Elad Schneidman

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.

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.

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.

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.

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.

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 populations 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.

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 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.

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 individuals 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.