Neural Variability and Sampling-Based Probabilistic Representations in the Visual Cortex.

Neural responses in the visual cortex are variable, and there is now an abundance of data characterizing how the magnitude and structure of this variability depends on the stimulus. Current theories of cortical computation fail to account for these data; they either ignore variability altogether or only model its unstructured Poisson-like aspects. We develop a theory in which the cortex performs probabilistic inference such that population activity patterns represent statistical samples from the inferred probability distribution. Our main prediction is that perceptual uncertainty is directly encoded by the variability, rather than the average, of cortical responses. Through direct comparisons to previously published data as well as original data analyses, we show that a sampling-based probabilistic representation accounts for the structure of noise, signal, and spontaneous response variability and correlations in the primary visual cortex. These results suggest a novel role for neural variability in cortical dynamics and computations.

Perceptual Decision-Making as Probabilistic Inference by Neural Sampling.

We address two main challenges facing systems neuroscience today: understanding the nature and function of cortical feedback between sensory areas and of correlated variability. Starting from the old idea of perception as probabilistic inference, we show how to use knowledge of the psychophysical task to make testable predictions for the influence of feedback signals on early sensory representations. Applying our framework to a two-alternative forced choice task paradigm, we can explain multiple empirical findings that have been hard to account for by the traditional feedforward model of sensory processing, including the task dependence of neural response correlations and the diverging time courses of choice probabilities and psychophysical kernels. Our model makes new predictions and characterizes a component of correlated variability that represents task-related information rather than performance-degrading noise. It demonstrates a normative way to integrate sensory and cognitive components into physiologically testable models of perceptual decision-making.

Spontaneous cortical activity reveals hallmarks of an optimal internal model of the environment.

The brain maintains internal models of its environment to interpret sensory inputs and to prepare actions. Although behavioral studies have demonstrated that these internal models are optimally adapted to the statistics of the environment, the neural underpinning of this adaptation is unknown. Using a Bayesian model of sensory cortical processing, we related stimulus-evoked and spontaneous neural activities to inferences and prior expectations in an internal model and predicted that they should match if the model is statistically optimal. To test this prediction, we analyzed visual cortical activity of awake ferrets during development. Similarity between spontaneous and evoked activities increased with age and was specific to responses evoked by natural scenes. This demonstrates the progressive adaptation of internal models to the statistics of natural stimuli at the neural level.

Statistically optimal perception and learning: from behavior to neural representations.

Human perception has recently been characterized as statistical inference based on noisy and ambiguous sensory inputs. Moreover, suitable neural representations of uncertainty have been identified that could underlie such probabilistic computations. In this review, we argue that learning an internal model of the sensory environment is another key aspect of the same statistical inference procedure and thus perception and learning need to be treated jointly. We review evidence for statistically optimal learning in humans and animals, and re-evaluate possible neural representations of uncertainty based on their potential to support statistically optimal learning. We propose that spontaneous activity can have a functional role in such representations leading to a new, sampling-based, framework of how the cortex represents information and uncertainty.

A structured model of video reproduces primary visual cortical organisation.

The visual system must learn to infer the presence of objects and features in the world from the images it encounters, and as such it must, either implicitly or explicitly, model the way these elements interact to create the image. Do the response properties of cells in the mammalian visual system reflect this constraint? To address this question, we constructed a probabilistic model in which the identity and attributes of simple visual elements were represented explicitly and learnt the parameters of this model from unparsed, natural video sequences. After learning, the behaviour and grouping of variables in the probabilistic model corresponded closely to functional and anatomical properties of simple and complex cells in the primary visual cortex (V1). In particular, feature identity variables were activated in a way that resembled the activity of complex cells, while feature attribute variables responded much like simple cells. Furthermore, the grouping of the attributes within the model closely parallelled the reported anatomical grouping of simple cells in cat V1. Thus, this generative model makes explicit an interpretation of complex and simple cells as elements in the segmentation of a visual scene into basic independent features, along with a parametrisation of their moment-by-moment appearances. We speculate that such a segmentation may form the initial stage of a hierarchical system that progressively separates the identity and appearance of more articulated visual elements, culminating in view-invariant object recognition.

