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.

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.

Dynamics of social representation in the mouse prefrontal cortex

The prefrontal cortex (PFC) plays an important role in regulating social functions in mammals, and its dysfunction has been linked to social deficits in neurodevelopmental disorders. Yet little is known of how the PFC encodes social information and how social representations may be altered in such disorders. Here, we show that neurons in the medial PFC of freely behaving male mice preferentially respond to socially relevant olfactory cues. Population activity patterns in this region differed between social and nonsocial stimuli and underwent experience-dependent refinement. In mice lacking the autism-associated gene Cntnap2, both the categorization of sensory stimuli and the refinement of social representations were impaired. Noise levels in spontaneous population activity were higher in Cntnap2 knockouts and correlated with the degree to which social representations were disrupted. Our findings elucidate the encoding of social sensory cues in the medial PFC and provide a link between altered prefrontal dynamics and autism-associated social dysfunction.

Generalization of Object Localization From Whiskers to Other Body Parts in Freely Moving Rats

Rats can be trained to associate relative spatial locations of objects with the spatial location of rewards. Here we ask whether rats can localize static silent objects with other body parts in the dark, and if so with what resolution. We addressed these questions in trained rats, whose interactions with the objects were tracked at high-resolution before and after whisker trimming. We found that rats can use other body parts, such as trunk and ears, to localize objects. Localization resolution with non-whisking body parts (henceforth, ‘body’) was poorer than that obtained with whiskers, even when left with a single whisker at each side. Part of the superiority of whiskers was obtained via the use of multiple contacts. Transfer from whisker to body localization occurred within one session, provided that body contacts with the objects occurred before whisker trimming, or in the next session otherwise. This transfer occurred whether temporal cues were used for discrimination or when discrimination was based on spatial cues alone. Rats’ decision in each trial was based on the sensory cues acquired in that trial and on decisions and reward locations in previous trials. When sensory cues were acquired by body contacts, rat decisions relied more on the reward location in previous trials. Overall, the results suggest that rats can generalize the idea of relative object location across different body parts, while preferring to rely on whiskers-based localization, which occurs earlier and conveys higher resolution.

Probabilistic models of individual and collective animal behavior

Recent developments in automated tracking allow uninterrupted, high-resolution recording of animal trajectories, sometimes coupled with the identification of stereotyped changes of body pose or other behaviors of interest. Analysis and interpretation of such data represents a challenge: the timing of animal behaviors may be stochastic and modulated by kinematic variables, by the interaction with the environment or with the conspecifics within the animal group, and dependent on internal cognitive or behavioral state of the individual. Existing models for collective motion typically fail to incorporate the discrete, stochastic, and internal-state-dependent aspects of behavior, while models focusing on individual animal behavior typically ignore the spatial aspects of the problem. Here we propose a probabilistic modeling framework to address this gap. Each animal can switch stochastically between different behavioral states, with each state resulting in a possibly different law of motion through space. Switching rates for behavioral transitions can depend in a very general way, which we seek to identify from data, on the effects of the environment as well as the interaction between the animals. We represent the switching dynamics as a Generalized Linear Model and show that: (i) forward simulation of multiple interacting animals is possible using a variant of the Gillespie's Stochastic Simulation Algorithm; (ii) formulated properly, the maximum likelihood inference of switching rate functions is tractably solvable by gradient descent; (iii) model selection can be used to identify factors that modulate behavioral state switching and to appropriately adjust model complexity to data. To illustrate our framework, we apply it to two synthetic models of animal motion and to real zebrafish tracking data.

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.

Towards the design principles of neural population codes

The ability to record the joint activity of large groups of neurons would allow for direct study of information representation and computation at the level of whole circuits in the brain. The combinatorial space of potential population activity patterns and neural noise imply that it would be impossible to directly map the relations between stimuli and population responses. Understanding of large neural population codes therefore depends on identifying simplifying design principles. We review recent results showing that strongly correlated population codes can be explained using minimal models that rely on low order relations among cells. We discuss the implications for large populations, and how such models allow for mapping the semantic organization of the neural codebook and stimulus space, and decoding.

