subjects' priors, and their reliance on high-order features even when learning a low-order rule. Further, we show that these models predict future individual answers to a high degree of accuracy. We then use these models to build personally optimized teaching sessions and boost learning.

Searching for collective behavior in a large network of sensory neurons.

Maximum entropy models are the least structured probability distributions that exactly reproduce a chosen set of statistics measured in an interacting network. Here we use this principle to construct probabilistic models which describe the correlated spiking activity of populations of up to 120 neurons in the salamander retina as it responds to natural movies. Already in groups as small as 10 neurons, interactions between spikes can no longer be regarded as small perturbations in an otherwise independent system; for 40 or more neurons pairwise interactions need to be supplemented by a global interaction that controls the distribution of synchrony in the population. Here we show that such "K-pairwise" models--being systematic extensions of the previously used pairwise Ising models--provide an excellent account of the data. We explore the properties of the neural vocabulary by: 1) estimating its entropy, which constrains the population's capacity to represent visual information; 2) classifying activity patterns into a small set of metastable collective modes; 3) showing that the neural codeword ensembles are extremely inhomogenous; 4) demonstrating that the state of individual neurons is highly predictable from the rest of the population, allowing the capacity for error correction.

Perceptual convergence of multi-component mixtures in olfaction implies an olfactory white.

In vision, two mixtures, each containing an independent set of many different wavelengths, may produce a common color percept termed "white." In audition, two mixtures, each containing an independent set of many different frequencies, may produce a common perceptual hum termed "white noise." Visual and auditory whites emerge upon two conditions: when the mixture components span stimulus space, and when they are of equal intensity. We hypothesized that if we apply these same conditions to odorant mixtures, "whiteness" may emerge in olfaction as well. We selected 86 molecules that span olfactory stimulus space and individually diluted them to a point of about equal intensity. We then prepared various odorant mixtures, each containing various numbers of molecular components, and asked human participants to rate the perceptual similarity of such mixture pairs. We found that as we increased the number of nonoverlapping, equal-intensity components in odorant mixtures, the mixtures became more similar to each other, despite not having a single component in common. With ~30 components, most mixtures smelled alike. After participants were acquainted with a novel, arbitrarily named mixture of ~30 equal-intensity components, they later applied this name more readily to other novel mixtures of ~30 equal-intensity components spanning stimulus space, but not to mixtures containing fewer components or to mixtures that did not span stimulus space. We conclude that a common olfactory percept, "olfactory white," is associated with mixtures of ~30 or more equal-intensity components that span stimulus space, implying that olfactory representations are of features of molecules rather than of molecular identity.

Stimulus-dependent maximum entropy models of neural population codes.

Neural populations encode information about their stimulus in a collective fashion, by joint activity patterns of spiking and silence. A full account of this mapping from stimulus to neural activity is given by the conditional probability distribution over neural codewords given the sensory input. For large populations, direct sampling of these distributions is impossible, and so we must rely on constructing appropriate models. We show here that in a population of 100 retinal ganglion cells in the salamander retina responding to temporal white-noise stimuli, dependencies between cells play an important encoding role. We introduce the stimulus-dependent maximum entropy (SDME) model-a minimal extension of the canonical linear-nonlinear model of a single neuron, to a pairwise-coupled neural population. We find that the SDME model gives a more accurate account of single cell responses and in particular significantly outperforms uncoupled models in reproducing the distributions of population codewords emitted in response to a stimulus. We show how the SDME model, in conjunction with static maximum entropy models of population vocabulary, can be used to estimate information-theoretic quantities like average surprise and information transmission in a neural population.

Sparse low-order interaction network underlies a highly correlated and learnable neural population code.

Information is carried in the brain by the joint activity patterns of large groups of neurons. Understanding the structure and function of population neural codes is challenging because of the exponential number of possible activity patterns and dependencies among neurons. We report here that for groups of ~100 retinal neurons responding to natural stimuli, pairwise-based models, which were highly accurate for small networks, are no longer sufficient. We show that because of the sparse nature of the neural code, the higher-order interactions can be easily learned using a novel model and that a very sparse low-order interaction network underlies the code of large populations of neurons. Additionally, we show that the interaction network is organized in a hierarchical and modular manner, which hints at scalability. Our results suggest that learnability may be a key feature of the neural code.

Retinal metric: a stimulus distance measure derived from population neural responses.

