The Geometry of Information Coding in Correlated Neural Populations

Jointly efficient encoding and decoding in neural populations

The efficient coding approach proposes that neural systems represent as much sensory information as biological constraints allow. It aims at formalizing encoding as a constrained optimal process. A different approach, that aims at formalizing decoding, proposes that neural systems instantiate a generative model of the sensory world. Here, we put forth a normative framework that characterizes neural systems as jointly optimizing encoding and decoding. It takes the form of a variational autoencoder: sensory stimuli are encoded in the noisy activity of neurons to be interpreted by a flexible decoder; encoding must allow for an accurate stimulus reconstruction from neural activity. Jointly, neural activity is required to represent the statistics of latent features which are mapped by the decoder into distributions over sensory stimuli; decoding correspondingly optimizes the accuracy of the generative model. This framework yields in a family of encoding-decoding models, which result in equally accurate generative models, indexed by a measure of the stimulus-induced deviation of neural activity from the marginal distribution over neural activity. Each member of this family predicts a specific relation between properties of the sensory neurons-such as the arrangement of the tuning curve means (preferred stimuli) and widths (degrees of selectivity) in the population-as a function of the statistics of the sensory world. Our approach thus generalizes the efficient coding approach. Notably, here, the form of the constraint on the optimization derives from the requirement of an accurate generative model, while it is arbitrary in efficient coding models. Moreover, solutions do not require the knowledge of the stimulus distribution, but are learned on the basis of data samples; the constraint further acts as regularizer, allowing the model to generalize beyond the training data. Finally, we characterize the family of models we obtain through alternate measures of performance, such as the error in stimulus reconstruction. We find that a range of models admits comparable performance; in particular, a population of sensory neurons with broad tuning curves as observed experimentally yields both low reconstruction stimulus error and an accurate generative model that generalizes robustly to unseen data.

The Determining Role of Covariances in Large Networks of Stochastic Neurons

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity.

Disentangling signal and noise in neural responses through generative modeling

Measurements of neural responses to identically repeated experimental events often exhibit large amounts of variability. This noise is distinct from signal , operationally defined as the average expected response across repeated trials for each given event. Accurately distinguishing signal from noise is important, as each is a target that is worthy of study (many believe noise reflects important aspects of brain function) and it is important not to confuse one for the other. Here, we introduce a principled modeling approach in which response measurements are explicitly modeled as the sum of samples from multivariate signal and noise distributions. In our proposed method-termed Generative Modeling of Signal and Noise (GSN)-the signal distribution is estimated by subtracting the estimated noise distribution from the estimated data distribution. We validate GSN using ground-truth simulations and demonstrate the application of GSN to empirical fMRI data. In doing so, we illustrate a simple consequence of GSN: by disentangling signal and noise components in neural responses, GSN denoises principal components analysis and improves estimates of dimensionality. We end by discussing other situations that may benefit from GSN's characterization of signal and noise, such as estimation of noise ceilings for computational models of neural activity. A code toolbox for GSN is provided with both MATLAB and Python implementations.

Probing the Structure and Functional Properties of the Dropout-Induced Correlated Variability in Convolutional Neural Networks

Computational neuroscience studies have shown that the structure of neural variability to an unchanged stimulus affects the amount of information encoded. Some artificial deep neural networks, such as those with Monte Carlo dropout layers, also have variable responses when the input is fixed. However, the structure of the trial-by-trial neural covariance in neural networks with dropout has not been studied, and its role in decoding accuracy is unknown. We studied the above questions in a convolutional neural network model with dropout in both the training and testing phases. We found that trial-by-trial correlation between neurons (i.e., noise correlation) is positive and low dimensional. Neurons that are close in a feature map have larger noise correlation. These properties are surprisingly similar to the findings in the visual cortex. We further analyzed the alignment of the main axes of the covariance matrix. We found that different images share a common trial-by-trial noise covariance subspace, and they are aligned with the global signal covariance. This evidence that the noise covariance is aligned with signal covariance suggests that noise covariance in dropout neural networks reduces network accuracy, which we further verified directly with a trial-shuffling procedure commonly used in neuroscience. These findings highlight a previously overlooked aspect of dropout layers that can affect network performance. Such dropout networks could also potentially be a computational model of neural variability.