A thesaurus for a neural population code

Information is carried in the brain by the joint spiking patterns of large groups of noisy, unreliable neurons. This noise limits the capacity of the neural code and determines how information can be transmitted and read-out. To accurately decode, the brain must overcome this noise and identify which patterns are semantically similar. We use models of network encoding noise to learn a thesaurus for populations of neurons in the vertebrate retina responding to artificial and natural videos, measuring the similarity between population responses to visual stimuli based on the information they carry. This thesaurus reveals that the code is organized in clusters of synonymous activity patterns that are similar in meaning but may differ considerably in their structure. This organization is highly reminiscent of the design of engineered codes. We suggest that the brain may use this structure and show how it allows accurate decoding of novel stimuli from novel spiking patterns.

DOI:http://dx.doi.org/10.7554/eLife.06134.001

Our ability to perceive the world is dependent on information from our senses being passed between different parts of the brain. The information is encoded as patterns of electrical pulses or ‘spikes’, which other brain regions must be able to decipher. Cracking this code would thus enable us to predict the patterns of nerve impulses that would occur in response to specific stimuli, and ‘decode’ which stimuli had produced particular patterns of impulses.

This task is challenging in part because of its scale—vast numbers of stimuli are encoded by huge numbers of neurons that can send their spikes in many different combinations. Furthermore, neurons are inherently noisy and their response to identical stimuli may vary considerably in the number of spikes and their timing. This means that the brain cannot simply link a single unchanging pattern of firing with each stimulus, because these firing patterns are often distorted by biophysical noise.

Ganmor et al. have now modeled the effects of noise in a network of neurons in the retina (found at the back of the eye), and, in doing so, have provided insights into how the brain solves this problem. This has brought us a step closer to cracking the neural code. First, 10 second video clips of natural scenes and artificial stimuli were played on a loop to a sample of retina taken from a salamander, and the responses of nearly 100 neurons in the sample were recorded for two hours. Dividing the 10 second clip into short segments provided a series of 500 stimuli, which the network had been exposed to more than 600 times.

Ganmor et al. analyzed the responses of groups of 20 cells to each stimulus and found that physically similar firing patterns were not particularly likely to encode the same stimulus. This can be likened to the way that words such as ‘light’ and ‘night’ have similar structures but different meanings. Instead, the model reveals that each stimulus was represented by a cluster of firing patterns that bore little physical resemblance to one another, but which nevertheless conveyed the same meaning. To continue on with the previous example, this is similar to way that ‘light’ and ‘illumination’ have the same meaning but different structures.

Ganmor et al. use these new data to map the organization of the ‘vocabulary’ of populations of cells the retina, and put together a kind of ‘thesaurus’ that enables new activity patterns of the retina to be decoded and could be used to crack the neural code. Furthermore, the organization of ‘synonyms’ is strikingly similar to codes that are favored in many forms of telecommunication. In these man-made codes, codewords that represent different items are chosen to be so distinct from each other that even if they were corrupted by noise, they could be correctly deciphered. Correspondingly, in the retina, patterns that carry the same meaning occupy a distinct area, and new patterns can be interpreted based on their proximity to these clusters.

DOI:http://dx.doi.org/10.7554/eLife.06134.002

Adaptation to changes in higher-order stimulus statistics in the salamander retina

Adaptation in the retina is thought to optimize the encoding of natural light signals into sequences of spikes sent to the brain. While adaptive changes in retinal processing to the variations of the mean luminance level and second-order stimulus statistics have been documented before, no such measurements have been performed when higher-order moments of the light distribution change. We therefore measured the ganglion cell responses in the tiger salamander retina to controlled changes in the second (contrast), third (skew) and fourth (kurtosis) moments of the light intensity distribution of spatially uniform temporally independent stimuli. The skew and kurtosis of the stimuli were chosen to cover the range observed in natural scenes. We quantified adaptation in ganglion cells by studying linear-nonlinear models that capture well the retinal encoding properties across all stimuli. We found that the encoding properties of retinal ganglion cells change only marginally when higher-order statistics change, compared to the changes observed in response to the variation in contrast. By analyzing optimal coding in LN-type models, we showed that neurons can maintain a high information rate without large dynamic adaptation to changes in skew or kurtosis. This is because, for uncorrelated stimuli, spatio-temporal summation within the receptive field averages away non-gaussian aspects of the light intensity distribution.

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.