The ability of an organism to distinguish between various stimuli is limited by the structure and noise in the population code of its sensory neurons. Here we infer a distance measure on the stimulus space directly from the recorded activity of 100 neurons in the salamander retina. In contrast to previously used measures of stimulus similarity, this "neural metric" tells us how distinguishable a pair of stimulus clips is to the retina, based on the similarity between the induced distributions of population responses. We show that the retinal distance strongly deviates from Euclidean, or any static metric, yet has a simple structure: we identify the stimulus features that the neural population is jointly sensitive to, and show the support-vector-machine-like kernel function relating the stimulus and neural response spaces. We show that the non-Euclidean nature of the retinal distance has important consequences for neural decoding.

Optimal population coding by noisy spiking neurons.

In retina and in cortical slice the collective response of spiking neural populations is well described by "maximum-entropy" models in which only pairs of neurons interact. We asked, how should such interactions be organized to maximize the amount of information represented in population responses? To this end, we extended the linear-nonlinear-Poisson model of single neural response to include pairwise interactions, yielding a stimulus-dependent, pairwise maximum-entropy model. We found that as we varied the noise level in single neurons and the distribution of network inputs, the optimal pairwise interactions smoothly interpolated to achieve network functions that are usually regarded as discrete--stimulus decorrelation, error correction, and independent encoding. These functions reflected a trade-off between efficient consumption of finite neural bandwidth and the use of redundancy to mitigate noise. Spontaneous activity in the optimal network reflected stimulus-induced activity patterns, and single-neuron response variability overestimated network noise. Our analysis suggests that rather than having a single coding principle hardwired in their architecture, networks in the brain should adapt their function to changing noise and stimulus correlations.

The architecture of functional interaction networks in the retina.

Sensory information is represented in the brain by the joint activity of large groups of neurons. Recent studies have shown that, although the number of possible activity patterns and underlying interactions is exponentially large, pairwise-based models give a surprisingly accurate description of neural population activity patterns. We explored the architecture of maximum entropy models of the functional interaction networks underlying the response of large populations of retinal ganglion cells, in adult tiger salamander retina, responding to natural and artificial stimuli. We found that we can further simplify these pairwise models by neglecting weak interaction terms or by relying on a small set of interaction strengths. Comparing network interactions under different visual stimuli, we show the existence of local network motifs in the interaction map of the retina. Our results demonstrate that the underlying interaction map of the retina is sparse and dominated by local overlapping interaction modules.

Synergy from silence in a combinatorial neural code.

The manner in which groups of neurons represent events in the external world is a central question in neuroscience. Estimation of the information encoded by small groups of neurons has shown that in many neural systems, cells carry mildly redundant information. These measures average over all the activity patterns of a neural population. Here, we analyze the population code of the salamander and guinea pig retinas by quantifying the information conveyed by specific multicell activity patterns. Synchronous spikes, even though they are relatively rare and highly informative, convey less information than the sum of either spike alone, making them redundant coding symbols. Instead, patterns of spiking in one cell and silence in others, which are relatively common and often overlooked as special coding symbols, were found to be mostly synergistic. Our results reflect that the mild average redundancy between ganglion cells that was previously reported is actually the result of redundant and synergistic multicell patterns, whose contributions partially cancel each other when taking the average over all patterns. We further show that similar coding properties emerge in a generic model of neural responses, suggesting that this form of combinatorial coding, in which specific compound patterns carry synergistic or redundant information, may exist in other neural circuits.

Smart swarms of bacteria-inspired agents with performance adaptable interactions.

Collective navigation and swarming have been studied in animal groups, such as fish schools, bird flocks, bacteria, and slime molds. Computer modeling has shown that collective behavior of simple agents can result from simple interactions between the agents, which include short range repulsion, intermediate range alignment, and long range attraction. Here we study collective navigation of bacteria-inspired smart agents in complex terrains, with adaptive interactions that depend on performance. More specifically, each agent adjusts its interactions with the other agents according to its local environment--by decreasing the peers' influence while navigating in a beneficial direction, and increasing it otherwise. We show that inclusion of such performance dependent adaptable interactions significantly improves the collective swarming performance, leading to highly efficient navigation, especially in complex terrains. Notably, to afford such adaptable interactions, each modeled agent requires only simple computational capabilities with short-term memory, which can easily be implemented in simple swarming robots.

Weak pairwise correlations imply strongly correlated network states in a neural population.

Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.

Reprogramming the topology of the nociceptive circuit in C. elegans reshapes sexual behavior.