Absence of paresthesia during high-rate spinal cord stimulation reveals importance of synchrony for sensations evoked by electrical stimulation

Electrically activating mechanoreceptive afferents inhibits pain. However, paresthesia evoked by spinal cord stimulation (SCS) at 40-60 Hz becomes uncomfortable at high pulse amplitudes, limiting SCS "dosage." Kilohertz-frequency SCS produces analgesia without paresthesia and is thought, therefore, not to activate afferent axons. We show that paresthesia is absent not because axons do not spike but because they spike asynchronously. In a pain patient, selectively increasing SCS frequency abolished paresthesia and epidurally recorded evoked compound action potentials (ECAPs). Dependence of ECAP amplitude on SCS frequency was reproduced in pigs, rats, and computer simulations and is explained by overdrive desynchronization: spikes desychronize when axons are stimulated faster than their refractory period. Unlike synchronous spikes, asynchronous spikes fail to produce paresthesia because their transmission to somatosensory cortex is blocked by feedforward inhibition. Our results demonstrate how stimulation frequency impacts synchrony based on axon properties and how synchrony impacts sensation based on circuit properties.

Information Geometry of the Retinal Representation Manifold

The ability for the brain to discriminate among visual stimuli is constrained by their retinal representations. Previous studies of visual discriminability have been limited to either low-dimensional artificial stimuli or pure theoretical considerations without a realistic encoding model. Here we propose a novel framework for understanding stimulus discriminability achieved by retinal representations of naturalistic stimuli with the method of information geometry. To model the joint probability distribution of neural responses conditioned on the stimulus, we created a stochastic encoding model of a population of salamander retinal ganglion cells based on a three-layer convolutional neural network model. This model not only accurately captured the mean response to natural scenes but also a variety of second-order statistics. With the model and the proposed theory, we computed the Fisher information metric over stimuli to study the most discriminable stimulus directions. We found that the most discriminable stimulus varied substantially across stimuli, allowing an examination of the relationship between the most discriminable stimulus and the current stimulus. By examining responses generated by the most discriminable stimuli we further found that the most discriminative response mode is often aligned with the most stochastic mode. This finding carries the important implication that under natural scenes, retinal noise correlations are information-limiting rather than increasing information transmission as has been previously speculated. We additionally observed that sensitivity saturates less in the population than for single cells and that as a function of firing rate, Fisher information varies less than sensitivity. We conclude that under natural scenes, population coding benefits from complementary coding and helps to equalize the information carried by different firing rates, which may facilitate decoding of the stimulus under principles of information maximization.

Cortical network and projection neuron types that articulate serial order in a skilled motor behavior

Skilled motor behaviors require orderly coordination of multiple constituent movements with sensory cues towards achieving a goal, but the underlying brain circuit mechanisms remain unclear. Here we show that target-guided reach-grasp-to-drink (RGD) in mice involves the ordering and coordination of a set of forelimb and oral actions. Cortex-wide activity imaging of multiple glutamatergic projection neuron (PN) types uncovered a network, involving the secondary motor cortex (MOs), forelimb primary motor and somatosensory cortex, that tracked RGD movements. Photo-inhibition highlighted MOs in coordinating RGD movements. Within the MOs, population neural trajectories tracked RGD progression and single neuron activities integrated across constituent movements. Notably, MOs intratelencephalic, pyramidal tract, and corticothalamic PN activities correlated with action coordination, showed distinct neural dynamics trajectories, and differentially contributed to movement coordination. Our results delineate a cortical network and key areas, PN types, and neural dynamics therein that articulate the serial order and coordination of a skilled behavior.

Common population codes produce extremely nonlinear neural manifolds

Populations of neurons represent sensory, motor, and cognitive variables via patterns of activity distributed across the population. The size of the population used to encode a variable is typically much greater than the dimension of the variable itself, and thus, the corresponding neural population activity occupies lower-dimensional subsets of the full set of possible activity states. Given population activity data with such lower-dimensional structure, a fundamental question asks how close the low-dimensional data lie to a linear subspace. The linearity or nonlinearity of the low-dimensional structure reflects important computational features of the encoding, such as robustness and generalizability. Moreover, identifying such linear structure underlies common data analysis methods such as Principal Component Analysis (PCA). Here, we show that for data drawn from many common population codes the resulting point clouds and manifolds are exceedingly nonlinear, with the dimension of the best-fitting linear subspace growing at least exponentially with the true dimension of the data. Consequently, linear methods like PCA fail dramatically at identifying the true underlying structure, even in the limit of arbitrarily many data points and no noise.