Adaptive colour contrast coding in the salamander retina efficiently matches natural scene statistics

The visual system continually adjusts its sensitivity to the statistical properties of the environment through an adaptation process that starts in the retina. Colour perception and processing is commonly thought to occur mainly in high visual areas, and indeed most evidence for chromatic colour contrast adaptation comes from cortical studies. We show that colour contrast adaptation starts in the retina where ganglion cells adjust their responses to the spectral properties of the environment. We demonstrate that the ganglion cells match their responses to red-blue stimulus combinations according to the relative contrast of each of the input channels by rotating their functional response properties in colour space. Using measurements of the chromatic statistics of natural environments, we show that the retina balances inputs from the two (red and blue) stimulated colour channels, as would be expected from theoretical optimal behaviour. Our results suggest that colour is encoded in the retina based on the efficient processing of spectral information that matches spectral combinations in natural scenes on the colour processing level.

High-order social interactions in groups of mice

Social behavior in mammals is often studied in pairs under artificial conditions, yet groups may rely on more complicated social structures. Here, we use a novel system for tracking multiple animals in a rich environment to characterize the nature of group behavior and interactions, and show strongly correlated group behavior in mice. We have found that the minimal models that rely only on individual traits and pairwise correlations between animals are not enough to capture group behavior, but that models that include third-order interactions give a very accurate description of the group. These models allow us to infer social interaction maps for individual groups. Using this approach, we show that environmental complexity during adolescence affects the collective group behavior of adult mice, in particular altering the role of high-order structure. Our results provide new experimental and mathematical frameworks for studying group behavior and social interactions.

DOI:http://dx.doi.org/10.7554/eLife.00759.001

All animals need to interact with others of the same species, even if it is only to mate. To date, social behavior has been studied mainly at two extremes: detailed observation of pairs; and studies of the collective behavior of large groups, such as flocks of birds. However, to gain an understanding of social behavior in mammals will require an approach that falls between these two extremes. It will be necessary to study animals in larger groups, rather than in pairs, but also to track individuals rather than looking at the activity of the group as a whole.

Now, Shemesh et al. have developed a system that can track the behavior of each of four mice with high spatial and temporal resolution as they move around freely in an arena containing ramps, nest boxes, and barriers. Because mice are largely nocturnal, Shemesh et al. dyed the animals’ fur with compounds that produced different coloured fluorescence under ultraviolet light, and then employed an automated system to accurately track each mouse during 12 hr of darkness, over a number of days.

Using these data it was possible to estimate the extent to which the behavior of the group is determined by the characteristics of individual mice and how much is determined by interactions between animals. Models based solely on the behavior of individuals could not accurately describe the behavior of the group. Surprisingly, neither could models that focused on interactions between pairs of mice. Only models that included interactions between three mice gave a good approximation of the observed behavior. This shows that, even in a small group, social behavior is determined by relatively complex interactions.

Shemesh et al. also found that the behavior of the mice depended on the environment in which they had been raised. Animals that had lived in larger groups and in more interesting enclosures were influenced more by pairwise interactions, and less by three-way interactions, than mice that had been raised in a standard laboratory environment. This suggests that being raised in a complex environment strengthens mouse ‘individuality’. The approach developed by Shemesh et al. could be extended to study larger groups of animals and could also be used to examine the interplay between genes, environment and other factors in shaping social interactions.

DOI:http://dx.doi.org/10.7554/eLife.00759.002

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.

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.

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.

The natural variation of a neural code

The way information is represented by sequences of action potentials of spiking neurons is determined by the input each neuron receives, but also by its biophysics, and the specifics of the circuit in which it is embedded. Even the "code" of identified neurons can vary considerably from individual to individual. Here we compared the neural codes of the identified H1 neuron in the visual systems of two families of flies, blow flies and flesh flies, and explored the effect of the sensory environment that the flies were exposed to during development on the H1 code. We found that the two families differed considerably in the temporal structure of the code, its content and energetic efficiency, as well as the temporal delay of neural response. The differences in the environmental conditions during the flies' development had no significant effect. Our results may thus reflect an instance of a family-specific design of the neural code. They may also suggest that individual variability in information processing by this specific neuron, in terms of both form and content, is regulated genetically.

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.

Many groups of organisms, from colonies of bacteria and social insects through schools of fish and flocks of birds to herds of mammals exhibit advanced collective navigation. Identifying the minimal features of biologically-inspired interacting agents that can lead to emergence of “intelligent” like collective navigation and decision making is fundamental to our understanding of collective behavior, and is of great interest in artificial intelligence and robotics. Previous models of collective behavior of agents, which relied on static interactions of repulsion, orientation (alignment), and attraction, have shown the emergence of collective swarming. Here we show the advantage of performance adaptable interactions for navigation of groups in complex terrains. Each agent senses the local environment and is then allowed to adjust its interactions with the other agents according to its local environment – by decreasing the peers' influence while navigating in a beneficial direction and vice versa. We found that inclusion of such adaptable interactions dramatically improves the collective swarming performance leading to highly efficient navigation especially in very complex terrains.

