Simultaneous silence organizes structured higher-order interactions in neural populations

Small-correlation expansion to quantify information in noisy sensory systems

Neural networks encode information through their collective spiking activity in response to external stimuli. This population response is noisy and strongly correlated, with a complex interplay between correlations induced by the stimulus, and correlations caused by shared noise. Understanding how these correlations affect information transmission has so far been limited to pairs or small groups of neurons, because the curse of dimensionality impedes the evaluation of mutual information in larger populations. Here, we develop a small-correlation expansion to compute the stimulus information carried by a large population of neurons, yielding interpretable analytical expressions in terms of the neurons' firing rates and pairwise correlations. We validate the approximation on synthetic data and demonstrate its applicability to electrophysiological recordings in the vertebrate retina, allowing us to quantify the effects of noise correlations between neurons and of memory in single neurons.

Uncovering hidden network architecture from spiking activities using an exact statistical input-output relation of neurons

Identifying network architecture from observed neural activities is crucial in neuroscience studies. A key requirement is knowledge of the statistical input-output relation of single neurons in vivo. By utilizing an exact analytical solution of the spike-timing for leaky integrate-and-fire neurons under noisy inputs balanced near the threshold, we construct a framework that links synaptic type, strength, and spiking nonlinearity with the statistics of neuronal population activity. The framework explains structured pairwise and higher-order interactions of neurons receiving common inputs under different architectures. We compared the theoretical predictions with the activity of monkey and mouse V1 neurons and found that excitatory inputs given to pairs explained the observed sparse activity characterized by strong negative triple-wise interactions, thereby ruling out the alternative explanation by shared inhibition. Moreover, we showed that the strong interactions are a signature of excitatory rather than inhibitory inputs whenever the spontaneous rate is low. We present a guide map of neural interactions that help researchers to specify the hidden neuronal motifs underlying observed interactions found in empirical data.

Emergence of functional information from multivariate correlations

The information content of symbolic sequences (such as nucleic or amino acid sequences, but also neuronal firings or strings of letters) can be calculated from an ensemble of such sequences, but because information cannot be assigned to single sequences, we cannot correlate information to other observables attached to the sequence. Here we show that an information score obtained from multivariate (multiple-variable) correlations within sequences of a 'training' ensemble can be used to predict observables of out-of-sample sequences with an accuracy that scales with the complexity of correlations, showing that functional information emerges from a hierarchy of multi-variable correlations. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Action potential-coupled Rho GTPase signaling drives presynaptic plasticity

In contrast to their postsynaptic counterparts, the contributions of activity-dependent cytoskeletal signaling to presynaptic plasticity remain controversial and poorly understood. To identify and evaluate these signaling pathways, we conducted a proteomic analysis of the presynaptic cytomatrix using in vivo biotin identification (iBioID). The resultant proteome was heavily enriched for actin cytoskeleton regulators, including Rac1, a Rho GTPase that activates the Arp2/3 complex to nucleate branched actin filaments. Strikingly, we find Rac1 and Arp2/3 are closely associated with synaptic vesicle membranes in adult mice. Using three independent approaches to alter presynaptic Rac1 activity (genetic knockout, spatially restricted inhibition, and temporal optogenetic manipulation), we discover that this pathway negatively regulates synaptic vesicle replenishment at both excitatory and inhibitory synapses, bidirectionally sculpting short-term synaptic depression. Finally, we use two-photon fluorescence lifetime imaging to show that presynaptic Rac1 activation is coupled to action potentials by voltage-gated calcium influx. Thus, this study uncovers a previously unrecognized mechanism of actin-regulated short-term presynaptic plasticity that is conserved across excitatory and inhibitory terminals. It also provides a new proteomic framework for better understanding presynaptic physiology, along with a blueprint of experimental strategies to isolate the presynaptic effects of ubiquitously expressed proteins.