The effect of the detailed connectivity of a neural circuit on its function and the resulting behavior of the organism is a key question in many neural systems. Here, we study the circuit for nociception in C. elegans, which is composed of the same neurons in the two sexes that are wired differently. We show that the nociceptive sensory neurons respond similarly in the two sexes, yet the animals display sexually dimorphic behaviors to the same aversive stimuli. To uncover the role of the downstream network topology in shaping behavior, we learn and simulate network models that replicate the observed dimorphic behaviors and use them to predict simple network rewirings that would switch behavior between the sexes. We then show experimentally that these subtle synaptic rewirings indeed flip behavior. Interestingly, when presented with aversive cues, rewired males were compromised in finding mating partners, suggesting that network topologies that enable efficient avoidance of noxious cues have a reproductive "cost." Our results present a deconstruction of the design of a neural circuit that controls sexual behavior and how to reprogram it.

Global features of neural activity in the olfactory system form a parallel code that predicts olfactory behavior and perception.

Odor identity is coded in spatiotemporal patterns of neural activity in the olfactory bulb. Here we asked whether meaningful olfactory information could also be read from the global olfactory neural population response. We applied standard statistical methods of dimensionality-reduction to neural activity from 12 previously published studies using seven different species. Four studies reported olfactory receptor activity, seven reported glomerulus activity, and one reported the activity of projection-neurons. We found two linear axes of neural population activity that accounted for more than half of the variance in neural response across species. The first axis was correlated with the total sum of odor-induced neural activity, and reflected the behavior of approach or withdrawal in animals, and odorant pleasantness in humans. The second and orthogonal axis reflected odorant toxicity across species. We conclude that in parallel with spatiotemporal pattern coding, the olfactory system can use simple global computations to read vital olfactory information from the neural population response.

The geometry of neuronal representations during rule learning reveals complementary roles of cingulate cortex and putamen.

Learning new rules and adopting novel behavioral policies is a prominent adaptive behavior of primates. We studied the dynamics of single neurons in the dorsal anterior cingulate cortex and putamen of monkeys while they learned new classification tasks every few days over a fixed set of multi-cue patterns. Representing the rules and the neuronal selectivity as vectors in the space spanned by a set of stimulus features allowed us to characterize neuronal dynamics in geometrical terms. We found that neurons in the cingulate cortex mainly rotated toward the rule, implying a policy search, whereas neurons in the putamen showed a magnitude increase that followed the rotation of cortical neurons, implying strengthening of confidence for the newly acquired rule-based policy. Further, the neural representation at the end of a session predicted next-day behavior, reflecting overnight retention. The novel framework for characterization of neural dynamics suggests complementing roles for the putamen and the anterior cingulate cortex.

Estimating the Unique Information of Continuous Variables.

The integration and transfer of information from multiple sources to multiple targets is a core motive of neural systems. The emerging field of partial information decomposition (PID) provides a novel information-theoretic lens into these mechanisms by identifying synergistic, redundant, and unique contributions to the mutual information between one and several variables. While many works have studied aspects of PID for Gaussian and discrete distributions, the case of general continuous distributions is still uncharted territory. In this work we present a method for estimating the unique information in continuous distributions, for the case of one versus two variables. Our method solves the associated optimization problem over the space of distributions with fixed bivariate marginals by combining copula decompositions and techniques developed to optimize variational autoencoders. We obtain excellent agreement with known analytic results for Gaussians, and illustrate the power of our new approach in several brain-inspired neural models. Our method is capable of recovering the effective connectivity of a chaotic network of rate neurons, and uncovers a complex trade-off between redundancy, synergy and unique information in recurrent networks trained to solve a generalized XOR task.

Social interactions drive efficient foraging and income equality in groups of fish.

The social interactions underlying group foraging and their benefits have been mostly studied using mechanistic models replicating qualitative features of group behavior, and focused on a single resource or a few clustered ones. Here, we tracked groups of freely foraging adult zebrafish with spatially dispersed food items and found that fish perform stereotypical maneuvers when consuming food, which attract neighboring fish. We then present a mathematical model, based on inferred functional interactions between fish, which accurately describes individual and group foraging of real fish. We show that these interactions allow fish to combine individual and social information to achieve near-optimal foraging efficiency and promote income equality within groups. We further show that the interactions that would maximize efficiency in these social foraging models depend on group size, but not on food distribution, and hypothesize that fish may adaptively pick the subgroup of neighbors they 'listen to' to determine their own behavior.

Author Correction: Dynamics of social representation in the mouse prefrontal cortex.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.