Neural activity at the human olfactory epithelium reflects olfactory perception

Organization of receptive surfaces reflects primary axes of perception. In vision, retinal coordinates reflect spatial coordinates. In audition, cochlear coordinates reflect tonal coordinates. However, the rules underlying the organization of the olfactory receptive surface are unknown. To test the hypothesis that organization of the olfactory epithelium reflects olfactory perception, we inserted an electrode into the human olfactory epithelium to directly measure odorant-induced evoked responses. We found that pairwise differences in odorant pleasantness predicted pairwise differences in response magnitude; that is, a location that responded maximally to a pleasant odorant was likely to respond strongly to other pleasant odorants, and a location that responded maximally to an unpleasant odorant was likely to respond strongly to other unpleasant odorants. Moreover, the extent of an individual's perceptual span predicted their span in evoked response. This suggests that, similarly to receptor surfaces for vision and audition, organization of the olfactory receptor surface reflects key axes of perception.

Odorant concentration dependence in electroolfactograms recorded from the human olfactory epithelium

Electroolfactograms (EOGs) are the summated generator potentials of olfactory receptor neurons measured directly from the olfactory epithelium. To validate the sensory origin of the human EOG, we set out to ask whether EOGs measured in humans were odorant concentration dependent. Each of 22 subjects (12 women, mean age = 23.3 yr) was tested with two odorants, either valeric acid and linalool (n = 12) or isovaleric acid and l-carvone (n = 10), each delivered at four concentrations diluted with warm (37 degrees C) and humidified (80%) odorless air. In behavior, increased odorant concentration was associated with increased perceived intensity (all F > 5, all P < 0.001). In EOG, increased odorant concentration was associated with increased area under the EOG curve (all F > 8, all P < 0.001). These findings substantiate EOG as a tool for probing olfactory coding directly at the level of olfactory receptor neurons in humans.

Role of eye movements in the retinal code for a size discrimination task

The concerted action of saccades and fixational eye movements are crucial for seeing stationary objects in the visual world. We studied how these eye movements contribute to retinal coding of visual information using the archer fish as a model system. We quantified the animal's ability to distinguish among objects of different sizes and measured its eye movements. We recorded from populations of retinal ganglion cells with a multielectrode array, while presenting visual stimuli matched to the behavioral task. We found that the beginning of fixation, namely the time immediately after the saccade, provided the most visual information about object size, with fixational eye movements, which consist of tremor and drift in the archer fish, yielding only a minor contribution. A simple decoder that combined information from <or=15 ganglion cells could account for the behavior. Our results support the view that saccades impose not just difficulties for the visual system, but also an opportunity for the retina to encode high quality "snapshots" of the environment.

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.

Redundancy in the population code of the retina

We have explored the manner in which the population of retinal ganglion cells collectively represent the visual world. Ganglion cells in the salamander were recorded simultaneously with a multielectrode array during stimulation with both artificial and natural visual stimuli, and the mutual information that single cells and pairs of cells conveyed about the stimulus was estimated. We found significant redundancy between cells spaced as far as 500 mum apart. When we used standard methods for defining functional types, only ON-type and OFF-type cells emerged as truly independent information channels. Although the average redundancy between nearby cell pairs was moderate, each ganglion cell shared information with many neighbors, so that visual information was represented approximately 10-fold within the ganglion cell population. This high degree of retinal redundancy suggests that design principles beyond coding efficiency may be important at the population level.

Synergy, redundancy, and independence in population codes

A key issue in understanding the neural code for an ensemble of neurons is the nature and strength of correlations between neurons and how these correlations are related to the stimulus. The issue is complicated by the fact that there is not a single notion of independence or lack of correlation. We distinguish three kinds: (1) activity independence; (2) conditional independence; and (3) information independence. Each notion is related to an information measure: the information between cells, the information between cells given the stimulus, and the synergy of cells about the stimulus, respectively. We show that these measures form an interrelated framework for evaluating contributions of signal and noise correlations to the joint information conveyed about the stimulus and that at least two of the three measures must be calculated to characterize a population code. This framework is compared with others recently proposed in the literature. In addition, we distinguish questions about how information is encoded by a population of neurons from how that information can be decoded. Although information theory is natural and powerful for questions of encoding, it is not sufficient for characterizing the process of decoding. Decoding fundamentally requires an error measure that quantifies the importance of the deviations of estimated stimuli from actual stimuli. Because there is no a priori choice of error measure, questions about decoding cannot be put on the same level of generality as for encoding.