Heterogeneity across neural populations: Its significance for the dynamics and functions of neural circuits

Neural populations show patterns of synchronous activity, as they share common correlated inputs. Neurons in the cortex that are connected by strong synapses cause rapid firing explosions. In addition, areas that are connected by weaker synapses have a slower dynamics and they can contribute to asymmetries in the input distributions. The aim of this work is to develop a neural model to investigate how the heterogeneities in the synaptic input distributions affect different levels of organizational activity in the brain dynamics. We analytically show how small changes in the correlation inputs can cause large changes in the interactions of the outputs that lead to a phase transition, demonstrating that a simple variation in the direction of a biased skewed distribution in the neuronal inputs can generate a transition of states in the firing rate, passing from spontaneous silence ("down state") to an absolute spiking activity ("up state"). We present an exact quantification of the dynamics of the output variables, showing that when considering a biased skewed distribution in the inputs of neuronal population, the critical point is not in an asynchronous or synchronous state but rather at an intermediate value.

Representation of foreseeable choice outcomes in orbitofrontal cortex triplet-wise interactions

Shared neuronal variability has been shown to modulate cognitive processing. However, the relationship between shared variability and behavioral performance is heterogeneous and complex in frontal areas such as the orbitofrontal cortex (OFC). Mounting evidence shows that single-units in OFC encode a detailed cognitive map of task-space events, but the existence of a robust neuronal ensemble coding for the predictability of choice outcome is less established. Here, we hypothesize that the coding of foreseeable outcomes is potentially unclear from the analysis of units activity and their pairwise correlations. However, this code might be established more conclusively when higher-order neuronal interactions are mapped to the choice outcome. As a case study, we investigated the trial-to-trial shared variability of neuronal ensemble activity during a two-choice interval-discrimination task in rodent OFC, specifically designed such that a lose-switch strategy is optimal by repeating the rewarded stimulus in the upcoming trial. Results show that correlations among triplets are higher during correct choices with respect to incorrect ones, and that this is sustained during the entire trial. This effect is not observed for pairwise nor for higher than third-order correlations. This scenario is compatible with constellations of up to three interacting units assembled during trials in which the task is performed correctly. More interestingly, a state-space spanned by such constellations shows that only correct outcome states that can be successfully predicted are robust over 100 trials of the task, and thus they can be accurately decoded. However, both incorrect and unpredictable outcome representations were unstable and thus non-decodeable, due to spurious negative correlations. Our results suggest that predictability of successful outcomes, and hence the optimal behavioral strategy, can be mapped out in OFC ensemble states reliable over trials of the task, and revealed by sufficiency complex neuronal interactions.

Neuronal responses can differ substantially during repetitions of the same tasks; however, they are often coordinated (shared) across multiple neighboring neurons. Such correlation between neurons has been related to the capacity of the brain to take decisions, but specifically how this relation is established is still under study. In this work, we address this question by focusing on an intriguing case study, the orbitofrontal cortex, since this brain area has been found in various studies to be useful for decision-making. Here, we question whether orchestrated groups of neurons encode sufficient information for optimizing their decision strategy; that is, whether the outcome of a choice can be predicted or not on the basis of previous experience. We thus designed a decision-making task for a rat in which some of the correct choices can be predicted. We found that only successful outcomes that can actually be predicted were robustly encoded over time. This finding was shown by analyzing sufficiently complex interactions between three neurons, whilst more complex orchestrations did not add further insights. Thus, we propose that coordinated responses of up to three neurons in the OFC could contribute to the capacity of the animal to take the optimal decision.