Modular Toolkit for Data Processing (MDP): A Python Data Processing Framework.

Modular toolkit for Data Processing (MDP) is a data processing framework written in Python. From the user's perspective, MDP is a collection of supervised and unsupervised learning algorithms and other data processing units that can be combined into data processing sequences and more complex feed-forward network architectures. Computations are performed efficiently in terms of speed and memory requirements. From the scientific developer's perspective, MDP is a modular framework, which can easily be expanded. The implementation of new algorithms is easy and intuitive. The new implemented units are then automatically integrated with the rest of the library. MDP has been written in the context of theoretical research in neuroscience, but it has been designed to be helpful in any context where trainable data processing algorithms are used. Its simplicity on the user's side, the variety of readily available algorithms, and the reusability of the implemented units make it also a useful educational tool.

Analysis and interpretation of quadratic models of receptive fields.

In this protocol, we present a procedure to analyze and visualize models of neuronal input-output functions that have a quadratic, a linear and a constant term, to determine their overall behavior. The suggested interpretations are close to those given by physiological studies of neurons, making the proposed methods particularly suitable for the analysis of receptive fields resulting from physiological measurements or model simulations.

What is the relation between slow feature analysis and independent component analysis?

We present an analytical comparison between linear slow feature analysis and second-order independent component analysis, and show that in the case of one time delay, the two approaches are equivalent. We also consider the case of several time delays and discuss two possible extensions of slow feature analysis.

On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields.

In this letter, we introduce some mathematical and numerical tools to analyze and interpret inhomogeneous quadratic forms. The resulting characterization is in some aspects similar to that given by experimental studies of cortical cells, making it particularly suitable for application to second-order approximations and theoretical models of physiological receptive fields. We first discuss two ways of analyzing a quadratic form by visualizing the coefficients of its quadratic and linear term directly and by considering the eigenvectors of its quadratic term. We then present an algorithm to compute the optimal excitatory and inhibitory stimuli--those that maximize and minimize the considered quadratic form, respectively, given a fixed energy constraint. The analysis of the optimal stimuli is completed by considering their invariances, which are the transformations to which the quadratic form is most insensitive, and by introducing a test to determine which of these are statistically significant. Next we propose a way to measure the relative contribution of the quadratic and linear term to the total output of the quadratic form. Furthermore, we derive simpler versions of the above techniques in the special case of a quadratic form without linear term. In the final part of the letter, we show that for each quadratic form, it is possible to build an equivalent two-layer neural network, which is compatible with (but more general than) related networks used in some recent articles and with the energy model of complex cells. We show that the neural network is unique only up to an arbitrary orthogonal transformation of the excitatory and inhibitory subunits in the first layer.

Slow feature analysis yields a rich repertoire of complex cell properties.

In this study we investigate temporal slowness as a learning principle for receptive fields using slow feature analysis, a new algorithm to determine functions that extract slowly varying signals from the input data. We find a good qualitative and quantitative match between the set of learned functions trained on image sequences and the population of complex cells in the primary visual cortex (V1). The functions show many properties found also experimentally in complex cells, such as direction selectivity, non-orthogonal inhibition, end-inhibition, and side-inhibition. Our results demonstrate that a single unsupervised learning principle can account for such a rich repertoire of receptive field properties.

Is slowness a learning principle of the visual cortex?

Slow feature analysis is an algorithm for extracting slowly varying features from a quickly varying signal. It has been shown in network simulations on one-dimensional stimuli that visual invariances to shift and other transformations can be learned in an unsupervised fashion based on slow feature analysis. More recently, we have shown that slow feature analysis applied to image sequences generated from natural images using a range of spatial transformations results in units that share many properties with complex and hypercomplex cells of the primary visual cortex. We find cells responsive to Gabor stimuli with phase invariance, sharpened or widened orientation or frequency tuning, secondary response lobes, end-stopping, and cells selective for direction of motion. These results indicate that slowness may be an important principle of self-organization in the visual cortex.