Network information and connected correlations

Entropy and information provide natural measures of correlation among elements in a network. We construct here the information theoretic analog of connected correlation functions: irreducible N-point correlation is measured by a decrease in entropy for the joint distribution of N variables relative to the maximum entropy allowed by all the observed N-1 variable distributions. We calculate the "connected information" terms for several examples and show that it also enables the decomposition of the information that is carried by a population of elements about an outside source.

Axons as computing devices: basic insights gained from models

Detailed models of single neurons are typically focused on the dendritic tree and ignore the axonal tree, assuming that the axon is a simple transmission line. In the last 40 years, however, several theoretical and experimental studies have suggested that axons could implement information processing tasks by exploiting: 1) the time delay in action potential (AP) propagation along the axon; 2) the differential filtering of APs into the axonal subtrees; and 3) their activity-dependent excitability. Models for axonal trees have attempted to examine the feasibility of these ideas. However, because the physiological and anatomical data on axons are seriously limited, realistic models of axons have not been developed. The present paper summarizes the main insights that were gained from simplified models of axons; it also highlights the stochastic nature of axons, a topic that was largely neglected in classical models of axons. The advance of new experimental techniques makes it now possible to pay a very close experimental visit to axons. Theoretical tools and fast computers enable to go beyond the simplified models and to construct realistic models of axons. When tightly linked, experiments and theory will help to unravel how axons share the information processing tasks that single neurons implement.

Ion channel stochasticity may be critical in determining the reliability and precision of spike timing

The firing reliability and precision of an isopotential membrane patch consisting of a realistically large number of ion channels is investigated using a stochastic Hodgkin-Huxley (HH) model. In sharp contrast to the deterministic HH model, the biophysically inspired stochastic model reproduces qualitatively the different reliability and precision characteristics of spike firing in response to DC and fluctuating current input in neocortical neurons, as reported by Mainen & Sejnowski (1995). For DC inputs, spike timing is highly unreliable; the reliability and precision are significantly increased for fluctuating current input. This behavior is critically determined by the relatively small number of excitable channels that are opened near threshold for spike firing rather than by the total number of channels that exist in the membrane patch. Channel fluctuations, together with the inherent bistability in the HH equations, give rise to three additional experimentally observed phenomena: subthreshold oscillations in the membrane voltage for DC input, "spontaneous" spikes for subthreshold inputs, and "missing" spikes for suprathreshold inputs. We suggest that the noise inherent in the operation of ion channels enables neurons to act as "smart" encoders. Slowly varying, uncorrelated inputs are coded with low reliability and accuracy and, hence, the information about such inputs is encoded almost exclusively by the spike rate. On the other hand, correlated presynaptic activity produces sharp fluctuations in the input to the postsynaptic cell, which are then encoded with high reliability and accuracy. In this case, information about the input exists in the exact timing of the spikes. We conclude that channel stochasticity should be considered in realistic models of neurons.

Two-point rapid palatal expansion: an alternate approach to traditional treatment

Rapid palatal expansion (RPE) causes separation of the lateral halves of the palate and traditionally has used four maxillary teeth as anchorage. The purpose of this study was to introduce a rapid palatal expander that requires only two anchor teeth (two-point RPEe) and to compare the expansion obtained with that from a Hyrax appliance. This study involved two groups of 25 children aged 7 to 15 years who were treated in a private orthodontist's office with either a Hyrax appliance or a two-point RPEe. Dental casts and occlusal radiographs were made before treatment and at least three months after stabilization of the appliance. Paired t-tests were performed to identify significant intragroup changes, and independent t-tests were performed to determine intergroup differences. The findings showed the two-point RPEe was as efficient as the Hyrax in obtaining dental expansion of the maxillary posterior teeth with less effect on the maxillary anterior and mandibular teeth. Therefore, the two-point RPEe may be useful in certain clinical situations.