Higher-Order Cumulants Drive Neuronal Activity Patterns, Inducing UP-DOWN States in Neural Populations

A major challenge in neuroscience is to understand the role of the higher-order correlations structure of neuronal populations. The dichotomized Gaussian model (DG) generates spike trains by means of thresholding a multivariate Gaussian random variable. The DG inputs are Gaussian distributed, and thus have no interactions beyond the second order in their inputs; however, they can induce higher-order correlations in the outputs. We propose a combination of analytical and numerical techniques to estimate higher-order, above the second, cumulants of the firing probability distributions. Our findings show that a large amount of pairwise interactions in the inputs can induce the system into two possible regimes, one with low activity (“DOWN state”) and another one with high activity (“UP state”), and the appearance of these states is due to a combination between the third- and fourth-order cumulant. This could be part of a mechanism that would help the neural code to upgrade specific information about the stimuli, motivating us to examine the behavior of the critical fluctuations through the Binder cumulant close to the critical point. We show, using the Binder cumulant, that higher-order correlations in the outputs generate a critical neural system that portrays a second-order phase transition.

Clustering of Neural Activity: A Design Principle for Population Codes

We propose that correlations among neurons are generically strong enough to organize neural activity patterns into a discrete set of clusters, which can each be viewed as a population codeword. Our reasoning starts with the analysis of retinal ganglion cell data using maximum entropy models, showing that the population is robustly in a frustrated, marginally sub-critical, or glassy, state. This leads to an argument that neural populations in many other brain areas might share this structure. Next, we use latent variable models to show that this glassy state possesses well-defined clusters of neural activity. Clusters have three appealing properties: (i) clusters exhibit error correction, i.e., they are reproducibly elicited by the same stimulus despite variability at the level of constituent neurons; (ii) clusters encode qualitatively different visual features than their constituent neurons; and (iii) clusters can be learned by downstream neural circuits in an unsupervised fashion. We hypothesize that these properties give rise to a "learnable" neural code which the cortical hierarchy uses to extract increasingly complex features without supervision or reinforcement.

Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties

The pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80-95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations.

Maximum entropy principle analysis in network systems with short-time recordings

In many realistic systems, maximum entropy principle (MEP) analysis provides an effective characterization of the probability distribution of network states. However, to implement the MEP analysis, a sufficiently long-time data recording in general is often required, e.g., hours of spiking recordings of neurons in neuronal networks. The issue of whether the MEP analysis can be successfully applied to network systems with data from short-time recordings has yet to be fully addressed. In this work, we investigate relationships underlying the probability distributions, moments, and effective interactions in the MEP analysis and then show that, with short-time recordings of network dynamics, the MEP analysis can be applied to reconstructing probability distributions of network states that is much more accurate than the one directly measured from the short-time recording. Using spike trains obtained from both Hodgkin-Huxley neuronal networks and electrophysiological experiments, we verify our results and demonstrate that MEP analysis provides a tool to investigate the neuronal population coding properties for short-time recordings.

A Moment-Based Maximum Entropy Model for Fitting Higher-Order Interactions in Neural Data

Correlations in neural activity have been demonstrated to have profound consequences for sensory encoding. To understand how neural populations represent stimulus information, it is therefore necessary to model how pairwise and higher-order spiking correlations between neurons contribute to the collective structure of population-wide spiking patterns. Maximum entropy models are an increasingly popular method for capturing collective neural activity by including successively higher-order interaction terms. However, incorporating higher-order interactions in these models is difficult in practice due to two factors. First, the number of parameters exponentially increases as higher orders are added. Second, because triplet (and higher) spiking events occur infrequently, estimates of higher-order statistics may be contaminated by sampling noise. To address this, we extend previous work on the Reliable Interaction class of models to develop a normalized variant that adaptively identifies the specific pairwise and higher-order moments that can be estimated from a given dataset for a specified confidence level. The resulting “Reliable Moment” model is able to capture cortical-like distributions of population spiking patterns. Finally, we show that, compared with the Reliable Interaction model, the Reliable Moment model infers fewer strong spurious higher-order interactions and is better able to predict the frequencies of previously unobserved spiking patterns.