Neuronal responses in mouse inferior colliculus correlate with behavioral detection of amplitude-modulated sound

Amplitude modulation (AM) is a common feature of natural sounds, including speech and animal vocalizations. Here, we used operant conditioning and in vivo electrophysiology to determine the AM detection threshold of mice as well as its underlying neuronal encoding. Mice were trained in a Go-NoGo task to detect the transition to AM within a noise stimulus designed to prevent the use of spectral side-bands or a change in intensity as alternative cues. Our results indicate that mice, compared with other species, detect high modulation frequencies up to 512 Hz well, but show much poorer performance at low frequencies. Our in vivo multielectrode recordings in the inferior colliculus (IC) of both anesthetized and awake mice revealed a few single units with remarkable phase-locking ability to 512 Hz modulation, but not sufficient to explain the good behavioral detection at that frequency. Using a model of the population response that combined dimensionality reduction with threshold detection, we reproduced the general band-pass characteristics of behavioral detection based on a subset of neurons showing the largest firing rate change (both increase and decrease) in response to AM, suggesting that these neurons are instrumental in the behavioral detection of AM stimuli by the mice.NEW & NOTEWORTHY The amplitude of natural sounds, including speech and animal vocalizations, often shows characteristic modulations. We examined the relationship between neuronal responses in the mouse inferior colliculus and the behavioral detection of amplitude modulation (AM) in sound and modeled how the former can give rise to the latter. Our model suggests that behavioral detection can be well explained by the activity of a subset of neurons showing the largest firing rate changes in response to AM.

Specialized structure of neural population codes in parietal cortex outputs

Do cortical neurons that send axonal projections to the same target area form specialized population codes for transmitting information? We used calcium imaging in mouse posterior parietal cortex (PPC), retrograde labeling, and statistical multivariate models to address this question during a delayed match-to-sample task. We found that PPC broadcasts sensory, choice, and locomotion signals widely, but sensory information is enriched in the output to anterior cingulate cortex. Neurons projecting to the same area have elevated pairwise activity correlations. These correlations are structured as information-limiting and information-enhancing interaction networks that collectively enhance information levels. This network structure is unique to sub-populations projecting to the same target and strikingly absent in surrounding neural populations with unidentified projections. Furthermore, this structure is only present when mice make correct, but not incorrect, behavioral choices. Therefore, cortical neurons comprising an output pathway form uniquely structured population codes that enhance information transmission to guide accurate behavior.

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.

Coding of object location by heterogeneous neural populations with spatially dependent correlations in weakly electric fish

Understanding how neural populations encode sensory stimuli remains a central problem in neuroscience. Here we performed multi-unit recordings from sensory neural populations in the electrosensory system of the weakly electric fish Apteronotus leptorhynchus in response to stimuli located at different positions along the rostro-caudal axis. Our results reveal that the spatial dependence of correlated activity along receptive fields can help mitigate the deleterious effects that these correlations would otherwise have if they were spatially independent. Moreover, using mathematical modeling, we show that experimentally observed heterogeneities in the receptive fields of neurons help optimize information transmission as to object location. Taken together, our results have important implications for understanding how sensory neurons whose receptive fields display antagonistic center-surround organization encode location. Important similarities between the electrosensory system and other sensory systems suggest that our results will be applicable elsewhere.

Computational methods to study information processing in neural circuits

The brain is an information processing machine and thus naturally lends itself to be studied using computational tools based on the principles of information theory. For this reason, computational methods based on or inspired by information theory have been a cornerstone of practical and conceptual progress in neuroscience. In this Review, we address how concepts and computational tools related to information theory are spurring the development of principled theories of information processing in neural circuits and the development of influential mathematical methods for the analyses of neural population recordings. We review how these computational approaches reveal mechanisms of essential functions performed by neural circuits. These functions include efficiently encoding sensory information and facilitating the transmission of information to downstream brain areas to inform and guide behavior. Finally, we discuss how further progress and insights can be achieved, in particular by studying how competing requirements of neural encoding and readout may be optimally traded off to optimize neural information processing.

The structures and functions of correlations in neural population codes

The collective activity of a population of neurons, beyond the properties of individual cells, is crucial for many brain functions. A fundamental question is how activity correlations between neurons affect how neural populations process information. Over the past 30 years, major progress has been made on how the levels and structures of correlations shape the encoding of information in population codes. Correlations influence population coding through the organization of pairwise-activity correlations with respect to the similarity of tuning of individual neurons, by their stimulus modulation and by the presence of higher-order correlations. Recent work has shown that correlations also profoundly shape other important functions performed by neural populations, including generating codes across multiple timescales and facilitating information transmission to, and readout by, downstream brain areas to guide behaviour. Here, we review this recent work and discuss how the structures of correlations can have opposite effects on the different functions of neural populations, thus creating trade-offs and constraints for the structure-function relationships of population codes. Further, we present ideas on how to combine large-scale simultaneous recordings of neural populations, computational models, analyses of behaviour, optogenetics and anatomy to unravel how the structures of correlations might be optimized to serve multiple functions.