Common inputs in subthreshold membrane potential: The role of quiescent states in neuronal activity

Experiments in certain regions of the cerebral cortex suggest that the spiking activity of neuronal populations is regulated by common non-Gaussian inputs across neurons. We model these deviations from random-walk processes with q-Gaussian distributions into simple threshold neurons, and investigate the scaling properties in large neural populations. We show that deviations from the Gaussian statistics provide a natural framework to regulate population statistics such as sparsity, entropy, and specific heat. This type of description allows us to provide an adequate strategy to explain the information encoding in the case of low neuronal activity and its possible implications on information transmission.

Computational Neuroscience: Mathematical and Statistical Perspectives

Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.

Bistability, non-ergodicity, and inhibition in pairwise maximum-entropy models

Pairwise maximum-entropy models have been used in neuroscience to predict the activity of neuronal populations, given only the time-averaged correlations of the neuron activities. This paper provides evidence that the pairwise model, applied to experimental recordings, would produce a bimodal distribution for the population-averaged activity, and for some population sizes the second mode would peak at high activities, that experimentally would be equivalent to 90% of the neuron population active within time-windows of few milliseconds. Several problems are connected with this bimodality: 1. The presence of the high-activity mode is unrealistic in view of observed neuronal activity and on neurobiological grounds. 2. Boltzmann learning becomes non-ergodic, hence the pairwise maximum-entropy distribution cannot be found: in fact, Boltzmann learning would produce an incorrect distribution; similarly, common variants of mean-field approximations also produce an incorrect distribution. 3. The Glauber dynamics associated with the model is unrealistically bistable and cannot be used to generate realistic surrogate data. This bimodality problem is first demonstrated for an experimental dataset from 159 neurons in the motor cortex of macaque monkey. Evidence is then provided that this problem affects typical neural recordings of population sizes of a couple of hundreds or more neurons. The cause of the bimodality problem is identified as the inability of standard maximum-entropy distributions with a uniform reference measure to model neuronal inhibition. To eliminate this problem a modified maximum-entropy model is presented, which reflects a basic effect of inhibition in the form of a simple but non-uniform reference measure. This model does not lead to unrealistic bimodalities, can be found with Boltzmann learning, and has an associated Glauber dynamics which incorporates a minimal asymmetric inhibition.

Networks of interacting units are ubiquitous in various fields of biology; e.g. gene regulatory networks, neuronal networks, social structures. If a limited set of observables is accessible, maximum-entropy models provide a way to construct a statistical model for such networks, under particular assumptions. The pairwise maximum-entropy model only uses the first two moments among those observables, and can be interpreted as a network with only pairwise interactions. If correlations are on average positive, we here show that the maximum entropy distribution tends to become bimodal. In the application to neuronal activity this is a problem, because the bimodality is an artefact of the statistical model and not observed in real data. This problem could also affect other fields in biology. We here explain under which conditions bimodality arises and present a solution to the problem by introducing a collective negative feedback, corresponding to a modified maximum-entropy model. This result may point to the existence of a homeostatic mechanism active in the system that is not part of our set of observable units.

Probabilistic models for neural populations that naturally capture global coupling and criticality

Advances in multi-unit recordings pave the way for statistical modeling of activity patterns in large neural populations. Recent studies have shown that the summed activity of all neurons strongly shapes the population response. A separate recent finding has been that neural populations also exhibit criticality, an anomalously large dynamic range for the probabilities of different population activity patterns. Motivated by these two observations, we introduce a class of probabilistic models which takes into account the prior knowledge that the neural population could be globally coupled and close to critical. These models consist of an energy function which parametrizes interactions between small groups of neurons, and an arbitrary positive, strictly increasing, and twice differentiable function which maps the energy of a population pattern to its probability. We show that: 1) augmenting a pairwise Ising model with a nonlinearity yields an accurate description of the activity of retinal ganglion cells which outperforms previous models based on the summed activity of neurons; 2) prior knowledge that the population is critical translates to prior expectations about the shape of the nonlinearity; 3) the nonlinearity admits an interpretation in terms of a continuous latent variable globally coupling the system whose distribution we can infer from data. Our method is independent of the underlying system’s state space; hence, it can be applied to other systems such as natural scenes or amino acid sequences of proteins which are also known to exhibit criticality.

Populations of sensory neurons represent information about the outside environment in a collective fashion. A salient property of this distributed neural code is criticality. Yet most models used to date to analyze recordings from large neural populations do not take this observation explicitly into account. Here we aim to bridge this gap by designing probabilistic models whose structure reflects the expectation that the population is close to critical. We show that such principled approach improves previously considered models, and we demonstrate a connection between our models and the presence of continuous latent variables which is a recently proposed mechanism underlying criticality in many natural systems.

Approximate Inference for Time-Varying Interactions and Macroscopic Dynamics of Neural Populations

The models in statistical physics such as an Ising model offer a convenient way to characterize stationary activity of neural populations. Such stationary activity of neurons may be expected for recordings from in vitro slices or anesthetized animals. However, modeling activity of cortical circuitries of awake animals has been more challenging because both spike-rates and interactions can change according to sensory stimulation, behavior, or an internal state of the brain. Previous approaches modeling the dynamics of neural interactions suffer from computational cost; therefore, its application was limited to only a dozen neurons. Here by introducing multiple analytic approximation methods to a state-space model of neural population activity, we make it possible to estimate dynamic pairwise interactions of up to 60 neurons. More specifically, we applied the pseudolikelihood approximation to the state-space model, and combined it with the Bethe or TAP mean-field approximation to make the sequential Bayesian estimation of the model parameters possible. The large-scale analysis allows us to investigate dynamics of macroscopic properties of neural circuitries underlying stimulus processing and behavior. We show that the model accurately estimates dynamics of network properties such as sparseness, entropy, and heat capacity by simulated data, and demonstrate utilities of these measures by analyzing activity of monkey V4 neurons as well as a simulated balanced network of spiking neurons.

Simultaneous analysis of large-scale neural populations is necessary to understand coding principles of neurons because they concertedly process information. Methods of thermodynamics and statistical mechanics are useful to understand collective phenomena of the interacting elements, and they have been successfully used to understand diverse activity of neurons. However, most analysis methods assume stationary data, in which activity rates of neurons and their correlations are constant over time. This assumption is easily violated in the data recorded from awake animals. Neural correlations likely organize dynamically during behavior and cognition, and this may be independent from the modulated activity rates of individual neurons. Recently several methods were proposed to simultaneously estimate dynamics of neural interactions. However, these methods are applicable to up to about 10 neurons. Here by combining multiple analytic approximation methods, we made it possible to estimate time-varying interactions of much larger neural populations. The method allows us to trace dynamic macroscopic properties of neural circuitries such as sparseness, entropy, and sensitivity. Using these statistics, researchers can now quantify to what extent neurons are correlated or de-correlated, and test if neural systems are susceptible within a specific behavioral period.

Clustering of neural code words revealed by a first-order phase transition

A network of neurons in the central nervous system collectively represents information by its spiking activity states. Typically observed states, i.e., code words, occupy only a limited portion of the state space due to constraints imposed by network interactions. Geometrical organization of code words in the state space, critical for neural information processing, is poorly understood due to its high dimensionality. Here, we explore the organization of neural code words using retinal data by computing the entropy of code words as a function of Hamming distance from a particular reference codeword. Specifically, we report that the retinal code words in the state space are divided into multiple distinct clusters separated by entropy-gaps, and that this structure is shared with well-known associative memory networks in a recallable phase. Our analysis also elucidates a special nature of the all-silent state. The all-silent state is surrounded by the densest cluster of code words and located within a reachable distance from most code words. This code-word space structure quantitatively predicts typical deviation of a state-trajectory from its initial state. Altogether, our findings reveal a non-trivial heterogeneous structure of the code-word space that shapes information representation in a biological network.